97


ΜΑΤΗ
4008.93
08.93


1
Math 4008.93

RISTO
ACADEMIAL
VER RO
WAS
TAS
ECCLESI
E IN
NOV
ONY
SCIENCE CENTER LIBRARY
FROM
The Author.
8 Sept. 18.93.



Complemente I own's
7
Horvard
89.
07. 2
A Math 4008.93
PRESENTATION
Com
OF THE
THEORY OF HERMITE'S FORM
OF
LAMÉ'S EQUATION
WITH A DETERMINATION OF THE EXPLICIT FORMS IN TERMS OF
THE p FUNCTION FOR THE CASE n EQUAL TO THREE.
CANDIDATES THESIS
FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
PRESENTED BY
J. BRACE CHITTENDEN, A.M.,
PARKER FELLOW OF HARVARD UNIV., INSTRUCTOR IN PRINCETON COLLEGE.
TO THE
PHILOSOPHICAL FACULTY OF THE ALBERTUS- UNIVERSITÄT
OF
KÖNIGSBERG IN PR.
PRINTED BY B. G. TEUBNER, LEIPZIG.
1893.





A.
$
PRESENTATION
OF THE
THEORY OF HERMITE'S FORM
OF
LAMÉ'S EQUATION
WITH A DETERMINATION OF THE EXPLICIT FORMS IN TERMS OF
THE PFUNCTION FOR THE CASE n EQUAL TO THREE.
p
CANDIDATES THESIS
FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
PRESENTED BY
گر
Jonathan
J. BRACE CHITTENDEN, A.M.,
PARKER FELLOW OF HARVARD UNIV., INSTRUCTOR IN PRINCETON COLLEGE.
TO THE
PHILOSOPHICAL FACULTY OF THE ALBERTUS- UNIVERSITÄT
OF
KÖNIGSBERG IN PR.
PRINTED BY B. G. TEUBNER, LEIPZIG.
1893.


JI, 80
8007.2
Math 400893,
COLLEGE
HARVARD
SEP 8 1893
LIBRARY
The Author.


DEDICATED
TO THE FIRST OF MY MANY TEACHERS,
MY
MOTHER
WHO MORE THAN ALL OTHERS HAS RENDERED THE REALI-
ZATIONS OF MY STUDENT LIFE POSSIBLE, FOR WHOM NO
SACRIFICE HAS BEEN TO GREAT IN FURTHERING THE
INTERESTS OF HER SONS.





Introduction.
The following thesis is practically a presentation of the general
analytical theory of Lamé's differential equation of the form known
as Hermite's. The underlying principles and also the general
solutions are therefore necessarily based upon the original work
of M. Hermite, published for the first time in Paris in 1877 in the
Comptes rendus under the title “Sur quelques applications des fonctions
elliptiques” and on a later treatment of the subject by Halphen in
his work entitled "Traité des Fonctions Elliptiques et leur applications",
Vol. II, Paris 1888. M. Hermite has employed the older Jacobian
functions while Halphen has used in every case the Weierstrass
p function, and not only the notation but the ultimate forms as
well as the complex functions in which they are expressed are in
the two works intirely different.
As far as I know, no attempt has before been made to
establish the absolute relations of these different functions. In
attempting to do this, I have developed the intire theory in a
new presentation, working out the results of M. Hermite in terms
of the p function, having principly in view a determination of
the explicit values of all the forms for the special case n equal
to three.
I may add that owing to the exceptional privilege granted
by the Minister of Education and the Philosophical Faculty of the
Albertus-Universität allowing the publishing of this thesis in English,


6
Introduction.
I am not without hope that this general presentation of the theory
of Lamé's Functions may prove a welcome addition to the literature
of the subject where in English Todhunter's “Lamé's and Bessel's
Functions” is the only representative. Finally I must acknowledge
my indebtedness to Prof. Lindemann not only for the direction of
a most valuable course of reading but for a general although, owing
to a lack of time, a by no means detailed review of the work.


Contents.
Introduction
page
5
Part 1.
History and Definitions.
The Problem of Lamé .
The Problem of Hermite .
Definitions
11
13
15
17
20
21
23
25
.
Part 2.
Hermite's Integral as a Sum.
The Function of the Second Species
Transformation of Hermite's Equation .
Development of the Integral
Development of the Eliment of the Function of the Second Species
Determination of the Integral
Part 3.
The Integral as a Product.
Indirect Solution
Solution for n = 2
The Product Y of the Two Solutions .
Direct Solution
Determination of Y for n = 3
28
30
32
37
40
.
.
Part 4.
The Special Functions of Lamé.
Functions of the First Sort
Functions of the Second Sort
Functions of the Third Sort
42
43
44
3”.
45
.
Part 5.
Reduction of the Forms "n
Identity of Solutions
Determination of x and v. First Method
x as function of $
Factors of $
Case 0 0
Definition of Y and p(v) as Function of Y
Definition of x and p' (v) as Function of x
Reduction of Lamé's Functions 0 = 0
Integral x
0
Case D 0
47
48
49
49
50
51
51
52
52
.


8
Contents.
2
0, Q2
0, 23
•
3
page
Relation of Y and C to the Special Functions of Lamé
: 52
Analytic Form of Y and y
53
Condition C = 0. Special Functions of Lamé
53
Condition P 0. Functions of First Sort
54
Condition Q 0. Functions of Second Sort
55
Absolute Relations of Q, and on
55
Determination of C
56
The Integrals Q1
0.
56
The Discriminant of Y
57
Resultant of Y and o(a).
57
Discriminant in terms of this Resultant
58
Discriminant in terms of P and Q
58
Special Results, n
59
Determination of x and v. Second Method
60
Reduction of the General Function
60
Development of (n = 3)
62
Development of ¥ (n 3)
64
Development of En 3)
65
Reduction of x and v from these Forms
66
General Forms for x, p (v) and p'(v)
66
Determination of Forms (n 3)
68
Reduction to the First Forms
69
Determination of v. Third Method
70
Value of the Constant ky
70
General Form as Product of 0, 0,,
71
The Functions Fi, F2, F
72
Forms for p(v) and p' (v) in terms of Fa and 02
72
Relation of F. to
x and the Factors of %
73
Reduction to the Forms of M. Hermite
73
General Discussion .
73
Review of the Theory
73
General Integral P 0
74
Integral Q , 0, ν =
0
74
Integral F.
0, v =
X + 0
74
Case v = 0
75
Functions of M. Mittag-Leffler
75
Relation to the Case x 0
75
Definition of the Functions.
75
Determination as a Special Case of the Doubly Periodic Function of
the Second Species
76
Determination of the Eliment, v 0
77
Integral (v = 0). .
.78
Table of Forms and Relations (n 3)
79
3
3
n=3
ܕ002
C =
0 or X
21
=


Thesis.





Part I.
Historical Development and Definition of the Equation of Lamé.
The Problem of Lamé.
In order to arrive at an understanding of the highly gener-
alized forms that have taken the name of Lamé it is adivisable
to return for the moment to the original problem of the potential
in which they claim a common origin.
Lagrange and Laplace (1782) in their researches with respect
to the earth regarded as a solid sphere developed the potential
function *) which led to the development of the theory of the Kugel-
function. From this date until 1839 the only name that need be
mentioned is that of Fourier (1822) who, in developing his theory
of heat solved the problem with reference to a right angled cylinder
discovering the series named after him.
In the following decade**) however Lamé ***) generalized the
work of his predicessors by solving the problem for an ellipsoid
with three unequal axes thus laying the foundation for the develop-
ment of functions of which the former are but special cases. He
used to this end the inductive method arriving at special solutions
through a study of the problem already solved with reference too
the sphere.
The problem of Lamé may be stated thus:
Let the surface of an ellipsoid be given by the equation u Uo;
it is required to find a function T which will satisfy the equation of
the potential and which for the value u=u, will reduce to a given
Uy
*) See note Heine, Handbuch der Kugelfunctionen, p. 2, Berlin 1878, and
Heine, 2d vol. Zusätze zum ersten Bande.
**) See also reference to Green Heine p. 1.
***) Memoire sur les axes des surfaces isothermes du second degree con-
sidérés comme des fonctions de la temperature. Journal des Mathématiques
pures et appliqués. 1re série. t. IV, p. 103. 1839.


12
Part I.
dº u
.
cz
a
d2T
d2 T
pv)
dus
.
2
where y
2
function of v and w, where T is the temperature at a point whose
elliptic coordinates are u, v and w.
The working eliments are then, the potential function, generally
written
[1] ·
0
doc
or transformed in terms of the p function
T
[2] (pv — pu) and + (pu — pulang
-
dv: + (pů – pv) dan
= 0
dw2
the relation,
[3] ·
T=f(u) f(v) f(w)
and the equation
day
[4] ·
du
[Apu + B]y
f(u) and A and B are constants.
If T is developed by Maclaurin's theorem with respect to the
rectangular coordinates, we may write: *)
[5] .
T= T. + T, + T, +...+In+...
Tn
where Tn in general is an intire homogenious polynomial of the
nth degree, it is observed that each of the functions In will also
satisfy [1], the equation of the potential, in which case [1] would
be an intire homogeneous polynomial of the (n − 2)d degree. This
-
polynomial must be identically zero which will impose (n 1)n
linear conditions. The quantities Tn will have in all - (n + 1) (n + 2)
constants, which leaves the difference 2n + 1 equal to the number
of constants that may be considered arbitrary.
Now the general expression for x2 in terms of p is known to be
(pu — ea) (pv — @c) (pw — en)
[6]
( - @p) (@a - ey)
being a constant, from which we see that by a change of variable
Tn may become an intire homogeneous function of the nth degree
with respect to the variables
[7]
Vpu — ez, Vpu ez
quantities proportional to the axes of the ellipsoid, and of the
1st degree, pu being of the second and p'u of the third.
We have then that T, the function sought, is composed of
similar functions In, where Tn is of the nth degree, is symmetrical
1
? 3.
α
· Vpu – ,
@y
27
*) see Halphen. Vol. II p. 466.


Historical Development and Definition of the Equation of Lamé.
13
u
.
with respect to u, v and w and having 2n + 1 arbitrary constants,
is capable of satisfying the equation [2] of the potential.
From the above relations we derive
d2T
[8] ·
=f"ufvfw
" = 1""
[Apu + B]T
du?
fu
with corresponding equations for v and w.
If then one can find 2n +1 systems of constants A and B of
such sort that for each of these systems there exists a solution y=fu
=
of equation [4] where y is an intire function of the nth degree each of
the corresponding products fu fv fw will furnish a term Tn of T and
the problem of Lamé will be solved. The value of A for all of these
systems is n(n + 1) where n may be considered as always positive,
since the substitution n (n + 1) does not alter the value of A.
The Problem of Hermite.
Continuing our review we find that one of the original forms
of Lamé's equation expressed in terms of the Jacobian function is
[9] ·
da — [n(n + 1)kºsn*x + h]y=0
)
corresponding to the form [4]*)
dạy
da y
[10] :
[n(n + 1) pu + B]y=0
d u²
where h is an arbitrary constant and n a positive whole number.
Lamé succeeded in finding the requisite number of values of h
to complete his solution for the ellipsoid and the solutions of [4]
corresponding to these values are known as the original special
functions of Lamé.
The problem then arose:
Required to determine a solution of Lamé's original equation
which shall hold for any values of h and n.
Except for the special values n = 1 and n= 2 no advance
was made towards a solution until M. Hermite **), making use of
the progress in the theory of functions inaugurated by Cauchy,
arrived at the solution and by so doing opened a new field for
*) See transformation p. 20.
**) Sur quelques applications des fonctions elliptiques. Comptes rendus
de l'académie des sciences de Paris. 1877.


14
Part I.
the application of the elliptic functions and leading later to the
integration of a large class of differential equations.*)
In this connection M. Hermite introduces the functions called
by him doubly periodic of the second species, which have the special
property, that save for a constant factor they remain unaltered
upon the addition to the argument of the fundimental periods.
The solution of M. Hermite developed in terms of snu and
for n odd may be written in the form
D2"-1f(u)
D21
2 -3
u
[11]
]
h, D%" – 3f(u)
t...thx-1f(u)
.
y = F(u)
+
r(21)
r(2v - 2
2v
into
O'(0)
® (w)
2 K
x(u) =
e
where n = 20 — 1, with a corresponding form for n even, where
=
f(u) is a doubly periodic function of the second species, namely,
f(u) = eica—ik') x(u)
where
H (0) H (4 + 10)
(u- i K')+
(u) (6)
That this shall be a solution the quantities w and a must be
determined to correspond with definite conditions and herein lies
the chief difficulty when explicit values of the functions are sought.
Moreover the above development fails as we shall find when seeking
to deduce the special functions of M. Mittag-Leffler from the
general form.
M. Hermite was thus led to a new presentation of the general
solution in the form of a product, namely
y=IT
*o (u + a)
e-usa
σα σω
a=a.b..
a form of solution suited to every case.
The general theory based upon the latter solution has been
lately perfected by Halphen**), who, confining himself in the main
to the use of the p function, presents the subject in an excellent
but highly condensed form.
*) Equations of M. Éimile Picard. Comptes rendus, t. XC, p. 128 and 293.
Prof. Fuchs, Ueber eine Classe von Differenzialgleichungen, welche durch
Abelsche oder elliptische Functionen integrirbar sind. Nachrichten
Göttingen 1878, and Hermite: Annali di Matematica, serie II, Bd. IX, 1878.
**) Traité des Fonctions Elliptiques et leur applications. B. II. Paris 1888.
von


Historical Development and Definition of the Equation of Lamé.
15
Definitions.
Returning to form [9] of Lamé's equation we observe that it
has the following properties:
It has a coefficient
m (1 + 1) kº snºx + 1
that is doubly periodic and has only one infinite x = iK' and its
-
congruents, and it is known to have an integral which is a ratio-
nal function of the variable. Conforming with these peculiarities
M. Mittag-Leffler *) defines the general Hermite's form of Lamé's
equation of the nth order as a linear homogenious differential equation
of the order n having coefficients that are doubly periodic functions,
having the fundimental periods 2K and 2i K' and everywhere finite
save in the point x = i K' and its congruents which alone are infinite
and whose general integral is a rational function of the variable.
The general theory of Herrn Fuchs **) then gives the form,
namely
[12] ·
Yn + 0,(x) y(n —2) + + On (x)y =0
where
Φ, (α)
do + Qysna
03 (20) = Be + Basnax + B2 D2 snax
Φ.(α) Yo + 715n-x + 72 Drsna x + 73 D snạ
2
2
1
1
1
1
dy
) +6+6+
[13] do
But a better generalization based upon a full representation
of the singular points is given by Prof. Klein ***) and later stated
as follows by Dr. Bôchert).
) First the ordinary form of the
equation of Lamé may through transformation becomett)
day
Ax + B
+
dx
y
4 (v — ez) (oc (2) (ac ez)
where the exponents of the zeros ez, ez, ez are 0 and and that
of the infinites
1+11 + 4A
From this generalizing by the intro-
duction of n zeros we have the following definition:
2
)
ei
X
eg
2-e.
2
4
*) Annali di Matematica, tomo XI, 1882.
**) Comptes rendus etc. 1880. p. 64.
***) Math. Annal. Bd. 38.
†) Ueber die Reihentwickelungen der Potentialtheorie. Göttingen 1891.
†t) See also transformation p. 20.


16 Part I. Historical Development and Definition of the Equation of Lamé.
„Mit dem Namen Lamésche Gleichung bezeichnen wir eine überall
reguläre homogene Differentialgleichung zweiter Ordnung mit rationalen
Coefficienten, deren im Endlichen gelegene singuläre Punkte en, l, .. en
sämmtlich die Exponenten 0, 1 besetzen, und in unendlichen nur einen
,
uneigentlich singularen Punkt aufweist.“
Lamé's equation becomes in accordance with this definition
and freed from the possibility of a logarithmic irrationality through
a determination of the coefficient of xn-3.
-3
dạy
f' (2) dy
[14] ·
+
daca
2fx dx
n
1
n(n — 4)
21—2 +
-2
- ["
(n − 2)(n — 4) E; 24–8+ Ar-4+.+M]y
-3
:0
4f (2)
where
xo
f(x) = (x – e) (x – ez)... (x – en).
ac
It is further evident that this form, like the Hermite form
and as previously developed by Prof. Heine, is but a special case
of a general equation of a higher order.
In speaking of Lamé's equation we will understand an equation
conforming with the above definition whose general form is given by
[14] and, if the order is higher than the second, distinguish by mention-
ing the order.
Forms [9] and [10] will then be called Hermite's forms of
Lamé's equation or simply Hermite's equation, where again the order
need be mentioned only if it be other than the second.
Any solution of any form of Lamé's equation will be a
function of Lamé and if the doubly periodic functions first deter-
mined by Lamé are mentioned they will be designated as the
special functions of Lamé.


