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(Cornell Unirmitg jibaro
BOUGHT WITH THE INCOME
FROM THE
SAGE ENDOWMENT FUND
THE GIFT OF
3Hcttrg ïîl. Sage
1891
k.&XOJJL
ALALIA





A
PRESENTATION
OF THE
THEORY OF HERMITE’S FORM
OF
LAMÉ’S EQUATION
WITH A DETERMINATION OF THE EXPLICIT FORMS IN TERMS OF
THE 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 OP HARVARD UNIV., INSTRUCTOR IN PRINCETON COLLEGE.
TO THE
PHILOSOPHICAL FACULTY OF THE ALBERTUS - UNIVERSITAT
OF
KÖNIGrSBERG IN PR.
PRINTED BY B. G. TEUBNER, LEIPZIG.
1893.
Ψ





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 Lame's deferential 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-Universitát allowing the publishing of this thesis in English,


6
Introduction.
I am not without hope that this general presentation of the theory
of Lame’s Functions may prove a welcome addition to the literature
of the subject where in English Todhunter’s “Lame’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.
page
Introduction............................................................ 5
Part 1.
History and Definitions.
The Problem of Lamé.....................................................il
The Problem of Hermite..................................................13
Definitions.............................................................15
Part 2.
Hermite’s Integral as a Sum.
The Function of the Second Species......................................17
Transformation of Hermite’s	Equation·. ................................20
Development of the Integral.............................................21
Development of the Eliment	of the	Function of the Second Species ...	23
Determination of the Integral...........................................25
Part 3.
The Integral as a Product.
Indirect Solution......................................................  28
Solution for n: = 2.....................................................30
The Product Y of the Two Solutions......................................32
Direct Solution.........................................................37
Determination of Y for n = 3............................................40
Part 4.
The Special Functions of Lamé.
Functions of the First Sort.............................................42
Functions of the Second Sort............................................43
Functions of the Third Sort............................................ 44
Part 5.
Reduction of the Forms u n = 3 ”.
Identity of Solutions...................................................45
Determination of x and v. First Method................................47
X as function of Φ................................................  48
Factors of Φ.......................................................49
Case Φ = 0.........................................................49
Definition of Ψ and p(v) as Function of Ψ........................50
Definition of χ and p (v) as Function of χ.......................51
Reduction of Lame’s Functions Φ = 0......................'....	51
Integral χ = 0.....................................................52
Case Ώ = 0.........................................................52


8
Contents.
page
Relation of Y and C to the Special Functions of Lamé....................52
Analytic Form of Y and y...........................................53
Condition (7	= 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 Qx and Φχ.......................................55
Determination of G....................................................56
The Integrals = 0, Q2 — 0,	= 0.............................56
The Discriminant of	Y.................................................. 57
Resultant of Y and Φ(α)...............................................57
Discriminant in terms of this Resultant...............................58
Discriminant in terms of P and Q......................................58
Special Results, n = 3.............................................59
Determination of x and v. Second Method.................................60
Reduction of the General Function..................................60
Development of	Φ(%	=	3)........................................... 62
Development of	Ψ(%	=	3)............................................64
Development of E (n — 3)............................................. 65
Reduction of x and v from these Forms.................................66
General Forms for .r, 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 Tct ............................................70
General Form as Product of Φ1? Φ2, Φ3..............................71
The Functions Fli P2, Fs...........................................72
Forms for p(v) and p	iv) in	terms of F? and Φ;......................72
Relation of Fn=i3 to	χ and	the Factors of χ.........................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, v =	ωλ, x	= 0.....................................74
Integral Fx = 0 or χ	= 0,	v = ωλ, x =4= 0.........................74
Case v = 0...........................................................75
Functions of M. Mittag-Leffler.............................................75
Relation to the Case χ — 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 (* = 0).....................................................78
Table of Forms and Relations (n = 3).......................................79


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 to
the sphere.
The problem of Lamé may be stated thus:
Let the surface of an ellipsoid he given by the equation u = u0¡
it is required to find a function T which will satisfy the equation of
the potential and which for the value u = u0 will reduce to a given
*) See note Heine, Handbuch der Kugelfunctionen, p. 2, Berlin 1878, and
Heine, 2d voi. Zusátze zum ersten Bande.
**) See also reference to Green Heine p. 1.
***) Mémoire sur les axes des surfaces isothermes du second degree con-
sidérés comme des fonctions de la temperature. Journal des Mathématiques
pures et appliqués. lre série, t. IV, p. 103.	1839.


12
Part I.
d]
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
or transformed in terms of the p function
[2]·	· (pv — pu)^ + (pu—pv)^ + (pu-pv)ÿ^r = 0
the relation,	
[3]		• T — f(u) f(v) f(w)
and the equation	
[4]		• g = [Apu + B]y
where 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:*)
Œ.............7=T0+ ZÍ + T2 + ... + r„ + ···
where Tn in general is an intire homogenious polynomial of the
nih degree, it is observed that each of the functions Tn will also
satisfy [1], the equation of the potential, in which case [1] would
be an intire homogeneous polynomial of the (w — 2)d degree. This
polynomial must be identically zero which will impose -|·(n — 1 )n
linear conditions. The quantities Tn will have in all y (w + 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
r61 .	. . T2X2 = (P*-««)(*»-‘«Hp*-*«)
(ea - ep) (ea - <V
x 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].............Ypu — ^,	Y pu — e2,	Y pu — e3
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 Tn, where Tn is of the wth degree, is symmetrical
) see Halphen. Vol. II p. 466.


Historical Development and Definition of the Equation of Lamé. 13
with respect to uy 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
[8] · ·	'	·	0 = f'ufvfw = Tfj£ = [Apu + B]T
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 fufvfw 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 -f- 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]..............^ — [w(w + 1 )k?sn2x -\-h]y = 0
corresponding to the form [4]*)
[10]...............τί~ [♦»(» + l)pu + B]y = 0
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 Lame's original equation
which shall hold for any values of h and n.
Except for the special values η — 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.
ethe 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
[11]
y = F(u) =
i>:
Γ(ϋν)
Vw + \»lr

p(2v — i
“f" · * * 4" hy _ 1 f(u)
where n — 2v— 1, with a corresponding form for n even, where
f(u) is a doubly periodic function of the second species, namely,
where
f(u) = eX(-a~~ir) %(u)
X(u) =
H' (0) H (u ω)
Θ{η) Θ (ω)
&'(ω)
Θ(ω)
(u — ÍK') +
i it ω
Υκ
That this shall be a solution the quantities ω and λ 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-Lefifler from the
general form.
M. Hermite was thus led to a new presentation of the general
solution in the form of a product, namely
i±A e-uca
* II 6Ü6U
a. = a. · h · ·
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 von
Göttingen 1878, and Hermite: Annali di Matematica, serie II, Bd. IX, 1878.
**) Traité des Fonctions Elliptiques et leur applications. B. II. Paris 1888.


Historical Development and Definition of the Equation of Lamé. 15
Definitions.
Returning to form [9] of Lame's equation we observe that it
has the following properties:
It has a coefficient
n (n + 1) Wsn2x + h
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. Conformiug 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 2iK' and everywhere finite
save in the point x = ÍK' 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]............Γ* + φ2(%<”-2> + · · + 9n(x)y = o
where
Φ2(χ) = cc0 + axsn2x
φ3<» --= βο + βι*η2χ + fiìDixsnix
<Si(x) = γ0 + γ^η2χ -f y%Dxsn2x + fsD2xsn2x
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. Bôcherf). First the ordinary form of the
equation of Lamé may through transformation becomeff)
rion +1/_J__ + -A-. + -1_\=___________________________axj^b__________
Li0J dx2 1	2 \x — ex 1 x — e% 1 x — ez) dx 4 (x — et) (x — e2) (x — e3) y
where the exponents of the zeros et, e2j e3 are 0 and y and that
of the infinites - ~	· From this generalizing by the intro-
duction of n zeros we have the following definition:
*) Annali di Matematica, tomo XI, 1882.
**) Comptes rendus etc. 1880. p. 64.
***) Math. Annal. Bd. 38.
f) Ueber die Reihentwickelungen der Potentialtheorie. Göttingen 1891.
ff) 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
regulare homogene Differentialgleichimg zweiter Ordnung mit rationalen
Coëfficiënten, deren im Endlichen gelegene singulare Punkte et, e2 · · en
sammtlich die Exponenten 0, ~ beset zen, und in unendlichen nur einen
uneigentlich singularen Punkt aufweist“
Lame’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—s:
&y_ \ f (χ)
dx2 ' 2fx dx
ΪΤφ-f-	+	+ ~+M]y= 0
where
f(x) = (x — ef) (x — e2) · · · (x — en).
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 Laméis equation we will understand an equation
conforming with the above definition whose general form is given by
[14J and, if the order is higher than the second, distinguish by mention-
ing the order.
Forms [9] and [10] will then be called Hermite5s forms of
Lamé5s 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 Lame’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 IL
Hermite’s Integral as a Sam.
The Function of the Second Species.
We have the problem required the integral y of the equation
[15]....................=	M y
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 ú 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:
[16]
F(u + 2 K) = μΈ(η)
F(u + 2iK') = p'F(u)
F (u) =
c(u — aí) a(u — aí) . . . a(u — an_f) ^
o(u — c(u —	a(u — 6n—1)
The factors μ and μ 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 ΞΞ o 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
*) Bd. II p. 495.
2


18
Part IL
itself and that therefore the quotient y" : y will not only be doubly
periodic but will have a single pole u εξ o of the second order.
This function then satisfies the necessary conditions and the
corresponding quotient may then be written equal to
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) arbitrarles 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
F(u= [iF(îi)j £& = 2-5Γ
f(u + a') — ¿F(u), a' = 2ík'.
Define:
[17]....................f(u) = Ae-^±^^*)
which function we will speak of as the Eliment the general form [16]
being a product of similar eliments.
We have the fundimental relations:
ö(u -|- a) = — a(ii)e2tiu+riS2
a(u + β’) = —
Ί~τ(τ)-Φ·
Whence
f(u -f· a) = A	6λ(ϋ + Ω) + 2ην
= f(u)QXS*+2ilv.
Choosing then v and λ correctly we may write
μ = βΐη + 2ην
*) Hermite, in the following analysis, employs the function given on p. 11,
namely the function χ expressed in terms of the Θ function.


Hermite’s Integral as a Sum.
19
with a corresponding value for μ and we may then write
F(A) = φ(η)
f(u) V[U>
where Φ is a doubly periodic function, that is
<i>(ti -J- M& + nao = φ(μ)·
Again
f(u — β) == i ƒ(«) and f(u — Sí') =	f(u).
Whence
f(u — z — a) = — f(u — z) where F(z &) = ^F{¿)
and we derive
[18]..................Φ(ζ) = F(z)f(u — z)
where Φ 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
Φ for the values of the argument that render it infinite and
equating the sum to zero as follows.
First f(ii) becomes infinite for the value u = 0 whence its
residual
EUrof{u) = [ufu]u=o = a------—----------=	= Aõ(v)
mu=o
and becomes equal to unity if we take
A
Whence
l
a{v)'
[19]
f(u) = a(u elu
' ^ ' g(u)o(v)
Again
= sl™u (B - «) φ(β) =	(0 - u)F(/)f(u - *)
and developing f(u — z) we have
ΕηΦ{ζ) = — F(u)
Again let a be any pole of F(u) in which case, developing by
the function theory, we may write
F(cl 4" í)e=o === As 1 4" A1Des 1 4- A^JDçS 1 4" * *
4“ ΑαΌ*ε~1 4“ ao “f“ aiε “1” a2“h * *
2


20
Part II.
and
f(u — a — s) = f(u — d) — jBuf(u — a) + ~Dlf(u — a)-
+ r=i^WM -«) + ··
where
We have then
KΦ = ,ΐο£ÍKe + £)/(M - « - *)
= Af(u — a) + A^ufiu — a) + A2D2uf(u — a) + · * *
+ AaDlf(u — a)
with similar expressions for Ebj Ec . . .
But Φ being a doubly periodic function we know that the
sum of its residuals with respect to u, a, δ . . equals zero whence
[20] F(u) =	[.Af(u — et) -j- A1 Duf(u — cl) -f- A2Duf (it — ci) -f- · ·
α=α, 6, c..
-f” AaJDuf{u — d)\
where At- 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 = F1 (u) may
be expressed.
Transformation of Hermite’s Equation.
We have written Hermite's equation in its original form
[21]...............= [»(» + 1)¿2 sri* x + /*].
That this is however but a special case of a more general
form is seen as follows.
Take the integral
	r dX X — I	= 1	rax
	J 1/(1 — *2)(1 - Jc*X2) * 0	J 0	yi
We have	dy dy don dy dX dx dX dx	1 y'A	
or	^ I*** 1^ II		


Hermite’s Integral as a Sum.
21
whence
a?y
dl*
or
d*y
dx2
d2y 1	i Λ' dy
dx2 Λ	2 J^dx
yidX* ^ dX
Substituting we derive the ordinary form of Lame's equation
[22] . . . · Λ	+ y gf- — Vn (n + 1) k2sn2x + h] = 0 .*)
The value of A gives as singular points + 1; +~ and oo.
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
ρ(ιή = e3 +
___ei____?8___
sn2u]/el — e3 ’
k2sn2(u + ik') =
and making the substitutions:
X — u'\/el —
e3 u oo u -j- i¥
we obtain:
d2y__________
du2 (β! — e3)
dx2 = du2 {ex — f3)
1	r../„ I *\PU — e3
[n(n + 1) ■
+ *]■
Define
[23]	.............B = h {el — e3) — n(n + 1)<?8.
Whence our equation may be written:
[24]	.............y"=[n(n+l)pu + B-}y.
Development of the Integral.
We observe, since snx reduces to zero only for the value x = 07
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 the form being
y = «” [y0 + &« +	—]
whence
y’ = vu*-xy0 + 0 + 1) yxuv + (v + 2)uv+1y2 + 0 + 3)uv+2y3 + · ·
y" = v 0 — 1) uv~2y0 + v 0 + 1) yiUv-x + 0 + 1) 0 + 2) ury2 + · ·
*) Compair general form [14] p. 16.


