«l
4:^
LIBRARY
OF THE
University of California.
GIF^T OF
Class
\J\f\JJL^^<^>^.
®l|e MttiDetBiti) of Cliicago
FOUNDED BY JOHN D. ROCKEFELLER.
On the
Reduction of Hyperelliptic Functions (p=2)
to
Elliptic "Functions by a Transformation
of the Second Degree.
A Dissertation
Submitted to the Faculties of the Graduate Schools of Arts,
Literature, and Science, in candidacy for the degree of
IDoctor of P*liilosopliy
(Department of Mathematics)
by
J. I. Hutchinson.
i UNIVEP.SITY )
or /
or
Gottingen 1897.
Druck der Dieterich'schen Univ.-Buchdruckerei
(W. Fr. Kaestner.)
,^^
, OF THE X
( UNJVERSfTY ]
On the Reduction of Hyperelliptic Functions (j) = 2)
to Elliptic Functions by a Transformation of the
Second Degree.
Introduction.
The problem that is the subject of the following paper is
not a new one. As early as 1832 Legendre^) had shown that
the two hyperelliptic integrals
J Sixil- x') a - x^x') ^^ J\lx(l-
X dx
six (1 - x^) (1 - x^ X') J\/x{l- x') (1 - x' x')
are each expressible in terms of two elliptic integrals of the first
kind by means of a quadratic transformation.
Immediately after, Jacobi'-^) in a review of Legendre's work
pointed ont that this property belongs to two linearly independent
integrals of a more general type than those discussed by Le-
gendre, viz.
/dx , C xdx
-==:- and /-
\lH{x) J\jR{x)
where Mix) = x{l-x)(i~k]ix)il + Jcx){l-i-Xx).
In 1867 Konigsberger^) proposed and solved the problem,
to determine the nescessary and sufficient condition on the roots
of the sextic in order that the associated hyperelliptic integrals
may be reducible to elliptic integrals by a transformation of the
1) Legendre— Traits des Fonctions Elliptiques, t. 3, pg. 333.
2) Jacobi — Werke, Bd. I, pg. 380.
3) Kouigsberger — Crelle, Bd. 67, pg. 58.
1
_ 2 —
second degree. Denoting these roots hy a^, . . . a^, oo, he found
the required condition to be
This relation expresses the fact that the roots are in involution,
and shows that Jacobi's integrals are the most general integrals
having the property proposed.
In obtaining the above result, Konigsberger made use of a
theorem communicated to him by "Weiers trass to the effect
that, whenever two hyperelliptic integrals of the first kind, linearly
independent, can be found, which are reducible to elliptic integrals
by an algebraic substitution it is always possible to transform
the corresponding ^-functions into a new system such that the
modulus TJg ^0- In a later paper *) Konigsberger gave a new
proof of the same result, and extended to the problem of redu-
cibility the algebraic transformation method employed by Jacobi
with reference to elliptic integrals.
In 1869 G r d a n ^) , in connection with his investigations on
the binary sextic, gave the algebraic condition for reducibility
in invariant form. He showed that when the skew invariant of
the sextic is zero, every associated hyperelliptic integral may be
expressed as an aggregate of elliptic integrals by substituting as
new variables the quadratic co variants I and m.
In 1876 the inversion problem was taken up by Her mite')
who expressed the odd hyperelliptic functions belonging to the
irrationality \Ui{x) in terms of elliptic functions. Immediately
following him Cayley*) accomplished the like result for the
even functions.
The solution of the inversion - problem is also contained
implicitly in the work of Pringsheim^), who, while studying
the general quadratic transformation of the •d-- functions, derived
as a special case the formulae connecting the elliptic and hyper-
elliptic ^-functions, which he made use of to obtain the reduction
of the integrals, previously effected by Jacobi by algebraic methods.
1) Konigsberger — fJeber die Reduction hyperelliptische Integrale auf
elliptische. Crelle Bd. 85, pg. 273.
2) Gordan — Applicazioni di alcuni risultati etc., Annali di Mathematica,
Serie II. — Tomo II., pag. 346.
3) Hermite — Sur un exemple de reduction d'int^grales abdliennes , aux
fonctions elliptiques. Annales de la Soci^t^ scientifique de Bruxelles, t. I (1876).
4) Cay ley — Comptes Rendus, t. 85, pp. 266, 426, 472 (1877).
5) Fringsheim — Mathematiscbe Anaalen, Bd. 9, pg. 445 (1875).
I
— 3 —
In 1882 Picard ') proved that the relation among the •8-
moduli previously found by Weierstrass may, by a proper trans-
formation , be expressed in the form ^[2 = l] ^'^d also that the
reduction from the hyperelliptic to the elliptic integrals can be
eifected by an algebraic transformation of the k*'' degree. This is
a special case of a theorem announced a little later by Weier-
strass^).
In the following pages I have attempted, at the suggestion
and with the kind assistance of Professor Oskar Bolza, a new
treatment of the reduction problem taking as my point of view
the CO variant character of the problem, first hinted at by Gordan.
This suggests the systematic use of another normal form for the
sextic, other than that employed by Jacobi, and adhered to by
most of the subsequent writers, and the solution of the problem
is greatly simplified thereby.
In the first part of the paper, the fact that the roots of the
sextic form an involution is dwelt upon , and some theorems
regarding involution, bearing directly on the subject in hand, are
deduced. Besides the two reducible integrals usually mentioned,
a complete list is given of those special cases of reducible integrals
characterized by more than one involution on the roots of the
sextic.
Following this, I enter upon a detailed examination of what
I may call the general case of reducibility (one involution) , be-
ginning with a study of the correspondence between the hyper-
elliptic and elliptic Riemann surfaces , and the derivation of the
reduction - formulae for integrals of the second and third kinds.
The remainder of the paper is chiefly devoted to a solution
of the „ inversion -problem". Conforming to our point of view,
the 0- functions introduced by Klein are used, and the three simple
reduction - formulae of Theorem VI, are among the most important
new results given in the present paper. They unite in an implicit
form the results of both Hermite and Cayley, and contain even
more, since the factors of proportionality which disappear when
the quotients are formed are here preserved. The simplicity of
these results and the ease with which they are derived clearly
show the advantage of the present mode of treatment ^).
