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% The Polynomial zeros of degree 2 of the 9-$j$ coefficient
% K. Srinivasa Rao, Sangita N. Pitre and J.Van der Jeugt
% Rev. Mex. Fis 42 (1996), 179-192.
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\begin{document}
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\begin{center}
{\Large \bf 
The Polynomial zeros of degree 2 of the 9-$j$ coefficient}\\[1cm]
K.~Srinivasa Rao$^*$, Sangita N.~Pitre$^{**}$ and 
J.~Van der Jeugt\footnote{Research Associate N.F.W.O. (National Fund
for Scientific Research of Belgium)}\\[1cm]
$^*$The Institute of Mathematical Sciences, C.I.T. Campus,\\
Madras - 600 113, INDIA\\[8mm]
$^{**}$Department of Applied Mathematics and Computer Science,\\
University of Ghent, Krijgslaan 281--S9, B9000 Gent, BELGIUM.\\[8mm]
(e-mail~: rao@imsc.ernet.in, Joris.VanderJeugt@rug.ac.be)
\end{center}
\vskip 1cm
\begin{abstract}
We generate and present, for the first time, the polynomial zeros
of degree 2 of the 9-$j$ coefficient from its representation as a
triple hypergeometric series. The first 355 degree~2 zeros of the
9-$j$ coefficient, upto $\sigma \leq 22$ are tabulated.
\end{abstract}
\vskip 2cm

\section{Introduction}

The study of ``non-trivial'' zeros of angular momentum coupling
(3-$j$) and recoupling (6-$j$) coefficients started with the 
tabulation of the first 1420 zeros of the 6-$j$ coefficient,
with any one of the six angular momenta being $< 18.5$, by
Koozekanani and Biedenharn$^{(1)}$, in 1974. Varshalovich
{\em et al\/}$^{(2)}$ gave a listing of the zero-valued 3-$j$
coefficients. Observing that neither of these contributions took
into account the Regge$^{(3)}$ symmetries of the 3-$j$ and the
6-$j$ coefficients, Bowick$^{(4)}$ tabulated the Regge inequivalent
zeros of these coefficients and naturally, Bowick's tables
are much shorter.\\

Since the 3-$j$ coefficient can be related to the Hahn or
dual Hahn polynomial$^{(5)}$ (cf.~Smorodinskii and Suslov$^{(6)}$),
and the 6-$j$ coefficient can be related to the Racah polynomial
(cf.~Wilson$^{(7)}$; Askey and Wilson$^{(8)}$), we prefer to call
the zeros of these coefficients as polynomial zeros and Srinivasa
Rao and Rajeswari$^{(9)}$ classified these zeros 
by their (polynomial) degree.\\

The triple sum series for the 9-$j$ coefficient, due to Jucys and
Bandzaitis$^{(10)}$, is the simplest known algebraic form for
that recoupling coefficient. This triple sum series has been 
identified as a special case of the formal triple hypergeometric
series of Lauricella-Saran-Srivastava$^{(11)}$, by Srinivasa Rao
and Rajeswari$^{(9)}$. 
This identification immediately led Srinivasa
Rao and Rajeswari$^{(9)}$ to show that there exist
polynomial zeros for the 9-$j$ coefficient and to study them, 
for the first time.\\

For a review of the closed form expressions for degree~1 zeros of
$3n$-$j$ coefficients; 
algorithms for generating the degree~1 zeros;
the connection between degree~1 zeros of $3n$-$j$ coefficients and
multiplicative Diophantine equations$^{(12)}$; 
the connection between
polynomial zeros and exceptional Lie algebras, 
we refer the interested
reader to Srinivasa Rao and Rajeswari$^{(13)}$. In this article,
we are concerned with the tabulation of the first 355 polynomial
zeros of degree~2 of the 9-$j$ coefficient. In section~2, we
define the degree of the polynomial zero of the 9-$j$ coefficient
and discuss the method of generating the degree~2 zeros; while in
section~3 we present the Tables of these zeros and in section~4 we
discuss the results and scope for further studies. \\