22
Part II.
We have also
P(u) -- 7.2 “f" clu2 + CZU\ + · · ·
These values in [24] give:
v(v — 1) y0ur~2 + · · · = n (n + 1) y0uv~t + ·
whence:
v(v — 1 ) = n (m + 1) or v — — w.
This value gives since the uneven powers fall out
[25].........y ~7 + ^ +	+ ' ' ' + 7^i + ' ' '
from which we again derive
y=n(n + 1)	+ (« - 2)(« - 1) ^ + (» - 4)(* - 3)-J^
+ (n — 6) (» - 5)	-1---h (» - 2*)(» - 2 i + 1)
Γ 1	7ï9	h. η
= fc + ^ + ^ + '“ + ^ +
[i (n + ij (^ϊ + c7 + c-i4(4 + ···) + -®j
,	s 1	h-t ,	h.n(n-\- 1)
= n (n + 1)	—^5	+ - (« +	1 ) n	-\--μ Λ-17Γ J. -\----
V 1 J	un +2	' vn K '	1	'	'	«-2,4-2	'
7^9
A
1 ι	B hχ	J5 h2
Bh,
4-5-J-----L _|---L j--J----_.L·. -L
1	n ' nn — 2 '	..n — 4 '	' Λ,η — 2i *
U	U
*	+»(»+i)c1-¿* + »(»+i)<iAi-¿iH-----
+ w(n + 1) cjii -^n—+ · · · + w (n + 1) c2 + * * ·
Equating the coefficients of like powers of u in this identity
one finds
7 I {n — 2) (η — 1)	= n (n + 1) \ + B
U
-77777 I (n — 4:) (n — 3) h2 = n (n + 1) Uh + cil + K B
7771 I (n — 6) (n — 5) \ = n (n + 1) (h3 -f- c-Ji^ + c2) + h2B
u


Hermite’s Integral as a Sum.
23
7=6 I (w — 8) (n — V]h = n (» + 1) 04 + CA + <A + A + hB
n 2Í-4-2 l(w — 2*)(n — 21 )/«,== w(« -f-1 )A + c1/i,_2 +1'2/*;—s -f* · · ·
U	1
+ Ci—i] -f- hi—i JB.
Whence we obtain all the coefficients in the development for
y by means of the recurring formula.
[26] 2 i (2 i — 2îi — 1) hi = u (u -j- 1) \jhh¿—2 “Ή C2h¿—3 -j- · · ·
+ Ci—^hi -f- c¿—1 ] -f·	1 JB.
Since then h¿ is determined we have when n is even and
equal to 2v
1	h-t	Ji __
y = -y-1-----i7=i + · ' · H—i~ +
u	u	u
and if n be odd and equal to 2v — 1
1	h\	/l 1
y — ~ 27=1 Η" · 2 r—2 Ά ' A------7----Η
u	u	u
where hi is given by (26).
Development of the Kliment f(u).
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 = Fx (u) be a solution.
We have then to determine the development of
/·(„)_ £ÖL+^ei,
1 v J g(u)g(v)
= u +
since	[nf{uj\u^ 0 = 1
To this end we develop first the function
[27].............9>W = /‘(«)e<2+e')“ = e—t'
We have:
d£u	d qu	1 ,	„ .	. .
PU— du ~ du' ou	u¿ + c1U + C2U +
d U QU u¿
whence
GU u
3
Ctu° — jC2u° — -CzU
7


24
Part IL
By Taylor’s theorem:
/	,	d — (*V)
i(w + v) = i.(v) + M «
, U* 6	,
> 1.2 *
du 1 1.2 du2
w2 , / N	W*
= % « - «1» M -rtP'^)-riti p" « · · ·
Passing now to logarithms we derive:
1	/ V μ2 , , V M3 /p"(v) οΛ
= -í-«PW-TíW-y(ir-i)
w4 fff / V w3 Γ ffff / V c21
ït^ (v)-5ib W-iJ —
= - \ + Au + ^ μ2 + jf u3 4-
4
2!
Integrating we have:
log φ = — log ÍÍ + Λ 27 + Λ *7 + Λ JT +
3!
4!
whence
i Γ**Ιπ+^!π+··Ί
[28]φ = ^βί21	31 J
= èt1 + Ij- + 4 jr + · ·] + i [λ |¡ + 4 h + · ·] H—
= ¿[1+P*£+P3~+Pi£+···]
where
P2 = A2 =	p(^) ; P3 = -43 == p (v);
P4 = _ 3*» +	- A,+ 3Λ2
P5 = — 3 pvp'v =	+ 10j42^3 etc.
showing that the coefficients Pi are intire functions of pi/ andp'i/.*)
*) The functions P» correspond to the functions Sí in Hermite’s treatis,
for example
P2 = — p(v) — SI = u2sn2u —	*
P3 = — p'w =	= %2snu cnu dnu
see p. 126 development of %.


25
Herinite’s Integral as a Sum.
From these forms we pass immediately to
[29]	f(u) = φ(ίί)^+^)“
= 9(u)[l + (λ + ξν) + (Λ + ξν)^ + · - ·]
=¿{[i+α+%u)u+(ρ,+α+w)
+ [ps + 3P2(λ + ζη) + (A + tu)·]	+ · · -1
= i + JÇ, + fii« +	+ Ρζ«8 + * · ·
Take
λ = X — ξν
whence
[30]	· ·	ƒ(*<)= ^±4 e(*-í*>«
L J	J	6 (u) σ(ν)
= i + * + (*s+P2)J + (*s + 3P2* + P3)^.
+ (s4 + 6 P2 z2 + 4 P3 * + P4)	+ ···
= i + P, + J3i («) + H2 (u)* + #,«».
Where in Hermite’s Notation
Ηϋ = χ.
fii = l(*2 + P2)
[31]	^2={(ít3 + 3P2ÍC+P3)
P's = ¿ (*4 + 6 P,*8 + 4 Ρ3ζ + P4)
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 IL
First we have:
ƒ(»)	=	¿	+ Ho + Ht» + HìU* + - - · + Η#* + · · ·
f (M) = — μ* “f“ -®1 “f" 2 HìU -{- 3 H¡ μ2 + · · · + iHi%l~x -j- · · ·
Γ0) = + ¿ + 2H + 2 · 3H3u + . · · + i(i -
Γ» = - ÿ + 2 · 3IÍ, + · · ·
+ *(» — 1)(» — 2)Si«i-H-----
/■((I ¡¡à! = +	^ + 2.3 .. · (» - 1) Hn _ X + · · ·
+ i (i — 1)· · · (i — n -f- 1) Hiui~n+1 -f- -
Again
1	\	hv___j
yn = 2r-l == ~2r-l +	2V—3 + *··“)-----------h Ku
1_
Ύν
yn~2v
And in generai
y = Fiu = AafW
Λι	K, *
+ -5^ + ··· + -^ + ^.
+	+··+ƒ
= 4, fi—1) +	H------h ƒ (» Odd).
Now substituting tbe values ƒ<*> 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
[32J · ·»-*■.(■)- çnhjï f"~" + çrbn ^("’> + · · *.-./■
(n odd and = 2 v — 1)
provided x and v le so taken that the constant term equal zero and
the coefficient of the next term equal hv and
roo , U=F9(u) —_____1__f <n -1)_-A_ .. fin-3)__5*__ ƒ(n — 5)	.
loo] V	(n-l)l'	(n — ZyJ	(n-6)1 T
--	^ f' (» even and = 2r)
provided x and v he so taken that the constant term equal hv and the
coefficient of the next term equal zero
or in general
[34] (- + + ^V<.-8 + .,.


Hermite’s Integral as a Sum.
27
where the last terms are obtained to accord with the above con-
ditions.
’ Substituting the values fW we find the conditions to be
(n odd)
[35]
H2V — 2 “t~ ihl^2r — 4 “1“ /^2S-2V — 6 ”f* * * * “f“ hr—iHq = 0
(2v — 1) H^v—i (2v — ÿ)Ji\S^v~z “h (2 v —	-)-···
hv-iHi — hv = 0
(« even)
[36]
H^V — I "4“ ^1 $2v — 3 “f"	2v — b -{-···-{” hv-.iH1 -j- Jlv = 0
2vH2v~\~(%V 2)/&χS2y — 2	(2 V — 4)/?2-fi^2r — 4 “-j” * * * -f" 2/¿v— 1JT2 ==s 0.
These conditions being satisfied y = jF(m) and we have two
forms
D* jPt(m) — [n (» + 1)+ E] F(u) = 0
since finite for u = ik' = y ·
A second solution being likewise obtained by making the sub-
stitution η ~ — n the general integral may be written:
y = cF(u) -j- c F (— u).
[37]


Part III.
Integral as a Product.
Indirect Solution.
It will be shown in developing the forms for the ease n = 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
[38]
y-fl
a=a'b‘
g (u + «) ρ-ηζα
a (u) 6 (a)	7
where the product is composed of n factors obtained by taking
α, b} c in place of a. The derivative of the logarithm is
y	[ξ (u + a) — ζ (u)
-£(«)] =21
p'u — pa
pu — pa
while a second differentiation gives
y - ar-2[pu -p(u+«)]·
From the first equation
yΊ2= s? _1_ /p'u —p’g\2 ■ i_
y J	4 \pu — pa) 2
p'u — pa
pu — pa
pu — p'b
pu — pb
But the addition theorem gives:
whence
y_
y
i pu — pa
4 pu — pa
= p {u + a) + pu + pa,

— p'a
pu — pa
p'u — p'b
pu — pb


Integral as a Product.
29
In order to decompose the last term in this expression we write:
l pu —pa p'u — p'b	.	. x\
£----r--V-----= 2 (pu + pa + po)
* pu—pa pu—po κ 1 ^ i χ' /
+ -pa -lb KC* + β) — S (« + δ)
ε« + δ&]·
Take the value μ = (α + δ), remembering the relations
ρ(— u) = + !>«; jp'(—«)—/(«); δ(—«)=---------δ(«)·
Writing then ƒ (α + 6) for the right hand member of the above
equation under these conditions we get
f(a + b) = 2np (a + b) + Σρα + 2p (a + b) -f- pa + ph
= 2 (»+ l)jpw + 227i»a.
from which we see that in general we would have
//
^- = w (w + l)_pw + (2w — 1) Σρα
the quantity in brackets being equal to zero.
If now we reunite the terms ζ(η -f- a) — ξα, ξ(η + b) — ζό etc.
in the general expression and make equal to zero the sum of their
coefficients we obtain n equations of condition, namely, writing
pa = a\ pa = a'; pl· = β; p'b = 0';
«'+ V' . <*' + y « a' + d' i__________ o
■ ^ — « * a — d '
[39]
β ■ a — y
f + <*' _j_ P' + y' i	+ i_Λ
f? — a ‘ β-γ'β — d'
y'+ i r'+ ft' i y'+ d ,
y — a y — β "* γ — d
= 0
[40]
If then we can solve the equations considered as simultanious
«'2 = 4a3 — g2a — g3
Γ = ^ - 92β - gs
together with the relation
(2n — 1) (a + β + γ Η-------) = Β
we will satisfy the necessary conditions to enable us to write:
V— = n(n -f- l)pu + jB.


30
Part III.
That is
9-Π
a (u + a)
6 (u) σ (a)
0 — uta
is a solution of Hermite's equation whatever be the value of B,
provided a, l·, c .. fulfil the above conditions.
Solutionrfor n = 2.
It is clear that, save for small values of ny 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
«'2 =p'2a = 4«3 — g2a — g3
[41]	—40s —	— &
pa-\-pb = ~B or a'2 -j- β'2 = 0.
Observe that by designating pl· by — β the above relations
remain unaltered and that we may therefore write
4«3 — g2a — g3 = — 403 + 9ΐβ — g3
or
4(«3 + βΆ) — g3(a + β) = 0
whence
But
β=«-\Β
whence the equation that determines the values of B.
[42]	· · ■ \B2-^aB + a2-\g3 = 0 and also B = 0.*)
If then n = 2 and a and h, the arguments of a = ( — pa2),
are so taken that B shall have the values of the roots of equation (42)
Hermite’s equation will have the solution
*) If in this result we take B =	£ we obtain the formula
r-af + «8-jÿ2 = 0,
see Halphen II p. 131.


