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Cremona Problems Related to Static Pion-Nucleon Theory.

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Annalen der Physik. 7. Folge, Band 27, Heft 2, 1971, S. 149-160
J. A. Barth, Leipzig
CREMONA
Problems Related to Static Pion-Nucleon Theory
By H. J. KAISER
With 3 Figures
Abstract
Certain mathematical aspects of the static pion-nucleon theory are investigated. We
start from the fact that the theory in its uniformized form (cuts transformed away) leads
to a system of functional equations for the 8-matrix. The nonlinear mapping involved in the
functional equations is a second-order CREMONAtransformation. After a summary of the
general properties of CRERIONAtransformations, the special transformations are studied
which arise in the symmetric scalar and the CHEW-LOW
theory respectively. The emphasis
is on the possibility to separate, by means of a finite-order CREMONAtransformation, the
functional equations into a set of uncoupled ones. For the symmetric scalar theory, the
separation is trivial. For the CHEW-LOW
theory, a proof of nonseparability is given.
1. Introduction
Considerable interest has been centered in recent times on the static theory
of pion-nucleon scattering as it provides a model to study the interplay of analyticity, unitarity, and crossing symmetry in 8-matrix theory. Unfortunately, in
theory), no general
the physically interesting case of three channels (CHEW-LOW
solution is known. The uniformization technique [l, 2, 31 allows one t o construct
the general solution for the two-channel case. I n the three-channel case, however, only a physically uninteresting class of solutions has been investigated so
far [a, 31. It is the particular case where the S-matrix elements are restricted by
cone”). The residue relation a t zero
the condition Sis = X,,S,, (“ROTHLEITNER
energy Sll(0) :#(, 0) :AY,J 0) = -4 : - 1: 2 demanded for a proper physical solution puts it outside the ROTHLEITNER
cone.
I n the present paper: an analysis of the static pion-nucleon theory is performed from the point of view of CREMONAtransformations [4, 51. I n section I1
a short summary about uniformization of the S-matrix is given. Section I11
provides an introduction t o the mathematics of CREMONAtransformations.
I n section IV the CREMONAproperties of the symmetric scalar and the charged
pseudoscalar theory are derived. Finally, in section V the problem of the existence of CREMONA
transformations separating the coupled functional equations
is discussed. It is proved that no “semiseparabel” CREMONAtransformation of
finite order is equivalent t o the “coupled” CREMONAtransformation met in
CHEW-LOW
theory.
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2. Uniformization of the S-matrix
We consider the static pion-nucleon theory with 2 and 3 channels i.e. the
symmetric scalar and the charged pseudoscalar theory respectively. The crossing
matrices for these two cases are
and
The branch cuts of theX-matrix X,(z) in each of the channels
formed away by means of the conformal mapping [2]
(x
can be trans-
z = cash ( u ) .
(3)
Indeed, this mapping effects uniformization such that the only remaining singularities are poles.
Expressed in both the x and the u plane, the conditions tjhe S-matrix
must fulfil are :
herm iticity
S,(Z*)
= X,*(Z),
X,(-u*)
= SZ(u),
unitarit y
To investigate the coupled system of functional equations (4)-(6) we will
t r y t o separate them b y means of a (nonlinear) u-independent transformation
replacing S , in a n one-to-onefashion by new functions V,. A necessary condition
for the simultaneous separability of Eqs. (5) and (6) is that
X;(u)
= S,(u
+ in) = 2B C,#,3/Sp(u)
(7)
must be separabel.
For our aim to solve the functional equations (4)-(6), complete separability
V:, = FdV,)
(8)
is not necessary. The weaker demand of semiseparability
V ; = Fl(V1)
v:,= F,(V,, 7,)
v::= F3(V1, v,, V3)
(9)
H. J. KAISER:CRE~WONA
Problems Related to Static Pion-Pl’ucleon Theory
151
(written down for the 3-channel case) is sufficient. I n order t o be equivalent to
(7), the equations (9) must have an unique reverse
3. CREMOXA transformations
Let us consider the most general one-to-onetransformations in two variables.
Whereas in the case of one variable we have only the broken linear transformat,ions
2’=
ax
cx
~~
+b
+d
as one-to-one mapping between 5 and X I , we are in the case of two variables not
restricted t o the projective transformations
(12)
There exist one-to-one transformations of much more general kind. They are
called CREMONAtransformations [ 3 , 51 and include the projective transformations (12) as CREMONAtransformations of first order.