1) Picard — Sur la reduction etc., Bulletin d. 1. Soc. Math. d. France,
t. 11, pg. 25. ;
2) Kowalevski — Acta Matliematica, Bd. 4, pg. 400 (1884).
3) Compare Cayley's remark : „Les reductions pour obtenir les values des
dix fonctions sent trds-penibles". Comptes Rendus, t. 85, pg. 426.
1*
_ 4 _
Finally, the Weber and the Gopel relations satisfied by the
(3-fanctions are derived, and attention is called to the way in
which the Kummer surfaces , represented by them , specialize for
the case of reducibility before us.
§ 1. Algebraic Reduction by JacoWs MetJiod.
Jacobi's method for the transformation of elliptic integrals ^),
can be extended at once to the investigation of the reducibility
of hyperelliptic integrals *) to elliptic integrals by a transformation
of degree n. If R {x^, x^ denote a binary sextic, and if
P(?7, F) =: {Y.U +f.' V){\V + I' V){)^U+)^' V) iy U -\-v' V),
then this method, for n = 2, leads directly to the result :
Theorem I, In order that the hyperelliptic integral of the first
Jcind,
'{A^x,-\-A^ x^) (x, dx^ - x^ dx^
r
SlR{x,, x^)
may be reducible to the elliptic integrcd
UdV-VdU
M
f
\IP{U,V) '
M being a constant, by the quadratic transformation,
U z= a^xl + 2a^x^x^ + a^xl
V = b,xl + 2b^x^x, + b,xl
it is necessary and sufficient that R{x^, x^ be decomposable into
three quadratic factors, (p, t^, x, such that
, / in
the form
(6) ^ _ ^^(^2_p.^a)
while the sextic l?(iP,, a?,) may, in consequence, be written in the
form
(7) f{z, , z^) = a^e\-\- Sa,z\el + Sa^ z\ + a^z\ ,
By the linear transformation (5), the integrals (4) are changed
into such as differ by constant factors only from
z^{zdz) r z^{zdz)
(8) (^0^' f
VA^x,^.) ' J^^fi^r,^^) '
1) See Reye — Geometric der Lage, dritte Auflage, p. 139.
2) Clebsch — Binare Formen, p. 208.
— 7 —
By the quadratic transformation Ci = -s'J, C2 = ^l we obtain
where
-R. (C, , CJ = 4C, K C^ + 3a, CI C, + 3a, C, Q + a, Q
^ B, (C, , Q = 4C, K C: + 3a, C: C, + 3a, C, C, + a, Q .
The normal form, (7), for the sextic, suggested as it is by
the properties of the reducible integrals, offers decided advantages
over the Richelot's normal form employed by Jacobi, and adhered
to by Hermite , Cayley , and Konigsberger , the Richelot's form
having no special relation to the reduction problem before us. As
already remarked in the introduction, one of the principle objects
of the present paper is a simplification of the problem by a
systematic use of this normal form ^).
As a first example of this simplification, we are now in a
position to establish as once a general theorem. For the greater
convenience we pass to non-homogeneous notation.
Theorem TTT. If F{g,^f{z)) denote any rational function of
the two arguments, s, \/f{z), then the most general hyperelliptic inte-
gral, jF{z,\lf{z)d3^ is expressible in terms of two elliptic integrals,
together ivith algebraic and logarithmic functions.
For, it is evident that F{z,^f{z)) can be separated into the
terms G^{z') + z ■ G,{z') + \^f{z)- H,{z') + z\Jf{i) ■ H,{z'), where G,,
G^, H^, H^ are rational functions of -£^^ When we make the sub-
stitution z^ == C, and integrate, the first two terms lead to algebraic
and logarithmic functions, while the last two are elliptic integrals.
By combining this theorem with the addition - theorem for
elliptic integrals it follows at once that the sum
Slf{g)dz'
is expressible in terms of two elliptic integrals together with
1) This normal form has already been employed by Pi card (Bulletin de la
Socidtd Mathematique de France, t. 11 (1883), pg. 45), and by Bolza (Inaugural-
Dissertation, Gottingen, 1886, p. 32).
— 8 —
logaritlimic functions , a theorem obtained by Hermite by means
of Abel's theorem ^).
The substitution (5) contains two arbitrary constants, Jc,7c'.
For our further purposes, however, it is convenient to determine
these by the conditions,
(1) Determinant of transformation =1,
(2) a, = a,.
The coefficients in (7) are now invariants of the sextic , and we
proceed to show how they may be expressed in terms of the
simultaneous invariants of 9, ^, /. These are -)
(11) A,^,^ = - 2 ^: [3^ ^ZT = - ^0 ^" (^' + *'>>
A^.^ = -2xW ^r^ = - ?o '!>. (a' + r ) •
The following useful combinations of these invariants are
easily derived:
Ai.^-A^^A.^^ = foylir-fy
(12) A'y^ - A.^^A^^ = x>: if-a'f
(13)
A^^A^^Ayy = Sal
A^yAy^A^,,^ = 9a,a^-al
(A^^A^y — A^y) (AyyA^^ — Ay,^){A^^A^^ — A^^)
= 27[at + 4a,{a\+al)-eala,a,-Sa\al].
A word is perhaps necessary in regard to the derivation of
this last equation. Comparing the left hand member with (12),
we observe that it is the negative of the discriminant of the cubic
form, a„ Cf + da^ C* C, + 3a, C^ Cl + a, C,. Referring to Clebsch's Binare
Formen, pg. 114, we see that it is equal to the bracketed ex-
pression in the right member of (13), except for a numerical
1) Annales d. 1. Soc. Scien. d. Bruxelles, 1876, pg. 10.
2) The notation is the same as that used by G r d a n , Invariantentheorie,
Bd. II, p. 147.
^ 9 —
factor. This factor is easily determined by assigning particular
values to the roots, say, a^ == i, [3^ = — i, -^^ ^ — 1,
From the equations (13) we can obtain ci,l,a,a^,a^{al+ al)
rationally in terms of A^,,, etc. Hence cil,al are the roots of a
quadratic equation whose coefficients are rationally expressible in
terms of V^^ A^,^, Vl A^,^, \J^ A^^, A^.^, A.^,^, A^,i, .
By combining equations (11) and (12) we obtain invariant
expressions for (p^^ (J^q, y^^, viz.,
while ({)o, ^0 are obtained directly from cp^ by the cyclic permu-
tations (
x^) (?X'^) respectively. The ambiguity arising from the
cube root is due to the separation of the sextic into three qua-
dratic factors. The cube root of unity may be chosen arbitrarily
for one of the functions, say (p^. The others are then determined.