Part II.
Hermite's Integral as a Sum.
The Function of the Second Species.
da y
We have the problem required the integral y of the equation
[15]
[n(n + 1) snu + h]y
d u²
where h is taken arbitrarily, n is any intire positive number and k is
the modulus of the elliptic function.
M. Hermite introduces to this end a function which he names
doubly periodic of the second species, which may be defined as a
product of a quotient composed of o functions, the number of zeros
being equal to the number of the infinites, and an exponential, having
the property of reproducing itself multiplied by an exponential factor
when the variable is increased by the periods 2K and 2iK.
It is defined then in general by the relations:
F(u + 2K) = u F(u)
F
[16]
F(u + 2iK') = u'F(u)
o(u — @) (u — a.)....(u — Q n-1)
ao , —
F(0)
o (u hy) o(u — h,)...0(u
(U - On-1)
The factors u and u are called Multiplicators.
M. Hermite might have been led to the employment of this
function by the following analysis which is essentially that given
by Halphen.*)
Consider for the moment that y be such a function of the
second species but having instead of the n different poles but one
pole u 30 of the nth order in which case the function will have
n roots. Upon developing the properties of this function one finds
that its second derivative has the same multiplicator as the function
n1
elu.
b
*) Bd. II p. 495.
2


18
Part II.
[17]
itself and that therefore the quotient y": y will not only be doubly
periodic but will have a single pole uo of the second order.
This function then satisfies the necessary conditions and the
y"
corresponding quotient may then be written equal to
y
n(n + 1) sn² x + h
where h is a constant. But we have taken this function with the
condition that it have but one pole of the order n subject to the
above conditions which affords n arbitrary constants and employing
also an arbitrary constant factor we obtain (n + 1) arbitraries in
all. That is sufficient to satisfy all the conditions and leave h to
be chosen at will. Hence we must conclude that there is no reason
why y should not be a doubly periodic function of the second
species and our problem reduces to the determination of a function
whose general form and properties [16] are known.
From this standpoint we have:
Required a function such that
Define:
Whence
Ω
2 K
F(u+2)=µF(u),
F(u+2)=u'F(u), 2' = 2iK'.
f(u) = A
o (u + 2)
6(u + 2')
η
which function we will speak of as the Eliment the general form [16]
being a product of similar eliments.
We have the fundimental relations:
-
u
σ (u + v)
σ (u)
-
==
6'
――
(2)
=
ehu *)
=
―
σ (u) e² nu+n
σ (u) e² n'u + n' '
Ω
f(u + 2) = 1 º (u + v)
σ (u)
= f(u) p¹ 2+2 ¹¹.
Choosing then v and λ correctly we may write
eλ2+217 v
e² (u +2)+2nv
*) Hermite, in the following analysis, employs the function given on p. 11,
namely the function x expressed in terms of the
function.


Hermite's Integral as a Sum.
19
[18]
with a corresponding value for u' and we may then write
F(u)
f(u)
Φ(u)
where is a doubly periodic function, that is
Þ(u+m2+n2') = Þ(u).
·
Again
and we derive
[19] ·
f(u — 2) = — f(u) and ƒ (u — 2′) — —, f(u).
Whence
f(u — z — 2) — — f (u — 2) where F(≈ + 2) = µF (2)
—
=
*
where is doubly periodic.
From this point the development of F(u) depends upon the
theory of Cauchy, as it is obtained by calculating the residuals of
for the values of the argument that render it infinite and
equating the sum to zero as follows.
First f(u) becomes infinite for the value u0 whence its
residual
Eu = of (u)
of(u) = [ufu]u :
Whence
=
Þ(2) = F(2) f(u —— 2)
—
=0
=
= A
and becomes equal to unity if we take
1
A
6(v)
f(u)
[o (u + v) e¹u],
-
Again
lim
Eu (2)
+ = u (≈ − u) Đ (z)
2 U
and developing f(uz) we have
[w]
Eu (2)
o (u + v)
o (u) o (v)
u = 0
-
u = 0
- ―――――
elu
= A
6(v)
'(0)
-
lim
z = u(≈ — u) F(z)f(u — z)
=
F(u)
Again let a be any pole of F(u) in which case, developing by
the function theory, we may write
A6(v)
+ Aa Da ε¬¹ + а + α₁ ε + а₂ ε² + ··
-1
1
1
1
F(a + ε)=0 = Aɛ¬¹ + A, D,ε~¹ + A½ D² ɛ¬¹ + · ·
2*


20
Part II.
and
2
f(u — a –
– €) =f(u — a) – Duf(u — a) + 1.D.f(u – a) -..
(-1)"
+
1 2
Difu – a) +..
• a
where
. 2
n
D%8-1 = (-1)"?
anti
We have then
lim
E-
Ε. Φ
'0 & F(a + 8)f(u — a — 8)
દ
Af(u -- a) + A, Duf(u – a) + A,Dif(u — a) +...
+ AqD4f(u - a)
with similar expressions for E., Ec ...
But ở being a doubly periodic function we know that the
sum of its residuals with respect to u, a, b .. equals zero whence
[20] F (u) = { [Af (u — a) + A, Duf (u
] Σ[- ) +
a) + A,Dif (u -- a) + ..
+ A Dif(u – a)]
a
where Ai is determined from the first development.
This important formula still further narrows our problem to
a consideration of f(u) in terms of which and its derivatives under
conditions to be determined it is now evident that y = F (u) may
be expressed.
a=a, b, c..
a
Transformation of Hermite's Equation.
We have written Hermite's equation in its original form
d'y
[21] ·
=[n
[n(n + 1)ka sna x + h).
d x2
That this is however but a special case of a more general
form is seen as follows.
Take the integral
2
• da
V(1 – 12)(1 – k 22) ул
2
X =
----
We have
dy
1
dy
da
dy dx
dx da
dx va
or
dy=17
dy
da


Hermite's Integral as a Sum.
21
whence
[23]
or
[22].
[24] ·
p(u) = es +
we obtain:
dy-ay-1-114x
dx²
Define
d22
day
du² (e₁
e1
sn²u Ve
and making the substitutions:
―
d2y
dx2
X =
Substituting we derive the ordinary form of Lamé's equation
dy
dλ
d2y 1
Λ + Δ' — [n (n + 1) k² sn² x + h] = 0.*)
dλ2 2
-
=
The value of 4 gives as singular points + 1; and ∞.
± k
For our present purpose however we need the equation ex-
pressed in terms of u and pu which is derived from (21) by
means of the relations
1
d2y
Λ + 1/1/13
d22 2
uve₁
Ve
dx²
eg)
eg
2
=
Δ'
es
=
dy
"
dλ
k² sn² (u + ik')
u ~u+ik'
l3
du² (e₁ — e3)
[n (n + 1) pu — es + h].
e1
=
· B = h (e₁ — е3) − n (n + 1) ez ·
―
-
1
Whence our equation may be written:
y' = [n (n + 1) pu + B]y.
1
sn² u
Development of the Integral.
We observe, since snx reduces to zero only for the value x =0,
that we have but one pole of the second order in Hermite's equation
and that we may therefore develop y within a cercle whose radius
is less than 2', the form being
=
y = u² [y。 + y₁u + y₂u² + ·· •]
whence
2
y = vu¹¹y + (v + 1) y₁ u² + (v + 2) u² +¹ y₂ + (v + 3) u¹ + ² y z + ··
-2
y' — v (v — 1) u' —²y。 + v (v + 1) y₁ u¹−1 + (v + 1) (v + 2) u² y₂ + ··
= -
*) Compair general form [14] p. 16.


22
Part II.
We have also
whence:
[25].
p(u)
These values in [24] give:
v (v — 1) yuv−2+
-[~
=
u'
u'
•
W
n
v (v − 1) = n (n + 1)
n.
This value gives since the uneven powers fall out
h2
+ +
n-4
un-
·
+
= n(n + 1).
n 2
y
+
1
=
+B +
n
W
+(n−6) (n − 5)
-
h₁
n- 2
u
1
un
+
1
un+ 2
from which we again derive
y' = n(n + 1) + (n − 2) (n − 1) ¹¹ + (n − 4) (n − 3)
1
un+ 2
h₁
un
-
+
tn(n+1) Ghi
h₂
+
un-4
Bhi
n-2
un
h₁
n
u'
+
h1
u2
n
2
u
un-
+ c₁u² + c₂u¹ +
hg
+
(n-4)
+ +
un-
Bha
n-4
(n + 1) n +
1
2i+2
2
› (n + 1) you²² +..
U
1
+ n (n + 1) C1 un—3 +n(n + 1) e̟h,
2
+
+· +
or
...
2 i
•
W
[n (n
(n + 1) (~ + qu² + cu¹ + · · ·) + B
+
2 =
+ (n-2i)(n2i+1)
h;
•
+]
Bh
n-2 i
+
h;
un-2 i
1/72 | (n − 2) (n − 1) h¸ − n (n + 1) h¸ + B
· h₁
-
=
un
-
+
1
N-
W
h; n (n + 1)
n-2 2i+2
W
+
4
+
..
...
1 | (n − 4) (n − 3) h₂ = n (n + 1) [h₂ + ç] + h₁ B
-
+ n (n + 1) c½ 2 − s +
c₂
n-4
u'
Equating the coefficients of like powers of u in this identity
one finds
+...
W
ha
n
u
h;
n―2i+2
| (n − 6) (n − 5)kg=n(n+1)(kstant)+h B
-
4
2


Hermite's Integral as a Sum.
23
1
w Ło | (n -- 8) (n — 7)h = n(n + 1) (hx+cho+cha+C3) + hy B
In
7
0 c )B
6
un
1
-3
U
n-2i+2(n-2i)(n-2i+1)h;=n(n+1)[Hitchi-2+cahi-3 +...
+ C-1] + hi-1 B.
Whence we obtain all the coefficients in the development for
y by means of the recurring formula.
[26] 2 i(2 i — 2n — 1) hi
- = n(n + 1) [cı hi-2 + cahi-3 +...
+ Ci-2h; + Ci-1] + hi-1B.
Since then hi is determined we have when n is even and
equal to 2v
h,
Y + + + th,
u2v
u?
-2
-3
--1
1
hy-1
2 V -2
w
and if n be odd and equal to 2 v - 1
- 1
1
hi
V-1
Y
utrit
h,
+
+
+
+ hou
2 V-1
2v-2
U
urgt
where hi is given by (26).
elu
6
1
U
Development of the Eliment f(w).
Having now a development of y we can, if we develop f(u)
and substitute in the development of F(u), find by comparison the
conditions necessary that y=F, (u) be a solution.
We have then to determine the development of
o(u + v)
f(u)
o() (v)
+ .
since
[uf (u)]u=0 =1
To this end we develop first the function
ou + v)
[27] ·
9 (u) = f(u)e(1+5v)u
ou) o (v)
We have:
deu
d
pu
+ Cu + Cqu4 t.
whence
23 — Ž czu• — c5 2?...
-
e-uζν
0
.
o'u
pu
du
au
ou
1
1
6U
u
3
7Cs u?.


24
Part II.
By Taylor's theorem:
d2
(v)
6
d
(v)
+
du
6
u2
1.2
du?
u2
1.2.30" (»)...
Ó
o'
0
6
1
u3
(u + v) = (v) +
+
= $ (v) — up (v) – 1. r' (v)
)
v
Passing now to logarithms we derive:
(W) (# + v) - (n) - )
u
u (v
- - - up (») - ***w -(OP)
p'(v)
*"(w),*"*») – ]-...
pv
+ Agu + 1 +
23 +..
Integrating we have:
log u + ,*+ A + +...
+ A2 Az A4
+
p” (v)
3!
C,
v
20
2
3
us
C2
!
67 [
A,
2!
u²
A,
3!
t
u
u2
u3
log 9
u"
4!
2!
3!
whence
1
u
u 3
A2 +A, +
2! 31
[28] 9
е
u
[49+4+...]
- [1+4+4+..]+:[4+425*+..14+...
1 , **."* +
= [1 + P + P + P +...]
u?
72
us
3!
us
23
.
-2 2!
+
A
.
U
2 2!
3
3!
u?
,
+
us
3!
24
4
4!
2
3
2
2
3
PA
where
P, = A, = - p(v); P = 4, = -p'(v);
, =
3p(v) + 92 = A4+3 A,
4
P. 3 pvp'v = A; + 10 A, A; etc.
showing that the coefficients Pi are intire functions of pv and p'v.*)
2
5
5
*) The functions Pi correspond to the functions 2 in Hermite's treatis,
for example
1 + x2
P,= - p(v)
zasnau
3
P. p'u
-
2
Son
hasnu onu dnu
see p. 126 development of %.


Hermite's Integral as a Sum.
25
9
u
1.2
From these forms we pass immediately to
[29] f(u) = q (u) e(2+$)u
= (N)[1 + (8 + $v) + (8 +Bv)**+...]
9 $?
= {[1 +(+ $u)u + (P. + (1 + $u)") ...]
+[P, + 3 P, (1 + $w) + (1 + $w"]...+.
[(() ...}
+ H. + H, u + H2u2 + Hyu+...
H , ,
u?
2
1
23
3
2
2
3
1
u
Take
λ = α - ζν
-
whence
[30] ·
f(u)
o(u + v)
e(x_{vu
o(u) (v)
0
u
u
+*+ (x2 + P,) + (x3 + 3 P x + P.) **
3) 2.3
+ (x** + 6 P.x2 + 4 P5x + P.), at
us
3
+
4
2
=
+ H. + H, (u) + H, (u)2 + H2u».
1
2
U
Where in Hermite's Notation
H.
X.
1
2
H = (x2 + P.).
H, = (23 + 3 P x + P3)
P,
1
[31]
6
1
H,
(24 + 6 Pzx2 + 4 Pz3 + PA)
3
24
Determination of the Integral.
We are
now enabled to determine the exact expression
for F(u) and the conditions necessary that it become equal
to y by a process of comparison of the several developments
obtained.


26
Part II.
First we have:
1
f(u)
+ H + H4 + H21° ++ H4 +..
u
f'(u)
1
u?
+ H+ 2 H2u + 3 H2u2 +
+ iH;u-1+...
f”(u)
+ + 2 H2 +2. 3 H3u +...
. ..tili — 1) H;ui2
2
u2
2.3
f'' (u)
u4
+ 2 · 3 H3 +...
2 +
tili - 1) (2 — 2) Hu—3 + ...
(n odd)
1
2 .3 .•(n-1)
+
+ 2 .3 ... (n − 1) Hn-1 t...
un
tili — 1)...(i — n+1) Hui-ntit...
Again
1
hy
In1
Yn=2v-1
+
to.it
+ hou
- ן 2
1
2 - 3
U
U
u
1
hr_1
+
2v
U
2v — 2
U
2
Y = Fiu
-
h,
Yn=lv
+
+ + hy.
u2
And in general
Aaf(a) + Aa-1f (a-1) +...tf
Aaf (n-1) + Aa-2f (n-3) + + f (n odd).
Now substituting the values f(x) found above and ordering the
coefficients so that the residual with respect to u will be unity we
find by comparison that we may write
1
1); f(n-1) +
3); hif (n—3) +
1
[32] . • Y = F1(11)
hif (n-3) + ...hr-if
(n
(n
(n odd and =2v — 1)
provided x and v be so taken that the constant term equal zero and
the coefficient of the next term equal hy and
ha
1)!f (n − 1)
f(n—5)
(n − 3)!/(1-3).
1
h
[33] y=F:(u)
(n − 1)!
(n — 5)!
- hr-1f' (n even and = 2 v)
provided x and v be so taken that the constant term equal h, and the
coefficient of the next term equal zero
or in general
1
[34] (-1)n-17=
(n − 3)!
h, f (n—3 + hef (n—5) +...
(n — 5)!
1
(n − 1)! f(n-1) +


Hermite's Integral as a Sum.
27
where the last terms are obtained to accord with the above con-
ditions.
Substituting the values f(x) we find the conditions to be
(n odd)
H2v-2+ h₁ H21-4+ h₂ H2v-6 + + hv−1 Ho = 0
[35] (2v 1) H₂v-1+ (2v - 3) h₁ H₂v-3 + (2v — 5) hq H2 v−5 + : · ·
+hy-1 H₁h, 0
0
—
forms
(n even)
H2v−1 + h₁ H2v−3 + h₂ H2v−5+ + hv−1 H1 + hv
[36] (2v H₂v+(2 v −2) h₁ H2 v−2+(2 v — 4) h2 H2 v−4+ ··· +2h,−1H2=0.
These conditions being satisfied y F(u) and we have two
...
=
Då F (u) — [n (n + 1) pu + B] F' (u) = 0
Ω
since finite for u
= ik'
=
2
=
=
A second solution being likewise obtained by making the sub-
stitution n~n the general integral may be written:
[37] ·
y=cF(u)+c'F' (— u).