Consider two planes P and P’. A transformation mapping almost every
straight line L of P into a curve L‘ of nth degree lying in P’ is a CREMONAtransformation C, of nth order provided any two of the curves L’ intersect in maximally
one variable point. Two curves L‘, and L.1 will, of course, intersect generally
in more than one point. To met the CREMONA
condition, the additional intersection points must be “non-variable” i.e. common t o almost all curves L‘.
Such a common point is called a fundamental point, in particular a fundamental
point A ; of multiplicity s1 if the curves L’ have there a n sl-fold point. The behaviour in the plane P is, of course, completely analogous. Let us denote the
multiplicities of the fundamental points A , in the plane P by r,.
Table 1
Fundamental points (other than zeroth multiplicity) of
the lowest-order CREMONAtransformations
Number and multiplicities
of fundamental points
0
3 single
1double
4 single
1triple
6
single
or
3 double
3 single
1 quadruple
8 single
or
1triple
3 double
single
or
G double
+
+
+
+
+
+
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*
19il
The multiplicities rk and s1 of the fundamental points belonging to an nthorder CREMONAtransformation fulfil the conditions
2
rk =
k
s1 =
3(n-l),
1
2 ri = 2 sf = n2 - 1,
k
1
(13)
where the sums run over all fundamental points of the respective plane. The
simplest cases are summarized in Table 1.
A fundamental point of multiplicity rk belonging to the plane P will be t,ransformed into a fundamental curve of degree rk belonging t o the plane P' and vice
versa. All other (i.e. "normal") points will be transformed into points. I n this
sense, every CREMONAtransformation is one-to-one. I n the plane P', a fundamental curve being the image of a fundamental point A , has in the fundamental
point A ; of the plane P' an a,,-fold point. The aklfulfil the equations
Cr,anl
= ns,
k
Ca;l
= s;
k
2
k
M,[LX,l'
1
xaEZ
+1
= ri
+1
1
(14)
2
akla,'l = r,rk*
1
= SPY/'
= 38, -
C & k l
~ s l a n l= nrk
Cakl
1
k
= 3r, -
1.
1
We may extend the scope of the equations (13) and (14) t o include normal
points by associating with each normal point the multiplicity rk = 0 and putting
'
0 if either A , or A ; is a normal point, the other a true
fundamental point
a,, = -1 if both A , and A; are normal points, mapped into each
other by C,
0 if both A , and A; are normal points but not mapped into
each other.
\
Consider two consecutive CREMONA transformations C: P -+P' and C' :
P' + PI'.Their product C" = CC' : P -+ P" is always again a CREMONAtransformation. C and C' will in general not commute, CC'
C'C. The order n" of
C" depends on the orders and n' of C and C' as well as on details of the coincidence of fundamental points in the intermediate plane P'. All properties of
C" can be expressed in compact form through the properties of C and C' if we
arrange the order n , the multiplicities rt and sz, and the wkZin the square matrix
+
ir,
ir,
is,
is,
-al1 -al2
-a2,
-a31
-0132
is,
-0113
*-.
-a23
-a33
................................
Each row of this matrix corresponds to a particular point of the plane P,each
column t o one of the plane P'. If we write t,he analogous matrix for the transformation C' and let their rows correspond to the same points of the plane P' as
used for the columns of C then we obtain the matrix for C" as the product of the
mat)rices C and C'. The number of points selected for representation as rows of
153
H. J. KAISER:CREMONAProblems Related t o Static Pion-Nucleon Theory
the matrix C must include a t least all true fundamental points ( r k > 0), those
for the columns all fundamental points (st > 0) of C in P' as well as all fundamental points (rk' >o) of
in P'. Furthermore, if any one of the normal
points in P is represented in a row of the matrix C then its image point in the
plane P' must be included as a column of the same matrix. The same conditions
must be met, of course, also with the roles of P and P' reversed.
We take as a n example the product of the two CREMONAtransformations
depicted in Fig.1.
.'i'
c'
Fig.1. Product of two CREMONA
transformations. 0,
0 , and
denote fundamental points of multiplicities 0, 1, and 2 respectively.