For, by eliminating a", P", y'^ from equations (11), the resulting
equations in tp^, (j>„, y^^ involve only the ratios of these quantities.
The signs of the radicals, Vi^ma; Vi^M? Viyv > ^^e entirely
arbitrary. To determine the signs of the remaining radicals, we
consider the three functional determinants of ?,4'>X> which we
will denote by ■6-x,., *y,., ■6- x. Thy satisfy the relation ^)
We find for their explicit expressions
X? ^^ *^ XT -^5(y-^cpcp ■ ^1^2
It is evident that the sign of one of the radicals may be
arbitrary, white the signs of the others must be chosen so as to
satisfy the above invariant relation.
If 9ii9z-i and g^^g^ be the invariants of i?i and i?2 respectively,
these may be expressed directly in terms of the coefficients a,,,
Oj, ttg \ and hence, by means of (13), in terms of ^ „, etc. Simil-
1) Clebsch — Biniire Formen, pg. 204.
2) See Klein - Theorie der EUiptische Modulfunctionen, Bd. I, pg. 15.
- 10 -
arly, if e^, e^, e^, and e^, e^, e^ be the roots of the two quartics, when
put in the Weierstrass normal form, they may be expressed in
terms of the roots, and hence in terms of the irrational invariants
of the sextic as follows :
e.-e, = ao(P'-f) = ?o^ \-\ = a,a'{^'-f) = ^^E
(15) e,-e, = a,if-a') = ^,F \-\ = a,^\f-a^) = ^!^^F
e-e, = a„(a*-p^) = x„G^ ~e-~e, = a„f (a'-p^ = X,fG
gj + e^ + eg = e^ + e^ + ~e^ =
where
F = \/A'y^-AyyA^^
^ — ^A'i^^—A^^A^^
§ 3. Special Cases in which there exist more than Two Beducible
Integrals.
It might happen that the roots of the fundamental sextic were
in involution in more than one way. In such cases there would
exist more than two hyperelliptic integrals of the first kind, each
of which could he directly expressed by a single elliptic integral
of the first kind. The determination of these can he reduced to
the study of those sextics that admit linear transformations into
themselves. For this purpose we make use of the following :
Lemma, A. If three quadratic forms, having no common
factors J whose discriminants are not zero, he in involution, then there
exists one and hut one binary coUineation ^) of period 2 which trans-
forms the three simultaneonsJy, each into itself, with a change of sign
and an interchange of the roots in each.
B. Conversely: When three quadratic forms are simultaneonsly
invariant for a binary coUineation of period 2, which interchanges the
roots of each^ then the three forms are in involution.
1) By a „binary coUineation" is meant a homogeneous linear transformation,
pZ[ = (MT, + bz.^
pg'i = czi + dZi
in which the factor of proportionality, p, is considered as non-essential, so that
two substitutions differing only by the factor p are regarded as identical.
- 11 -
The truth of A appears immediately when we consider that
the three forms may he expressed as the sums of squares by a
linear transformation. Then, if the coefficients of the general
linear transformation be determined on condition that it shall
leave the three forms invariant , it will he found that the
collineation
(16) '' ^ '^'
is the only one having the required property. It will also he
found that this operation interchanges the roots, and reverses the
sign of each form.
In order to prove B, let a, a' ; p, p' ; y> y' ^^ ^^^ three pairs
of roots. There exists only one linear transformation that in-
terchanges a and a', p and p'. Written in the non-homogeneous
form for convenience, it is
0z' (a + a'- p - p') + (^ + ^') (PP- aa') + aa' (p + p') - pp' (a + a') = 0.
According to our assumption this equation must be satisfied
by 4f = Y) ^' = f'j Hence
YY' (a + a' - P - P') + (y + y') (PP'- aa') + aa' (p + p') - pp' (a + a') = 0.
In determinant notation this relation may be written
1 a + a' aa'
1 p + p' PP' =0,
which is the well-known condition for involution.
From this lemma follows: To every decomposition of the sextic
f into three quadratic factors , «p, ^, X , belonging to an involution,
there corresponds a collineation of period 2 ivhich transforms f into
itself at the same time interchanging its roots in pairs. And vice
versa : To every collineation of period 2 which transforms f into itself,
accompanied by an interchange of all the roots in pairs, corresponds a
decomposition of f into three quadratic factors forming an involution.
Hence, in order to solve the problem proposed at the be-
ginning of this section, we have
(1) to determine all sextics with linear transformations into
themselves, and
(2) to select from the automorphic transformations, belonging
to each case, those of period 2, having the properties mentioned
- n -
in the lemma. To eacli such transformation correspond two
reducible integrals.
The solution of the first part of this problem has already
been given by Bolza in the American Journal, Vol. 10, pg. 50.
The result there enunciated is :
Any binary sextic, f, with linear transformations into itself,
(except of course the transformations, ^/= ± i^J, ^.^ = ± i^'^) , and
wJiose roots are all distinct, is reducible by a linear substitution, of
determinant 1, to one of the following canonical forms ^),
I /• = a^^l + 3a, ^\ ^l + 3a, ^\ s\ + a, 4
{Cyclic group, w = 2)
n. f= a,z,{z\^z\)
{Cyclic group, n = 6)
in. f = z,z^ {a, z* + 2 a, z\ z\ + a„ z"^
{Bihedron group, « = 2)
IV. /■= a,z\ + 2aX< + a,z\
{Dihcdron group ^ w = 3)
V. /•= ao(^x + ^")
{Dihcdron group, w = 6)
VI. f = a^z,z^{z\ + z\)
{Octahedron group).
The six roots of the sextic may be interpreted as points on
the surface of a sphere of radius unity, and those substitutions
that transform it into itself, as rotations of the sphere which
leave the configuration of points as a whole unchanged ^). "We
have, then, merely to select those rotations through the angle tt,
whose axes do not pass through any of the six points. To each
such rotation corresponds a pair of reducible integrals whose
numerators are represented by the two points on the sphere
through which the axis of rotation passes. The total number of
pairs of integrals for each case is respectively 1,0, 2, 3, 4, 6.
Case I is identical with the canonical form of § 2. For the
remaining cases, the hyperelliptic , together with the equivalent
elliptic integrals and the reduction formulae are given in the
following tables.