Integral as a Product.
31
[43]
[44]
c (u — a) 6 (u + b)	\
V = C —-------\ V 1---- e(Qa — i,b)u
υ	Ü2U
= c’ — fg(M ~ α + fc) iCf«-C6). 1
(itt L (»M	J
« AIAA+A
dtt L (»w	J
where v — a + &.*)
That our solution given above be complete we must obtain
the corresponding values of x and v as follows:
Λ [*(“ + .*) e(« —f»)a1
L GU	J
We have also
i p’a + p'b
y-du
pv + pa + pi
since
Again we have
or
Hence
whence
4 pa — pb
p'a = p'b = a'.
£2 - αξ + «* - \gt = 0
£ — f ± t’KVs — 3«2.
!>(«) = f +
P(*0 =
/ JP %	\
\ρα — p6/
pa—pl = Ÿg2 -	3 «2
i>a +	= a
= — 4a3 + #2a — 03.
These values in (44) give:
w...........ιή-Α-ΙΓ’·-
*) The last is the form given for the expression cos CX -f i cos CY in
the solution of the pendulum problem in the direct investigation of which one
arrives at the expressions
d2X
dt2
NX;
d2Y
dt2
= NY;
d2z
dt2
= NZ + g
where N is found to be 3z2(2pu — ρα2) which causes the solution to depend
upon Lame’s functions.


32
Part III.
If we take a — 2b we have
[46]
p (V) = 8i3 + &
12 62 — g2
[47]
where
φ = 4 δ3 — g.2b
For X we have:
X = ξ(α — V) -f %a
__£ ρ'φ — α) —p'(a)  p (h — a) -f- ff'fr — p'a
2 p (6 — a) — p (a)	2p(b — a) — pi) — pa
9z and <p'= 12b2 — g2
tb = 1 p (~h ~	+Ì*'6
*	2 pQ> — a) — p&
P V
2pv — a
since
— y
p'a —ρ'δ = 0
pa -f-ρδ = a.
Combining these relations we obtain:
P'V I	τ
^+pv = b
and
i)V = 2(6 -pv)	= 2 (6 -	]/3lÿ^
=	1/3 fe φ ' φ
φ' r φ' '
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
[48].................3 =JJ\ (a
6 {a) 6 (u)
gii L,a
X and V remaining as before.
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=ye
which we will proceed to develop as follows:
We would find from the integral s as in the case of y
= Σ^ (β — u) — %u + ζά\
and combining with
ΐΓ=2Τ[ξ(α + ω)_ξ(Μ)~ξα]
we obtain
y - 7	+ α) - 50“ - α) -	= ~Σ^¥~α
But
β (a + u) g (a — u)
Whence
or
y =17
'	'	p a
yz — zy = > —---------·
J	σ ¿mipu — pa
=]Γ[ (pu—ρα)-*)
yz
-Σ^ΊΓα -Tl·*" - *β) =
2C
2 G
pa ____
pu — pa Π (pu—pa) *
G being a constant or expanding and writing t = pu we have
[50]
+
+
+ =
2 G
t — a 1 t — β 1 t — y 1 (t — a) (t — β) (t — y) . ..
an identity independant of the value of t.
To determine a\ β'.. . multiply both members by (t — a),
(t — β) ... and take t = a7 β ... for example
'i β’ (t — «) , νΊβ — <*) ,	_______20
■r (ί_β) -r t__y -r··· {t _ β) {t_y) _
whence making t = a we have
[51]
2 C
(α — (3) (a — y). ..
and in a similar manner we find
2 C

(β — «)(β — y) ■ ■ ■
*) see theory of p and g functions.
3


34
Part Iti.
These values of a and β'... determine the constants a, b...
provided we can find the value of the constant C. It is also clear
that C muet be a constant involved in the relation
Y=f
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 = ±ΫΤ.
I expand these principles of M. Hermite*) (Annali di math.)
and Halphen**) as follows:
Lame's equation may be written
[53]	................... y"= Py
where	_
P = n (n + l)pu + B and y=ŸY.
Seeking the equation in terms of Y we write
whence
Y'= 2 yy
Y" = 2y'2 + 2yy"= 2y2 + 2Pif- 2*/'2+.2ΡΓ,
also
(y'f-2P Y)' — 4y'y"= 4Pyy'= 2PY
whence
[54]................ r"- 4Pr— 2Ρ'Γ = 0
[55]
a linear differential equation in Y of the third order.
From the theory of the linear differential equation, if y and z are
solutions of (53) yy + qz will also be a solution y and q being
arbitrary constants, and we derive also as distinct solutions of the
transformed (54) t/2, y$ and z2 obtained from the complex form
(yy + qz)2	,
P = n(n + l)l? u
and the transformed may be written:
•	·	F"- 4[n(n + 1 )pu + B] T— 2n(n+ 1 )p'uY= 0
where
This value indicates that (55) has n solutions in terms of p («)
*) Bd. Π. 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] · · · · F=¿” +	+ a2tn~2 H-------h 1< + 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
φ = φ(β) = 4¿3 — g2t — gs = p'¡u
Di u = φ
and
DtU = —· -- φ 2 φ'; D¡ u =
3 -
4<P
2φ'2 —
φ ‘ φ
DUY = D( YDu t
9>*D«
Di Y =
DtutfY - DtYD]u
(Dtuf
rf V (Τ)*ΗΤ Df Y - I)( u ΐή wIJtY SD(uDluOtY+ s(D*uy DtY
u '	φ,*γ~
= φ·Σζ Y + }	φ'31 Y- j- φ"Dt Y
These substitutions give:
[57] (4ί3- Λί-Λ) ~ + 3 (W-}g2) ~ - 4 l(w2-f- n - 3) t + B] ^
— 2«(w + l)T=0.
From [56] we obtain the values of these derivatives, namely
= ntn~1 -f- «J (» — l)iB~2 + a2 (w — 2) tn~3 + aB (n — 3) t"-4
+ aA (n — 4) tn~6 + · · ·
da Y
_ = η (η — 1) t”-2+ Oj(» — 1)(« — 2) tn~3-\- a2(n—^)(w — 3)t*-4
+ α3 (ή — 3) (w — 4) ί*-5 -|--------------------
d*Y
— = n(n — 1) (« — 2) <B-3 + ax (n — 1) (» — 2) (n — 3) i»-4
+ a2 (w — 2) (w — 3) (n — 4) tn~’J 4-
and equating the coefficients to zero we have:
3


36
Part IÎL
η — 3: 4α3 (« — 3) (« — 4) {η — 5) — (ra — 1) (ra — 2) (ra — 3)
- ÿ3w(ra — 1) (» — 2) + 18α3 (μ — 3) (η — 4)
—	1 &«1 (w — 1) (ra — 2) — 4 (η- + » — 3) (η — 3) α3
—	4-Β«2 (μ — 2) — 2« (ra -f- 1) α3 = Ο
η — 4: 4α (μ — 3) (η — 4) (η — 5) — g2a2{n — 2) (ra — 3) (« — 4)
—	9ζαι (w — 1) (w — 2) (ra — 3) -f· 18α4 (ra — 4)(w — 5)
---1-	2)	(μ B) — 4 (w2 -(- w — 3) (w 4) α4
—	42? (w — 3) α3 — 2η (η -f- 1) α4 = Ο
η·— k: 4α* (η — k) (η — k — 1) (η — k — 2)
—	gÿak-2 (η — k + 2) (η — k + 1) (w —
—	gsak-3(η — k + 3) (ra — k -f 2)(ra —	+ 1)
+ 18a* (w — k)(n — k—1)
—	I g^-t (n — k -f 2) (n — k + 1)
—	4 (n2 -f- n — 3) (n — /,·)«* — 41! (n — k -f-1) a*_i
—	2n (n -f- 1) a* = 0.
From the last value we pass to the ratil by writing
k = n — μ
%
whence the recurring formula:
[58]	2 (n — μ) (2 μ + 1) (f* H” w Η** 1) αη—μ -f- 4 (μ -f- 1) Βαη—μ—\
+ 7 Λ (" + ^ (** + 2) (2 μ + 3)
+ #3 (f* + 1) (f4 + 2) (μ + 3) αη_μ_3 = 0
from which equation we find the unknown coefficients by making
μ *=> n — 1, w — 2,... or k= 1.2...
These results are simplified by employing the notation intro-
duced by Brioschi, namely:
8=t—b:	—	<p(t) = 4t3—g,t—g?i = <p. t=pu,
by means of which the above forms are expressed as follows:
[59]	[4£8 + I φ" S2 + φ’Β + *>]g + (l8S2 + ± φ" 8 + I Ψ') §£
~[402 + « — 3) 8 +	ψ"] g— 2ra(ra + 1) F = 0
[60]
F — S* +	+ AsSn~* +-+A„


Integral as a Product.
37
[61] 2 (n — μ) (2 μ -f- 1) {μ + w -f- 1) Αη—μ
= 12 (ft + 1) (μ + 1 — n) (ft + 1 + η) bAn-fl- i
+ y (f4 + 1) (ƒ* + 2) (2μ + 3) φ' (6) An—μ—2
+ (f* + 1) 0* + 2) (f* + 3) φ (6) Αη-μ-ί.
Taking μ = n — 1 we find Al = 0
μ = n — 2:	A0 =
w (w 1)
(δ)
8 (2 η — 3)
μ = η-
η (η — 2	/, X
12 (2« — 5)
Μ (η — 1) (« — 2)
2 (2 μ — 3) (2 η — 5)
0φ'(6).
And the term containing the highest power of B is obtained as
follows:
μ = η — 2: 2-2 (2n — 3) (2w — 1) a2 = — 4 (η — 1) B
__	(η — 1) I?
0r	(2 η — 1) (2w — 3)
0	(w — 2) B2
(l—n 3:	tt3— 3 (2* _ 1) (2w _ 5)
u = n — 4·	(w 2) (w — 3) jB3_______ . ...
ft At ■*. it4 2 ·3·(2» — l)(2w — 3)(2w — 5)(2« — 7)	'
(n — 3) (n — 4) B*	.
^ = n 0: ®5 —	2 · 3 ■ 5(2n — 1) (2 « — 3) (2w —5)(2n- 7)(2w — 9) ■
rtíoi	1	·	(- l)’*·»*	i
[62J ft = 1 :	α»-ι=[3.5·7···2«-ΐΤ + ·'·
Direct Solution.
Having Y = yg, 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' = y s' + sy' and	yg — = 2(7.
whence	2 ys’^= 20+ F,	0' 20 + Y' 0r 2= 27
and	— 2sy'=2C~T,	y · Y'—2C or 2F
whence		
	yy"—y'2_y'i /	2/'\2 ΥΓ- Y2
	y‘ y '	.y) ~~ 2P
or	y"	2 YY"-	- r,2+ 4C2
	y	4 Y2


38
Part III.
This value in Hermite’s equation gives:
[63J .... 2 YY"~ F2 + 4C2 = [n(n + 1 )pu + B] 4 Y2.
Whence we derive the value of G sought, namely
[64] . - - - 4(72= F2 - 2 ΓΓ+ 4[n(n + 1 )pu + B] Γ2.
Let a, β, γ · * · = pa, pb, py · · · be roots of Y.
Then
Yu=a· 6·· = tn -(- a1tri~~1	0
Κ = β·6.. =	-- 1)£Λ”2ί'+ · · · = 0
or
Whence
and
dT
du

v, , ár
4C>-/>(«) [i£-L„- r.	*>···
But from algebra we have
[S=* ~	r) ·· ·
Whence
[65]...............2C = cc' (a — β) (cc - γ) . . .
with like expressions for the other roots which we observe are the
values obtain before (see [51]), namely
2 G
a —
β' =
(« — P) (cc — y) . . .
_____________2 G
(β — α) (β -t y) . . .
To obtain Y we have:
2G = Y'a= Y'b — Y'c = - · ‘	.
+ a, +6, + c being the roots of Y = f(u).
We have also:
2C = yz — £?/'
= Y,Ù(U + «)’ — SO — «) - 2£(a)]ys
or
2a
r
= ^[SQ + «) — SO — a) — 2 g (a)].
2 (pu - pa) “ T [£(« + «) + É(« — «) - 2 ga]

But


Integral as a Product.
39
whence
- T =2ΐξ(Μ + ®) - 2{pu^-pàj -	- H
= ƒ- ^[logg(w + a) — log |/pw — pa — log eu — wg«]
=¿log]7
= Æ 1ο%Πα
a(u + a) c_MÍ:a
]/pw — pa ■ cu
+ «) g-κζα_______íL
σΐί	d ft
l°g]7	— pa.
But
1
2
a
Y
J^jYpu — pa = y2z
; lü«J7
0 (m “4“ öt)
£
2
55 ke»**
Whence
= —	+ ¿ logjf7
+ «) ¿-«Ça _ 1 ^' + ^'
σ (w 4" Λ)
dft ƒ I au e	% yz
2C
2 Y
2 Y
2/
dft
A
dft
g— uça '
g(u)
l0*IJ—,e-·!"
or
log y = logJJ^ “ loS c
0 = /7σα.
Whence the value of «/ is obtained directly, namely
[661 · · ·. · · · · · y	<r“te·
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
2 C	,7	2 G
p a —
p'b =
(oc — β) (cc — y) · . · ’	^ " _ (β — cc) β — y)' ·
wheat a = γα, B — yb . . . are ¿Ae roofe of Y.
These relations determine the arguments a b · c ..., having which
the solution is
y = j Jf ° —e~~M A
If we take the second root of C2 we obtain the integral & obtained
also from y by changing u into — ft.