\section{Triple sum series and polynomial zeros}

The simplest known algebraic form for the 9-$j$ coefficient, due to
Jucys-Bandzaitis$^{(10)}$ is the triple sum series given by~:
\bea
&& \ninej{a}{b}{c}{d}{e}{f}{g}{h}{i}\,=\, 
  (-1)^{x5}\, {(dag)(beh)(igh) \over(def)(bac)(icf)}  \nn\\
&&\times\,{\displaystyle \sum_{x,y,z}\, 
\frac{(-1)^{x+y+z}}{x!y!z!} }
\, {\displaystyle \frac{(x1-x)!(x2+x)!(x3+x)!}
{(x4-x)!(x5-x)!} } \nn \\
&& \times\,{\displaystyle \frac {(y1+y)!(y2+y)!}
{(y3+y)!(y4-y)!(y5-y)!} }
 \,{\displaystyle \frac {(z1-z)!(z2+z)!}
{(z3-z)!(z4-z)!(z5-z)!}} \nn \\
&& \times\,{\displaystyle \frac {(p1-y-z)!}{(p2+x+y)!(p3+x+z)!}}  
\eea
where
\beq
\begin{array}{rccclcr}
0 & \leq & x & \leq & \min(x4,x5) & = & XF \\
0 & \leq & y & \leq & \min(y4,y5) & = & YF  \\
0 & \leq & z & \leq & \min(z4,z5) & = & ZF \ ,
\end{array} 
\eeq
and
\beq
\begin{array}{lll}
x1=2f        &     y1=-b+e+h   & z1=2a              \\
x2=d+e-f     &     y2=g+h-i    & z2=-a+b+c          \\
x3=c-f+i     &     y3=2h+1     & z3=a+d+g+1          \\
x4=-d+e+f    &     y4=b+e-h    & z4=a+d-g            \\
x5=c+f-i     &     y5=g-h+i    & z5=a-b+c         \\
p1=a+d-h+i   &     p2=-b+d-f+h & p3=-a+b-f+i\ ,
\end{array}
\eeq
\beq
(abc)=\left[{(a-b+c)!(a+b-c)!(a+b+c+1)!
\over (-a+b+c)!}\right]^{1/2}\, . 
\eeq

Replacing the factorials by Pochammer symbols, and akin to Appel's
procedure$^{(13)}$ to get double series, with the following
replacements for the three pairs of products~:
\bea
(1+p2)_x(1+p2)_y &\hbox{ by } & (1+p2)_{x+y} \nn\\
(1+p3)_x(1+p3)_z &\hbox{ by } & (1+p3)_{x+z} \\
(-p1)_y(-p1)_z &\hbox{ by } & (-p1)_{y+z}, \nn
\eea
where
\bea
 (\al)_n=\al(\al+1)(\al+2)\cdots(\al+n-1)=
\Ga(\al+n)/\Ga(\al),
\eea
is the Pochammer symbol, Srinivasa Rao and 
Rajeswari$^{(9)}$ were led
to identifying the triple series as a special case of the triple
hypergeometric series of Lauricella, Saran and 
Srivastava$^{(11)}$.\\

The triple sum series does not exhibit any one of the 72 symmetries
possessed by the 9-$j$ coefficient, which manifest when it is
expressed as a sum over the projections of a product of six 3-$j$
coefficients (cf.~Edmonds$^{(14)}$). We define the degree of the 
polynomial zero of the 9-$j$ coefficient as that given by the
value of $XF+YF+ZF$. Srinivasa Rao and Rajeswari$^{(9)}$ studied
the polynomial zeros of degree~1 of the 9-$j$ coefficient 
(Viz.~$XF+YF+ZF=1$) by generating them using either closed form
expressions or through sets of homogeneous 
multiplicative Diophantine
equations of degree~3. Here we are concerned with generating and 
tabulating the polynomial zeros of degree~2, Viz.~$XF+YF+ZF=2$.\\

Six cases arise. Of these, the three cases corresponding to~:
$XF=2$, $YF=ZF=0$; $XF=0$, $YF=2$, $ZF=0$; $XF=YF=0$, $ZF=2$,
give rise to the 9-$j$ coefficient being expressible in terms
of either a 3-$j$ (or Clebsch-Gordan) coefficient or a single
non-zero term$^{(15)}$. 
This would imply that the 9-$j$ coefficient
even if it is a zero, it is only as a consequence
of the polynomial zeros of the 3-$j$ coefficient. These cases are
not of interest to us here and will be dealt with elsewhere.\\