1) The invariant criteria that characterize each case are given in the paper
just referred to, pg. 70.
2) See Klein — Vorlesungen iiber das Ikosaeder, pg. 31.
- 13 -
o
I— t
+
CO •*
e"
I
+
+
CO
iO
+
CO
+
1
^
^
«sr
1
«»"
6i
^r
tsT
1.
^>-
+
§
1
t»
15 -
Case V,
z^ {zds)
^2)
^2)
{2^+Z^){0d0 )
^f^a
m)
VauC2(C!+CD
m)
(C!+q)
/
\/2 (WC)
V2(WC)
VaoC,(C, + CJ(C: + 14C,C, + CD
Co = ^2
C. = (^x-^J
C2 = (^x+^2)'
/
/
{z^-^ pz^){zdz)
(WC)
9 J V«oC2(c.+g(c!+i4c,c,+q)
(C^C)
V2
P
J V^oC.C,
+ y(q+i4c,c, + CD
Cx =
(^.-P^2)^
C2 =
(^I+P^J
2iTi
6
{z,-fz^{zdz)
(WC)
ya r
pV V«oC.(c,+g(C!+i4c,C3+a)
/
vf(^^J
p^
i VaoC.(C.
m)
(^-P^^J
+ g(q+i4c,c.+CD
16 —
VJ> ij>
+
«4
«4
iW^
1
+
tH
1—1
•'^i.
■<^
+
+
OJ
1
l(M
o
1
CO
■^
a
+
CO
+
CO
+
*♦
,**
"^
... — ,^
'^
t4
«si
v^
^
e<
«*
^Js*,
«4
1
•■+>.
+
,.»r
>
,**"■
^ ^<^ ^> ^ ^°^ ^ ^^^ paired
iu the inTolution.
— 19 —
A Ti r*
Three involutions , t) j^ ^ >
ABC ABC /TTi- TTT^
FED' EDF' ^^^^' -^^^•
V. The sextic consists
of the six vertices of a re-
gular hexagon inscribed in
a circle.
Four involutions j^y^ ,
ABC ABC ABC ,^. ^^
FED' EDF' DFE' ^^^^- ^''
VI. The sextic consists
of three mutually harmonic
point-pairs, and is represented
by the vertices of a square
inscribed in a circle, together
with the two imaginary cir-
cular points at infinity, Coo,
F oo.
Six involutions , T\nv> >
ABC BAD BAD CAD
DFE ' ECF ' EEC ' FBE '
^^g, (Fig. VI).
Fig. IV
Fig. V
Fig. VI
§ 4. Periods of the Integrals of the First Kind.
Introducing non- homogeneous notation, we proceed to a more
detailed consideration of Case I, beginning with a study of the
periods of the integrals,
dz
and their corresponding paths in the Riemann surface represented
by the function
(19) V/S) = Va„^«+3a,^*+3a,^^ + a„ = \JcLjz'-a.'){z'-^'){z'-f).
If B,{0 = ^%^' + Sa,C + 3a,C + a J = 4a,(C - a^) (C - ^ (C - f)
(20)
R,{0 = 4C(a„C-f 3a,C+ 3a,C+aJ = 4a,C(C- a^) (C-p^) i^-f)
2*
20 -
ihen, by the transformations
(21) (a)j .jri,Y= ''
^^1 V^(c) = 2^V7?)
we have
(22)
r 2ds _ r d: r a^ ^ r
d:
\lR,{r) J \Jaz)
V^.(C)
By these two transformations , the hyperelliptic Riemann
surface T is brought into a two to one correspondence with
each of the elliptic surfaces belonging to the functions , \jli^{ C)
and SjRJJl.) , which may be designated respectively by T and T.
Lines radiating from the origin (i. e. the point .s- = 0) , and
concentric circles about the origin in P correspond to similar
paths in T and T.
Fig. VII
Fiff. VIII
Pig. IX
— 21 -
Let the two - leaved surface , T (Fig. VII) , be regarded as
cut into pieces along the straight lines joining the origin with
the branch-points, a, p, y, etc., and extending to infinity. In the
elliptic surface T, the branch - points are a'*, [3^, Y^ °°. We choose
the branch-lines so as to join a^ to p^ and f to oo. If the branch-
line a? [3^ be the conform representation of the line a p in T, and
if, further, we agree that the point at the origin in the upper
sheet of T shall correspond to the like point in T, Then the
strip a^ p^, extending to infinity , corresponds exactly with the
strip aOp , upper sheet with upper , and lower with lower. A
similar correspondence holds between the strips P^Oy' and pOy-
The upper sheet of -fOct.^ corresponds to the strip between
yO — a , beginning with the upper sheet at and passing into the
lower sheet beyond the branch - line. The remaining sheet of
yO— a corresponds to the lower sheet of •f — OcL^. In a similar
manner, the upper sheet of — aO — p corresponds to the continuous
sheet in a^ [3^ that lies above from to the branch - line , and
below beyond it. The remaining sheets then correspond. The
upper sheet of [3^0y'^ corresponds to the sheet of — pO — y that is
above from to the branch -line, and below beyond it. The
remaining sheets then correspond. Finally, the strip — YOa cor-
responds to -fOcf? without any inversion in the order of the sheets.
In the surface j > the branch - point replaces oo of T , and,
accordingly, the second branch -line runs from to y^- A.s in the
previous case, the strips aOp, POy , —YOa correspond respectively
to a'Op', P'Oy', fOcf.\ But that sheet of YO-a which is above at
0, and below beyond the branch -line, corresponds to the lower
sheet of y^O^^- The upper sheet of — aO — p corresponds to that
which is below at , and above beyond the branch - line in the
strip a'^Op^; while a similar correspondence of sheets holds in
regard to the strips — pO — y and p'*OY^
Having established this correspondence between the hyper-
elliptic , and the two elliptic Riemann surfaces , it is easily seen
that to a path in the former corresponds unambiguously a certain
path in either of the latter ; but to a path in an elliptic surface
corresponds simultaneously two paths in the hyperelliptic.