Part III.
Integral as a Product.
Indirect Solution.
It will be shown in developing the forms for the case n = 3
= 3
that the original solution of M. Hermite as a sum will not be
applicable in the forms given in the last chapter, when B is so
taken as to give a value, v equal to zero, which leads to a second
development in the form of a product, the eliments being as in
the first case doubly periodic functions of the second species.
Assume that
o(u + a)
[38]:
Y
e-uka,
o(u) o (a)
n
.
.
II W0
a=a.b..
1 2
U
where the product is composed of n factors obtained by taking
a, b, c in place of a. The derivative of the logarithm is
y'
p'a
15(u + a) — $ (u) – $(a)] = 2
[) -
ри - ра
while a second differentiation gives
y"
2
-
Y
2
- (*) - (pu – p (u + a)].
y
From the first equation
2
1
1
[=> 1 (uma) +
-Σ:( + :Σ=
pu
pu
p'u – p'a pu – p'b
pu pa pu
4
2
y
ра,
pb
But the addition theorem gives:
1 p'r
p'a
ри - ра
p(u + a + pu + pa,
4
whence
2npu + pa+; } pu – p'a p’u -- po
++
p'b
pb
y
ри
- ра
ри -


Integral as a Product.
29
2
pu
ра
+
- pb
In order to decomposé the last term in this expression we write:
1 p'u – p'a p'r
p'b
2 (pu + pa + pb)
ри - pb
p'a + p'b
pa - po 15(u + a) – $ (u + b) – $u +86].
[
& )
Take the value U= (a + b), remembering the relations
p(- ) + pu; p'(-u) – p'(u); $(-u) = - $(u).
—
Writing then f(a + b) for the right hand member of the above
equation under these conditions we get
f(a + b) = 2np(a + b) + Epa + 2p(a + b) + pa + pb
)
+
p'a + p'b
[$(-6) - $(-a) – $a + $b]
pa
pb
2 (n + 1) pu + 2 Epa.
from which we see that in general we would have
y"
=n(n + 1)pu + (2n — 1) Epa
у
a; p'a =
OC
the quantity in brackets being equal to zero.
If now we reunite the terms $(u + a) — &a, $(u +b) – $b etc.
- (
in the general expression and make equal to zero the sum of their
coefficients we obtain n equations of condition, namely, writing
ра
a'; pb = B; p'b = B';
a' + B' d'+y' a' + d'
+ 0
В
B' t. a B' + g' ß' td
+ +
+
0
B
B
B
g' to' y' + B'
y' + d
+
+ 0
B
.
+
+
α -
Q -
n
OC -
d
[39]
a
d
.
+
g
a
7
g
d
12
OC
[40]
93
If then we can solve the equations considered as simultanious
403
920 93
B'2
483 — 92B -
together with the relation
(2n — 1)(a + B + y + :) - В
we will satisfy the necessary conditions to enable us to write:
y"
= n(n + 1)pu + B.
y


30
Part III.
y=
e-uta
That is
6(u + a)
Y IT
6 (u) o (a)
is a solution of Hermite's equation whatever be the value of B,
provided a, b, c.. fulfil the above conditions.
Solution for n = 2.
It is clear that, save for small values of n, an attempt to solve
the above equations by the ordinary methods would give rise to
insurmountable difficulties. The case n=2 however, which is famous
=
as affording a solution to the problem of a pendulum, constrained
to move upon a sphere, can be readily solved as follows:
Given n
= 2: we have the conditions
12
a
12
[41]
[
p'?a= 4ą%
92a
93
B'2
483 – 92B – 9:
pa + pb=B
B
a'? + B’? = 0.
p'26
1
2
or
3
Observe that by designating pb by — B the above relations
remain unaltered and that we may therefore write
403
92a
93
48% + 92B — 93
or
4 (a+ ) - 92(a + B) = 0 ).
whence
? – αβ + β2
a? — «ß + B2 – 192 = 0.
=
But
B=a -B
whence the equation that determines the values of B.
[42] : · Bº – ja B +a? — 192 = 0 and also
B2 şa
O and also B=0.*)
If then n= 2 and a and b, the arguments of a=(- pag),
(,
are so taken that B shall have the values of the roots of equation (42)
Hermite's equation will have the solution
1
.
9
4
1
*) If in this result we take B
Ś we obtain the formula
3
& 2 — as ta ?
-
1
92
4
- 0,
see Halphen II p. 131.


Integral as a Product.
31
O(u — a) o (u + b)
[43] ·
Y =
elta-
6u
Cou
a + b)
C
elša–5)u]
du
би
c' los
a so (u + ») ere–50)]
d
[
Šv)
du
6U
6 u
where v = a + b.*)
That our solution given above be complete we must obtain
the corresponding values of x and v as follows:
d Co (u + v)
y
du
[ ele-50u
()]
We have also
[44] :
1 p'a + p'b
pv + pa + pb=
p'a,
6
pa pb
pb)
since
p'a =p'b=a'.
Again we have
ξ2 – αζ + α? 4 92 = 0
4
pa
1
or
α
1
2
α
2
α
1
2
2
- + V92 – 30%.
-
Hence
p(a) = + V 9. - 302
P (b)
V9,- 3a2
whence
pa pb =V9. – 30
pa + pb
p'a, = – 40% + 9,0 – 93.
These values in (44) give:
403 + 9,0 – 93
·p(v)
92
3 2
92
=
[45]
.
a
Q3 + 93
3a
92
*) The last is the form given for the expression cos CX + i cos CY in
the solution of the pendulum problem in the direct investigation of which one
arrives at the expressions
d? X
d2 Y
NX;
NY;
dta
dt2
dt?
where N is found to be 3 r2 (2 pu — pa,) which causes the solution to depend
upon Lamé's functions.
d2%
NZ + g


32
Part III.
-
o'
· 92
If we take a = 2b we have
8b3 + 9
bo' - 0
[46] ·
P (v)
12b2 92
9
where
ф
433 92b -- 93
and
1262
For X we have:
1
p' (b -- a) + p'b
[47] x = $(a - b) + $a – $b
= S
=
2 p (b − a) – pb
1 p' (b − a) -p'(a)
a p' (b − a) + p'b - p'a
p (b a)
2 p b -- a) - pb - pa
( — —
=
-
-
p'v
2pv
2
- P (a)
CZ
3 bo'
9
-1
since
p'a — p'b=0
pa + pb
Combining these relations we obtain:
= a.
p'v
+ pv=b
2 x
9
9
.
1
9
29
9
-
I
and
/369'
bo'
3 bo'
p'v=2(b – pv) V 201 - 9) V
9
90
3 bo'
V
*)
9
Finally we observe that if -- u is substituted for u in Her-
mite's equation it remains unaltered which gives us the second solu-
tion, namely
6 (a u)
[48] ·
-IT.
o(a) o (u)
X and v remaining as before.
.
2
euta
2
Product of the Two Solutions.
It becomes evident from the illustration in the previous para-
graph that while in general the theory involved in the solution
just given holds it is practically inapplicable for other values of n
than two or at most three whence one is led to a study of func-
tions of the integral in the hope of discovering inherent properties
*) Compair results obtained by M. Halphen and obtained in a different
manner, II p. 131 and 527.


Integral as a Product.
33
-
that will lead to a more practical result. The first of such func-
tions to command attention would be the product of the two
integrals
[49] :
Y
Y%
which we will proceed to develop as follows:
We would find from the integral % as in the case of y
z'
= t; (a — u) — 6u + $a]
-Σεξ (α - — ga
and combining with
y'
- a 1
[5 (a +u) — $(u) – ça]
we obtain
y'
is (v + a) — $(u — a) — 25a]
p'a
But
co (a + u) o (a u)
Y Y? -IT
=17(pu-- pa).*)
Whence
p'a
· Y Z
ра
p'a
-Σ, При ) 2C
ра
Y
pu-pa
y
2
62a 62 u
y e' – By=pu?" pa
-
pur-“ra 17(pu – pa) = 20
or
>
+
a
t
t
y
2
=
p'a
20
Σ
ри pa
II (pu
- pa)?
C being a constant or expanding and writing t=pu we have
a' B' g
2 C
[50]
+ +
B
(t a) (t B) (t – y)...
an identity independant of the value of t.
To determine a', B'... multiply both members by (t - a),
(t – B) ... and take t = a, B... for example
a) n'at a)
20
a'+
+
+
(t B)
(t - B) (t - y)...
—
whence making t= a we have
=
2C
[51] .
a'r
(a B) (a — y)...
and in a similar manner we find
B' (t
t
.
a
20
B'
(B a) (B
y)...
*) see theory of p and o functions.


34
Part III.
These values of a' and B'... determine the constants a, b...
provided we can find the value of the constant C. It is also clear
that C must be a constant involved in the relation
Y=ya
.
2
2
2
and we are thus led first to a development of Y according to the
powers of t and to the finding of the relation between the coeffi-
cients. Thus y becomes available in a practical form and C being
determined as a function of Y and its derivatives we have our
relation in a new form
[52]
Y = +VY.
I expand these principles of M. Hermite*) (Annali di math.)
and Halphen**) as follows:
Lamé's equation may be written
[53]
g”= Pu
where
P=n(n.+ 1) pu + B and y=VY.
=
Seeking the equation in terms of Y we write
Y' = 2yy'
whence
Y'' = 2y'? + 2yy” = 2y'? + 2 Py = 2y'? + 2PY,
also
(Y” – 2PY) = 4y'y" = 4 Pyy' - 2PY
y" ' "
=
whence
[54]
Y"" – 4PY' – 2P'Y=0
a linear differential equation in Y of the third order.
From the theory of the linear differential equation, if y and 2 are
solutions of (53) vy+will also be a solution y and q being
arbitrary constants, and we derive also as distinct solutions of the
transformed (54) y, yx and m2 obtained from the complex form
(ry + 92)
P' =n(n + 1) p'u
and the transformed may be written:
[55] · Y''' — 4 [n(n + 1) pu + B] Y' — 2n(n + 1) p’u Y = 0
where
To(a + u) (a - u)
Y=IT
o
ua).
=II (pu
gº a gau
This value indicates that (55) has n solutions in terms of p (u)
:
*) Bd. II. p. 498.
**) Bd. II. p. 498.


Integral as a Product.
35
from which it follows also that Y may be written as an intire
polynomial of the nth degree in t
=
pu. That is
[56] · ... Yt + α₁tn−1 + αtn-² +
-1
-2
·+an-it + an.
Equation [55] is written in terms of derivatives with respect to
u whence to determine the coefficients in (56) we must express (55)
also in terms of derivatives of t = pu and equate the coefficients
of like powers in the two identities thus obtained.
Take
whence
D₁u = q
Ꭰ
and
Du Y DYDut
1
„³; Diu
2
Då
Da Y=
=
D³ Y —
=
d2 Y
dt2
d³ Y
dt3
2
9 = 9 (t) = 4t³ — J₂t — 93 = p² ² u
[57] (4t³ — g₂t — 93)
---
3
u = 19 29; Du
Ф
ዎ
-
=
1
92 D, Y
Du DY - D, Y Du
3
(Du)
These substitutions give:
...
3
1
3
1
2
"
— 9 ¹³ D; Y + 9 ³ q´ D; Y — — 9 ³ 9″ D, Y
2
=
2
2
2
u = ዎ
3
4
5
2
Φ
―
/2
▬▬▬▬▬▬▬▬▬▬▬▬▬
(D¸u)² D³ Y — D¸u D¾ u D¸ Y — 3 D¸u D¾ u D¸ Y + 3 (D² u)² D¸Y
t
t
(D,u)
d2 Y
d³ Y
dt3
+ 3 (61² — 1 92 ) 2 − 4 [(n²+ n − 3) t + B] dr
dY
dt
-
2
dt2
2n (n + 1) Y = 0.
From [56] we obtain the values of these derivatives, namely
dY
dt
1
2
9
= ntn−1 + a₁ (n − 1)tn—² + α, (n − 2) tn−³ + az (n − 3) t”—4
a₂
2
-3
-4
-
3
-
4) t−5 + ..
+α (n − 4) tn−5
n(n−1)!"-+a(n −1)(n −2)!"-sta (n−2)(n−3)-4
-3
+ as (n − 3) (n − 4) tr−5 + ···
"
=
a
n (n − 1) (n − 2) t−3 + a (n − 1) (n − 2) (n − 3 ) -
-3
—
·
-5
-
ta (n-2) (n − 3) (n − 4) tt
and equating the coefficients to zero we have:
3*


36
Part III.
n
3
n – 3: 4a3 (n − 3) (n — 4) (n – 5) – 9,0(n 1)(n − 2)(n − 3)
3
n
5— 92a
— )
Izn (n − 1) (n − 2) + 18ag (n − 3) (n — 4)
+
* , a ( – 1)(x – 2) - 4 (x +m– 3) (x - 3)a,
g(°+ n n
4 Bag (n — 2) – 2n(n + 1) ag = 0
- +
n --- 4: 4a (n − 3)(n — 4)(n — 5) — 920,(n -- 2)(n -- 3)(n — 4)
—
3
93a, (n − 1)(n − 2)(n − 3) + 18a, (n — 4)(n — 5)
- 920,(n − 2)(n-3) — 4 (n + n-3) (n — 4),
nº 0 – 0 – a,
–
4B(n — 3) az — 2n(n + 1) aq = 0
= )
-
3
2
.
-
-3
3
-
2
k =
= n
1
n — k: 4ax (n – k) (– k – 1) (n — k — 2)
– 920x-2 (n – k + 2) (n - k + 1) (n — k)
- 9304–3 (n -- k + 3) (n – k + 2)(n - k + 1)
k 2
1
+ 18ax(n = k) (n -- k – 1)
k
- 920x-2 (n — k + 2) (n — k + 1)
-
- 4(n2 + 1 - 3)(n – k)ax — 4B (n-k+1) ax--1
-
n —
- 2n(n + 1) ax = 0.
-
From the last value we pass to the nth by writing
- M
whence the recurring formula:
[58] 2 (3 -4)(2n + 1)(x + m + 1) a = + 4 (4+1) Ban-4-1
] (n u1u n an.
Inx u 4—1
+92 (n + 1) (u + 2) (2 u + 3) An—u—2
+ 93 (u + 1) (u + 2) (u + 3) an—^—3
from which equation we find the unknown coefficients ai by making
u
2,...
k 1.2...
These results are simplified by employing the notation intro-
duced by Brioschi, namely: :
Set-b:
B. 9(t)=4t— 92-93 = 9. t=pu,
n (2n 1)
by means of which the above forms are expressed as follows:
703 Y
d? Y
[59] [48% + $9"S? +0'S + glas + (1882 +30"S + 9
oʻS +9 9')
d82
[4(n2 + n — 3)S+
-
ds
2n(n + 1) Y=0
[60]
Y Sn + A, Sn—2 + Ag Sn—3+ + An
=0
n
-1,
n
or
1
SC
1
3
2
d
2
2
n?
1
2
9") ay
-