The three remaining cases corresponding to $XF=YF=1$, $ZF=0$;
$XF=1$, $YF=0$, $ZF=1$ and $XF=0$, $YF=ZF=1$, are the ones which
will give rise to genuine degree~2 polynomial zeros of the 9-$j$
coefficient. For instance, $ZF=0$, when substituted in the
triple sum series for~(1), yields~:
\beq
\ninej{a}{b}{c}{d}{e}{f}{g}{h}{i}\,=\, 
  (-1)^{x5}\, {(dag)(beh)(igh) \over(def)(bac)(icf)} 
{z1!z2!\over z3!z4!z5!} S_{x,y},
\eeq
where $S_{x,y}$ is a double sum series which can be looked
upon as a product of two $_4F_3(1)$ series with one parameter
contracted, as in the case of the single variable Appell
series $F_3$. Explicitly~:

\newpage
\bea
S_{x,y}&=& 
\sum_{x,y}\, \frac{(-1)^{x+y}}{x!y!} 
\, {\displaystyle \frac{(x1-x)!(x2+x)!(x3+x)!}
{(x4-x)!(x5-x)!} } \nn \\
&& \times\,{\displaystyle \frac {(y1+y)!(y2+y)!}
{(y3+y)!(y4-y)!(y5-y)!} }
\,{\displaystyle \frac {(p1-y)!}{(p2+x+y)!(p3+x)!}}  \nn\\
&=& {x1!x2!x3!\over x4!x5!}\,{y1!y2!\over y3!y4!y5!}\, {p1!\over
 p2!p3!}\nn\\
&&\times {1\over
 x!y!}{(1+x2)_x(1+x3)_x(-x4)_x(-x5)_x\over(-x1)_x(1+p3)_x}\nn\\
&&\times {(1+y1)_y(1+y2)_y(-y4)_y(-y5)_y \over (1+y3)_y(-p1)_y}
 {1\over (1+p2)_{x+y}} \nn\\
&=& {x1!x2!x3!\over x4!x5!}\,{y1!y2!\over y3!y4!y5!}\, 
 {p1!\over p2!p3!}\nn\\
&&\times F^{0,4}_{1,2}\left[
 \begin{array}{cccccc}
 - & : & 1+x2,\, 1+x3,\, -x4,\, -x5 & ; 
& 1+y1,\, 1+y2,\, -y4,\, -y5&; \\
 1+p2 &:& -x1,\, 1+p3 & ;& 1+y3,\, -p1&;
 \end{array}
 1,1 \right], \nn\\
&&
\eea
where we have used the compact notation for the Kamp\'e de F\'eriet
function$^{(15)}$ devised by Burchnall and Chaundry$^{(16)}$, Viz.~:
\bea
&&F^{A:B}_{C:D} \left[
 \begin{array}{cccccc}
 (a)&:&(b)&;&(b')&;\\
 (c)&:&(d)&;&(d')&
 \end{array} X,Y\right] = \nn\\
&&= \qquad \sum_{x,y}
 {\ds\prod_{j=1}^A (a_j)_{x+y}\prod_{j=1}^B (b_j)_x (b'_j)_y \over
  \ds\prod_{j=1}^C (c_j)_{x+y}\prod_{j=1}^D (d_j)_x (d'_j)_y}
 {X^x Y^y\over x!y!},
\eea
where, in the shortened notation due to Barnes$^{(17)}$, it is
understood, that there are $A$ of the $a$ parameters, $B$ of the
$b,b'$ parameters, etc.\\