Of particular interest are the period -paths. In Fig. VII is
represented a canonical system of such paths , so chosen as to
represent the reduction properties of the periods in the simplest
form. The images of K^, K^, K^ may be made to coincide with
2K (i.e. Z" twice traversed), K',K respectively, by simple
deformations which respect the connectivity of the surface, while
- 22 -^
the image of K^ can be contracted into a point. Likewise , in T
the images of K^,K^,K^ can be made to coincide with 2K,K,^',
while the path corresponding to K^^ can be contracted to a point.
By means of these correspondences we may at once express
the hyperelliptic periods in terms of the elliptic. Let 2(o,
— — -" — along
_V^i(C)
the paths K and K' respectively. Also let 2w, 2(o' be similar
results for the integral u
.=/,
/^~77\ ? when referred to the
paths K, K'. Then we find for w^ , w^ the following table of
periods :
If we put
Kr
K,
^3
K,
w.
4(1)
2(0'
2(0
w.
4(0
2w
2w'
^1 = r- ^1
* 4(0 ^
4(0
then Vj, ^2 are normal integrals of the first kind with the table
of periods,
K
K
■^3 K,
^l
1
1(0' 1
2(0 2
v^
1
1 1(.^'
2 2a^
— 23 —
This table establishes the truth of the Picard-Weierstrass
theorem^) for our case, viz., t,2 = ^.
§ 6. Integrals of the Second and Third Kind.
In the present section we propose to discuss the reduction
to elliptic integrals of Klein's commutative integral ^) of the
third kind
x'y>^ r" r'' \lf{^) \lf{z[,z[) + a'X ' {^d,){z'd,')
xy
V -^y' 2(^^') • \Jf{,^,z,) V//K,<) '
and Weierstrass' „ Normal integrals of the second kind", w^, w^,
associated ^) with the two reducible integrals of the first kind and
x'y'
with the integral Q^ „ .
A. Integrals of the second hind. The explicit expressions for
W3, w^ are given by Wiltheiss (1. c. pg. 276). For our canonical
form (7) they are
- -A /-
^Wa^.,^j
'6a^z\){zdz)
T
Similarly, in the two elliptic fields we find that to the system
of normal integrals of the first and second kind.
(27)
1) Picard — Bull. d. 1. Soc. Math. d. F., 1. 11, pg. 43; Kowalevski —
Acta Mathematica, Bd. 4, pg. 400.
2) Klein — Hyperelliptische Sigmafunctionen, Mathematische Annalen, Bd.
27, pg. 443.
3) Cf. Wiltheiss — Ueber eine partielle Differentialgleichung etc., Math.
Annalen, Bd. 29, pg. 275; and Bolza — On Weierstrass' Systems of Hyper
elliptic Integrals etc. (A paper read before the Mathematical Congress, Chi-
cago, 1893).
24
belongs the covariant commutative integral of the third kind,
(28) ^.,^ J J 2(-T ^B:^Q^IiMA)^
7) 7)
and to the system,
M, =
(WC)
(29)
belongs
«^ = _ ri^^o C^ + 3a, C. C, + ^3 CD (CdC)
qvJ2.(c„g
(30) v.^ j^j^, 2(ccT V^MgyV^^g'
where -^^(Ci, Cal C'„ Q and -F, (Ci, Ca; Ci, Q are the second polars
of i?i (Ci, C2) and B,^ (Ci, C2) respectively.
By inspection we are easily led to the result:
Theorem IV. The hyperelliptic integrals w^, w^, w^, w^ admit
of reduction to elliptic integrals as follows :
(31)
w, = u,
«^4 = iW2 + ?«2Wl
If the periods of Wj along the paths K, K' are 2-/], 2"/)', and
the periode of u^ along the paths K, K' are 27], 27]', then the
integrals w?, , w^ have the following table of iutegrals
^^
^.
^.
Jfi
w.
27j + ta,w,
V+I«i<^'.
7] + fa,o)
w^
,
2^ + fa,w
?+|«aW,
7]'+|a,a>'
J5. Integrals of the third hind. Passing to non - homogeneous
— 26 -
notation for convenience, and introducing the integrals
J^ J^, «' V^ (C-C')v/iJ.(C) ''
an easy calculation establishes the relation,
(32) S, + S, = 2S+log^-f^ + 2mni
where m is an integer depending for its value upon the paths of
integration xy, x'y'.
Returning to homogeneous notation, the integrals
x'y' rri' l'r{
xy St) 5t]
are expressible in terms of 5, 5^, S^ by the formulae*)
Q = S — ^ w '^w ^
xy a=l,2 " 2 + a
From this follows at once :
Theorem V, The covariant commutative integral Q satisfies
xy
the reduction-formula,
(33) I ^y ^i«/2+^2«/i Vi^l+vX ^n %
1) Wiltheiss — Math. Annalen, Bd. 33, pg. 269—270; Bolza — Oq
Weierstrass' Systems etc., pg. 11.
— 26 —
The hyperelliptlc integrals of the second kind can now be
readily expressed in terms of elliptic c-functions.
Fori)
o referring to ^r^, ^3, and o referring to ^„ g^.
Hence, from (31) follows
Also, since ^)
^'^ '-o(»^')<«f)'
we 'obtain from (33)
s^'^'^'+ioff^i^^^t^ . Vii^+y^y'i. = 1 q(^ )qW )
' ^2/ ^ ^1 2/2 + ^2 2/1 2/1 ^; + 2/2 ^i (wj'^') a (uf)
(34) { __ _ ' '
a(w )a(Wi )
§ 6. The Hyperelliptic a- Functions.
"We now propose to express Klein's hyperelliptic 0- functions
in terms of elliptic o - functions , and for this purpose we make
1) See, for instance, B 1 z a — On the First and Second Logarithmic De-
rivatives of the Hyperelliptic c-functions, American Journal, Vol. XVII, pg. 11.
2) Klein — Hyperelliptische Sigmafanctionen , Math. Ann. Bd. 27, pg. 466.
— 27 —
use of their definition in terms of the integral of the third kind ^)
(f^ , in which the path of integration xy is the conjugate of the
xy
path xy, point for point. When we place this restriction on the
path x'y', formula (33) reduces to
(36) 2/^+iog J^^Mi ^ qS Ql'+naM^y+ «.(^^/]
xy (^12/2 + ^2^2)" ^n e^]
the paths of integration now being such that m = 0.
Since the o- functions are covariants ^) of the sextic /'(^j, ^2)?
we may perform our computations on the canonical form (7), or,
as we may also write it,
to which the original sextic has been reduced by a linear trans-
formation of determinant 1.