Integral as a Product.
37
1
2
[61] 2 (n – u) (2 u + 1) (u +n + 1) An-M u
= 12 (u + 1) (n + 1 – n) (v +1 + r) An—-—1
1
nu nb u
+ (u + 1) (u + 2) (2u + 3) 9' (b) An—u—-2
+ (u + 1) (u + 2) (u + 3) 4 (6) An—-—- 3.
Taking u 1 we find A = 0
n In
1)
u
2: A2=
8 (2n
3)
nan
nan 1) (n 2)
3: Az =
M
bo'(b).
12 (2n
5)
2 (2n
3) (2n — 5)
= n
I
En
Φ' (0)
2
n
3
9 (6)
=
1) B
(2n
-
And the term containing the highest power of B is obtained as
follows:
и u = n — 2: 2.2 (2n — 3) (2n — 1) ag 4 (n − 1) B
-
(n
or Az
1) (2 n 3)
4 = 0 – 3:
(n
и
–
1) (2n - 5)
(n 2) (n
un - 4:
2 · 3 · (2n 1) (2 n 3) (2n 5) (2n
3) (n
u
+...
2.3.5(2n-1)(2n- 3)(2n- 5)(2n -- 7)(2n-9)
- 3: Az
2) B2
3 (2n
4: 24
3) B3
.
-
7) X
(n
5: A5
4) B4
[62] u = 1:
An-1=
[3
(-1)" B”
5 7 2n
+.
.
.
1]
ya
C
Direct Solution.
Having Y=ys, we are enabled to obtain a rigid and direct
solution of Hermite's equation in the form of a product as follows:
In addition to Y we have:
Y' = ys' + xy'
= yz
and yz' – zy'=2C.
=
whence
2C+Y
2 yz'= 20 + Y',
=
2 Y
and
y
Y' – 20
— 2 zy'=2C –Y',
whence
yy" - y'?
YY" Y'2
y?
Y
2 Y
2
or
2
or
y
2 Y
2
2
y"
(O)=
2
or
2 YY"
y"
Y
Y'? + 4C
4 Y2


38
Part III.
[63]
[64]
.
.
This value in Hermite's equation gives:
· 2 YY" — Y'? + 4C = [n(n + 1)pu + B]4Y?.
Whence we derive the value of C sought, namely
4C2=Y'? – 2YY" + 4[n(n + 1)pu + B] Y.
Y'2
(
Let a, b, p ...=pa, pb, py ... be roots of Y.
=
Then
Yu = a... = " + a, th-1 +
= tn
0
Yu=a.... = nt" – 1t' + a, (n
ntn—1t' + ay(n − 1)tr — 2% +.
(...= 0
0
-1
=
=a.b
or
'
U
Y: -p'2**
4C° =p"?(a) ["Y I_ = Ye = p“()(h) x = Y;...
12
dY
np(un-1p'u.
du
Whence
d Y
dt
and
dY72
d Y/2
:
dt
dt
=
But from algebra we have
dY
= (a — B) (a – v)...
—
dt
Whence
[65] ·
2C=a' (a - b) (a – v)...
a'la
with like expressions for the other roots which we observe are the
values obtain before (see [51]), namely
20
(a – B) (a - y)...
2 C
B' =
(B a) (B — v...
(
= a
a'
Y.
To obtain Y we have:
2C= Ya Y;
+a, +1, +c being the roots of Y=f(u).
We have also:
2C = yz' --- zy'
=>15(u + a) – $(u — a) - 28(a)]yz
15(v + a) – $(u — a) — 28(a)].
n + ) -
&
.
But
p'r
[$(u + a) + $(u — a) — 28a]
pa)
or
20
-
2 (ри


Integral as a Product.
39
whence
С
2 (pú – pa) — śu — $a
]
p'u
-
d
du
- pa
log ou
- į => [s(v + a)
Σ(+ ) -
[logo (v + a) – log Vpu
u
uça]
IT
log II
au log|TVpu -- pa.
d
du
log
e-uζα
o(u + a)
Vpu
pa • си
co(u + a)
d
d
du
e-uga
σι
But
1
1
Пури
TTVpu – pa =
yon
2
2
1
с
Y
d
d
du
()
1
2
log[/ºu + a)
e- wζα
log ya 2
би
du
o (u + a)
d
du
log[]
TI
é uša
1 yz' + zy'
Y?
6U
2
Y'
6
d
+
du
logIT
(u + a)
0(u)
- «ζα.
2 Y
Whence
d
Y' – 20
2 Y
co )
=
e-uζα
log[T•(-+ a)
у
du
σι
or
6
logy
log/T"(u + a)
e-uša
log C
σω
.
.
бо а
C = 16 a.
Whence the value of y is obtained directly, namely
co(u + a)
[66]
y-IT e-uζα
The third method of integration is then the following:
Calculate the polynomial Y by the aid of the relation [58] or [61]
from which derive the Constant C2 by means of equation [64] extracting
the square root to obtain C and finally obtain the constants
20
20
p'a =
(a B) (a y).
(B - a) B - v...
— y)
when a = pa,
B yb ... are the roots of Y.
These relations determine the arguments a·b.c..., having which
the solution is
co(u + a)
y=IT:
If we take the second root of C2 we obtain the integral obtained
also from y by changing u into – u.
p'o
>
-
e-usa.
σω
u.


40
Part III.
Determination of Y for n = 3.
The foregoing solution while complete and rigid from a theo-
retical standpoint needs to be greatly perfected before it becomes
practically applicable. It is indeed but another example, the in-
variant theory being a second of the fact that it is often an easier
task to obtain a general than an explicit form. Having determined
the explicit forms for n 2 let us attempt to apply the above
rule to the next case n = 3.
=
Given n = 3
where
A₂
As
From (60) and (61) we obtain.
-
[67] ·
n(n
8(2n
[68] ·
Hence
•
n(n
12 (2n
:: S3
-
--
Again St b
=
―
=
=
-
Yn=3
:. (t) = 4(S+ b)³ — g₂ (S + b) — 93
1)
--
· 1 (44b³ — 3g₂b + 93).
3)
2)
5)
1
1
g'(b) — — 9′(b) = ¦ (126² — — 92)
4
9 (b)
=
Yn=3 = S³ + A‚S + ø
=
=
2)
n(n 1) (n
2(2n-3) (2n — 5)
-
483126S2+1262S+4b3gSbg293
19 (t) — 3b82 -
---
12bS²
-
48 +12682 + (126292) S+4b³ — bg — 9s
4S³ + 12bS² + ❤´S + 9.
3 =
= S³ + A₂S + Ag
-
- 19'S - 19.
4
S³ + 1 9'S + 1 9 — bq'
ዎ
-
4
1
= S³ + (3b² — — 92) S — ¦ (446³ — 392 b + 93)
-
❤(t) − b(ø′+ 3S²)
bq' b = = q(b) — bq'b
-
1
ø (t) = b[' + 3 (t − b)³]
4
Whence
Y' = 3S² + A2, Y"= 6S, 2 YY":
and substituting in (64) we have
1
1
= t³ — 3bt² + (6b² —9%)t (156³ — gab + 1 93).
=
12S (S³ + A₂ S+ Ag)
C2 (382 + A,)² - 3S (S³ + A, S + A3)
+ 3(4S+ qb) (S³ + A, S + Ag)².


Integral as a Product.
41
To attempt to extract the square roots of this equation in
accordance with the theory, C² being expressed as an equation of
the 7th degree in S or t were clearly impossible without some
further knowledge of the properties of C. To arrive at such know-
ledge we are led ultimately back to a study of the special functions
of Lamé.


Part IV.
The Special Functions of Lamé.
Functions of the First Sort.
>
[69]
.
.
Lamé derived originally functions of three different sorts,
values for y, depending on the value of n and corresponding in
each case to a specific value of B, the chief peculiarity being that
for these values y is doubly periodic.
The functions of the first class are characterized as developable
in the form
Y = pln—2) + ay pine — 4) + azpín—6) +..
and that such an integral may exist is seen from the following:
Writing the corresponding function of the same sort y.p(u)
we have
n(n + 1)yp(u) =pin) + A4p(2-2) + A2p(n—4) +
whence by subtraction
y” – n(n + 1)yp = (a, – A,)pn—2) + (az — A,)pn—4) +
Ву
B : 02
that is a function of the first sort will be a root of Hermite's
equation provided
01 - A
- A2 - Baz: 0g – Ag = Ba, etc.
Az
Where the quantities (A) are linear functions of the quantities (a).
But since the number of these condition equations is greater by
unity than the number of unknown (a) it follows that upon their
ellimination we obtain an equation in B whose degree will equal
the number of equations, that is n +1 if n is even and ;(n 1)
-
if n is uneven:
For example take n = 2, whence y=p + a, and y"=p" and
we derive
p" -- 6(2 + a)p Bp – Bay
or
1
Ba, + 9 = 0,
Ź 92
also
611 + B=0
2


The Special Functions of Lamé.
43
- B
6
1
6
whence
A1
6
and we find
y =P B where B4 – 392 = 0.
B2
=
Again let n = 3 in which case the equation in B would be
-
of degree i(n − 1)
1) = 1, that is B 0, for which value we have
at once
y=p'(u).
Substituting indeed this value in Hermite's equation for n = 3
we derive at once
p" — 12p'p=0
a well known identity.
Define (P=0) equal to the equation in B of degree } (n − 1)
that in any case determines the values of B giving rise to an
integral of the first sort.
We have then that when P=0 the general solution of Her-
mite as a sum has in place of f(u) the p(u) and may be written
[70] · (-1)"y
pin — 2) (u) + h, pin — 4)(u)
1)!
(n 3)!
1
2
a
1
1
in
1
+ họpl2-6)(u) +.
(n 5)!
the coefficients being the same as in the corresponding general
development.*)
Functions of the Second Sort.
.
Q=1. 2.3
To attain a function of the second sort assume that n is odd
and that the solution has the form
[71] ·
Y y=%Vpu – este
la
where may be developed in the form
%=pin - 3) + apin-5) + a,261 — 7)
equation in p differing from (70) in the degree of the deri-
vatives only. Proceeding as in the former case by substituting in
Hermite's equation one finds that the solution holds provided B
be now taken equal to any one of the roots of a perfectly deter-
mined equation of degree (n + 1), the right hand member of
which we will define as Qu which is equal to zero.
1
*) see (34) and (26).


44
[73]
Part IV. The Special Functions of Lamé.
Writing for convenience Hermite's equation in terms of the
derivatives of z with respect to pu by aid of the identity
p²
4p³
92P 93
we have*)
[72] ·
0
or
and (72) becomes
(4p³ — I¿P — I3)
and
whence
=
•
[75] ·
•
= [(n −
and differentiating we have
4 la
=
d² z
dp*
Take now for example n = 3. whence
=
z = p + a
―
dp
+ (10p² + 4e¸p + 4c² − 292) dz
1) (n + 2) p + B − €a] ≈.
•
=
Qa
an equation whose degree is
-
3
10p²+4eap+4c292 (10p+ Bea) (p + a₁)
dz
dp
-
=1
-
a₁ = 1/2 ea - 11 B
αι
10
10a₁+ Bla
z = p + 1/1 ea - 1/1 B
2
10
=
d2z
dp²
B2-6ea B+45e-1592
— (n + 1) = 2,
2
compair transformation p. 35.
=
and as a may have the values 1, 2 or 3 we have in all six values
of B giving a doubly periodic solution of the second sort and
determined by an equation of the sixth degree defined as
[74] ·
Q
Q1 Q2 Q3
0
Functions of the Third Sort.
We have finally solutions that are doubly periodic of a third
sort the integral being written in the form:
y = z√(pu — es) (pu Cα)
--
where n is restricted to an even member and z has the form
2 =
...
p(n−4) + α¸p(n−6) + α¿p(n−8) + ·
and a similar analysis to the former cases shows that this solution
holds when B is the root of a determinate equation whose degree
is n.
-
15g₂ = 0


Identity of Solutions.
Having developed in the foregoing the necessary underlying
principles we return to the case where n equals three, that is to
a determination of the integral of the equation
[76] ·
y"= [12p(u) + B]y
[77]
where B is to be arbitrarily chosen.
The first form obtain from (32) is
y = 1/2 f" + h₁ f
and from the first of equations (26) we have
B
10
•
Part V.
Reduction of the Forms when n equals three.
where
h₁
-
Hence disregarding the constant the integral is
y = f" — 3bf
σ (u + v)
o (u) o (v)
and x and satisfy the conditions (35)
where B
f:
←←
e(x-5v) u
=
[H₂+ h₁ H₁ = 0
Ho
2
156
[78]
where
\ 3 H₂ + h₁ H₁ = h₂
ζα
x = Ev — Ea — Eb — Ec
v = a+b+c.
(p. 17 and p. 16.)
1st Solution.


46
Part V.
[79]
euša
y=17°(u + a)
11. (a) 0 (0)
eu ša
σα σε
The second form obtained from (66) is
u
(4 a)
) u
o(u a) o(u b) o(u c)
elsa +56+$cu
o(a) (b) o(c) 03 0
where
20
a'rp'(a)
(a - b) (a y)
20
B' =p'(b)
(B a) ( y)
20
y'rp'(c)
(y a) (y – B)
and
(= +VY; Y=S% + A,S + Az
S=t—6; 4, = + (1262 – į 92); Ag = – 4 (446* — 39,6 +93).)
bA=
The transformation of form (79) to form (77) may be accom-
plished as follows. Taking the eliments we have
2d Solution.
C=
(p. 40.)
-
4
0 (u + a) e-
1
u
e-usa
σω σα
pa +
2
u
o(u + b)
1
u
e uto
pb +..
(u) ob
u
2
U
o(u + c)
σασο
e-usc
pet..
u
2
whence
0
6
ou + a) ou + b) (u + c)
o(a) (b) (c) o'r
e-(Sa+$6+50) u
Y
1
u
[mt- (pa + pb)+]6 - PCC) +)
(
2
u
2
Take
(11 + a + b + c)
f=
o(a + b + c) ou
0 (u + v) p(x->») u
e-u(sa+56+50)
e— šv
σι σν
1
u
U
2
(pa + pb + pc)+.
- į (pa + pb + pe) +...
.
1
f'=
1
u
2
2
f" =
+
u3
Whence we observe that we may write
y
y=[f"u — (pa + pb + pe) fu].
But
pa + pb + pe=B=31
pc В
1
5


Reduction of the Forms when n equals three.
47
and, disregarding the factor, we obtain the first form:
y = f" — 3bf.
Having then a method of reduction the determination of
abc is involved in the determinate of v.
Determination of x and v. First Method.
To this end we have from (31) and (26)
1
H₁ = ½-½ (x² + P₂);
2
and also
set
H₁ = x;
Ho
h₂
whence relations (78) become
1
B
— (x³ + 3 P₂ x + P₂) —
3
2
ι
-
=
1
5
and the useful relation
B2
120
1
B
¦ (x² + 6P¸x² + 4P¸x + P₁) − ²² (x² + P₂)
2
=
H₂ = (x² + 3 P₂x + P3)
B or
B
10
1/2 x = 0
92
20
h₁
=
ι
3
and take from (p. 24)
P 3
PA
P₁ = 3p² v + $98
+92
P₂ = pv;
pv;
Which values reduce our relations to the form
=
(a) | x³ — 3p(v)x — p′(v) — 3lx — 0
[80]
(b) | x¹ — 6p(v)x² — 4p′ (v)x — 3p³ (v) — 21 + 2lp (v) =
-
=
B2
30
H₁ = = (x² — p(v)), or p(v) = x²
·
2
92
5
572
3
92
which are reduced forms of the equations of condition that y = F(x)
be a solution in addition to which we have the identity
p′(v)² = 4p³(v) — J₂P (v) — I3
2 H₁.
The product of equations (80) is an equation of the seventh
degree in x the roots of which are functions of v and B and hence
the values of B that will reduce x to zero are in number not more
than seven.
But when x equals zero (and v=wa), y is in general a doubly
periodic function and the doubly periodic special functions of Lamé