The numbers 1, 2, ..., l', ..., 6"
serve only t o identify particular
points of the planes P , P', and
P". The akl cannot be read off
the figure, they are specified i n the
matrices (16) and (17)
P
P 'I
I..0.
I n accordance with Fig. 1, we choose the matrix for the second-order CREtransformation C as follows
MONA
c,=
i -1
0
i -1 -1
i
0 -1
0
0
0
0
0
0
0
0
0
1
0
0-1
0
0
0-1
0
0
0
0
Here, the points of the planes P and P' 'corresponding to the rows and columns
of the matrix are indicated explicitely. For example, we find a t the crossing of
the 5t*1row with the 6th column the figure 1. It tells us that a45= -1, i.e. the
CREMONAtransformation chosen transforms the (normal) point A, of the plane
P into the (normal) point A ; of the plane P'.Likewise, we specify the thirdorder CREMONAtransformation occuring in Fig. 1 by the matrix
1"
2 ! 1 3"
4" 511 6,) p t ~ / p t
~
~
3 2i
i -1
o
i
2 i -1
i -1
0 -1 -1
0
0
i -1
i -1
1
0
0
0
0
0
i
~~~
i
-1 -1
1 2'
0
0
0
0-1
0
0
0-1
5'
' G'
I
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*
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The matrices (16) and (17) satisfy all the conditions (13) and (14) characteristic
of CREMONAtransformations. The product C,Ci is the third-order CREMONA
transformation
1"
y , 3" 4" 5" 6" p"lp
-
1 3 . 2i
'%a-1
i - I
i - 1
,i-1
i
_ _ ~ . ~ _ _
i
0
i
1
i ) II
-1 -1
0
0
0-1
0
0-1
0
0
0
'4
0-$5
0
0
, G
as shown in Fig. 1. It should be stressed again that (16) -.. (18) are only examples
of CREMONAtransformations of second resp. third order, not the general ones.
To every CREMONAtransformation, there exists an inverse. I t s matrix fulfils C-1 = CT j . e. it i 9 identical with the transposed matrix.
4. The CREMONA transformations occuring in nN theory
We begin with the case of symmetric scalar theory. Inserting the crossing
matrix (1)into Eq. (7) we get
with the inverse
This is a second-order CREMONAtransformation. Ignoring for the moment the
fact that it can be easily separated by putting x = SJX,,y = S,,we will study
the transformation in its present form (19), (20). It has (0, O), (00, c), (c, 00) as
fundamental points in the S,- X, plane and (0, 0), (00, -am), ( 4 0 0 , ~ ) in
the S;- 8.3 plane. Note that the points a t infinity are to be understood in the
sense of projective geometry, i. e. for example (00, -200) = &lim
I-.- ( M , - 2 M )
is regarded t o be distinct from (w, c ) = lim ( M ,c o n s t ) . Furthermore, a circle
M-f-
with infinite radius is regarded as limiting case of a straight line. Fig. 2 shows
the fundamental points and fundamental curves of the CREMONA
transformation
(19), (20).
I B
Fig. 2. Fundamental points and fundamental
curves of the CREMONAtransformation related t o symmetric scalar theory. The three
J-;fundamental points i n the plane P and the
corresponding fundamental curves in the
plane P' are denoted by a, b, c. Analogously,
A, B, C are fundamental points in P' and
fundamental curves in P
155
H. J. KAISER:
CREMONAProblems Related to Static Pion-Nucleon Theory
The simplest possible matrix description of the transformation is
0
oo 400 s;l
0 -2.200
00
s.;1
s, s,
_ _ _ ~
~-
i
i
2
~~
0
-1
-1
0
c o o
But to render possible the calculation of iterated transformations (C,)P, we
extend the matrix t o include the successive images of normal points which become
finally transformed into fundamental points (i. e. the predecessors of fundamental points). Furthermore, we let the jth column of the matrix denote the
X,) = (w,ul).The fact that the
same point (Si, 8;) = (w,w)as the jth row (IS,,
transformation (19), (20) has in both planes fundamental points a t infinity
compels us to distinguish carefully between points close t o the origin. Thus we
find the infinite-dimensional matrix (22). I t s structure is simple enough to allow
the calculation of its powers.
0
0
00
-26
c
E
00
-200
...