§ 7. The Odd a -Functions.
To every decomposition of the sextic into the product of a
linear and a quintic factor corresponds an odd o-function. If,
for example f = m- w^jj/, where m = mJ^z^—cLZ^, n = nJ^2^-{- az^,
and ^, X have the same meaning as in (6) , the corresponding
function we shall denote by ci^(w^, iv^ , and , in accordance with
Klein ^), it may be defined by the equation
where
^ __ ;^ ,,{zdz) ^„ _ r ^.(^^^)
r z,{zdz) r
y ^f{^v ^2) -4 \lf^v ^2)
On the other hand, let Oi(w), <:ij^u), aj^u) be the three even a-
functions belonging to the invariants *) g^, g^, and g^{u), ci^{u), Oj(w)
those belonging to g^, g^, where
1) 1. c, pag. 449.
2) 1. c, pag. 439,
8) 1. c, pag. 449.
,^ 4) Cf. § 2 of the preceding.
— 28 —
Then, in accordance with Klein's definitions (1. c. pg. 455),
and in view of formulae ^) (15), we write
(u) = fe y^x-'^^g.)(->]-p'7),)(ir]-T'iri,)+\/4a„Y],(y3 - «\)(6-p-g(g-f ^J \\
^1
(36)/ 2Vi2.(^)i2,(7,)
e CI
^
,,(w) = V4ao^.(e.-«^aY],-p\)(7j,-Y^^+V/4^o-ri.(T],-a\)(S,-13^$,)6-Y'y .^^g^
while o,(m), 02(«/) are obtained directly from o,(m), a,(w) by inter-
changing a and p, and 03(w), ^.(w) are similarly obtained by inter-
changing a and y.
The odd elliptic o- functions are
(37) I tX«)(B.(il)
I * = «««•'<
It may be remarked, in passing, that the sign ambiguities in
the above expressions need not be taken into account, since the
functions in which they occur will hereafter be used only in com-
binations from which these ambiguities virtually disappear.
In order to express a„(tfi. w^ in terms of the elliptic o-
functions, we observe, in the first place, that oK^i, w'g) is the
product of an exponential e^xy ^ and an algebraic part whose
denominator is ^\lf\x)f{y) and whose numerator is rational in
^ii^vVvVv Further, if each of the four products (i^{u)o^{u), <32(m)o,(u)j
1) If we denote the left hand of (36) by ai(M), and let 5^(u), a,,(tt) be ob-
tained from Ci{u) by the permutations (ap), (a-jf) respectively, then, since
cfjj(«) = 1 — I e;iM'-f . . . , by applying to this last equation, and also to (15), the
permutations (ap), (ay) and comparing results, we find X = 1, p. = 2, v = 3.
— 29 ^
"sW^aC^)* o(w)a(M) be multiplied by the exponential
c J
which will be denoted briefly by s{u,u), the resulting quantities
will each consist of the product of an algebraic part together
with the exponential factor
(38) e(u,u).e ^^i ^].
By means of (35) the exponential (38) can be replaced by
2 \Jx^x,y,y, ^ Q^y
When we make the substitution
the four products e(M, u) o^(m) o^Wj ^^^' > will each consist of the
exponential factor ,
and an algebraic factor, rational in x^,x.^,y^,y^ except for ^f{x)f{y)
which occurs in the denominator. We see , on inspection , that
o^(«(;„ w^) cannot be expressed in terms of any one of the above
four products alone. The next simplest expression being a linear
combination, we proceed to determine whether constants A,B,C,D
can be found such that
(39) ol{ti\,w^) = e(M,M)[^o,(M)aXM) + Ba,{u)(i,{u) + Ca^uXiu) + 2>o(w)o(w)J
When we make the substitutions indicated above, divide out the
common factors, and clear of fractions, we have the relation
/ i^iV, + ^22/,) (^1^2 - ^22/i)' • <{^, - a^ J (2/1 - a2/2)
,= 2JaJ^,^./^^a^a;D(t/:FP^2/D(2/^T'2/D+M2(2/!-a^2/D(^:-r^D(^:-TX^)]
(40)<;+25a,[a;,^,(a;:-p^^^)(l/^-aVD(2/:-YVD+2/x2/2(2/^P^^2)(^!-aX^)(^:-TX)]
+ 20aJ;r,a;,(a;^Y';rD(y^r2/D(y:-«VD+2/.y.(y^YVD(^;-r^D(^!-a'^D]
+ 4{A^B + CXx,y, + x,y,) \JMf'(y) + ^Di^y.-Kyiy
which must be an identity for all values of x^, x^^ y^, y^.
•— 30 —
In this equation we make the two specializations
x^ = a. a;, = Y
1) y, = p '• 2) y, = a
^2 = 2/2 = 1 ^2 = 2/2 = 1
and obtain the conditions ^
Aa,^{f-a.*)-Ba,a{^'-f) + B =
Aa.-da'- p^) - Ca, a (p«- f) -B = 0.
If, further, we make w^^w^ = u^u = in (39), we have
the additional condition
A+B+G = 0.
These three equations of condition give for the ratios of the
constants A, By C, D the values
By putting iCj = — a, j^^ = p, a;^ = 2/2 = 1, D is found to
have the value , D = — | »»^a .
If, now, these values be substituted in (39), the relation is
found to be, in fact, an identity. Accordingly we have
(41) oj. (m;„ w^ = B b(u,u)[<5^{u)o^{u) - ap^Xiu) + a,a(P' - fHu)o{u)] ,
where
B= *^"
2aor-T') '
It only remains to express the coefficients of (41) in terms
of the simultaneous invariants of m, n, (}>, x- For this purpose let
m ^ m, = m[ = m^z^-\-m^e^
n = n^ = n', = w, e^ ■\-n^z^.
Then, in the symbolic notation, we have '
(42)
(wm) = 8w„n(,a
- 31 -
from which we obtain
_g ^ i(^^)
(t{>n)(x«')((l;X)
Also, from (12) and (42), remembering the notation of (15),
§ 8. The Even a -Functions.
The even a -functions are defined by the equation
xy
where ^3, (jjg are two cubies such that 93 ^^ = f. In the present
special case the ten functions divide themselves into two classes ac-
cording to the way in which the linear factors of f are distributed
in ipj and ^^. The linear factors of f may be denoted by m, n, p,
q, r, 5, where mn = (p, pq = (jj, rs = /. The two classes of 0-
functions are, then,
I- ?») '^a of the form
The corresponding o-functions may be designated by the notation
There are six such functions.