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.
From (60) and (61) we obtain.
Given n = 3
Fw = 3 =	“f* -AgS A3
where
A2
A3
n(n — 1)
8(2 n — 3)
<p'(V) — T 9>'0) — T (12&2 — T ft)
n(n — 2)
12 (2 n — 5)
ΨΦ)~
n{n — 1) (n — 2)
2(2n — 3) (2n —~5)
btp'b = ~φ(ί>) — bq)'b
-τ(44δ3-3^δ + ^.
Again 8 — t — b
.·. φ(ί) = 4 (£ + δ)3 - &(S + δ} - ÿ, ·
= 4S3 + 12δ32 + 12δ2£ + 4δ3 — g2 S - bg, — g3
= 4Ss + 12bS2 + (12δ2 - gt) S + 4δ3 - hg2 - g3
= 4 S3 + 12bS2 + φ’8 + φ.
··· S’ - T *(<) - 36S* - ! φ'-S-i φ.
Hence
[67j · ■ În=z==S* + A2S+As
= S3 + jg/S + \φ-1φ'
*= S3 + (3δ2 - ¿</2) 8(44 δ3 - 3ft δ + Λ)
= 1φ(ί)_δ(φ' + 3«2)
= χ?>(0 ~	+ 3(ί — δ)2]
= *3 - 3δί2 + (6δ2 - t - (Ι5δ3 - Λδ + \g3).
Whence
Γ'=3£2 + Λ, Γ"=6£,	2 ΥΥ"= 128(83+ Α28+Α3)
and substituting in (64) we have
[68] .... C2 = Ì(3/S2 + ¿2)2 - 3/S(/S3 + Λ £ + A,)
+ 3(4S-+ 3δ)(δ’3 + Λ^+Λ)2·


Integral as a Product.
41
To attempt to extract the square roots of this equation in
accordance with the theory, G2 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 IY.
The Special Fan étions of Lamé.
Functions of the First Sort.
Lamé derived originally functions of three different sorts,
values for y7 depending on the value of n and corresponding ‘in
each case to a specific value of B7 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
[69]	.... y =p(n—2) -f-	—4) a2p(rl~G) + · ■ ·
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) + ^iP(*“2) + A2p^n-^ + · · *
whence by subtraction
y"— n(n + 1 )yp = (at — AJp^-® + (a2 — A^p^-^ + · ·
= By
that is a function of the first sort will be a root of Hermite’s
equation provided
at — At = B : a2 — A2 = Bax : a3 — A3 = Ba2 etc.
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 γη + 1 if w is even and γ(η — 1)
if n is uneven:
For example take n = 2, whence y = p -f- at and y"= p” and
we derive
p” — 6(p + <^i)P — Bp —
or
Bax -f \g2 = 0, also 6^ -f- B = 0


The Special Functions of Lamé.
•	43
whence
and we find
y — p — j B where P2 — 3¿/2 = 0.
Again let n = 3 in which case the equation in B would be
of degree y (n — 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"— 12 pp = 0
a well known identity.
Define (P = 0) equal to the equation in B of degree y (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] · · · (— l)’y — fft , ƒ<■-»(») + fn :i)1
the coefficients being the same as in the corresponding general
development.*)
Functions of the Second Sort.
To attain a function of the second sort assume that n is odd
and that the solution has the form
[71]..........................y = #Ÿpu — ea	« = 1.2.3
where z may be developed in the form
z =p(n—s) -f-	+ a2p^a~~7)
an 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 y(w+ 1), the right hand member of
which we will define as Qa which is equal to zero.
*) see (34) and (26).


44	·
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
i>* = 4P3 — 9sP — 9s
we fiave*)
[72] · · (4pB — <jtp — gA) ddy + (lOp2 + 4 eap + 4e2 - f &) ^
*=[(»— 1) (» + 2)p -f B — ea] λ
Take now for example w = 3. whence
.'“■P + ® Jÿ-1 ^ = °
and (72) becomes
lOp2 + 4e«|) + 4e2 — \g2 = (10ji + B — ea) (p + «J
and differentiating we have
4βα = 10 öq “j- B — ea
or
whence
and
1	1 T)
«i = Te« - Ϊ0ΰ
P + T«e—
[73]
....	Qa==B2_ 6eaB + 45e2 - 15& = 0
an equation whose degree is
γ(η + 1) = 2?
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
[^4].........................Q— Φ1Φ2Φ3·	*
Functions of the Third Sort.
We have finally solutions that are doubly periodic of a third
sort the integral being written in the form:
[75]................... y = eY(pu —efi)(pn — ett)
where n is restricted to an even member and s has the form
S == jp(»—4)	¿y)(»-6> 4-	+ · ‘ ·
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 4-w.
*) compair transformation p. 35.


Part V.
Reduction of the Forms when n equals three.
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
where B is to be arbitrarily chosen.
The first form obtain from (32) is
y = jf" +
and from the first of equations (26) we have
■β
.	ht — — tq ’ where B = 156
Hence disregarding the constant the integral is
[77]........................y^f'-Sbf
where
'	6{μ)α{ν)
and X and v satisfy the conditions (35)
Jtf2+ K =

where
X = ξν — ξα — ξό — ξο
v == ci -j- b -f- c.
(p. 17 and p. 16.)
Ist Solution.


46
Part V.
The second form .obtained from (66) is
[79]
» = ΓΤσ (’* + α) e- « t« = Γ7°---- ç«
^ JL 1 6a au	χ x ~'-A e
where
and
«(<*) e(«)
gfr - a)	- c)^8 + tt+c
®(a) e(6) <r(c) σ3α>
a' = p’(a)—	2C
*0' =ρ'(δ) =
y — P (c) — (y _ tt) (γ^~β)
c=±VT·, γ=53+λ« + Λ
(a — β) '(a — y)
2 C ^
(β - a) (β — y)
2(7
(p. 40.)
^ = ί — δ; Λ = T (126‘¿ - T&); Λ = - 7 (44δ3 — 3&δ +
The transformation of form (79) to form (77) may be accom-
plished as follows. Taking the eliments we have
g(M +<*)r-uta
au aa
u
Y pa 4-
whence
y ■■
6J^±3 e-utb_±._!Lnh .
e ~ u 2 Pb +
G(u + c)	u 1 u
6U6C	U ’ 2 PC
Take
^_±d)a(^±b) a(u + ^ e_(^+?6+fc)„
a(a) 6(b) 6(c) o3u
= [ω5 τ(Ρα H"pV) +] (v — yP(c) +)
f =	C-“^“+f6 + fe) = - (_“+jO (x-iv)n
e(« + ö + c)öM °	auav e
~Y~Y (Pa + pb-\- pc) H---------
f Y¡ — I (pa + ρδ + pe) -------------
ƒ"___ 2	?
/ — 55 + ···
Whence we observe that we may write
y ~ ~2 if"u — (pa + pi> + pc) fu\.
But	*
pa + pb pc = \ B = 36
• 2d Solution.


Reduction of the Forms when n equals three.
47
and, disregarding the factor we obtain the first form:
y = f"-m
Having then a method of reduction the determination of
a : δ : c is involved in the determinate of v.
Determination of x and v. First Method.
To this end we have from (31) and (26)
H0 = X] Ht = y (x2 + PO; S2 = y(æ3 + 3 P2x + P8)
and also
ht = -
B
10

2?2
120
whence relations (78) become
L·
20
1(*3 + 3P2* + P3)-|z = 0
±(x*+6P2x*+4P3x + P4) - f (** + P2) = f - f
set
Í-T.B or
and take from (p. 24)
P2 = P3 = — pv; p4 = — 3p2v + |&¡
Which values reduce our relations to the form
(a)	lx3 — 3p{v)x — p'(v) — 3lx = 0
Γ 80]	J	c72
(b)	he4 - 6p(v)x2 — 4p (v)x - 3p2 (v) — 21 + 2 Ip (v) = ~ — //2
I	o
which are reduced forms of the equations of condition that y = Fx (x)
be a solution in addition to which we have the identity
p'O)2 = 4ps(v) — g2p(y) — g3
and the useful relation
H1=\ (a;2 — p(v)). or P(v) = X? — 2HV
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 = w%\ 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 æ as a function of Φ where Φ 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
3 a:4 - 6p(v)x2 — 10lx2 - 2lp(v) + 3/0) = — ~ + g2
whence the relation
p(v) — X2 — 2 H1
gives
(c)	36 Hi — 3 Ux2 + 12 lHt + 5Z2 — 3 g2 = 0.
Again from (b) and the identity
p'(y)2 = (3hx -j- 3p(v)x — χ3)2 = 9ϋ2χ2-\-9ρ2(ν)χ2-\-χ(ί-\- 18bp(v)x2
—	6bx4 — 6jp(v)x4
= 9b2x2 -f 9íc204 - 4a;2iT, + 4Hf) + a;0 + \Ux\xì — 2IQ
—	Qbx4 — 6x4(x2 — 2-íTj)
— 4(íc« - 6xiHl + 12a2Hi - 8HÎ) - g2 (x2 - 2HJ — g%
or multiplying by 9
(d) 81 l2x2 - 108x2H\ + 108lx* — 9 · 36ZH, a:2 + 9 · 32H\
+ 9 g2x? — 18	+ 9g3 = 0.
From (a), (b) and the value for p(v)
a:4 — &x2 (x2 — 2 H, ) — 4a:(a;s — 3 p (v)x — 3 bx) — 3 ([x4 — 4:X2HÍ+4 Hf)
- 2lx2 + 2l{x2 - 2HJ = ^	&
or
(e)	12Za:2 - 12Hj - 4bH, — ψ- g2
and multiplying (e) by 3 and 85^ it becomes
(f)	36 · 8lx2B* - 36 · 8H? - 961HÌ = 40Z2H, — 24g2H,
whence from (c) elliminating BX
(g)	81 Z2*2 - 108x2H¡ + 108Za;4 - 36lHtx2 - 96IBI = 40Z2H,
— 6g2H1 — 9 g2x2 — 9gs.


Reduction of the Forms when n equals three.
49
Whence a further combination with (c) gives
(h) 72i2a;2" - 721H\ - 32PH1 + 6~ + 9gt - 2lg2 - 0
and again
(i)
Whence
where
8 l2Ht - Sg.H, - ^ - 9Λ + 8 lg2 = 0.
10 Z3 — 6Z^2 + — #3
~ 6(22 —
10 Z3 — 8 ax l — b1
■ 6(Za — ax)
and bi =
2Z
T?8·
Prom this value of we have by substituting in (c)
125Z6 — 210a1Z4 — 22&jZ3 + 93af Z2 + 18a, ^ Z + b\ — 4af
® =	36 Z (Z2 — aj2 ~
__ 4(Z2 — a,)3 + (11Z3 — 9a, Z — &x)2
—	~	" 36Z(Z2 - aj)2
-
— SD2
where
Φ(0 = 125ï6 — 210a,£4 - 226^* + 93a2i2 + 18aV + δ2 — 4a\
S = m, _D = (Z2 —ax), l = y B = 3b
«x=3f=p (!	+ fc4); = - 2^s=¿ (1 + *!) (2 - F)(l - 2P).*)
Φ(Ζ) = 0 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.
Φ(1) must then be Q(l) times a constant and as we have seen
that Q is separable into three factors of the second degree it
follows that Φ(ΐ) is a reducable equation of the sixth degree.**)
Moreover if we make the transformation
ï-6f*
*) The expressions used here are essentially the same as those of
M. Hermite in his celebrated Memoir. The following reduction of the function
φ(Ζ) 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.