...
800 7~ 600
5~ 400
500 4 F 300
2E
oo
c
E
00
48
200
3~ 400
...
500 68 700 ...
9:
9;
4 &
... 0 0 0 0 i
0 0 0 0 o...
0
0 0 0 o...
... 0 0 0 0 - 1
... 0 0 0 0 - 1
0 0 0 . ..
0
0
... 0 0 0 0 0 1 0 0 0 0 o . . .
.....................................................................................
0 0 0 0 0 ... 0 0 0 0 0
0 0 0 0 0 . . .
0
0 0 0 0 ... 1
0
0
0
0
0 0 0 0 0 . . 0
0
0 0 0 ... 0 1 0 0 0
0 0
0
0 o...
0
0 0 0 0 1 . . . 0
0
1
0 0
0 0 0 0 o...
0 0 0 0 0 ... 0 0 0 1 0 1 0 0 0 0 o . . .
i -1
0 0-1
I...
0 0 0 0
0
0 0 0
0 0 . ..
0 0 0 0 0
0
0 0 0 0
1
0 0 0
o...
0 0 0 i : : : o
0
0 0 0
0
0
~ 10 0 0
0 . ..
0
0 0 0 0 .'.0
0 0 0
0
0
1 0 0 ...
0 0 0 0
0
... 0 0 0 0 : i u
0
0 1 o...
...................................................
1 ..........................
2 i
i
0
i -1
0 0
'I, _
I
0
0
0
1
0 i
0-1
0 0
0 0
I
i
0
0
m
c
-28
E
-200
00
..........
554
800
7E
4F
600
300
5E
2E
400
00
c
o
8
200
500
3E
6&
700
400
..........
(22)
One may add to t,he description of the transformation (19), (20) that the
points (1,1)and (-1, - 1)as well as the line X, = 8, remain invariant under the
transformation, i. e. they are fixed points resp. a fixed curve.
For the CHEW-LOW
theory, we reduce the system (2), (7) by introducing the
new variables
w, = s,
w2 = slIs2
w, = X,/S,.
o
48
(23)
156
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*
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*
1971
and obtain
We will concentrate on the problem posed by the CREMONAtransformation
between the W , - W , and the W i - Wg planes i. e. Eqs. (24b) and (24c).
The fundamental points are located a t (-718, -1/2), (114, 1/16), and (-2, 4)
in the W , - W , plane and a t (00, c ) , ( c , m),and (0, 0) in the W i - W ; plane.
The iterable CREMONAmatrix given in Eq. (25) is again infinite dimensional.
...
...
-6 5z
0 5 - 2
2
--.
0 -1 4
36 25 1 6
4
c
5
4 ....
2
25
00164
7 14 -7
...
-.. 5-2 i7 8
. . . 3_ _2 --1
22 17 2
- 8 1 77 14
17
c -2 2
00-
...
.v;
...
y',
Wz
2 . . . 0 0 0
i
o
o
i o o . . .
................................................
0 ... 0 1 0 0 0 0 0 0 0 ...
0 ... 0 0 1 0 0 0 0 0 0 ...
i ... 0 0 0 - 1 0 0 0 0 0 ...
o * . . o o o 0 1 0 o o o . . .
o . . * o o o 0 0 1 o o o . . .
i ..- 0 0 0 0 0 0 -1 0 0
0 ... 0 0 0 0 0 0 0 1 0 ...
o . . . o o o 0 0 0 O O l . . .
0 ... 0 0 0 0 0 0 0 0 0 ...
................................................
................................................
o . . * o o o 0 0 0 O O O . . .
o * * . o o o 0 0 0 o o o . . .
i ... 0 0 0 -1 0 0 -1 0 0 ...
o . . . o o o 0 0 0 o o o * * *
o . . . o o o 0 0 0 o o o . . .
o . . . o o o 0 0 0 0 0 0 - . *
................................................
. * a
~~
~
...
0 0 0 i 0 0 ...
................................
... 0 0 0 0 0 0 ...
*.. 0 0 0
* * a
0 0 0
0 0
-1
0
0 0
0 0
-1 0
0 0
0 0
0 0
0
0
0
0
0
0
0
0
*.*
**.
... 0 0 0
...
... 0 0 0
...
... 0 0 0
...