48
Part V.
are in all seven in number for n equals three one being of the
first sort and six of the second.
It follows then that by elliminating p(v) and p´(v), we should
obtain x as
a function of where is a function of B the
vanishing of which will be the condition for the special functions
of Lamé.
This complicated ellimination, suggesting the practical use-
lesness of this method for any higher value of n is performed as
follows.
Multiplying the first equation by four and subtracting we obtain
3x4-6p(v) x²-101x2 - 2lp(v) + 3p³ (v) = +92
whence the relation
gives
(c)
p(v) == x²
or
2 H₁
2
36 Hi-36bx² + 121 H, + 512 —
Again from (b) and the identity
-
p' (v)² = (3bx+3p(v) x − x³)²=9b²x²+9p² (v) x²+x6+18bp(v)x²
-6bx¹ - 6p(v)x4
3g₂ = 0.
=96²x²+9x²(x² - 4x² H, +4H₁) + x + 18bx² (x² - 2 H₁)
-6bx46x4 (x² - 2 H₁)
-
=
— 4(x6 — 6x¹H, + 12x² H — 8H³) — 9₂ (x² - 2 H) — 93
-
--
or multiplying by 9
1
(d) 817²x² - 108x² Hi+ 1087x¹ — 9. 361 H₁x²+9 · 32 Hi
+992x²-1892 H₁ +993 0.
From (a), (b) and the value for p(v)
x¹ —— 6 x² (x² — 2 H₁ ) — 4 x (x³ — 3 p(v) x − 3 b x) — 3 (x² - 4x² H₁+4H;)
-
21x² + 21(x² − 2 H‚) — 377-
312
3
92
572
3
―
(e)
12lx² - 12H, - 4b H₁
572
3
and multiplying (e) by 3 and 8H₁ it becomes
(f) 36.81x2H, 368H961H4012 H₁-24g, H₁
=
92
whence from (c) eliminating Hi
――――――――
993.
(g) 817²x² - 108a2H+ 1081-361H, x²-961 H} = 401² H₁
-69, H₁ - 992x²


Reduction of the Forms when n equals three.
49
Whence a further combination with (c) gives
(h) 721²x² - 721H - 321 H₂+ 6g₂H₁ +
1073
3
and again
(i)
Whence
[81] ·
where
[82] ·
where
Φ(0)
=
•
812 H₁
392
=
a₁ = 3 9 2 —
4
――
Φ(0)
SD2
12576
69½ H₁
a₁
H₁
=
=
392
4
210 a₁ 74
-210a, 14
-
―
D
4073
27
107³ — 67g½ + 93
1
4
1073
and
b₁
From this value of H₁
we have by substituting in (c)
1
12576
-
x²
22b¸ 1³ † 93 a} 1² + 18 a, b₁l + b² — 4a³
361(1a)
2
S=361,
1
1 ( 1 — k² + k´¹); b₁ =
(1
22
=
3
—
6 (12 — 11 92)
4
-
-
6 (12
4(a)(11739a, l b₁)²
361 (12
a₁)2
ι
993 +8lg₂ = 0.
-
8 al
-
-
a₁)
1
BY
226,1 +93 a 12+18ab₁l + b² - 4a³
(1² — a₁), l =
b₁
27
493.
1
23
+9g3-21g2
-
−
= B
5
27
4
93 (1+k²) (2 — k²) (1—2k²).*)
=
= =
Þ(1)
O is then the condition for the existence of the special
functions of Lamé the seventh value of B, as we have already
seen (p. 43), being B = 0.
(1) must then be Q(7) times a constant and as we have seen
that is separable into three factors of the second degree it
follows that (1) is a reducable equation of the sixth degree.**)
Moreover if we make the transformation
3b
= 0
*) The expressions used here are essentially the same as those of
M. Hermite in his celebrated Memoir. The following reduction of the function
() is also indicated by Hermite.
**) It is interesting to note that it is not given under the head of
reducable forms of the sixth degree by either Clebsch or Gordan.
4


50
Part V.
the coefficients of
variant of the fourth degree
=
α1
[83] · Þ(§₁)
•
[84] ·
bi
1
and we have the form:
[85] ·
[86] ·
=
and c³
Define
4 (1)
210c§¹
+1-4c³ 0.
=
If then this equation be written in its expanded form in terms
of the modulus k it will not be difficult to see by inspection (for
rigorous proof see p. 56) that if we write
Φ
-
D (b3 §)
b2
-
•
D3
=
=
=
all reduce to functions of the absolute in-
=
1 2
these factors of
ظ н г
corresponding to the special functions of the
second sort are, as given by M. Hermite:
Φι
572 2 (k²
2)1 374
Φ — 572 — 2(1 — 2k²)l — 3
-
=
-
3
α
b2
12586
P = 512 — 2(1
=
=
When O we have x
O whence, as before stated, Ø 0
is a necessary condition for the existence of a doubly periodic
function. But in order to be a sufficient condition it must involve
a definite value of v, that is v must be a half-period. That this
is the case, although the reverse as we shall find later does not
hold, is seen by a determination of v as follows:
We have (p. 47)
p(v):
x²
=
-
1 92
108 2
93
=
-
-
Whence we write
p(v):
Returning to (80, a) we have
k2 sn² w
(1) 127 (12a) (1073-
516+6α₁l
(1k2k4)3
(1) (2 k³)* (1 — 2k³)*
·
―――
2 H₁
Þ(1) — 127 (12 — a₁) (10788a₁l-b₁)
367 (12a) 2
xx
--
+ k²)l — 3(1 − k²)².
8 a₁l — b₁)
10b,1-3a² + 6a₁b₁l + b² — 4a³.
-
22393c²² + 18c§
1+k²
3
187(1
a₁) 2°
―
p' (v) = x(x² — 3pv — 31)
-
367 (12
4(1)
¸P(7) — 3 4 (7) — 10872 (72 — a₁)²
Þ(1)
-
-
-
XC
367(1a)
a₁)2


Reduction of the Forms when n equals three.
51
[87] ·
[88].
Where we define
Where
or
y
.
x = // [Þ(1) — 34(1) 10812 (12 — a₁)2]
= 16 — 6a, 1² + 4b₁13 — 3 alb² + 4a³
=A.B.C.*)
•
Refering then to note (p. 24) we have:
k¹su²v · cn²v · dn² v =
p' (v) =
=
A = 12 — (1 + k²) — 3k²
72
l
་
= 2
B — l² — (1 — 2k²)1 + 3 (k²
12
C = 12 — 2) — 3 (1 —
(k²
That is p(v) vanishes where x vanishes
a semi-period, and in consequence, when
e-us (w₂)
――――
―――――――
――――――
=
Hence
1
y = [2p(u) + ca − ¦ (1 + k²) —
ea
f₁
o (u + w₂)
ou o (wx)
The value of the function of Lamé corresponding to any
value of B giving rise to the condition
= 0 is then deduced as
follows.
From P
O we derive:
-
B = 5l = 1 + k² + 21/19(1 — k²)² + k²
and the special equation of Lamé becomes
y' = [12p (u) + 1 + k² + 2 √19(1 — k²)² + k²] y
-
=
· [2 pu + e − / (1 + k²) −
—
5
and from the general form of the integral [77]
1
y = f'{' — } { 1 + k² + 2 √/19 (1 − k²)² + 1;² } f₁•
But differentiating f₁ we have
f₁ = [2p(u)+p(w₂)] fi
[2p(u) + ea] f1.
―
-
· -
— k²)
k²).
.
x(1) · x
187 (la)
=
= 2 [p(u) + 1 la − ¦ ¦ B]Vpu — ea
10
°
which gives v w₂
= 0, f reduces to
α (u)
6(u)
•
2
(u)
¦ √/ 19 ( 1 − k²)² + k²] a)
5
V pu
la
where
(p. 44).
*) Compair [161] p. 73.
has the value determined by the elimentary consideration
α=1, 2, 3
(w)=n₂
3 √19(1 − k²)² + k²]Vpu — Ca
4*


52
Part V.
If x
Case
X
= 0.
( we have a second case in which the p'(v) vanishes,
v taking the value of a semi-period, but as this may occur without
reducing x to zero the eliment will not be doubly periodic since it
will contain an exponential factor eit u. If then
x = 0) we will
have from (87) six values of B for which the integral will take
the form
o(u + wn)
y=f- Bf where fa
e+ (x − ¢wn)
2
σε σωλ
ба и
u
exu.
σα
Moreover the second integral will be
са и
f
-mu
6U
a
or
4
3
the form remaining unchanged which is not as we have seen in
general the case.
Case D=0.
The only remaining case to be considered is where D = 0, or
72 — ay = 12 – 1+ ka - k4 = 0
: —
l=+(1 — "* +:$4)% = + V39,
— ka +
since aga = (1 — k+ k4). Also l = 36 whence
b=V%
12 62
92 = 9'(6)=0.
That is D = () and g'(b) : 0 are conditions for one and the
same function of Lamé. In this case p(v) and also the p'(v) become
infinite which gives v= 0 or the congruent values 2 mw + 2 m'w'.
The general form of our integral will not hold for this exceptional
case and we are obliged to return to the treatment of the subject
from the standpoint of a product.
-
92
3
2
or
Relation of Y and C to the Special Functions of Lamé.
Returning first to (Part IV, p. 42), the elimentary determina-
tion of the special functions of Lamé, we there found with reference
to B that, first, if n be odd, it is determined by two sorts of
equations, one of degree (n --- 1) giving rise to functions of the
>
1
2


Reduction of the Forms when n equals three.
53
2
3
1
>
1
2
v...
first sort, and the other, three in all, of degree (n + 1) giving
rise to functions of the second sort; whence combining we have, n
being odd, B determined by an equation of degree (n+1)+(n-1)
= 2n + 1. If n is even we find but one equation, degree n +1,
+
for functions of the first sort and three equations, degreen, for
those of the second sort making a single equation whose degree
as in the first case is 2n + 1. If then these roots are all different
we have in all 2n +1 special functions of Lamé.
Returning now to the forms (65)
2C = a(a – B) (a — »).
a'la (
we have the half periods or values of the roots a, B that will
reduce them to zero. Moreover they will not be double roots, for
consider t en as a double root of Y in which case all the terms
of equation (57) will reduce to zero save the second which will
be identically zero, which is a condition that the root be tripple.
Differentiating we find an analogous equation and a similar course
of reasoning shows that the root must be quadruple and so on
which is absurde. Hence the roots that are half-periods are not
double. On the other hand any other root of Y may be double
but as a similar course of reasoning shows it could not be tripple.
If then C=0 all the roots will be double unless they are
semi-periods and we may write
[89] ·
Y= — *
(pu - e) (pu - ex)'(pu — ez)?" II (pu - pa)?
e
.
whence
.
.
င်
[90] : · y=V(pu – e)'(pu- - ey)" (pu — ez)"" II (pu - pa)
where
E,
', 0 or 1.
But this form we observe at once is that assumed in every
case by the special functions of Lamé where we found y always
equal to a (
polynomial in p(u) times some one or more of the
factors (pu – ea)%.
ex%
That is C =0 is a condition that the integrals be the special
double periodic functions of Lamé.
By a transformation similar to that on p. 35 we may write
equation (64, p. 38) in the form:


54
Part V.
4 c* = (4 ** – 9:t – 9:)[(a) – 2 ro] – (12ť – 9.) Y Y
;
)
2 n
.
as
d
dY
dY
2 Y
dt
dt
dt
+ 4[ ( + 1)t + B]Y?
and we have (62, p. 37)
(- 1)"B"
Y
t...
[3 · 5
7
1]”
from which relations we see that the highest power of B in c is
2n + 1 and that the condition C= 0) gives rise to an equation of
O
the 2n + 1st degree in B which is as the number of the special
functions of Lamé.
Refering to (68, p. 40) we see that C² = 0 has been found
an equation of the seventh degree in B as required by the
above theory.
Functions of the First Sort.
Following the notation of M. Halphen designate by P the first
member of the equation that determines B corresponding to func-
tions of the first sort. Refering again to (Part IV) we observe
that if n is odd each of these functions contains the factor pu.
For example we have:
n = 3: Y= p where
: = p where B=0,
the degree in B being unity.
n=5:y=p" – Bp' =p (12 p — B) where B’ — 2792 = 0
yp
'
the degree being two, etc.
But p' (u)=4(pu - e)(pu — ez) (pu – es) whence for n odd
--
or equal to three, &, é, " are all equal to unity.
Moreover we have obtained Y (67, p. 40) expressed as a poly-
nomial in t and b in the form
Yn=3 = * $(t) b[g' + 3 (t — b)?]
– 0
)]
and since
p' (en) =ť (e) = 0
we derive
[91] ·
Yn=3(en) b [0' + 3 (en — b)].
Hence Pn=3 =B=15b is a factor of Yn=3(en) times a
3
constant.
If on the other hand n be even none of the functions of the
first sort contain a factor Vpu en and Pn=2x will not be a
factor of Y=%(ea).
3
3
E
2
1
en
.
=
n2x


Reduction of the Forms when n equals three.
55
1
2
1
2
>
•
08
•
2
2
1592
=3
2
=3
9(b)
n
Functions of the Second Sort.
We have found three equations each of degree (n + 1) or
n as n is taken odd or even, that give values of B that, if n
be odd, correspond to functions of the second sort, or, if n be
even, to functions of the third sort. Designate the first members,
by Q1, Q2, and Q3. Refering again to Lamé's special functions
we see that if Q1 O the function of Lamé corresponding contains
the factor Vpu - & if n is odd and the two corresponding factors
Vpu - €, Vpu – ez if n is even. In the first case Q. is a factor
ез
of Y(21) and in the second case of Y (ez) and of Y (Cz), while in
the second case we have also Y(e) contains the factor Q2 23.
Returning to n=3 we have (see (73) p. 44)
- p
[Q1]n= B2 - 64 B + 45e,? — 1592
[92] ·
[Q2]n= B2 6 ez B + 45 e,
[23]n= B2 6 6 ez B + 45 ez? — 159,
or in general writing B 15 b and o
468 — 92b — 93
--
[93]
· [Qı]n=3 = 32.5 [9' + 3(e2 – )?].
-
Also from (91).
[94] ·
Y(ez) b [g' + 3(e – )]
- )]
b [15 b2 + 30, 6e4b — 92]
ве, В
+ 3e;" – 9
cB [B?
6e, B + 45e," – 15 92]
g
cʻQ1P
where in general
[95] ·
The quantities Q are also necessarily the functions 0 times a
factor as is shown by taking the substitutions
ei (2 — k“),
k“), O2
(2k” – 1), ez
ез
(1 + ka),
gº (1 – k2 + k)
ka
whence:
[Qi]n=3
B2 (2 – k2)B-
[Q2]n=3 B2 À (1 — 2 ) B
ka
3 : 5 (1 – 12)
[Q3]n=3 B2 À (1 + k?) B
2
В ГВ?
15 L 15
15
с
1
с
3
. 5
2n
1
1
31
32
32
4
32
2
2
ܘܬ ܐܚܝܢ
<
5 · 3 K4
22
5. 3
22
2
=3
2
-
2
22


56
Part V.
,
n3
5 03
Q1 Q2Q
1
3
S
3
1
108
2
Hence making a = constant, equal 1 and B-57 156
we obtain
[Q1]n=3 50 = 5 [512 — 2 (k— 2)1 – 3]
ki
[96] [Q2]n=3 50, = 5[512 – 2 (1 – 2 k)1 – 3]
2
[Q3]n-S
5 [512 -- 2(1 + k4)1,– 3(1 — kº)?].
–
Hence also:
[97] Q = k1122: = 50 (1) = 5° 0, 0,03
)
5
: 53 [4 (72 — a) + (11 73 – 9al - 6.)']
b?
53 [125 G 210 0,84 - 2258 + 93 c252 + 18 C +1 - 40,*]
c
etc.
where
ai
(1 — kº + k4)3
q=-=108
b, 2
93
(1 + ka) (2 - k) (1 - 2 k):
kº)2 – k22
We have moreover that the conditions that the integrals be
special functions of Lamé are that Q1, Q2, Qg and P vanish. But
C2=0 was also found to be a condition and we note that the
sum of the degrees of Q, and P is equal to the degree of Cwhich
Qa
equals the number of the functions of Lamé. We must have then
the relation
c' Q1 Q2 Q3 P.
But we have shown that the highest power of B in the de-
velopment of 4C2 is (p. 38)
4B-B2n
402 = 4BY? +
+
[3 · 5 (2n
whence
1
C
[3 · 5
(2n
which for n= 3 gives as before taken c
We have then in general
[98]
C2
C4 PQ1Q2Q3
and when n=
.3
[99] ·
C2 QP=3175 PQ.
(15)
34.5
If then we take
Q, =0:B=3e, +134–12c; +592)=h? — 2+1(162 — 2)2 + 15k4
=(-12e -
y={p + 1 - 1 (3e, +63(592 – 12cz))} Vp – en
p
en
ei
={P+ is (— 2) + "V (k2 — 2)*+ 1544} Vp- }(k? — 2)
ka 152 -
C2
1)]*
A
.
1)]4
1
.
3
5
.
1
2
10
1
15
10
3