50
Part V.
the coefficients of Φ all reduce to functions of the absolute in-
variant of the fourth degree
, 3 _ <4 i $ _	(1 — ¥ + le4)3
bi	C b\ 108 g\ (1 + ¿\>2 (2 - Vf (1 - 2 Icy
and we have the form:
[83]	·	Φ&) =	= 125ξβ — 210c|4 — 22|3 + 93c2|2 +· 18c|
+ 1 _ 4c3 = 0.
If then this equation be written in its expanded form in terms
of the modulus h it will not be difficult to see by inspection (for
rigorous proof see p. 56) that if we write
[84]	.........................Φ = Φ,Φ2Φ3
these factors of Φ corresponding to the special functions of the
second sort are, as given by M. Hermite:
Φ, = 5l2 - 2{h2 - 2)1 - U4
[85]	· · ·- · ■ Φ2 = bl2 2(1 — 27c2)? — 3
φ3 = 512 — 2(1 + lc2)l — 3(1 — Jc2)2. '
When Φ = 0 we have x = 0 whence, as before stated, Φ = 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) =	— 2 H,
_ Φ(Ζ) — 12Z(Z2 — αχ)(ί013 — SaJ — bt)-
~	36 Z (Z2 — axy
Define
ψ(0 — Φ(0 — 12?(I2 — ax) (10Z3 — 8axl — \)
= 5 ?6 + 6 aj — lOb^—Sall2 + 6 ajtj + h\ — 4 a¡.
Whence we write
t86].............^(v) = Tc2 sn2 ω —	·
Returning to (80, a) we have
p(v) ==» x(x2 — 3pv — 3?)
_ Φ(Ζ) — 3τρ(ΐ) — 108 Z2 (Z2 — axf
— χ	mnp-ayy
___·>' · χ__
18 Z (Z4 — axY


Reduction of the Forms when n equals three.
51
Where we define
X = y [Φ(1> — 3ψ(0 - Í0SP(P— αχ)2]
= P — GaJ* + áòjZ8 — 3α*Ζ — h\ + 4 α»
=* A · E ■ C.*)
Where
A = P — (1 +#·)* — 3¿2
B = l2 — (1 — 2A2)? + 3(ft2 — fc4)
[87]	.............C — P — (IP — 2)i — 3(1 — ¿2).
Refering then to note (p. 24) we have:
[88]	·	■	·	· p'(v) = — IPsvPv ■ cn2v -drPv =
That is p'(V) vanishes where ¿r vanishes which gives v — wi
a semi-period, and in consequence, when Φ = 0, f reduces to
= C(M + W*) e_ .C(ei) e!îW. ·
<y(w)	^w?)==tix
The value of the function of Lamé corresponding to any
value of B giving rise to the condition Φ = 0 is then deduced as
follows.
From Φ3 = 0 we derive:
E = 5? — 1 + IP + 2 J/Ï9(l — Vf + ¥
and the special equation of Lamé becomes
y"= [l 2_p (u) + 1 + ¥ + 21/19Jï^¥f~+¥]y
and from the general form of the integral [77]
y = n — y {1 + W + 21/19(1 — F)2 +T2} /].
But differentiating ƒ* we have
K = [2jp(«) +iNlf,
= [2p(«) + ea] fv
Hence
y = [2p(«) +	- i (1 + TP) - f l/Ï9(l - A2)2 + fc2] ^
= [2pM + ea - |(1 + Λ*) — 41/19(1 — ¿2)2 + ¿2] 1/pu — ea
= 2 [K**) + 4e« — ¿ -B] W — e«
or
y = s|/f>M — ea
where 0 has the value determined by the elimentary consideration
(p· 44).
*) Compair [161] p. 73.
4*


52
Part Y.
Case χ = 0.
If % = 0 we have a second case in which the p (y) 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 e?u. If then χ = 0 we will
have from (87) six values of B for which the integral will take
the form
y = f" — ~ Bnf, where f2 = ^	(* —
v 1 2	5 2/2	i ¿	eu a ωλ
c7u
__ A qXU
6U
Moreover the second integral will be
the form remaining unchanged which is not as we have seen in
general the case.
Case Ό = 0.
The only remaining case to be considered is where D — 0, or
P -, a1 = V — 1 + k* —	= 0
or l = ±(l — ¥ +	=
since X2g2 — y (1 — k2 + ¥). Also l = 3& whence
or 12 l·2 — 9% = φ'(Ρ) = 0.
That is D = 0 and φ'(δ) = 0 are conditions for one and the
same function of Lamé. In this case p(v) and also the p'{y) become
infinite which gives v = 0 or the congruent values 2mw -f* 2m 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.
Helation 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


Reduction of the Forms when n equals three.	53
first sort, and the other, three in all, of degree γ (n + 1) giving
rise to functions of the second sort; whence combining we have, n
being odd, B determined by an equation of degree γ (n -f- 1) + y (n ~~ 1)
= 2n + 1. If n is even we find but one equation, degree 
-\- 1,
for functions of the first sort and three equations, degree γ n9 for
those of the second sort making a single equation whose degree
as in the first case is 2n -f- 1. If then these roots are all different
we have in all 2 n -f- 1 special functions of Lamé.
Returning now to the forms (65)
2 G = a (a — fi) (a — γ) · · ·
we have the half periods or values of the roots a, β that will
reduce them to zero. Moreover they will not be double roots, for
consider t = e% 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 — ef)s (pu — e2Y (pu — e3)*" Π(pu -	CM 1
	whence	
[90]	• · · y = V(pu — ety {pu — e2y (jm — e3)°" Π(pu -	-pa)
where
£, s , b" = 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 —
That is C = 0 is a condition that the integrals he 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 c2 = (4 ís — g2t
93)[(^
dt)
2 ¥
d* r
dt*
] — (12i2
Λ)ΓΪ
and we have (62, p. 37)
Γ =
+ 4 [n(n + 1 )t+ B] Y2
(__ xyiBn
[3 · 5 · 7 · · 2 w — 1]
12 +
from which relations we see that the highest power of B in c2 is
2 w + 1 and that the condition (7=0 gives rise to an equation of
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 fornid-
as 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 B = 0,
the degree in B being unity.
n = 5 :y = p" — y Bp = p ( 12p — J- B) where B2 — 21 g2 = 0
the degree being two, etc.
But p' (u) = 4 (pu — ef) (pu — e2) (pu — e3) whence for n odd
or equal to three, f, ε, έ' are all equal to unity.
Moreover we have obtained Y (67, p. 40) expressed as a poly-
nomial in t and h in the form
Γη=ΐ={φ(*)-δ[φ' + 3(*-&)2]
and since
p�i) = t' (βχ) = 0
we derive
[91]..............Yn=z(ef) — — b [φ + 3 (βχ — δ)].
Hence Pw==3 — P = 15 δ is a factor of Yn=$(ex) times a
constant.
If on the other hand n be even none of the functions of the
first sort contain a factor ]/pu — βχ and PM==2* will not be a
factor of Yn== %x(ex).


•Reduction of the Forms when n equals three.
55
Functions of the Second Sort.
We have found three equations each of degree ~ (n -f- 1) or
Y	n as n is taken odd or even, that give values of B that, if n
he odd, correspond to functions of the second sort, or, if n be
even, to functions of the third sort. Designate? the first members,
by Qu Q2, and Q3. Refering again to Lame's special functions
we see that if Qt = 0 the function of Lamé corresponding contains
the factor Ypu — et if n is odd and the two corresponding factors
Y	pu — e2, Y pu — e3 if n is even. In the first case Qt is a factor
of Y(e±) and in the second case of Y(e2) and of Y(e3), while in
the second case we have also Y(ef) contains the factor Q2QS.
Returning to n = 3 we have (see (73) p. 44)
[&]*=3 =	- 6 ^ B + 45 c,2 - 15 #2
[92] ........[&U3 = B* - 6 e2B + 45e22 - 10g2
[Q3
=3 = B2 — 6es B + 45 e32 — lbg2
or in general writing B = 15& and φ = <p(b) = 4&3 — g2b — g3
[93] ...........[Qx]n=3 = 32.5 W + 3 (ex - ft)2].
Also from (91).
[94]	· . . .	Γ(0«-δ[φ+ 3	— &)2]
— - b [lbb2 + 3e,2 - 6etb - g2]
B ΓΒ2
15 L 15 '
6etB
15
+	— ft]
= - cB [B2 — 6^B + 45e2 - 15ft]
= - c*&P
[95]
where in general
__	l
C 3 . 5 · · . 2n— ï'
The quantities Q are also necessarily the functions Φ times a
factor as is shown by taking the substitutions
ƒ = à C1 - ** + ^
[&].=» = B*-\{2- Tc2)B -
[&]„=s = B2 - \ (1 - 2k2) B - ~
[&].=. = B2 - \ (1 + ¥) B - 8-*(1jl-*y.
whence:


56
Part Y.
Hence making λ — constant, equal — 1 and B = 61 = 15 δ
we obtain
mn=* = 5 Φ, = 5 [bl2 - 2 (h2 - 2) l - 3
[96]	· . - [ρ2],==3=5Φ2 = 5[5ϊ2-2(1 -2A2)i^3]
[^3]n=3 - 5.Φ3 - 5 [51® - 2 (1 + V) l - 3 (1 - V)*].
Hence also:
[97]	Q-QXQ%Q,-^Φφ-δ3®^,®,
= 53 [4 (ï2 - aty + (11 i3 - 9 V - δ,)2]
= 53 [12516 — 210^|4 —22|3 + 93q2|2+ 18 01 + 1 — 4^®]
= etc,
where
3__V_____!	_ 10Q	(1-F + fc4)3____
Ci &t» “ 108 * ^ - iUO (1 + ¿2)2 (2 — ¿2)2 (! __ 2 ¿2)2
We have moreover that the conditions that the integrals be
special functions of Lamé are that Qlf Q2J Q3 and P vanish. But
C2 — 0 was also found to be a condition and we note that the
sum of the degrees of Qi and P is equal to the degree of G2 which
equals the number of the functions of Lamé. We must have then
the relation
C2 = c'Q1Q2Q3P.
But we have shown that the highest power of B in the de-
velopment of AC2 is (p. 38)
4/72 = ABY2 4- ...__ __ 4#(— B)2n i . . .
40 Λ-f-	— [3 . 5 . . . (2n _ i)]* +
whence
C' =_______________= é
[3 · 5 · · . (2w — l)]4	6
which for n = 3 gives as before taken c — ~-=■ ·
0*0
We have then in general
[98] ··.··'......CP — àPQiQiQ»
and when n = 3
to............c'-m-QF-^FQ·
If then we take
Q, = 0 : B = 3c, + Y$(—12^+5¿¡) = 7c2 — 2 ± V(h2 — 2)2 + 15**
y~{ï,Jr jei - ¿ (3^ ± Ÿ3 (5^2 — Ï2e|})}
= {p + ά (fc2 - 2) ± ¿ νζν-2γ+ΐδ&} VP j(k2 2)


Reduction of the Forms when n equals three.
57
[10Ò] Q2 = 0 : B = 3e2 + ]/3 (5g2 - 12e,) = 1 - 27c2 + ]/(l — 2¿2)2 + 15
y =	+ i «* - fo ± ν'» (5λ- í2^1)) } V*3 *
= {*> + è (1-2J*) + ¿ y(ï=2^+Ï5} l/p-fíl-2F)
Ç3 = 0:B==3e3 +]/3(5</2 — 12e|) = 1 +	+ 2 V(2 — F)2 - 3fc
2/ = íi> + T«b - ¿ (3ea ± y3(5^-12e·))}
“ h + ¿ (1 + m ± i V(2-k*f-M} YP-l(l + ¥)
all of which are special functions of Lamé of the second species,
the general form being
y = ζγρη — ea
where
Z = p(*~3) 4“ «iP(?î — 5) +··.· + C,
and as given (p. 43) the general form for n = 3 including the
above is
[101]· ·■···· y-(p + Tee-¡5B)yp=^
where
B = 3ea± γ3(δg2 - 12e*).
The Discriminant of Y.
From (65) p. 38 we have
2(7= a (a — β) (cc - γ) ■ ■ ·=β'(β — a) (β — γ) ··· = ·■ ·
= Ϋφ (α) (α — β)(α — γ) ■ ■ ·]/φ (b) (β — α)(β - γ) ··· = ·· ·
where ·
Ψ («) = 4 (l>« — e,) (#μ — e2) (pu — ¿3)
Γ" = (pw — ei)' (ρΜ — e2Y (pu — e3)f" Π (pu — pa).
The roots of
Ψ (a) = 0 are e¡, e2, es.
The roots of
Y= 0 are el7 e2, es, α1βί · · ·
Whence the resultant of φ(α) and Y written as the product of
the differences of the roots is
H = n{a — ex), where α = α1/31·· · to n letters and λ = 1, 2 or 3
= [(« — ei) 0 — e2) (« — e2Ji [(β — β,)(β — <h) (β ~ «3)] * ' '
=^ΤΊφ^·


58
Part V.
But again
T(et) = [(« - e,) (β - (y - e,)] · · ·
r(c2) = [(« - e2) (β - e2) (y -e,)]···
*&)-[(«-0.(0-'»)(r-<%)]···
whence
Β=Π(~α-^==^Πφ(α) - (~-1)'ÍTr(ei)·
Again we have shown (94; p. 55) that for w — 3; and the
same method gives in general for n odd:
w odd:	r(0 = -c2P(?i;	Y(e2) =—c2 PQ2¡ Y(e3) = - c2PÇ3
and likewise
» even:	Y(et) = c2Q2Qs·, Y(e2) = c*<?s&; Γ(tv) = c2#^
Whence we finally derive
[102] R -π » - «)=- (- ')·ΐ7 ^>- lí £ v *
α. λ	λ	7
Now the discriminant of Y equals the product of the squares of
the differences of the roots and may be written:
A == (« — β)2(« — γ)2· ■ ■
whence from (65)
22 (72 2 2C2	22w£2»
Δ2
φ(α) φ (b)
Πφ(α)
Πφ (a) = 4riP
C2n
Δ2 =

But we have first found
whence
Again
C2 = c4P$
and we derive from these n being odd
λ 2 _ c2n _ (CY _ c4w PnQn _
^ ~ Β ~ B ~~ cQP*Q~
(from 99)
c2(2n — 3)pn—3Qn — l
or
n — 1	n — 3 n — 1
Δ = (— 1) 2 c2n~~BP s Q 2 : n odd
[103] and in like manner we derive
(Sign ambiguous)
Δ = (—i)2C2^p~2 ρ2
n even
and we have also Δ = 0 since Y has at least one double root.