*.. 0 0 0
..0 0 0
... 0 0 0
...
................................
................................
... 0 1 0 0 0 0 ...
-.*
0 0 1 0 0 0
... 0 0 0 0 0 0 ...
... 0 0 0 0 1 0 ...
. a *
* a *
* * *
*-.
0 0 0
0 0 0
0 0 1
0 0 0
................................
a * .
**.
**.
w3
...........
9/36
4/25
1/16
0
114
316
215
114
0
-112
-2
c
4
o
o
4
16
512 2514
...........
...........
7/22 3/22
14/17 2/17
-718 -112
0
0
c
-2
17/14 1712
-a17
............
(25)
The transformation has the fixed pojnts
(26)
and the fixed curve W , = W ; (this parabola corresponds to the ROTHLEITNER
cone). No other fixed curves of finite order do exist. I n Fig. 3, the main characteristics of the CREMONAtransformation CCL are shown.
H. J. KAISER:CREMONAProblems Related to Static Pion-Nucleon Theory
157
Fig. 3. Fundamental points, fundamental curves, fixed points, and fixed curve of the
CREMONAtransformation Coz related to CREW-LOW
theory.
5. The separability problem
We begin by asking under what circumstances an nth-order CREMONAtransformation C,: (z, y) -+ (%', y') is semiseparable in the sense that it may be put
into the form
x'
=
ax
cx
~
+b
+ d'
Two immediate consequences of (27) are that all parallels t o the y axis will
p
be transformed into parallels to the y' axis and that all straight lines y = Ax
are transformed into one-valued functions y' = y'(z'). Now, a CREMONAtransformation has in general a lot of fundamental points of high multiplicity. Consequently, straight lines will be transformed by them into curves having multiple
points. To secure, nevertheless, one-valuedness, all fundamental points of (27)
with multiplicity > 1 must be situated a t (c, 00). For, the multiple points of
y' = y'(z') then will lie a t ( c , 0 0 ) too, and this means these curves will have
several asymptotes all parallel to the y' axis.
Now, we know t h a t a CREMONAtransformation C, maps the straight lines
of the plane P into curves of nth order of the plane P'. More precisely, this holds
for straight lines avoiding all fundamental points of P. A straight line L passing
through some fundamental points A , with multiplicities r, of the plane P will be
transformed by C , into a curve L of order
-+
m'=n-
2' 'k
(28)
(sum over all fundamental points on L).L' has in the point A ; of P' a &-fold
point where
q; = SL - 2' OLkl.
(29)
Now we concentrate on the semiseparable case i. e. we demand that all straight
lines passing through the fundamental point A, = (6,~)
and no other fundamental point should be mapped exclusively into straight lines having a t the same
point Ad = (6,~)
a fundamental point but passing through no other fundamental point. Accordingly, we have t o put
= 0.
m' = 1 q; = 1 q' = q: =
(30)
...
158
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*
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From (28) and (29) we conclude
ro = so = n - 1.
(31)
According t o the arguments given above we introduce the notion of CREMONA transformations of type S .
D e f i n i t i o n : An nth-order CREMONAtransformation belongs
t o type X if it possesses in both planes P and P' a t the same
place ( E , q ) a fundamental point of multiplicity n - 1.
(32)
The properties (13) valid for all CREMONAtransformations are sufficient t o
secure that the only other fundamental points (with multiplicity > 0) of a
type X CREMONAtransformation are 2n - 2 fundamental points of multiplicity
1.CREMONAtransformations of type S attain the semiseparated form (27) as soon
as (E, 7) = (c, 0 0 ) .
From (13) and (14) we recognize that a CREMOWA
transformation of type 8
exists in every order n. Its matrix reads up to permutations among the last
2n - 2 columns and/or the last 2n - 2 rows and up to the possibility to introduce any number of normal points
n
(n - 1 ) i
i
i
i
(n - 1)i -n
2 -1 -1 - 1
i
-1
-1
0
0
i
-1
0 -1
0
=
i
-1
0
0 -1
+
c:
i
...
. a *
* a -
.................................................
i
-1
0
0
0
-1
0
0
0
... -1
6 7
x1
Y1
x2
Yz
x2 Y3
...........