^' ?3} ^3 of the fofm
'I's (^1, ^2) = w(^^, ^j • q (^1, ^2) • s (^x) ^2) •
The corresponding a-functions will be denoted by O/mpM^v ^2)-
They are four in number.
OF THl
UNIVERSITY
% OF
— 32
"<''-^ Class I, The six functions of this class must he expressible
in the form
(43)
^)^p\ i^v ^i) = s («j «) [-4 Oi (w) <3, (m) + -B Oj (m) o, (m)
1x2/
+ Co3{tt)a,(«) + 2)o(u)o(u)].
For, by the addition, in equation (41), of a properly chosen
set of half periods to w^, w^, and the corresponding half periods
to u, u, the equation may be made to take the form of (43). This
process would, in fact, lead us directly to the values of A, B, C, D,
but, in the absence of sufficiently explicit formulae for effecting
such a transformation, it is easier to determine these values by
the method of indeterminate coefficients.
Substituting the explicit expressions for the o - functions
dividing out common factors, and clearing of fractions, we have
mi
+ Ba, [x^x, {x]-^X) (y!-a VD if-fyl) + y.y. iy'-^yl) K-^'^l) (^'--(X)]
+ Ca, [x,x, {xl-fxD (yl-^Y,) {y]-«.Y,) +y,y, {y\-fyl) {x\-^X) (•*!-«'a:J)J
+ {A + B-^C){x,y, + x,y,) \IMM + JD{xlyl-xlyy.
Making the four specializations.
Ix^ = a
1) k = P
'X^ = —Cf.
I^. =^3 = 1
X — Y
(^, = 2/2 = 1
u = u =
we obtain the following conditions,
Aa, p (T'-a*) - Ba, a (p'-r') + D =
Aa,^{f-a') + Ba,a{^'-t') + D =
J5a„ Y (a'-p^) + Ca, p (T'-a') + i) =
A + B + C = 1, '
from which me find
B = 0, A == C = h ^ = -i«oP(T-a') = -^{pq)F.
Class II, The four functions of this class are expressible
in a form similar to (48). By the same method as in the prece-
— 33 —
ding, we derive the equation
^1^2 + ^2^/1
(45).
2 L«o(^i-a-^2) (^ -P^s) (^•1-1^2) {y^+ ci.y,) {y,+ ^y,) {y^+ yyj
+a^{y~cLy,) (y-^ (2/1-T2/2) (^i+a-^^) (^i+P^a) (^1+ T-^i,)+2V^/'(a;)/"(^)]
= Aa,[x,x, (x^clX) if-K) iyl-fyl)+y^y. {yl-<^'yl)i^l-?'^l){^\-f<)]
+ Ba, [x,x, {xi-^-'xi) (2/:-aVD {y\-fyl)+y.y. {y\-m) (^^-aVj (x^-T^^D]
+ c\ [^,^, (a^!-TX') (2/:-I3'>:) { y:-^'yl)+ y.yM-fyl){^\-^'K) {<-<^'^\)]
+ {A + B + C){x,y, + x,y,)\lflx)f(y)+Dix\yl-xlyl)\
By making tlie following specializations,
ix^ = a Ix^ = a. Ix^ = p
yi = P 2) L^ = Y 3) 3/^ = Y 4)
^2 = 2/2 = 1 la^2 = 2/2 = 1 1^2 = 2/2 = 1
we obtain the conditions,
M? if- a") - -Sao* (P - t') + ^ =
^«oT (« - P') - ^«o« (P - f ) - D =
A + B + C = 1
from which we find
_ a(P^-f)
(a-P)(P-T)(Y-a)
jg _ P(f-a")
|t(;j^ = tt;^ =
'w = M =0
C =
(a-P)(P-T)(T-a)
t(«-P^)
(a-PJ(P-T)(T-a)
D = 0.
These constants are readily expressed in the invariant form
when we observe that
«^oa(P-f) = ifnn)E
«oP(T-«^) = ipg[)F
«oT(a-P') = M^
«o(«-P)(P-T)(T-a) = «o«(P-f)+«oP(T-a^)+«^oT(a^-P^)
= {mn)E+{pq)F+{rs)G.
The results of the present and the preceding section we will
summarize, for convenience, in the following theorem.
3
- u -
Theorem VI, Let tp = mn, ^ = pq^ -^ = rs he three qita-
dratic forms in the variables z^^ ^„ whose skew invariant R is zero,
and let m, n; p, q; r, s be their linear factors. Further, let J.,, A^;
A[, A[ he constants such that cp, ^, )^ are each linearly expressible m
terms of {A^z^ + A2Z^f,(A[z^ + A'^z^Y. If, besides, we put
r {A^z, + A,z,){zdz) ^ w,= r
V94'X ' " Jy
V/(p.
- 36 —
written as the product of its linear factors as follows,
f{,,,,,) = (^a)(^p)(^Y)K)^13')(n').
Then, ommiting the arguments of the o- functions for the sake of
brevity , we will denote by o the function corresponding to the
decomposition of the sextic into the linear factor (^a) and the
remaining quintic, and by oi ,^, ,1 the function corresponding to
The Weber equation, in its general form, will be a relation
between the functions
p = a^,, q = ^\, r = a^, s = o^(^,P,^,).
From the relation^)
we eliminate the even functions by means of the formulae '^)
(pT)a'(pY^') = (pT)5 + {aa')(Tp')(TT')2-K)(PP')(PTV
(46) {(Ta)^\ppY) = (Ya)5-(Pa'){Tp')(T7')P + (M{ap')(aT')r
and obtain immediately the relation sought, viz.,
\l'mPm)s + (aa')(Tp'r(n')2- K) W(Pt')^J
(47) + V(Ta) g [(Ya) s - (^a') (tP') (ttO P + (?«') («?') {ay') ^]
+ V{aP) r [(ap) s - (ya') (a^') (ay') g + ha') (PP') (Pt') 3] = 0.
If we compare this with Kummer's equation^)
1) Burkharidt — Systematik der hyperelliptischen Functionen, Math, An-
nalen, Bd. 35, pg. 242 (1890).
2) 1. c. pg. 241.