Reduction of the Forms when n equals three.
59
Case n = 3.
[104]	B = (a — et) (a—e2) (a — e3)(β—ο,)(β—ε2)(/3—e3)(y—ex)(y—e,)(y-e.¡)
= ¿ («3 —·Λ« - 9s) (β3 - 92 β~ 9s) (73 —927 — 9s)
= ~ VW+Z fe - ¿fe] [φ'+ 3 fe - ¿fe] [φ'+ B fe - ¿fe]
[105]	A = ¿P°<? = g)01<?2<?3
= 33[φ' + 3(6, - h)2] [φ'-f 3(e2 — b)2] [φ' + 3 fe - l·)2] (see (94))
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 λ has been determined equal to (— 1) we
have from (97)
Q = 53[4 {l2 - α,γ + (11 Is ~ 9al - b)2]
and the relations:
I = 3b:al —
== f 9s ■ A = T (12 δ2 — £2) : A = — j (44ft3 —	+ </3)
4fe — a,)3 = 4 · 27A| : 11Z3 — 9atl — ¿>fe = — 27As
[106]	· · ·...........Ç = 3s-53[44 + 27^]
[107]	.............A = [4Ja + 27^Ì]
which latter value we would have derived directly from the form
y=s* + a2s + a3.
Writing A2 = y<p' and Α3 = ~φ— b<p' we derive still
another form for Q namely
[108]· ·	<? = -^[φ/3 + 27φ2- 8·27δφφ' + 16·27δ2φ'2].
Again we find
[109]· · ·
.2 _	Φ(0
36Ϊ (ϊ2“— a¡)
4233Δ
36 · 336φ'2
_{ 2 ΐ/4ΑΪ+27Α|Γ
13φ'Κ δ~ J
from which value we again see that the vanishing of ψ’ is equi-
valent to the Vanishing Of Ώ.	(compair p. 49 and 52.)


60
Part V.
Determination of x and v. Second Method.
We have the general theorem: every rational function of pu
and pu can be written in the form:
Ac (u — vj a (u — v%) · · · a (u — Vu)
<Pl W G(u — 1//) o(u — V%') · · * 6 (u — v'u)
where the number of 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:
[U0] (_ irii _ 0(pu) _ w (pu)
where Φ and Ψ are intire polynomials in pu and p'u, \ a con-
stant to be determined and the relation exists « + b + c =
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 Φ and Ψ are thus determined as follows:
φ	ψ
n odd:	| (w + 1)	~ (n — 3)
' n even	~n	— 1
n = 3	2	0.
The n roots of the first member in the general case being
a, 6, c ... we have:
[111]	................Φ(α’) — ^1α'Ψ(α) = 0
where
u—p'(a)7	a=p(a).
From (p. 38)
dY 2C f '	t ' \
~dt = «'=(“- β) (“ - r) ·■ ■
whence
J_ , _ ( dt\
2 C®. \dYJi=a
and [111] becomes
[112]	.............Γφ^Ι-ψΊ	=0.
But a, β, y,... are also roots of Yf whence the. relation
[113] ·...............- Ψ = ΕΥ
where E is also in general an intire polynomial in t whence
t114J...................#4?=^+?	'


Reduction of the Forms when n equals three.
61
We have also
[«"-ίΊ·.-0
etc. for the other roots of Y. The degrees of [114] are
7JS
Y	.	Φ
n odd:	,-----------------i---------r	Ί-----------
|(H-8)-» = -i(» + 8)	} (n + 3) -1
n even:	γ (n) — 1 — n = — (γ n + l),	(j« + l] — 1.
We have
. Y = tn -f- altn~1 -f- a2tn~~2 -f- · · · -f- an-xt -f- un
Y' = ntn~x + (m — 1) a±tn~2 + (n — 2) a2tn~3 + · · · + an—i
and
Y_ = nt"-1 + ax (η — 1) tn~2 -\-_ h> i h JL h 4. .
Y f1 + α,Γ"1 + a2tn~2 + ·. -	^ * i2 '	”*
or
ntn~1 + ax (n — 1) tn~2 -f- a2 (n — 2) ¿w—3
+ · · · =A	+ *·’·) + (*”~2 + Mn~3 + · * 0
and equating the corresponding coefficients we obtain:
\ = n
at (n — 1) = nat + ^ or = — «1
[115]·.................δ2 = — 2ft + aî
g
= — 3	-{— öq $2 —
etc. . —	—	—	—	—
Proceeding in like manner we write:
Φ = B0V + JBji—1 -1------h + -Br
where
V	=[}(w+ 1), 4W]
whence
(Τ + Ψ + F + · · ·) W + B^~l + ' ' '■ + Μ + *)
= 60 (ΰοί’-1 + η,#»-* + ■·· + B,_ ! + *’-)
+ δ, (-B0¿”-2 + B**»-» H---l· -B. -2 + Br-i*"1 + -Β,ί-2)
+ \ (-B0¿r—8 + Bt £’~4 -f- ···) + ·· ·


62
Part V.
[11.6]
^+VU1+iiW+W^+"'+f'7+'·1^
+ \Bvir* + b.Bv-it-1 + \Br-i + hB,-»t + · -
+ ’ . ,
+ hv-1Bvt~v+	-----H ®*-i
=	+ h0B,-iP + b0Bv-^+l-\-l· KBJ2
+ b0B0t**-1+t>iBvt''-2+b1Br-it*-1-\-hBv-i!t''+ ■·
from whence the relations:
\Βν + \Bv-i + b2Br-2 + · · · + b>vB0 = 0
\Βύ + \Br-1 +	H----l· ^r+l-^O = 0
V — 2
bv-XBv + bvBv-1 + br+tBv-2 H---------1- &2,-i#c = 0
We will define:
[117]
dm =
w
&1 &2
*	* ’ bm
*	‘ * .^m + l
^m^ra + 1 ^m-f-2 * * * ^2m
We will define
B, = ^r—1
and we will then have from the above conditions, all the coeffi-
cients B1B2 , .. as intire functions of W ... which are in turn
functions of a19 a2 ... which finally are expressed as functions of
B,g2 and g3.
That is we have obtained Φ, of which the first coefficient
shall be âv^x intire in terms of t,B,g2 and g3.
Case n = 3 we have:
B
5
from	(p. 36)				
μ =	2: 2 ·	1 ·	5 · 6at + 4 · SB = 0		or
μ =	1: a2	=	2*’ Λ 3 · δ2 4		
μ =	0: as	=	B3 i 3* 52 1” 3 -5	93 4	
and	from (115)				
tn-1	n = b0				
¿«-2	al (n —	.1)	= Kaû \ =	‘■ — Οχ	
	a2(n —	■2)	= b0a2 -f- l>1 al	+ ^2;	h-
tn ^	as (M —	3)	— �3 +	+	+ V>


Reduction of the Forms when n equals three.	63
The conditions (116) become:
b0B2 + \Bi + b2B0 *= 0
\B2 + b2 B1 + bsB0 = 0
whence
(6Λ-^)^-(δΛ-^)50
(Po^2 hi) -®i — (bxb2 b0b3) B0.
But
B0 = &Λ - bí = \g- I B* = 4 g2- 18 δ2 = - 6A2 = - f φ'2
whence
jB2= (— %) φαχα2— 3α3— α®) — (4α| — 4α^α2 + <ή)
= α^α2+ 3%^— 4α|
=	I ft·5*	,	_ 1 2
3*. 53	3·4·5*	'	4 · 5	4 y2
= 32 · 5δ4 + τ ^ + T — Τ ^2
Β1 = (— %) (α2 — 2 α,) — 3 (3 α, «2— 3	— α®)
= 2α® — Ία^α^ + 9α3
7Β° g,B	3 V
3· 53 "τ"	4	4
= - 3276®+^2δ-|5ί3
= 9^3+ 126 Α2 — -J- φ — 6δφ\
We derive then, finally
Φ = 5*» + Btt + B2
— — 6(362— ■!· <?2) (£2+2δ£ + δ2)
+ (636»+“&6--|&)(S + 6)
+ 32·52δ4+ -|#2δ2-|-|-δ#3— ^^2-
Coef. S2 is - 6' (3 δ2 -· | &) = - 6
„ 5Ϊ8-9(-116»+|Λ6-|Λ)=9Λ
» S°„ -4(3δ2-|Λ)2 = -4^2
Hence
[118] Φ = - 6 (3δ2— jg2) S2+9 (- 11δ®+ ±gib-l.gt)S-4(SP-±gt)‘
= - 6 J.2S2+ 9Aa8 — 4 At
Having obtained Φ, the calculation of Ψ and E is simplified
by the following considerations:


64
Part Y.
Let n be taken odd and take for B a root of the equation
Qi = 0. In this case (see p. 54) we have Y as a· product of t — ei
by a polynomial TP 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	·
Φ ~ — Ψ = EY
at
we find that U must also be a factor of Ψ. This, however, we
know to be impossible since the degree of U is γ (n— 1) and
that of Ψ only y (n — 3) (p. 60).
The only conclusion possible then is that Ψ contains a zero
factor. We know also that B being any value whatever, Ψ con-
sidered as a function of B contains the factors Ql7 Q2 and Q3 and
it follows that we may write
*Pnote=Q®
where Q = 0, if B be taken as a root of Qt = 0, ()2=0, or
Q3 = 0, and Θ = f® (t).
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
V of degree γ n where Ψ is only of degree γ n — 1, and that in
consequence one has
Uneven = P@
Hence we write:
wodd: 0~-Q@ = EY
n even: Φ — ΡΘ = EY
dt
where all the functions are intire in t.
As we have before determined the first coefficient of Φ is the
determinant	and in like manner we find the first coefficient
of Ψ to be
[120].............dv = bvBv -f~	i -{-···-[“ ^2vBq.
Hence if we divide δν by Q, n being odd we will have γ7 the
first coefficient of Θ.
[119]


65
Reduction of the Forms when n equals three.
To find Έ, n = 3.
The degree of Φ is — (n + 1), the degree of Y is n — 1 and
the degree of Ψ is ~{n — 3) less than Y. Hence from the rela-
tion on p(64), the degree of EY must be
j(«+l) + (W-l) = ί·(3«-1).
But the degree of Y is n and hence the degree of Έ is -~(w — 1).
We have then
[121]	ÆU3 = ηί+ηι
and Ψ reduces to a constant, namely:
[122]	ψη==3 = γρ
We have:
r3 =s* + a2s + a,
Γ3' = 3^ + Λ
Φ = — 6A2S* +	— Â22
and substituting we derive
(3^+ΛΧ-6Λ«2+9Λ^4Λ2)=(^ + ι?0(^+Λ^ + Λ)+^« .
and from these we have
r¡ = — 18 Δ2·7 ηί = 27 A?,
whence
[123]	Es = - 9[2A2S—3A3].
Returning to our original form we find that when n is three
we may write:
[124] (-	+ *) _ Φ(ρη)
L	J 1	eaobec{6uy	*
2 c
pu W(pu)
= φ— YçS'yQ=(- 6Λ S2+ 9Λ5 - 4 Λ*) + ioS’ (4 AJ+21A*).
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 = Ci [£(w — a) + g(w — l·) + g(w — c) · ■ · ξ(η + ν)—(η + 1 )ζιι.
But the developments are known:
ξ(ιι + v) — %u = ζ(ν) — Ì — up (v) - 2
g (w — a) — g (m) =» — ga — Ί — «pa — p a ■ ■ ■
g (« — δ) — g (w) = — g6 — ^	— ~pb···
5


66
Part V.
and we may write
Α-*($ν — ξα — ζό — ξο··) —	1 _ (pv + « + β + γ + ..)«
- ΐτ (ί»'ν + pa + p'b + · ·) · ■·
But
ζν — ζα — ζό — ζο — — X and _ρ'α'+ p'b p'c -1- · · = O
(see pages 25 and 45); whence
[125] A = —	+ a: — (a + /3 + y + · + pv) u — \pv -f · ·
The degree of Φ is γ (w + 1), °f B’, 3, of Ψ, \ (n — 3), and
of p, γ which gives the degree of the second member as γ (w -f- 1),
also
1 .	.	-<* + »>	1	i	_ 1	,
*«-*+··· whence p* =	‘
and developing the second member (B) we write, disregarding the
constant factor
B
<h ,	I___?8----L .
+ 77 + ,,η-1 ^	^
„"+1
whence
[126] rf log B =
w + 1
,7”+2
_
¡5+Ϊ
(η — 1] ___ (m — 2) qs
~	u
„«+1
+ -^ +
I —2	»
1 (n + 1) + *Jigi_±i!L±i^,+,(V: 2)-^^
- · i---- ! 4- W + 3ΐ« + 9c “■ + · · ·
= _ » + i i s^(2gt—g/)u + (3g3—3gtg,+&>?+..·
Again:	,	...
Φ = I>y2
1
.}(»+» _f_ ^ 2
. ÍB — 3>
Θ = y t2
p’ = (4 f tfft + &)
+
_L (rt — 5)
3)+1ί*2	+
¿ 2
'2 == — ¡Te +
whence
/	Ì(*+1)I 7}iï("_1,J. )_γγ(4^—Í#g-
Β=(ΰ0ί2	+ ^1*	+··7	26
r,),/:(ytl (η -j-yí2<M 5)
-.«/d
+
_ υ0 , Or.+ -&-.+ττΓ~,
— y+ï ^ CuK “(η~ι)
Οϊι~ 4-
£7^