X2n-2
Y2n-2
(33)
It is easy t o derive from (33) that the product of any two CREMONA
transformations of type S belongs again to type S and that the square of a
transformation C," of type S has its order within the limits
1
order
((C2)2)
5
2n - 1.
(34)
Consequently, the pth power of C," fulfils
1 5 order ((C,S)P)< pn.
(35)
Within these limits, the order depends on the relations among the fundamental
points of multiplicities 0 and 1 i.e. on the akl.
Let us now apply this formalism t o the CREMONAtransformations (22) and
(25) of symmetric scalar and CHEW-LOWtheory respectively.
The CREMONAmatrix (22) is of type S because the fundamental point (0, 0)
is common t o both planes and a t the same time of order n - 1 with an 01 =
n - 2. We shift the point (0, 0) to (c, 0 0 ) by the substitution
x
= X,/Sz
Y = 8,
(36)
H. J. KAISER:
CREXOXA Problems Related to Static Pion-Nucleon Theory
159
and arrive a t the semiseparat,ed transformation
42 - 1
x’
__
x
22‘
1
4-x‘ ’
x+2’
+
y =---
(22’
(37)
3
+ 1) y’.
This is a CREMONAtransformation of third order having fundamental points of
multiplicity 1 a t
(x,y)
(x‘,y’)
= (-2
+
- E , 01, (-2
E,
(0, -1,
01,
(-1
c)
(38)
O), (-V, a), (4, 0)
and, of course, the common fundamental point of multiplicity 2 a t (c, 00).
On the other hand, the CREMOXA
transformation of the CHEW-LOWtheory
(25) is not of type S. This does not exclude, a priori, that it could be transformed into type S. Our problem is therefore: Is the matrix CCL given in E q . (25)
equivalent t o a semiseparated one! I n other words, we ask whether there
exists a (finite order) CREMONAtransformation T such that
= (00. 0).
(00,
SI
TCCLT-’ = CA(39)
is an Nth-order CREMONAtransformation of type S. Suppose (39) holds, then the
s
same T will transform the pthpower of CCLinto the pth power of Cn-
T ( C ~ ~ T) -’ l = (c$)..
(40)
We know already from Eq. ( 3 5 ) that (CL$)’ is necessarily of less than
order. Furthermore it is not too difficult t o derive from the explicit form (25)
of Ccz the structure of its powers. For our present purposes, it is sufficient t o
notice that the recursion formula
M , = 2M,-1 - M D . 4
(41)
holds for the order
of the pt” power of CCJ,.The first M , are given in (42).
p
l
~-
M,
1
2
3
4
3
G
7
8
9 1 0
-
1
2
4
8 15 28 52 ‘36 1 7 7 32G
11
~~
GOO
1104
(42)
The asymptotic behaviour of the 31, for large p is easy to derive from ( 4 1 ) .
Denoting by I , the root with largest absolute value of the equation
~4
-
we find
M,
+ 1= 0
21~3
-
A17 w 1.35 * 1.8393’.
(43)
(44)
If we insert the (Cf-)”of a t most linearily increasing order (35) and the ( C C L ) ~
of exponentially increasing order (44) into (40) and let p tend to infinity then we
recognize that no finite-order CREMONAtransformation T can bring CCL t o
type S . And, consequently, CCLis not equivalent to a finite-order semiseparated
CREMONAtransformation. This excludes the possibility to solve the functional
equations of CHEW-LOWtheory by the methods developed for static theories
with two channels.
160
Annalen der Physik
*
7. Folge
*
Band 27, Heft 2
*
1971
References
[I]
[2]
[3]
[4]
L5]
WANDERS,
G., Nuovo Cim. 23 (1962) 817.
ROTHLEITNER,
J.,Z. Phys. 177 (1964) 287.
MESCHERYAEOV,
V. A., Dubna preprint P-2369 (1965).
CREMONA,
L., Opere Matematiche, vol. 2, Milano 1915.
MLODZEYEVSKY,
B. K., Matem. Sbornik 31 (1922) 7.
Z e u t h e n , Institut fur Hochenergiephysik der Deutschen Akademie der
Wissenschaften zu Berlin.
Bei der Redaktion eingegangen am 4. November 19iO.
Anschr. d. Verf.: Dr. H. J. KAISER,Institut fur Hochenergiephysik der DAW
DDR-1615 Zeuthen, Platanen-Allee 6:
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