Reduction of the Forms when n equals three.
57
-
1
2
10
ez
1
15
3
1
2
10
1
15
5
3
[100] (2=0:B=3e, +V3(592 — 12e,) = 1 - 2k+V(1— 2k?? + 15
Q2
)
y={p + e, - 1 (3e, +13 (59. — 12cz))} Vp — €,
={p+1 (1–22) +-V(1—2k+)? +15}Vp- }(1–21:-)
Q3ez V3 ;
=0:B=363 +13 (59. – 12c) = 1+**+2V(2 - k*) — 3k
- km
y={p + 5e3 – (3e3 + V3 (592 — 12e;)} Vp - e;
y
)
={p+ (1+2k“) +5V2 — :2) 2 — 3k} Vp -- 1 (1+%*)
+
all of which are special functions of Lamé of the second species,
the general form being
9
Vpu –
where
p(n-3) + a,pin—5)+...+c,
and as given (p. 43) the general form for n= 3 including the
above is
[101]
Y
(+ - B) Về -
)
where
B=3ea + V3(592 – 12c).
la
Ca
ea
10
The Discriminant of Y.
a
From (65) p. 38. we have
2C = a' (« — B) (a — »)... = B'(B – a) (B — y).
v
7.
=V9 (a) (a – B) (a — v)...V9 (6) (8 — «) (B – v).
-
(B a
where
9 (a) = 4(pu – e) (pu – ez) (pu – es)
Y = (pu - e)(pu — €)*'(pu – ez)*" II (pu - pa).
. eg
”
The roots of
9 (a)=0
en, ez, ez.
The roots of
Y=0
C1, C2, C3, Quß...
Whence the resultant of q (a) and Y written as the product of
the differences of the roots is
R=1l (a — ea), where a=a,B,.. to n letters and 1=1, 2 or 3
II
n a
[(a — e) (a – e) (a — e)] [(B -- e) (B – en) (B – ez)]...
ΠΦ (α).
(
are
are


58
Part V.
1
4"
a.
2
2
-c
:
1
[](« — c) = 119(a) = (– 1)"]] Y(en)={**,
n even.
But again
Y(e) = [(a --- e) (6 -- e) (Y --- )]...
e
e.
Y(e) = [(a — ey) (B — ey) (y = e)]...
.
Y(e) = [(a – ez) (B – ) (– ex)]. .
e3
whence
R-I[(« —e) = [() – (-1)"]]Y(c).
ΠΦ α) [TY
Again we have shown (94, p. 55) that for n = 3; and the
same method gives in general for n odd:
n odd: Y(0)=– cº PQ; Y(en) = -6% PQz; Y(e) = -6 PQ:
e. - -c) -
and likewise
n even: Ye) = c*Q2Qz; Y(e) = cºQ3 Qı Y(0) = cQ.Q2
(+
Whence we finally derive
[102]R-II
1cm PQ, nodd
Now the discriminant of Y equals the product of the squares of
the differences of the roots and may be written:
A=(a – B)'(a – v)...
whence from (65)
22 C22202
22n cºn
A?
9(a) (b)
II 9 (a)
But we have first found
II 9 (a) 4" R
whence
Can
A2
R
Again
C2 =PQ
(from 99)
and we derive from these n being odd
C2n (C)"
c4n P"Q"
A?
c2(2n—3) Pn—3 Qn-1
R
R
c PsQ
Q.
2
n
2
-1
or
n-
n-3
n-1
A=(-1)"7*cºn-5 p
3
3
2
:n odd
2 - Q
[103] and in like manner we derive
(Sign ambiguous)
=(-
1) ? can—3 P 2 "Q : n even
and we have also A ( since Y has at least one double root.
1
1
1.
n
n-1
a


Reduction of the Forms when n equals three.
59
1
43
as
1
3
153
(see (94))
Case n = : 3.
[104] R=(a —e) (a—ex)(a — ez)(e)(B-ex)(-es)(y-e)(y-2)(y--es)
- ( -,q – ) (– (0
920 93) (23. 92 B- 93) (73 – 927 – 93)
63
13 [9' + 3(e1 — b)] [9' + 3 (0, -. b) ] [9' + 3(ez — b)?]
?e ?
]
(3) 3
[105] 4.
PºQ
(15)
Q
21 22 23
38[' + 3(ez – b)] [' +.3(e—b)?] [9' + 3(en -- b) ]
9
which for the special case n = 3 furnishes the interesting relation,
Q differs only by a constant factor from the discriminant of Y.
Remembering that a has been determined equal to (- 1) we
have from (97)
Q 53 [4 (72 — a,) + (1113 — 9al — b)”]
and the relations:
1 = 3b:a, = * 92
b = *93 : A,= 1 (1262 — 92): Az = (4463 — 392b +93)
a)} = 4.27 A%: 1113 — 9a, 1 — b) = -- 27A,
3
· ?
[106] :
Q = 38.53 [4 A3 + 27 A}]
[107]
A = [4 A3 + 27 A3]
which latter value we would have derived directly from the form
Y = $$ + A,S + Az.
S +
Writing A, = ' and Ag = 1 g — bo'we
A 9
–
derive still
another form for Q namely
*1108]: Q [9'3 + 2792 – 8.27b90' + 16.27620'2].
3
4
ܝ
27
1
4(1
.
1
4
4
(15) 3
16
Again we find
Φ (0)
X2
367(72 — a)
2
4233
36 · 38b9'
-
[109] ·
(VA
2
39'
.
2
2
- {
V 44% + 274
"
39
b
from which value we again see that the vanishing of g' is equi-
valent to the vanishing of D.
(compair p. 49 and 52.)


60
Part V.
V
Pi(u)
Ao
6 (u
2
Determination of x and v. Second Method.
We have the general theorem: every rational function of pu
and p'u can be written in the form:
A. (u - v.) (u -- vy)... (u – »)
o
6 - 'M
vi') o (u ve') o(u )
where the number of o functions in the numerator equals the
number in the denominator, making the number of zeros equal to
the number of infinites. The reverse theorem is also known and
we may write:
) o bc
[110](-1)”k, (t – a) 0 (u — 6) (u — C)o(u + v)
0
oa ob oc (ou)
(pu) — ac p'up (pu)
20
where Q and I are intire polynomials in pu and p'u, k, a con-
stant to be determined and the relation exists a +b+c=v.
Also, from the general theory, the degree of the right hand member
is four, p (u) being considered as of the second degree and p' (u) of
the third. The degree of $ and Sare thus determined as follows:
Φ
Y
(n + 1)
an 3)
-
-
1
1
n odd:
2
1
n even
n
n
1
2
2
n
1
.
a
3
0.
The n roots of the first member in the general case being
a, b, c... we have:
[111]
0(a) — 20 a' F () = 0
Φ (α)
a
where
a'rp' (a),
a =p(a):
From (p. 38)
dY 20
(a — B) (a - y)...
dt.
whence
a'=(ai),
dY
and [111] becomes
dY
[112]
[
= 0.
dt
Jt = α, β, γ,
But a, b, y,... are also roots of Y, whence the relation
dY
[113]
- =EY
Y
where E is also in general an intire polynomial in t whence
Φ dΥ
Y
[114]
Y dt
Y
1
dt
20
t = Q
v]_
dt
-E+


Reduction of the Forms when n equals three.
61
We have also
dY
dt.
Y
= 0
tB
etc. for the other roots of Y. The degrees of [114] are
Y
Y
Φ
1
1
n =
2
2
2
n odd:
(n − 3)
(3)
(n + 3)' 1 )
(n + 3) — 1
n even: § (n) - 1 -- =-(; n+1),
n
, (; n + 1) — 1.
1
We have
1
n :
2
2
C
Y = t" + a, ta-1 + a, tu-+...+ an-1t+ an
Y
Y' = ntn-1 + (n -- 1) a, tn -2 + (n − 2) a, tu-s +
0 2) a, tn^3 +...+ An-1 .
and
Y'
nth-1 + a, (n 1) t»—2 +.. b. bi b
+ + en + +
Y
to ta, th-
t.
-1
1
t
+ a, th—2
or
-1
-
-
1
-2
2
-3
n'
ay (n
ai
b2
nta-1 + a, (n − 1) {r^2 + az (n --- 2) {n—3
- + (–
+=b, (in-1 + antr-+ ...) + b (tn–2 + axtn—+ ...)
and equating the corresponding coefficients we obtain:
bo
- 1)
na + bi or
[115] ·
2a2 + ai
bz 3az + az az
ai
etc.
Proceeding in like manner we write:
Q = B, t + B, t"-1+...
" . + Bn-1t+ B,
where
v = [] (n + 1), į n]
=
whence
be
or + + + ..
+ ...)(B.t" + B,8-1 + + Bt + B)
bo (B,t=1+3,8-2+...+By-1+
)
+b, (Bot”-3+ Bt-3+ + B-2+ By-17-1+ B,t2)
+ b2 (Botr=3 + B, tr–4 + ... +
2
>
2
b
1
be
t3
t
ta
BN
0
1
-
t
-2
1
-
V
-3


62
Part. V.
0
V-
-3
0
V -
V-2
-3
-1
-1
and
6, B,+1+b,B,-1+b, B,-2t+b, B,—32 + ... +b, B, tv-2 +1, B, tr-1
+6, Beta + b, B,-1+1+bB,-- + 6, B,-st +...
+
+ br-1B,tv + br-1B,-1t(n-1) + ...+ br-1
b. B,t—1 + b, B,-1t" + b, B,-2t+1+ + bB, t2v-2
+ b B.t2v-1 + b, B, t"=2 + 6, B,-14"-1 + b BY-24 + ..
from whence the relations:
b, B, + b, B,-1 + 1, B,-:+...+b, B = 0
[116]
b, B, + b, B,-1+ b3 B,-2+. + bx+1B, - 0
--2
-
0
-1
22
-
-
V-2
V
0
V-
-
+ ber-1B
= 0
V
1
br -1B, + b, B,-1+ bx+1B,-2+
We will define:
bob
b, b, b,
by be bg
[117]
...bm
... bm+1
om
т
dr-1
2 •
-1
>
bmbm+1bm+2 bam
We will define
B =
and we will then have from the above conditions, all the coeffi-
cients B, B, ... as intire functions of b,bı ... which are in turn
functions of an, a, ... which finally are expressed as functions of
B, 92 and 93.
That is we have obtained ø, of which the first coefficient
shall be dr-, intire in terms of t, B, 9, and 93:
92
Case n
- 3 we have:
from (p. 36)
B'
M
2: 2.1.5. 6a, + 4.3B=0 or ay
2 B2
93
u = 1:
A2
3. 52 4
B3
93
M
0:
03
+
3252 3.5 4
and from (115)
tn-1:
tn-2: a, (n − 1) = b,aj; by
:
ay
--
th-3: a, (m - 2)
2) = b, a, + by an + ba; b2
b, - 2a, - ba
—
tn–4: az (n
3)=+ ba,
b, az + 6,4, + b,a, + b3; b3 = 3a,4,- 3az - a
-
5
B92
n =
bo
n


Reduction of the Forms when n equals three.
63
S
0
2
2
=
2
62
,
0.
3
3
3
-
12
6 A,
2
2
2
The conditions (116) become:
b, B, + b B. + b, B. =
b, B, + b B. + b3 B=0
=
whence
(bob, — bî) B. — (— b) B.
,
–
64bz
(1,5, -- bi) B = (1,62 — 6,63) By.
But
B. = b,by --- = 9:- B’ =9:— 181=
=
B2
92 —
-0
whence
B,= (-a) (3a, 0,— 3az — a;) – (4a; — 4a;az + af)
a a2 + 3a; Qg — 4a
- 19
g
= 32.564+* 9962 +93b- g
B=(-a) (a} – 2a) — 3 (3a, 0, – 3az - a;)
— )
2a; – 7a+ 9az
3?g
2
B4
32.53
92B2
+
1
+
393 B
3.4.52
4 · 5
4
3
9
1
4
4
4
7B3
3. 53
+
9, B
4
4
9
15
4
926
9
15
4
4
3
9
1
2
bg
32763+ * 995 93
-- 943 + 12b A,=;9 — 6bq'.
Az
We derive then finally
Φ
Q=Bť+ B,t + B,
6 (36? – À 92) (S. +268 + B2)
+(6363+ * 996 – 193) (S +b)
+ 32.5264 + *99b2+ i b93-3 gå.
Ꮽ+
Coef. S is – 6 (362 – 1 92) -
9(- 1169 + $ 936 93) – 943
So
– 4 (362 – 192) = – 4 A
Hence
[118] Φ
- 362
6(31 – 1 92) S? +9(--116+ * 936–193) S-4(362–192)
Å 92b —
6A,S?+ 9 A,S -— 4 A
9A3S 44
Having obtained ø, the calculation of Y and E is simplified
by the following considerations:
642
4
3
S is
4
1
2
442
3
2
4
4


64
Part V.
[119]
―
-
1
Let n be taken odd and take for B a root of the equation
Q₁ = 0. In this case (see p. 54) we have Y as a product of t
by a polynomial U² where U has the degree (n 1). Moreover
U enters as a double factor and is therefore also a factor of Y',
whence, from the form
2
=
Φ
Ψ EY
=
1
-
we find that U must also be a factor of T. This, however, we
know to be impossible since the degree of U is (n − 1) and
that of only (n − 3) (p. 60).
1
2
dY
dt
V of degree n where
2
consequence one has
The only conclusion possible then is that contains a zero
factor. We know also that B being any value whatever, ¥ con-
sidered as a function of B contains the factors Q1, Q2 and Q, and
it follows that we may write
Hence we write:
n odd:
T
where Q0, if B be taken as a root of Q₁ = 0, Q₂ = 0, or
0, and fo (t).
=
n odd Qo
By a similar course of reasoning we show that if B be taken
as a root of P, n being even, Y will be the square of a polynomial
1, and that in
n even:
1
-
is only of degree n
Yn even
Φ
dY
dt
dY
dt
=
ΡΘ
Q0=
PO
=
-
ܡܘ
e1
EY
EY
...
where all the functions are intire in t.
As we have before determined the first coefficient of is the
determinant d,. and in like manner we find the first coefficient
of T to be
v-1
[120]
8,b,B,+by+1By-1+
Hence if we divide d, by Q, n being odd we will have y, the
first coefficient of .
+bạ, Ba
ν


Reduction of the Forms when n equals three.
65
2
1
2
1
2
To find E, n = 3.
The degree of 0 is (n + 1), the degree of Y is n — 1 and
1
the degree of T is ;(n − 3) less than Y'. Hence from the rela-
Ys
tion on p(64), the degree of EY must be
1 (n + 1) + (n − 1) = { (3n – 1).
1
But the degree of Y is 'n and hence the degree of E is į (n − 1).
We have then
[121]
En=3 = = nt + ni
and reduces to a constant, namely:
[122]
We have:
Yz S + A,S + Ag
Y;
3 S2 + A2
6 A,S? + 9 A,S – A,
and substituting we derive
(382+ A,)(-64,S?+ 9 Az S-4 A,*)=(n8+ n)(S3+ A,S+ A2)+pQ
and from these we have
7
n3
P Q
Φ
3
η :
18 Ag;
mi
27 Az
ER
-
2
1
2 c
1
Ф.
A
c
20
whence
[123]
9 [2 A, S — 3 A3].
Returning to our original form we find that when n is three
we may write:
[124] (— 1)"", "(« — a) o co chocolate)o (16+») = (pu)
u (u — b) (u — u v)
Φ()
ga ob ()
p'u F (pu)
0-2 S'YQ=(-6A, SP+9A,S—44,9+, CS'(44,9+274,2).
+9 )+
.
Having this development, the determination of x and v is made
possible as follows: – Taking the derivative of the log. of the
first member, A, and developing according to the powers of u we
write in general
A = [$(u — a) + $(u -- b) + $(u —c)...$(1 + v)-(n + 1)&u.
c, [૬ (
au
But the developments are known:
$(u + v) — gu = $(v) - - up (v).
v ६0 v)
$ (u – a) – $(u)
६ (
ça
upa
1
u?
0)
p'v..
u
2
uz
pa...
w
2
1
& (u — b) — § (u)
६०
upb
u?
pb...
2
U
5