Reduction of the Forms when n equals three.
67
and
[127]	dlog. B=B0[- î±i + %y + (2B, - ^-)u+(^p- - ^2?1+2?1S)]M2+ ·
Prom developments [126] and [127] we find
[128]	· · · · 2, =^jr; &=§‘; qs==BWo’ nhein9oM>
and from developments [125] and [126]
:r —n — 9.t
X qi~ UBa
[129]	« + β + γ Η-----1- pv) = q* — 2 q<¡
p'v = 2(3qiq2 - 3qs + q*)
* These forms are transformed by the aid of the relations
C = <?yPQ (p. 56); (2η — 1)(cc + β + γ + · · ·) = B (p. 29)
(2η - 1) α, = — B (p. 86) whence (« + β + γ + ···) = — α, = 2 χ
giving as result:
_ Qy_  y i/Q
CB0	c¿B0 V P
Qyt¿ 2ΒΛ	Β
«V	■ 2 Bl
	¿V
6QyBi	sQyi i 2<?V OB, ‘ 6'3P„3
OB,*	
[130]
x
η odd.
PV c*PB,
2
2Bj
Β~
2 η — 1
1 ν c2 le* PP*3	Β02 ^ Ρ0 I K Ρ
and from these the combined forms arise
3yx
p V
2x
<?y*_ 4_
c*PB* ’ Bñ
p'v	Bx 3 yx
+ pV = w-
2x 1 Γ B0 y 2 n — 1
These formules are perfectly general for n odd and the corre-
sponding forms n even obtained in like manner are
[131]
X	II $|8 II	ΨΫΪ
pv	c4 QB,2	_ 2γχ Β
	Py2 '	y 2 η — 1
p'v	50 (M 1 II	í c4W 3 BoYl [ Pyà y2
p' V	1 Yl	2 Bx B
> 2x	+ Pv = -j	B0 2w —
n even.
+
ψ\ν
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.
For case n = 3.
We have found all the eliments except γχ which is derived
from development of Θ, or more easily as follows.
From (106, p. 59)
« = (15)» [4 4+ 27 4]
and from (p. 65)
Ψη=* = Qy = (3S* + A) (— 6 A¿S2 + 9A^8 - 4 Λ3)
+ 9(2Α,8 - 3+3)(S3 + Λ « + Aÿ )
= -(4J32 + 27^)
and a comparison gives immediately
I132!.......................y=~(Í5)s'
The other values for the eliments have been found, namely		
c = k	yi== 0	φ'=12δ2-<72
P= 156	Λ-τφ'	ï = 36
J5o=-l«P'	Λ= j9> ~	è9>' «i = χΛ
Βί=^φ — 6bq)' φ == 463 —		— h =—”ít3.
We have then for	w equals three	
■ X =	/Q 2 (15)2 Ί P f (15)a * 3φ/ 1	1 /(lñ)3(443+2742) r 156
-Ì7Ì	/¿Λ3 + 27	(compair 109, p.
1 ί Ψ 3	27φ2 — .8(27) 1)φφ'	16(27)b2 φ' 2 \ 2
6φ' I	b	J
Squaring we have:


[134]·
Reduction of the Forms when n equals three.
;== 4(3b*- |g,)8+ 27(1163,
36b(3b* -\9i)'¡
_ 4(P — a,)3 -f (116* - 90,6,— 6,)2
69
36 l(P — a,)2
Φ(1)
361 (P - a,)2
Φ, Φ, Φ3
(compair 82, p. 4.9.)
[135]
_______________
36Z(/2 — o,)2	5362Z (Í2 — a,)2 ~ ,S’JJ2 —’ '
Again we have:
_	<2y2	2P,	B
1>V ~ c4PJB„2 B„ 2w — 1
4[4Λ3 + 2742]	4(| φ — 66φ')
9φ'2δ
+
3φ'
u
whence
„οβη	7	^[¿Φ'3+ 27(¿V2 - }6φφ'+ 62φ'2) I + ^ψφ'6 -·7262ψ'2
[Í3b]pv - b = L---------------------—--------- --------------------
__ φ'3 + ‘27φ2 — 108δφφ'
36 δ φ’2
Writing φ and φ' in terms of g2- g3 and l· we have:
φ ·’= 17285*— 4325^ + 36b*g\ - g\
27 φ- = 432 6® — 216 54</2+ 27 52¿/‘2—216 bsg.ò -[- 27 í/2 4- 54 </2 y¿ b.
- 1085φφ'=-----518456+172854</2 —108 52^2 +1296 63^3 — 108bg<¿g3
whence
21606e +-2166‘(/s + 108063<73 — 962</2 - bibg^g, - g\ + 27gi*
[137] pv =
36δ(ΐ44δ4 — 24δ2#2 +
Again from the first method
if; (3b)
where
pv = — SI = Li*’ - hHn*v = ioW5i _ ai)2
ψ = 5(35)6 + 6α1(3δ)4-1051(35)3—3a2(35)2 + 6a161(35)-f52-4af
or expanding we again obtain
[138] pv =
21606e + 21664</2 + 10800, 63 — 96202 — 546<kg, —+ “7^a
36&(l4464 — 246202 + flr|)2
φ'3	27φ2 — 108δφφ' + 36δ2φ'2
36δφ'2
*) Compair Hermite whereP=#)1,y=s<2al ■^== $3» S=3&liD=^(l2 w ' αι·


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
pv .	Bx 3γχ B
-+ρν = Ά- — -¥-Γ—
= l? - 66φ' _ u = Μφ - 3φ
Whence
_Φ
φ'
p'v = (b —	— pv^2x = — \{pv — b) +
1626φφ' — 27φ* — φ'"
3 φ
2a;
108 φ’3όΐ
Ϋφ'3 + 27φ2 - 2Ϊ6Ϊφφ; +"4326*φ78.
Determination of ν. 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
use] (-1)-*,·<“—>»<«-»>·	_ ¿„w (*,„).
caco · · cv(pu) “
Also
rc(4t - a)~j _ _ j
L ca Jm=o
whence it follows that the left hand member of (139) depends for
its value on the terms
(- O"*i .
c{u)n + 1
But we have again
Γ—1	= i
L w Ju = o
whence we may write, taking n odd
[(_	6(u + a)-'~c(u + v) Ί =	\	,__
]}	1 <>(ά) βψ) · · · 0(.v)(tftt)""h1JM=o ww+1
and from p. 66	K
=	4-
-	c
That is n being odd
kt = B0.
And a similar investigation gives n being even
k - Py-
. i-----C-
Qv i
+


Reduction of the Forms when n equals three.
71
Since V == a b c we may write
p—a-\-b-f-c-j--v) ^	^
and multiplying by this factor we can separate the left hand
member into factors of the form
1140!.................. +
ou g a	g a
for U = wv
But for this value p\w^) = 0 and our relation becomes
and
h
o1aoxb	• · · V
6aeb ·	• - GV
t in a	similar
<>2 a a2 b	• · · 6t V
aacb	• · · GV
αΆ ao3b	■·■ 6.0V
6aob	• · · GV
= Φ(Ρ™2) ■= Φ<Δ) = Φ2
= Φ(ρν)Ά) = Φ(%) = Φ8.
Recalling the known relation
t	^ 6,U 69U 6„U
p u = — 2 -—I——
L	60U
we have upon taking the product of the above equations
[142].	.	.	·	. tfp'apb · · · pv = (— 2)”+1 Φχ Φ2Φ3.
Again from the relations (65)
“-(*-(I) (« - y) (« - Í) . . . etC·
to n terms and we obtain the product
a"e\- !)»·*···—»_(_ 1)1
[143]	α'/Τ/------(α _ (})*■(* _ y)* (« _ d)* ■ ■ ■ (β — y)2 (β-d)* · · · (y - d)2 · :
~	Δ
Δ being the discriminant of Y. Substituting this value in [142]
we derive
[144]	· · · · (— l)T”(”-,)2“cVv = (— 1)’,+12Φ,Φ2Φ8Δ.
Again squaring we get
a cib · · <>
or (see [89])
k*	= φ*(βι) = x)*W(pa — ei) 0& — e 1) · · (pv — e,)
o*a <r 0 · · · <> v
[145]...............(— l)“/c2 Ffa) (pv - e,) = ®2(e,)·
and we have also the two corresponding expressions.


72
Part Y.
We have shown (see p. 58) that when t = et we have
Y(e 1) = — c2PQ whence it follows from this and relation [145]
that Φ(βj) is divisable by Qt and in general Φ(βχ) by Qx. We
thus derive the relations
[146]. . . .	Φ^&Ρ] : Φ2 = ÇÿF2 : Φ3 = ÇSF3.
We have also found n being odd:
n — 1
k = B0 : C =	: J = (- 1) 2 c*»-»P*<M ”Q2 (* W
rfa) = — c2PQt
These values in [144] give
(- iÿ(n~1)n BynP^ Q^pv
= (- lf+*+—	Q^n~l)QF^F^
or
[147]
p V
; 2 QFtFtFa __ 2 FtFtFt yQ
e'P^B/çfi c3PB0a V P
and from [145]
BypQ, (pv - et) = Φ\ = Q\F\
Whence we have in general
[148].
[149].
pv — ex =
QxFÎ
C2P02P'
The corresponding expressions for n even are
2ο3ΦχΦ2Φ3 -yfQ
p V =-----
jpv “
y3P
y2 P
Again from [130]
2 P1
~c3J
f £y3	3 γΒ, 1 |/v
lc4PP03	¿7 ) Γ P
ϋΐ/§	
p K p	
■ l£ys	8yBie*BeP|
,3P 1 c3	c3P03P 1
2y
q^-SB^PcM j/|
Compairing the second and fourth forms we have
[150].........F.F.F, = - i (Qy>- 3 P0P, Pc*).
c3P03P
w odd.


Reduction of the Forms when n equals three.
73
Substituting the values n — 3 (p. 68) and refering to the value
of χ (p. 51) we find the relation
[160] · · · · F,=1 = [fjilFJ,», -	= ¿456’.
It follows then that χ, if expressed in terms of the modulus k
and δ or as a function of δ, βχ, g2 and g3f will be separable into
three factors which from the expressions for Φ are seen to be of
the same degree in 6, namely, the second.
The factors of χ which we before obtained by. inspection
(see p. 51 [87]) are
A = l2 - (i + k2)l-3k2
[161]	............B = l2 — {\ — 21c2) l + 3(4* — £4)
C=l2 — (ik2 —2)1 — 3(1 — k2)
and we find the relations:
[162].
Ft = ^A·,	^ = ¿2*; Fa = lQ.
Taking now S = 361 and D = l2 — a1 = l2
find the following relations of M. Hermite
2	Φ(Ζ)	PQB
x 36Z(Z*—SB2
—pv = £li = k2 snu cnu dnu —
1 + k2 — Ί& we
[163]
zd)x
ABCx
SB2
361^*— at)
1 + 4*	*ψ	_12ϊ(Ρ — α1Υ(1 + &) — 'ψ{1)
whence
3	36 Ιψ — α,Υ	·36Ζ(Ζ* —«Λ*
121 (P— α{γ (2k'¿— 1) +	(¿B2
361 {V¿ — axY
(2 - **) + j,® _ BC>
.	®	36 Î (?*—«,)*	— fin*
where x = λ and at = a and ω = v.
SB2
SC2
¿>’Z)2
PA2
£Z)2
(see also note p. 69)
General Discussion.
Reviewing the foregoing theory we have found that when n = 3
y=f'—Uf
and that in general y is a function of f where we write
ƒ = c(u + v) e{x_K„)u
the one exception occurring where v equals zero.


74
Part V.
We find further, that where Q or Φ 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
a;==_L_|/V
e2B, V P
from which form we see that x will be zero when γ and Q vanish
and will be infinite where B or P vanish. But from the form
Qy2	2 B1 B
Ï>V c*PB0* B 2w + l
we observe that pv is also infinite where x becomes infinite through
the vanishing of B0.
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
[164] (— 1 )ny=j^è-^^n~ì)u + (¿3)! M>(b-4)m + (^ZTg)! Î41>(B-6)W + · · ·
the values of B conforming with the above cases being roots of
the equations P = 0, Qx = 0, Q2 = 0, Q3 = 0.
Moreover when Q vanishes x and p'v will vanish simultaniously
which makes v one of the semi-periods ωχ, and f may be written
Μ· · ...................../«-«±Sr
Again, observing the last forms obtained, we see that v can also
be a half period if Fx9 n being odd, or Φχ, n being even, vanish, but
it does not follow that x will also reduce to zero. That is the
integral will in general have the form
[166]............./j = ΰ(Η	e(*—£(«>*))“ = aJ^L ¿cu
L J	11	6U	6U
when
Fx == 0, or Φχ = 0, or χ = 0, or A = 0, or B0 = 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
• /* =
axu
GU
e-XU
a form which does not differ from ƒ* a peculiarity which does not
appear in the special functions of Lamé.


Reduction of the Forms when n equals three.
75
We have finally but one more case to consider, namely when
V — 0, a condition arising when B0 or γ, common to the functions
xf pv and p'v, vanish, in which case the integrals become functions
named after their discoverer.*)
Functions of M. Mittag-Leif 1er.
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
/i 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
Γ1β7Ί	ίρf \	<?(w - ax) a(u - a,)
[16 ] ■ · F(u) — a(u _	_ bj
■ G(« — a„-l)
----.------,-----(,·'·’ "	(808 (16) p. 17.)
• «(U - &,_!)
and as 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
μ =e2yi(B — A)-\-2qw
μ == e2ìi' (B — A) + 2qw'
when
B = bi	bn— i
A =	“I- ftg · · “|“ (In — 1	(see P· 17·^
and η and η are constants.
The factors μ and μ' are general and we may if we choose
take them at pleasure and then seek the corresponding function.
Doing this we have μ and μ' given and also ρ to determine
B — A from the relations [170].
Solving we have
log μ = 2η(Β — A) + 2qw
log μ' = 2η (B — A) -|- 2qw'
*) See Mittag-Leffler, Comptes rendus t. XC, 1880, p. 178.