3) Kummer — Ueber die Fiachen vierten Grades, mit sechzehn singularen
Punkten. Monatsberichte der Berl, Akad., 1864.
— 36
SJpiH + V + 8s + W(a'i> + 'i'r + S's) + \Jr (a> + ^"q + h"s) =
it is found that the constants satisfy the required conditions
a'Y + a"p - Py =
a'Y + P"T-aT = 0.
Referring to the reduction formulae (§§ 6 — 7), we observe that
a new set of variables a?,, a;,, x^^ x^ may be introduced by means
of a linear transformation such that
X, = a (u)a (u)
^8 = <5^(w)a^(w)
x^ = (m) a (m) .
The resulting equation is that of the "Wave Surface ').
B. Borchardt's Equation. Borchardt's equation ex-
presses the relation between the four functions of a Gropel's
quadruple, such as, for example,
aPY\ /ap'YX /a p t'\ /a p Y\
a'pYJ' U'PtV' Wt/' WPtJ'
Starting with the first equation of (46), we derive two others by
permuting : 1", a with p , and a' with p' ; 2"^, a with y , and a'
with 7'. Solving these three equations with respect to 0', we
obtain
(48)
\m in') KM M («P') + (P't) (t'«) (^'P)] o« = (P't') (t^) («?) <'^(^,^,J,)
(PY0(Ta)(aP0a'(^,^'^,)-(P'T)(7'*P)c»'(^,^,;[)+(pT)(T
and from this, by permuting a with a', p with p', and 7 with y',
we have
(48')
(-(P'7)(T'a')(a'P)a'(^,P'.^,)- (PT')(Ta')(a'P')o'(^|:[') + (P'T')(T«')(a'?)<'.(^|'j')
1) Study — Spharische Trigonometrie , orthogonale Substitutionen und
elliptische Functionen, pg. 225.
— 37 -
Burkhardt's formula (94), (1. c. pag. 242), gives us
Pt7 Va'P'T
(49)
Squaring each member and eliminating al, o^ , by means of (48)
and (48'), we obtain, after a little reduction, tbe equation
sought, viz..
In the special case in which a, a'; p, P'; Y) Y' ^^^^ pairs in an
involution, the 15 Gopel quadruples composed of even functions
divide themselves into three classes.
Class I. One quadruple,
„(<'?i\ „('='P't\ J^^A J'^^'A
Class II. Six quadruples of the type
/a^yX /otp'yN (aa'^\ Wp'\
HarT'J'Ha'pW'HTT'PrHTY'pJ'
- 38 -
Class III. Eight quadruples of the type
/a,3Y\ m'a\ /aa'T\ M'^\
°Vp'Tr lnvj'Hi3?YJ' 'wrr
I. In the equation of Class I (see formula (49)) one of th?
terms disappears, since
(PT')MW) + (?'T)(A)(a'p) = 0,
as we may immediately see where we take the sextic i^ the
canonical form of § 6. Equation (49) then easily reduces to
where
< - \/(P'T)(T«)(a?0(?T'){TV)Wa'(^,^'^,) = ^^^ta'pT')
x\ = V(?T')(T'«)(--P)(i3'Y)F')W) -'{^J^?^) = ^a^ta'^'T)
It is to be observed that the sign of one of the radicals may
be chosen arbitrarily. The signs of the others are there deter-
mined by the conditions
\ = -^-(y'P«P')
A3 = -A,.(YaYp)
A, = -A,.(p'aYa')
where , for brevity , (7'p a p') is used to denote the anharmonic
The Kummer surface, in this case, becomes a doubly - covered
quadric surface.
n. Using the quadruple given for illustration un^r Class 11,
the equation connecting these functions may be obtained at once
from (49) by permuting a' and y'- If in this we make the sub-
stitution
— 39 —
^. = v'G3T)(Ta)(aP)(i3'T')(TV)(a'P') a (^, J,^^,) = ^K ^Qi^]
X., = v/(P'T)(Ya)(a.3')(P7')(Y'a')(a'P) o(^,^'^,) = VA; a (^,^^';^,)
^, = (/(P'a') (a'a)(ap') (S)(Y7)Wr ^ (^'^^,') = "^^ 'i^^''^^)
the resulting equation takes the form
x\-\-xl + xl + xl + Xx^x^x^x^ + {1 (ic^ic* + ;r|ii7^)
The roots of unity involved in the above radicals we may
choose arbitrarily, subject only to the conditions
When we make the linear transformation
^1 = 2/1 + 2/2
^2 = 2/3 + 2/*
^z = 2/1-2/2
^4 = 2/8-2/4
the equation assumes the simpler form
2/t + 2/2 + 2/3 + 2/4 + ?^'(2/Jy^ + 2/8 2/D
+ P-'(2/i2/I + 2/2 2/D + v'(2/>: + 2/l2/D = 0.
This reduces to the well known equation of the Wave Surface
— =
when we make the substitutions
DC ^ ca ^ ah
a» — ^^2/2 .,^ _ ca«/| _ aby\ ^
r" = ic'^ + «/^ + ^».
— 40 —
The equations of Class III represent a surface more properly
associated with the reduction problem for a transformation of
the fourth degree , and will not he considered further in this
connection. It is what Humbert calls an Elliptic Kummer Surface
of index 4*).
1) G. Humbert — Sur les surfaces de Kummer elliptiques. Americaa
Journal, Vol. XVI, pg. 221.
University of Chicago, August 1895.
^Vita.
I, John Irwin Hutchinson, was born at Bangor, Me.,
on the 12*^^ of April, 1867. I completed my preparatory coarse
at the Edward Little High School, Auburn, in 1885, and in 1889
was graduated from Bates College , Lewiston , with the degree,
Bachelor of Arts. During the two years 1890—92, I was Scholar,
and Fellow in Mathematics at Clark University, where I attended
lectures by Bolza, Perott, Story, Taber, and White. The
two following years I was Fellow in Mathematics at the Uni-
versity of Chicago, attending lectures by Bolza, Hale, Laves,
Maschke, Moore, and Young. Also for a considerable portion
of the time during one year I was occupied in laboratory ex-
periments and astronomical observations at the Kenwood Astro-
Physical Observatory of the University of Chicago.
To the above professors and lecturers, and in particular to
the Professors Bolza, Hale, and Moore I gratefully acknowledge
my indebtedness.
("- Of THL \
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