66
Part V.
and we may write
A ($у — a — 0 — с. :)
n+ 1
(pv + a + B +p+..) u
u
22
(p'v + p'a + p'b+..)...
+ + +
2
But
=X
2
-
U
2
Су — $a — fb — fc - and p'a + p'b + p'ct..=0
(see pages 25 and 45), whence
n+1
u ?
[125] A
+ x (a + B +pt. + pv) u p'v t...
The degree of Q is (n + 1), of S', 3, of , (n — 3), and
'
of p', , which gives the degree of the second member as (n + 1),
also
+ whence pero
pā
+
+.
1
1
2
2
3
1
2
1
1
(n+1)
1
1
ри
2
u
2
u
nt
unti
and developing the second member (B) we write, disregarding the
constant factor
1
B =
+ 21
+
92
u”
+
93
2
t..
n
n -
1
N-
unti
u
U
whence
n + 1
nai
1)
92 (n
(n
[126]d log B
2) 93
unt2
unti
u"
un-1
1
+
91
u"
+
92
n-1
+
93
n - 2
u
+
unti
un-
1
2) 93 u3 +.
(n + 1) + nuqi + (n + 1)92 u+ (n
1 +au+ 9. u^ + 93 23 +
u
n + 1
+4+(2qp-q°)+(3q– 3q, q+g°)(*+...
U
.
Again:
1
1
Botza
1
(n+1)
(n-1)
+ Bit? t.
=yt (n-3)
(n −5)
?
+
p = (
(4 të — tg2 + 93)
t..
+ Y1
t2
2
u3
whence
1
1
1
(n+1)
B.t2
B=(5,60+P+By tě (n="+...), 2 (46–194–95)"btico-
__
:(
n-1)
(n
3
BE
-5)
+
Lytzen
+....
20
Bo
B
Qx1
+
Qy
Cu"
+
+
an
unti
un
- 1)
+
Cun - 2


Reduction of the Forms when n equals three.
67
and
2
Qv
127]d log. B=Bo[ – **1 + $8+ (2B, – QX*)u+($%! _ 3QYB,+B;')]uº+..
+1
B
Qy ?
C2
?y
1
C
Qy
СВо
CB'
0
X =
CB.
2
2 B,
2
2
6 Q.
+
2
3
(
21
1
γ
X
c"B,
From developments [126] and [127] we find
B
Qy1
128]
91
; 92 ; 93
n being odd,
B.
and from developments [125] and [126]
Qy
91
Q?y ?
129] {a + B +y +..+ pv) = 9,’ – 292
CB,
B.
6QyB
2 Qy
p'v = 2 (34192 – 393 +91%)
сво? CB C'B'
These fornis are transformed by the aid of the relations
C = (VPQ (p. 56); (2n — 1)(a + B + y +.. ) =B (p. 29)
B
(2n – 1)a, =-B (p. 36) whence (a +B+r+...)=-a,
p-
giving as result:
Oy
Vio
Q
n odd.
св.
P
Qy 2 B, B
pv
c PB.
B
2
3 y B
p'v
130]
Le PB.
}V?
B. B.
and from these the combined forms arise
pv
Qy? 3B
+
CPB B. y
B
B.
1
These formules are perfectly general for n odd and the corre-
sponding forms n even obtained in like manner are
Q
Py
P
C4 QB. 27.
B
pv
Py?
B Q
p'v 2 c2
}
Py8
P
p' v
2 B
B
2
2n
1
I
3 Yi
+
3
2
P
371
23
2
p'o
BY
3 Y1
+ pv
2x
2 n
СВо
c2B.
х
V
n even.
n
2 n
131]
1
C4 QB.
3
0
3
3B, 71
+ 3V
2
71
+ pv
2 x
Во
y
2n
-
1
The superiority of these forms over those first derived, showing
as they do at a glance the synthetic relations, is unquestionable
5*


68
Part V.
and the explicit forms for our case n equals three and also for n
equals four and to some extent for yet higher values, are obtainable
with greater easy than by the first method. Even here however
the forms increase in complexity so rapidly that n is practically
restricted to the lowest values.
>
3
2
For case n = - 3.
We have found all the eliments except 7ı which is derived
from development of ®, or more easily as follows.
From (106, p. 59)
Q=(15)* [4 4 + 27 4]
and from (p. 65)
Qy (38? + A) - 6 A, S2 + 9 A,S – 4 12)
4
+ 9(2A, S - 3 Ag) (S8 + 4,8 + 43)
3 3 A2S
- (4 + 27 A)
and a comparison gives immediately
1
(132]
(15)
The other values for the eliments have been found, namely:
Yn=3
2
3
2
r
3
2
1
с
P=0
o'= 1262
15
92
1
P
156
1
4
7
- 30
3
A,=
=
A = 19 – bo'
463
3
-92
a
4
2
B =
,=
B=9
o'
6bg'
9
27
99
b92 — 93
ba
4
4 93
We have then for n equals three
VA
[133]
Q
(15) 1/(15)9 (4 A3 + 274)
)
+
P (15)3 39'
V
-2
c2B.
156
0
3
2
V
44% + 2743
(compair 109, p. 59.)
3 Ф
1
3
1
19"+ 2792
(
2
8(27)b90' + 16(27)629
b
Myny
60'
Squaring we have:


Reduction of the Forms when n equals three.
69
for
Tere
T
Tea
MISS
E
[134].
[135].
whence
[[136] pv — b
[138]
whence
[137] pv
Again we have:
Qy2
CPB
x²
where
pv
―
pv
=
Writing
ዎ
272
108b❤❤
=
=
'3
1
3
3
2
4(36² — — 92)³ + 27 (116³ — — bg. + 193)²
--
4
4.
4
/
4 (1² — a₁)³ + (1163 - 9a,b,—b)2
361(13-a)2
$(1)
361 (1a)"
Φ, Φ. Φ.
1
3
367 (la)
pv =
=
-
= ―
'3279² - 108bqo'
36bq 2
1
2
36b (30 — — 92)
*
-
1
9
-
4[4 43 + 2743], 4(† 9 — 6 b 4
+
99'2b
39
= Ω
2B,
Bo
-
―――
'3
2
4 [1, 9'³ + 27 ( 17, 9² - 1 bq q' + b² q' ³)] + 10° q q′b — 72b² q¹²
--
16
16
2
4
99'2b
Q1 Q2 Q3
53627 (12 a₁)²
Again from the first method
1 + k²
3
―――
-
=
B
2n
and ' in terms of 99 and b we have:
€172866. 432b¹g₂+ 36b2g2 — gå
36bo
-
43266-216b¹g₂+27bg-216b³g +27g3+549293b.
·5184b6+1728b¹g₂-108 b² g²+1296b³g — 108bg29 3
21606 +2166*g₂ + 1080b³g — 9b² gå — 54b9, 93 - 93 + 27 g
36b (144b24b²g + 93)².
12
=
――――
PQR
SD2
3b
k2sn² v
(compair 82, p. 49.)
etc.*)
2
¥=5(3b)6+6a, (3b)¹ — 10b, (3b)³ — 3a² (3b)²+6a,b₁(3b)+b²−4a³
or expanding we again obtain
2
2
-
2160b6+216b¹g + 1080 g, b³ — 9b²g — 5 4 b J₂ 9 3 — 93 + 27g3
g 2 g 92
36b (144b24b2g½ + g²)²
2
9'32792 108b36b2q'
y (3b)
1086 (9ba₁)²
*) Compair Hermite where P= Þ₁, Q=Q₂, R= Q3, S=361, D = (1² — a), a = a,.
19
21


70
Part V.
It is, finally, evident from the general forms that if it be
required to determine p'v it will be easier first to find
p'v
B
Bu
371
+ pv
2 x
B.
v
2n
1
9
.
6 bo'
36
2bg' – 39
29'
3
.
2
90
= b
3 Ф
2
818
Whence
3 p
p'v= (6 - - pv) 2x =-{(pv - b) + * 2
= -P
– }
3 P
2
Q
x
P
162 οφφ΄
2702
9'3
'2
V8'3 + 27 g?
216 690' + 432o'?
108 9'37
Determination of v.
Third Method.
The formulae may be obtained by a third method and in yet
different forms as follows:
Starting anew with equation [110] we write
a) o(u b) ... (u + v)
[139] (-1)"k
0(pu) – ap'u' (pu).
σασ .
Also
(u
n
2c
ov(ou)+1
force cas l_o
)
1
u=0
whence it follows that the left hand member of (139) depends for
its value on the terms
(-1)"K
n
(u)* +1
But we have again
ve buro
1
U
ki
whence we may write, taking n odd
(u + a)...(u + v)
”k
(a) (b) ... 6(v) (ou)"+1
o)
0 n
and from
p.
66
[(- 1)^2;
Del
+
6
unti
u=0
Bo
1
Qy
+Boy+
unti
n
c
That is n being odd
k = B..
And a similar investigation gives n being even
Pγ.
k
С


Reduction of the Forms when n equals three.
71
Since v= a + b + c we may write
e-a+u+c+...--")ni 1
and multiplying by this factor we can separate the left hand
member into factors of the form
o(a + u)
o, a
(140]
- ani =
.
e-a
би ба
6 a
for u = W1
But for this value p'(w) = 0 and our relation becomes
p
Φ(φω,) Φ(e,) = Φ.
k 9 b
9, ao, b
o aob
"
σν
li
σν
63 v
03
••• σν
[141] And we obtain in a similar manner
k
0, a 6, b
Og v
()
Φ(pw,) = Φ(c.)
() Φ,
σασή
and
63 63
0 (pw3) Φ(e.)
σασο
Recalling the known relation
о, иб, иб, и
2
p'r =
63 u
we have upon taking the product of the above equations
[142]
kp'ap'b ...p'v=(- - 2)"+1 0,0,0
Again from the relations (65)
20
á
etc.
(a B) (a v) (a d)
to n terms and we obtain the product
- )
[143] a'B'y'...=
2"c" – 1,1.2.- -*— 1) = (– 1)£u (n − 1)(2)"C"
[] '.
" ()
(a B)2 (a y)2 (a d)? ... (B ) 2 (B – d) ... (y -- d)?
...
(-1){n(n— 1) (2)"C"
3.
-
n
n
n-
n
n
Д
1
- 1
n
n
.
A being the discriminant of Y. Substituting this value in [142]
we derive
[144]
(1)ğmen–12" cºp'v = (– 1)*+120,0,0,0.
Again squaring we get
of a ob... 0
22
0”(@)=(-1)"kº(pa – e)(pb – en)... (pv —e)
σα σου.
or (see [89])
[145] ·
(-1)"k? Y(en) (pv – e)
kề
- 04(e)
and we have also the two corresponding expressions.
62 v
.
.


72
Part V.
.
02
.
:
2
We have shown (see p. 58) that when t= e, we have
ei
Y(e) = = -cPQ whence it follows from this and relation (145)
that 0(e) is divisable by and in general °(en) by Qu.
Φ(,)
Qi
0)
We
thus derive the relations
[146]
0,= Q.F
Q.F : 0, Q.F, :
: Ф. 23 F3.
We have also found n being odd:
n-3)
k
- - B.:(= VPQ:A=(-1)7c2n–3 pz (a–
? . (-1)
Qzon-
Y(G) - cPQI
These values in [144] give
n-1
1
1
(
1)
3
n
n
(n-1) n
)
B3c2n P2 Qp'v
(-15>
--(-1,6+0+*3*2*3–1 pro-*-"QF,E,F,
-3
1
(n)
n-1
2
(n
in-
)
= -1)(19
1
3)
02
2 c2n-
P2
F
2
3
or
2F, F,
FED
1
2
1
3
V
P
n odd.
1
2 QF FF.
F Q
[147]
p'v
c* p B.'q
Q?
cºPB,
and from [145]
B%cº PQ1 (pv — 4)
(1) = 0 = OF
Whence we have in general
2. F?
[148]
pv la
cB:P
The corresponding expressions for n even are
2c0, 0,3 Q
yP
[149]
c* Q,03
Ca
2
0
2
va
1
3
?
ש 10 |
y?P
3
p'v=-(-V
3
2
.
2
Va
2
Again from [130]
21 Qys 3 y B Q
=
c2
a lc PBS
B.
P
2 F F F
Q
c'B'P
IQy 3 y B, c'B, PU
C” B.P 1 03 c'B.'P
P
2 y (Qx8 =
Qy-- 3B, B, Pet Q
c: B.'P
cº
P
Compairing the second and fourth forms we have
[150] ·
F,F,F, (Qy?
4
0
3
3
3
V
V.
1
3
3
(2 – 3 B,B, P4).
3


Reduction of the Forms when n equals three.
73
Substituting the values n = 3 (p. 68) and refering to the value
of x. (p. 51) we find the relation
[160].
Fn=3
[161] ·
[163]
•
ABC.
It follows then that x, if expressed in terms of the modulus k
and b or as a function of b, ez, 92 and gз, will be separable into
three factors which from the expressions for are seen to be of
the same degree in b, namely, the second.
The factors of x which we before obtained by inspection
(see p. 51 [87]) are
[162].
A = 1² − (1 + k²) l — 3k²
B
12
12
and we find the relations:
k² sn² ∞ =
whence
=
k² cn² ∞
dn² a
с
――――
[F₁ F₂ F3]n=3
2
=
➖➖
FC.
72
·
Taking now S 361 and D = 1² — a₁ = l² − 1 + k² — k¹ we
find the following relations of M. Hermite
x²
F₁ =
Φ(0)
367 (12
a₁)
— p'v=Q₁ = k² snu cnu dnu
45
=
A;
―
where x and a₁
―
(1 — 2k²)1 + 3(k² — k¹)
(k² — 2) 7 — 3(1 — k²)
=
8
33153 X =
F₁ = B;
F2
P Q R
SD2
=
ψ
1+k²
3 361 (1² — a₁)²
---
2
45
8
3653
X\1) x
367(-a)
――――
127(a)
a₁)³ (2 k² — 1) + 4 (l)
367 (1— a₁)
127 (1² — a,) (2 − k²) + 4 (1)
367 (12
α₁)²
2
-
-
=a and ∞ = v.
127 (l² — a, )² (1 + k²) — 4 (7)
367 (1² — a₁)²
QB2
SD2
ABCx
SD2
RC2
SD2
PA2
SD2
(see also note p. 69)
General Discussion.
= 3
Reviewing the foregoing theory we have found that when n =
y=f" — 3bf
-
and that in general y is a function of f where we write
o (u + v)
f=
e(x-5v) u
συ
the one exception occurring where v equals zero.


74
Part V.
VE
X =
CB
c?B.
B
1
2
B
We find further, that where Q or vanish in which case x
and p'v also vanish, our integrals, six in number (n = 3), become
doubly periodic and are in fact the original special functions of
Lamé of the second and third sort.
We have found for x the general value
@
P
from which form we see that x will be zero when y and Q vanish
and will be infinite where B or P vanish. But from the form
Qy? 2 B
PV =
c'PB.
2n + 1
we observe that pv is also infinite where x becomes infinite through
the vanishing of Bo.
We have further that in case P vanish the integral becomes
a function of Lamé of the first sort in which p takes the place
of f in the general solution the form being
1
[164] (- 1)"y=
2) u +
92p(x-4u+
(n-
(n − 3)!
(n – 5): 94p(n—6) u t...
the values of B conforming with the above cases being roots of
the equations P = 0, y = 0, 1, = 0, 03 = 0.
= R2
Moreover when Q vanishes w and p'v will vanish simultaniously
which makes v one of the semi-periods wa, and f may be written
1
1
)0(
ou
OU
a
[165]
fQ=0 +
Again, observing the last forms obtained, we see that v can also
be a half period if Fr, n being odd, or On, n being even, vanish, but
it does not follow that will also reduce to zero. That is the
integral will in general have the form
0(u + wa)
[166] ·
fi
ele-$(wa))u
exu
when
F2=0, or 02 0, or x = 0, or A = 0, or B. = 0, or C = 0.
In this case as in general two distinct integrals exist which are
doubly periodic of the second species the second integral being
6
onu
σω
6U
oqu
u
f2
σι
a form which does not differ from f, a peculiarity which does not
appear in the special functions of Lamé.