76
Part V.
[171]
[172]
[173]
whence
η log μ' — η' log μ = 2g(yw' — r¡'w) — ρπί] (r¡w' — y¡'w =
w' log μ — w log μ = 2(B — A) (r¡w' — r¡'w) = (B — A) m.
This solution however becomes indeterminate when F(x)
becomes doubly periodic, for then ρ = 0 and
B — A — 2mw + 2m w\
This gives
in(2mw + 2m'w') = w' log μ — w log μ'
whence
w ___ log a — 2inm
w’ log μ + 2iπm'
which means that the logs of the multiplicators are proportional
to their corresponding periods.
Returning to the form
f _ g(u + v) U£9
'	a(u)
we observe that when
we have
v = ai + a 2 + * * “ 0
y = 2mw + 2m w'
and f vanishes showing that this eliment can not be utilized in
this case. Written as a product however and for u = 3 we have
where
= *(* + V) *(« + b) C(U + C)	+
J	gBu
a -1-6 + 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$u. It is moreover known
that any function f(u) of p and p may be resolved in the form
f(u) = L + P
where
L = 11ξ(η — vt) + (w — v2) + 13ξ(u — v.ò) -f- · ·
P = c + ΣηιΡΜ (u — v)
where
h + h 4“ h + * ' = 0.
This property being general, we have, f being doubly periodic,
but to multiply by e$u to find a development for the eliment
required in [172] namely
• · · ...............Φ(μ) = tfu£(u)


deduction of the Forms when n equals three.
77
We have then
ξ(η) = 0(u)e~vu
ξ' (u)	Φ'(η) e~^u — ρ Φ (m) <r~'Qu
ζ" (u) = Q>"(u)e—Qu— 2ρΦ\η)€Γ^Η -f- ρ2Φ{ιι)ο~^η
ξ(3)	= φ"'(ιι) e~ — 3 ρ Φ "(u) e~~ Qu -f- 3 ρ2 Φ\η) e~~Vu — ρ3 Φ (w,) e~~ c u.
Whence
[174]	βΡ“6<·>(«) = Φ<»)(μ) — j ρφί»-1)^) + 5Í!LzJÍ. ρ*φ(»-2)(Μ) _|-
We have then a decomposition in the form
[175]	......../í(m) = c^“+22’a.v®w(m - V.)
•	m V
where vn stands for the several infinites of fx (u) and φ(*> 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 fx (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]	.................fi(u)ce#u + ζη · eüu.
This function when v is zero, in which case Φ = 0 and D = 0,
takes the place of f(u) and hence the general solution is
Vi = f" M — 3 bftu
= (ce$u +	· e8u)'f — 3 δ (ζη · (#u + ceeu)
fi(u) = ççeeu + £'uevu + Qi(u)e$u
— ç2ce£M + ζ’u&u -f- 2ρξ’ (u)a#u -|- ç2Ç2(ii)eQu
whence
(&u&u)" = g’u(#u + 2ρο<,Ηξη + ρ V Ηξη
and we have
[177]	· · · · yx = (ζη · eeu —· 3b£uc#u -f c evu
= rf* [É"w + 2 ρ£'^ + (ρ2 — 3 ft) git + c\.
But from the foregoing theory in this case we have the coeffi-
cients of ξ(η) equal to zero, i. e.
or	ρ2 — 3 ft = 0
ρ2 = 3ft.
[178]


[179]
78	Part V. Reduction of the Forms when n equals three
To find c we proceed as follows: —
S'« = — A
*:« =i + p« + ^=- = ^e- +
1	92	
u	20	T
1		u2
u2	20	lAs
2	&	
uA	10	It
Hence
,_[i + „ + ö: + !s + ..]([J-S._]
“ 2ρ[έ + í)u2 Η----] + c Η---1
and taking c so that the constant term equal zero we have
.................c = T03 = 2 ρδ.
The general solution (v = 0) is then:
yi = (£ueeu)" — 3 b(£u · eQu) -f- 2çbeQu
where
1/36.
Finis.


Table of Forms
η = 3.





Forms for n = 3.
where
and
The complete Integral is
yt = CF(u) + C'F(- u)
y — F(u) = f" (w) — 3 bf(u)
°(u + v) .fa-i
f(u)
e(x — Çv)u
the ordinary form of the equation of Hermite for n = 3 being:
g = [l 2p(u) + B]y.
A second form of the integral is: —
y-Tl^
CL 0
g (u + a) _
ν,ζα
=U-
a—a, by c
_____al ρΐιζα
•where
_ G(u-a)e(u-b)G(u-c) β(ζα+ζ,+ζ(:)ΐί
g a cb oc(gu)3
c = ζν — la — ζό — ξ0 v = er+ b + c
and B = 15 b which is intirely arbitrary and is originally expressed
in the form
B — h(et — e3) — n (n + 1) e3
in which case the equation of Hermite is
g = [12 *»«»** + *].
We have also the general form: —
y = + VY = 1/(pu — e±y (pu — e2Y(pu — e3)s" J J (pu —pa)
e, e, e
0 or 1.
6


82
Table of Forms n = 3.
The functions developed in the general theory have values as
follows:
9	= 463 6^2—9i	c =	1 15	A2	1 Λ = Τ9	
9	= 1262 	g2	P =	15 6	A	1 = Τ9 -	- btp'
Z	=-36 = }P	J5o =	3 y	φ ζ	= ρ(«)	
a.	=	P,=	9 ï*-	- 66φ' t’	=ρ U =	OS 1
&i	SH II	ri =	0	8	= ¿ δ	
9 (Í) = 4£3 + 12 bS2 + (12 62 - Λ) g + 4δ3 - bg2 - gs
= 4 S8 + 12 6S2 + φ'^+φ
5=|φ(ί)-3δ^2-ίφ'5-|φ.
Γ=ηΛ^+4=η|φ'S+ |·φ - δφ'
= Sì+(M*-\g2)S-\(Ub*-3g2b+gs)=±9(t)-b(Tp'+3S*)
= τ ψ (0 — δ [ç>'+ 3 (ί — δ)2]
= ¿3_ 36ί» + (6δ2- {Λ) ί - (156s -g2b + ± &)
F(e,) = - b [φ'+ 3 (e, - δ)2] = - b [15δ2+ 3e,2- 6e,δ - Λ] ·
1? ΓΒ2
15 L15
6et -B
15
+ 3ei2~ λ]
= — cl? [B2— 6e, P + 45e,2— 15<¡r2]
= -c2^P
125Z3— 210V4— 226, Z3+ 93a,2Z2-f 18a,6, J + δ,2— 4a,3
—	^	361 (Z2— a,)2
4 (Z2- a,)8+ (HZ3— 9 a, Z-δ,)2
—	36Z (Z2— a,)2
Φ(Ζ)
~' SD2
where
Φ(Z) = 125Z6 - 210a,Z4 - 226,Z8 + 93a,2Z2 + 18a,6,Z + 6,3 - 4a,3
or
φ(ξ,) = 1251e— 210c|4— 22|3+ 93c2£2+ 18c| + 1 — 4c3
8 = 36Z P = Z2 — a, ξ=δ~τΖ
3 a,3  1 V_	(1 — fc2+Z;4)3
C	δ;2	108 V	(1 + fc2)2 (2 — A2)2 (1 - 2 fc2)2


Forms for n — 3.
83
Also:
„ _ Qv______riA? _ 2 ι ΑΛ8+27A,8	j2_ /p	
x CB0	c2B0K P 3 φ r	b	3 φ' �/
JL_
_	1 ίφ'3 + 27φ2 — 8(27) &φφ' + 16(27) 6Vfl"*
6 ςρ ι	δ	J
where
y<2 = -(4 +28 = 27 +32) = -^	C^àPQ^Q, .
Q-Qi QtQ»	Φ(0 = ®i φ2φ3	y = - ¿3
= (15)3
= [4Α28 + 27 Λ*]
& = 32 · 5 [<*>' + 3 (ft — δ)2] = 5Φλ
<?χ = B2-6eiB + 45ex2—15&,= 5 [öl2—2(¥—2) 7 - 3^] = 5ΦΧ
Q2 = Β2 — 6e2£ + 45 e22—15^ = 5 [572— 2 (1 — 2¥)1 — 3] = 5Φ2
ρ3 = 52_ 6esB + 45e32 — lòg2 = 5[572- 2 (1 + ¥) I — 3 (1 — F)2]
= 5Φ3.
ei = 31 (2 - *)	% = à ^ - !)	% = -	(! + fc2)
Λ = ¿i (1 —	+ &4)
Ρθ) =
öΖ6 —(— 6α, 7 — 10&χ73— 3α,272 + 66Χï + 6,2— 4α,3
367 (72 — α,)2
= k2sn2w — î-iA
O
y>(i)
367 (72 - α,)2
Çy2 2Β, P _ Qx^x2
c*PB2	-B» 2w — 1 c2P2P + e,î
21606°+ 816b«gl+ 1080¿r363 - W‘g2-hibg.g., - g,3 + 2Ίg¿
36 6(144 6*— 24 62í?2+ír22)2
φ'3 + 27 φ2 — 1086φφ'+ 3662φ'2
36 6 φ ' 2
βΥν) = — Λ2 sw2v cw2v dn2v — -0 7 *7γ-Χ—-
ν '	18Ζ (Ζ2 — αχ)
= 1β26ίρφ' —27φ2—φ,:ί|/φΜ + 27φ2 - 21()δφφ' + 432¿ V*
108φ'36τ
= 2 /'■, η 7·; ι/Ι
c3PB® ' -Ρ
6


84
Table of Forms n — 3.
pv —l:
φ '3 —27qp2— 108qpg/
36 φ'*ϋ
pv — e¿=
Qx*Y
c*BÍP
__ |φ'+ »('* - ψ] [12(&-e,)(2& - CX)~ ψ-γ
36 φ '2 b
ρ'ν	7	3 φ
^--pv~b-Tf
where
Ψ(1) = Φ(Τ) — 12l(jP — flj) (io?3 — 8aJ — &x)
= 5Z6 + 6«XZ — lOiJ* — 3αχ2ί2 + 6axfcx7 + ?>x2 — 4ax3
z(l) =|[Φ(0 - 3ψ(0 — 108Ζ2(Τ — αχ)2
= Ie — Qa XZ4 + MJ» — 3atn — V + 4ax3 = A · J5 · C
A = P-0. + V¡)1-SV-*Fi
* - (1 - 2¥) l + 3(7c2 - ¥) = f F,
B
C-=P— (Λ*— 2)7 - 3(1 — &2) = fi;
8
8 = 3653 Λ 3653
F=F1F2Fs = ¿χ =	■ B ■ C
Case 1. P = 0.
Integrai a special function of Lamé of the first sort.
y = p', B = Ö.
Owe & Ç = 0; Φ(Ζ) = 0; & — 0; Q2 = 0; Ç3 = 0; x = 0;
29'î; = 0; 7; —
Integrals, six in number, of the second sort.
' f— I	= *(“+”*) g-«£(«*)
'	— σ μ σ vt g
a = 1, 2, 3
■■ z y pu — ea
where
£ = jUM —
-■ÌJ3
(a) Q, = 0	' · ___
= 3ex + /3(^Ï27f T^/a) = W — 2 ±	2f + \h¥
y = U> + y ex - ¿(3ex + V3 (5& ~ 12ei)) f Vp - ex
= {ï5 + à (fc2 — 2) ± è^ - 2)2 + 15^} Vp2).


Forme for η — 3.
85
(b)	& = 0
B = 3e2 +1/3(5Λ - 12ea) = 1 - 2 F + yCÍ^FT+Iõ
y = [ÿ +	— Ì (3e2 + ys (bg2 — 12 e*))}
= {p + ¿ (1 - 2F) + f0 y(ï-2Ff+Î5} ]/7-171-2/?)
(c)	'Qì= o
i? = 3e3 ± y3 (5</2 -12e2 = 1 + F ± 2 y(2^F)*'-_~3jfc
y = ii» + -7,-fo(3e3 + y»^,. i2<§)} yy^7
= {* + ¿ (1 + 2F) ± I ŸI^W -ΰ} Vp—îq+w).
Case 3.
Ή = 0;	* = 0;	¿ = 0;	£ = 0; C = 0; v
a? =y= o
6; w	ΰ (u — ωΛ
f =----exu = —-----— e(*—ί·(ω^>Μ)
1	6U	6U
Six values of this form corresponding tó the roots of ^4 = Ο-
Ι? = 0; C — 0, namely
B = 4 (1 + F) + ? y(T+ 7c*)*'+ 6F
or
B = 4(1 -2F) + f y(l - 2F) — 6(F^= F)
ór
B = 1 (F — 2) + -|y(F - 2) + 6 (1 — F)
which determine corresponding values for x.
Case á. Conditions as in case (3) with the additional con-
dition of the functions of M. Mittag-Leffler.
The integral is:
Vi = (g«e?“)"— 36(gMe?“) -f- 2p6eiJ“
where	__
Q=yu
Φ == - 6^S2 + 9AsS —Al
E= — 9[2A2S—3A3]
W‘ = yQ.