Reduction of the Forms when n equals three.
75
ν --
0
We have finally but one more case to consider, namely when
0, a condition arising when B, or y, common to the functions
X, pv and p'v, vanish, in which case the integrals become functions
named after their discoverer. *
2
6
a,)
6
elu
(800 (16) p. 17.)
Functions of M. Mittag-Leffler.
As M. Hermite observes (p. 28) the vanishing of A, B, C and D
are necessary conditions that the integrals shall be functions which
he first called functions of M. Mittag-Leffler, but they are not
sufficient conditions. The functions are in fact special cases of
fi and f2 having the additional property that the logarithms of
the so called multiplicators are proportional to the corresponding
periods. In this case the integrals assume a special form where
the elimentary function is a function of p and p' multiplied by a
determinate exponential having the above property. We can show
that these are but special cases of the general doubly periodic
function of the second species of M. Hermite as follows:
We have as the general form
(u a) o (u
q(u an-1)
[167] · F(u)
o(u b) (u b) (u – 0,-1)
o b
a function of the second species upon the addition to the
arguments of the periods 2w and 2w' the function remains
unchanged save in the exponential factor which takes the forms
respectively
u
[170]
200
c*n (B − A) + 2 ou
u' = 62''(B - A) + 2ow'
M
when
B=b, + b + : + br-1
b
bn
A a + da +..+ (–1
and n and n' are constants.
The factors and u' are general and we may if we choose
take them at pleasure and then seek the corresponding function.
Doing this we have u and u' given and also o to determine
B - A from the relations [170].
Solving we have
- 2n(B – A) + 2ow
log u' = 2n(B -- A) + 2ow'
'( +
and as
e2
1
(see p. 17.)
log u
*) See Mittag-Leffler, Comptes rendus t. XC, 1880, p. 178.


76
Part V.
[171]
whence
n log u'
n' log μ
w' log μ
w log μ
2(B — A) (ŋw' — n' w) — (B — A)ñi.
=
This solution however becomes indeterminate when F(x)
becomes doubly periodic, for then 0 and
B― A
2mw + 2m'w'.
whence
This gives
we have
[172] ·
-
we observe that when
where
where
where
=
[173].
2º (nw' — n'w) —— o̟xi;
· (nw' — n'w = xi)
=
y
in (2mw + 2m'w')
w
W
which means that the logs of the multiplicators are proportional
to their corresponding periods.
Returning to the form
=
f
=
-
w' log μ
log u 2inm
log u'+2inm'
-
o (u + v)
6
6 σ (u)
2 = а ₁ + a₂+
v = 2mw+2m'w'
this case.
=
and f vanishes showing that this eliment can not be utilized in
Written as a product however and for u 3 we have
o (u + a) o (u + b) o íu + c)
63 u
=
e=u&a+b+c)
•
―
=
· 0
a+b+c=v = 0
and our eliment may be taken as a rational function of pu and
p'u multiplied by a factor of the form e". It is moreover known
that any function f(u) of p and p' may be resolved in the form
f(u) = L + P
w log u
-
L = 1, ¿ (u — v₁) + 1½§ (U — v₂) + 13 § (u — v3) + · ·
(u
P= c + Σm P‹¹) (u — v)
4 + 12 + l3 +
This property being general, we have, f being doubly periodic,
but to multiply by eeu to find a development for the eliment
required in [172] namely
D(u) = eeu {(u)
=
0.


Reduction of the Forms when n equals three.
77
S.
من من مد
We have then
$(u) Ф(и) е- ри
&' (u) O'(u)equ
equ — 00(11) e-on
Š" (u) o”(u)equ 200'(11)cou + p'D(u)cou
Ç(3) (u) 0"(u)e–94—300"(u)e-qu +3p2 Ó'(u)c-gu— 08 0 (u)e-eu.
и
Whence
n
Φ(()
non — 1)
12
o° p”- 2)(x)+..
1
[174] eru çin) (u) = O(n) (u)
((
i pon 1)(u)
ọ -) +
We have then a decomposition in the form
[175] · fi (u) = ce** + 2X4 GOM (
= c +ΣΣΑ,
Q(v) u
.
Vn)
m v
where Vn stands for the several infinites of fi(u) and Q() for the
derivatives where v must be of an order one degree less than the
multiplicity of the infinites. The coefficients A will be determined
in general by developing fi(u) according to the powers of (u — vn)
while c will be a fixed value depending upon the given conditions.
In our case then we may write
[176] ·
f1 (u) celu + fu elu.
This function when v is zero, in which case O=0 and D=0,
takes the place of f(u) and hence the general solution is
y = fi" (u) — 3 bf, u
Yı - —
= (celu + çu.epu)" – 3b ($1.epu + ceer)
+
fi' (u) pcelu + Sueou tos () elu
fi” (u) pcepu + Sueou + 2 08' (u)epu + 9%84(u) eeu
whence
(Suceu)” = 6" uctu + 2 congu + p*ccugu
uçu
and we have
[177]
Yı (&u - elu 3 bucou + c'eru
= 4 *[S" u + 2 08'u + (o? — 36) 81 + c).
But from the foregoing theory in this case we have the coeffi-
cients of $(u) equal to zero, i. e.
U
or
02
36
0
[178]:
e?
3b.


78
Part V. Reduction of the Forms when n equals three.
To find c we proceed as follows:
-
1
Eu
92 u 3
u
20 3
t'u
11
u2..
2
u?
20
&" u
U
92
U
10
.
us
p?u?
0°43
Clu
6
3
2
92
10
U
3
ga
20
u²
.
1 + ou +
t..
Hence
ou"
y = (1 + ou +
pu?
+ + -]{[
%+ -]
6
20 [+ + 0*20* +...]+c+...}
and taking c so that the constant term equal zero we have
[179]
- 03 = 2 pb.
2ob
The general solution (v = 0) is then:
)
yi (su eo u)" – 36(su •efu) + 2 obegu
where
V3b.
2
C =
.
3
Finis.


Table of Forms
n
3.





where
and
The complete Integral is
where
y
y
=
=
y₁CF(u)+C'F' (— u)
F(u) = f'' (u) — 3bf (u)
e(x-(v)u
the ordinary form of the equation of Hermite for n
II
a=a, b, c
o (u
Forms for n = 3.
=
-
A second form of the integral is:
o (u + a)
ба би
f(u)
x = Ev —
=
d2y
du2
σ (u + v)
συσν
= [12p(u) + B]y.
―
a) o (u
ca ob oc (ou)3
-
e—u$a = ]] º (u — a)
II
баб 0
a=a, b, c
b) o (u
Ea — Eb — Ec
-
ફ્
-
=
v=a+b+c
and B 156 which is intirely arbitrary and is originally expressed
in the form
-
B = h (e₁ — e3) — n (n + 1) ez
=
in which case the equation of Hermite is
d2y
dx2
- c)
We have also the general form:
y = ±VY=√(pu — e¸)* (pu
e, é, é"
[12 k² sn² x + h].
-
e($a+56+5c) u
――
=
ρυζα
3 being:
€3)º' (pu — €3)º″ II (pu —pa)
: 0 or 1.


82
Table of Forms n = 3.
The functions developed in the general theory have values as
follows:
ዎ
ø'
ዎ
ι
a1
b₁
=
=
=
x²
=
=
1262
3b==/ B
4bs-bg2-93
Y(e₁)
-
3
4
where
Φ (1)
92
27
4 93
or
Ø (1)
= p(u)
6bq' t'p'u[4 t³ — tg₂-g3]2
21=0
S
t—b
❤
(t) = 4S³ + 12bS² + (1262 — g2) g +46³ — bg2
= 483+12682 + ¢´S + ❤
92
=
=
P
――――
C =
Bo
Φ (0)
SD2
=
— 9 (t) − b [ø' + 3 (t — b)²]
4
=
=
B₁
--
Y = S³+ A‚S + A¸= S³ + ¦ 9' S + ¦ ❤ — bø′
ዎ · bo'
4
=
1
15
=
· S³ + (3 b² — — 9 ½ ) § — 1 (44b³—3g₂b+93) = 19(t) − b (œ'+3,8²)
――――
4
C3
15 b
=
t³ — 3bt² + (6b² — — 92) t — (156³ — g₂b + 1—1 93)
•
9
1
S=
S = 1 9 (t) — 3 b S² — — ¢´S — — 9 . ·
-
-
2
3 /
2 ዎ
b [ø'+ 3 (e̟₁ — b)²]
В ГВ
6e, B
15
15 L 15
S=361
a₁s
b₁
ዎ
4 (1² — a₁)³ + (117³ — 9 α, l — b₁)²
361 (12 — a₁)²
-
+36₁2-92]
2
cB [B² — 6e̟₁B + 45e̟₁² — 15g]
c² Q₁P
=
-
1257º — 210a, l¹ — 22b, 1ª + 93 a, l² + 18 a, b, l + b₁²— 4a,³
361 (12-a₁)
A₁ =
A2
D
3
1 925
108 93
2
-
-
A₁ = 19 - bø
A3
4
t
1
=
3
12576210a, 226,193a, +18a,b,l + b₁³ - 4a,3
9'
=
12-
1256210c¹- 22 §³ + 93 c²²+18c§ + 1 — 4c³
1
-
a1
b [15b²+ 3e̟²— 6e̟b — 9½]
93
§ — b₁
= 31
(1 — k² + k¹)³
k4)3
(1 + k²)² (2 — k²)² (1 — 2 k²)²


Forms for n =
83
= 3.
Also:
where
γρ
Q2=
Q₁
Q₂
Q3
=
e₁
=
=
=
Q = Q1 Q2 Q3
= (15) 4
=
=
X=
p (v) =
-
1
32
=
=
(2
Qr
CB。
Ф
-
=k²sn² w
Qy2
ePB
/
=
=
— 1:2)
―――
Ύ
* Bo
3
(4A,³= 27 Ag²) — — 4
= -
Δ
-
√ 121
P
1 '³+ 27 p² - 8 (27) bøy′+ 16 (27) b²ø′
6 9
b
Þ (1)
2 B₁
Bo
Δ
1 + k²
3
2 F, F, FS V
1 2
c³ PB3
=
-
=
2
C2
92
P
=
-
=
32.5 ['+ 3 (c₂ — b)²] — 5 Þa
B² — 6e̟₁B + 45e2—15g,=.5 [5 1² — 2 (k² — 2) 1 — 3k¹] = 5 Þ₁
B² — 6e, B+45e,2-15g-5[57² — 2 (1 — 2 k²)l — 3] = 5º,
B² — 6e3 B+45e32-15g=5[57² — 2 (1 + k²) l − 3 (1 — 7²)²]
-
2n
1
(15)3
3 q
B
2
Þ₁ н г
1
-
4
32
2
3
516a, 110b, 1³ — 3a, 1º + 6a,b,l+ b, ² - 4a, ³
V
Q
'3
162 bpp' — 27 ❤² — ❤'
108´³½
3
[4A₂³+ 27 Ag²]
2
3
p'(v): k2 sn²v cn2v dn² v =
1
1
32 (212
=
=
361 (12 — a₁)².
-
ข (1)
361 (12
=
3
4 A¸³+ 21 A¸²
2
3
b
1
(5)3
50g.
―――――
(1 — k² + k¹)
---
- 1)
a₁)2
Qi F₂2
ん
​c2 B2 P
=
C = c¹ PQ1 Q2 Q3
У
(15)3
Qi Qy Q3
2
+ ex
---
39
C3
g
21606216b¹g + 1080 g, b9b2g54b92 93 923+27 932
36b (144b24b9₂+923)*
'32792-108bq q'+36bq' 2
b²
36b 2
7
1
2
710
1/11
32
2
x() x
187 (72
a₁)
V9'³ +279² - 216bøø′+ 432b²ø
(1 + k²)
12
6*


84
Table of Forms n= = 3.
3
pν -
q'+ 27q? – 108 99
90'
36 9'26
Q. F?
2
2
PV — 22-
cºB P
[0'+ 3(€– 6)?] [12 (6 — ex) (2b – en) — '?]?
36 9'2]
pv = b
p'o
3 99
2
9
.
2 x
()
2
1
)
3
where
♡ (1) 0(l) — 121(1» – az) (1073 — 8a71 — b)
)
l —
= 576 + 6a,1 – 106,7 – 3a,+ 6a,6,7 + – 4a, .
212 12 bi
x(1) = [0(1) - 34(1) — 108 14 (72 — a,)?
[Φ( ()
= 18 – 6a74 + 46,78 -- 3a,41 – 6,2 + 4a' = A·B.C
= A
À = 12 – (1 + k?) 1 — 37.* = F
72
B = 12 — (1 – 2k) 7 + 3(kº -- 14) = F,
72
)
C 72 — (k? — 2)2 – 3(1 — k?)
(— = Fs
F
F=F, F, F3 = 30 ga X
A.B.C.
45
2
1
45
2
45
-
2
8
8
3653
1
2
3653
Case 1. P = 0.
Integral a special function of Lamé of the first sort.
y =p'.
B = 0.
Case 2. Q ; Φ(0)
0; 0(1) = 0; Q.
Q1
0; Q2 = 0; 23 0; x = 0;
p'v = 0; v = Wa.
Integrals, six in number, of the second sort.
ба и (u + w)
a=1, 2, 3
f=+
u & (w?)
$(w2)=12
10(W)
Vpu ea
би
= 2
where
1
1
2
la
B
10
B
- ри —
-
(a) Qı= 0
- 3e, + V3(- 12e + 592) = kº — 2 + V (12 — 2)? + 15k4
y = {p + ja - (34; + V3(592 — 12c)) } Vp – en
+ {P+ 1 (1:4 — 2) + V (1:2 — 2)2 + 1574} Vp - (k2 -- 2).
2 - 2 to ? —
k2
1
1
10
-e1
ey
2
1
pt
15
3


Forms for n =
85
= 3.
(c)
(b) Q₂ = 0
Q2
=
-
B=3e₂ ±√3 (59₂ — 12е) — 1 — 2k² +√(1 − 2k²) + 15
y = {p + 1 e, —
1
— (3e, ±√3 (5g, — 12c;)) } Vp — ez
-
2
10
F₂
or
Q3
or
= { p + 1/3 ( 1 −
15
=
=
0
y = {p +
Case 3.
where
B=3eg±√3 (5g - 12c - 1 + k² + 2 √ (2 — k²)2 3k
= {p +
0;
1 e -
2
1/
x = 0;
(1 + 2 k² ) +
2k²)±
62u
би
Y1
――
Φ
E
2 k²) ± 1%√(1 − 2k²)² + 15} Vp — 1 (1 — 2k²)
10
3
f=
e(x-(0%)u).
Six values of this form corresponding to the roots of A = 0;
B= 0; C = 0, namely
-
Ψ
1/1 (3e, ± √3 (5g, — 12€;))} Vp — es
-
10
5
B = ½/ (1 + k²) ± √ (1 + k²)² + 6k²
2
=
5
5
B = 1 / (1 − 2k²) + ½ √(1 − 2k²) — 6 (k² — k¹)
-
-
2
2
which determine corresponding values for x.
=
A = 0; B =
ex u
5
B = 1 / (k²2 − 2) + ½ √(k² − 2) + 6 (1 — k²)
-
·
2
==
Case 4. Conditions as in case (3) with the additional con-
dition of the functions of M. Mittag-Leffler.
The integral is:
—
} √(2 − k²)² − 3k } V p − } (1 + k²) ·
x = 0
o (u — w₂)
би
-
0;
(Sue)" — 3b (gueou) + 2gbeou
-
VQ.
C = 0; v == 02;
-
-
୧ V36
6 A S² + 9 AS - A²
9 [2 AS3 A]
2


















DUE JAN 1 1911



Math 4008.93
(A presentation of the theory of He
Cabot Science
003006941
3 2044 092 008 168