If in~(7) we further stipulate $XF=1=YF$, 
then the double sum reduces
to~:
\beq
S_{x,y}=1-{N_1\over D_1}-{N_2\over D_2}+{N_1 N_2(1+p2) \over
D_1D_2D_3}
\eeq
and obviously, degree~2 zeros occur 
whenever the nine angular momenta
$a,b,\ldots,i$ satisfy the condition~:
\beq
((D_1-N_1)D_2-N_2D_1)D_3+N_1N_2(1+p2)=0
\eeq
where
\beq
\begin{array}{ll}
N_1=(1+x2)(1+x3)x4.x5, & D_1=x1(1+p2)(1+p3),\\
N_2=(1+y1)(1+y2)y4.y5, & D_2=(1+y3)p1(1+y2),\\
D_3=p2+2. & 
\end{array}
\eeq
Eq.~(11) stands for a set of eight conditions, 
since $ZF=0$ represents
$z4=0$ or $z5=0$; $XF=1$ represents $x4=1$, $x5\geq 1$ 
or $x4\geq 1$, $x5=1$;
$YF=1$ represents $y4=1$, $y5\geq 1$ or $y4\geq 1$, $y5=1$.\\

Similarly, when $YF=0$, the 9-$j$ coefficient will 
be proportional to
the function~:
\beq
F^{0,4}_{1,2}\left[
 \begin{array}{cccccc}
 - & : & 1+x2,\, 1+x3,\, -x4,\, -x5 & ; 
& -z3,\, -z4,\, -z5,\, 1+z2&; \\
 1+p3 &:& -x1,\, 1+p2 & ;& -z1,\, -p1&;
 \end{array}
 1,1 \right],
\eeq
and when $XF=1=ZF$, polynomial zeros of degree~2 arise when
\beq
((D_1-N_1)D_2-N_2D_1)D_3+N_1N_2(1+p3)=0
\eeq
where
\beq
\begin{array}{ll}
N_1=(1+x2)(1+x3)x4.x5, & D_1=x1(1+p2)(1+p3),\\
N_2=z3.z4.z5(1+z2), & D_2=z1.p1(1+p3),\\
D_3=p3+2. & 
\end{array}
\eeq
Eq.~(14) stands for a set of eight conditions, 
since $YF=0$ represents
$x4=1$, $x5\geq 1$ or $x4\geq 1$, $x5=1$; 
$ZF=1$ represents $z4=1$, $z5\geq 1$ or
$z4\geq 1$, $z5=1$.\\

Finally, when $XF=0$, the 9-$j$ coefficient will be proportional to
the function~:
\beq
F^{0,4}_{1,2}\left[
 \begin{array}{cccccc}
 - & : & 1+y1,\, 1+y2,\, -y4,\, -y5 & ; 
& -z3,\, -z4,\, -z5,\, 1+z2&; \\
 -p1 &:& 1+p2,\, 1+y3 & ;& -z1,\, 1+p3&;
 \end{array}
 1,1 \right],
\eeq
and when $XF=1=YF$, polynomial zeros of degree~2 arise when
\beq
((D_1-N_1)D_2-N_2D_1)D_3+N_1N_2.p1=0
\eeq
where
\beq
\begin{array}{ll}
N_1=(1+y1)(1+y2)y4.y5, & D_1=p1(1+p2)(1+y3),\\
N_2=z3.z4.z5(1+z2), & D_2=z1.p1(1+p3),\\
D_3=p1-1. & 
\end{array}
\eeq
Eq.~(17) again stands for a set of eight conditions, 
since $XF=0$ represents
$y4=1$, $y5\geq 1$ or $y4\geq 1$, $y5=1$; $ZF=1$ represents $z4=1$,
$z5\geq 1$ or
$z4\geq 1$, $z5=1$.\\

A simple and straightforward procedure to generate the polynomial
zeros of degree~2 of the 9-$j$ coefficient is then to program the
set of 24 conditions given by (11), (14) and (17). 
The first degree~2
zero is~:
\beq
\ninej{3/2}{2}{3/2}{3/2}{1/2}{1}{2}{3/2}{1/2}
\eeq
with $\sigma=a+b+\cdots +i=12$, being the sum of the nine angular
momenta. In this case since $XF=0$, $YF=ZF=1$, the degree of this
zero is~2, but is has four terms. This can be looked upon as a
``4-term'' degree~2 zero. Unlike the polynomial zeros of the 3-$j$
and the 6-$j$ coefficient it is to be emphasized that the inherent
lack of symmetry of the triple sum series 
implies that the symmetries
of (19) will {\bf not}
 all be polynomial zeros of degree~2. To illustrate
our point, consider the symmetry of (19)~:
\beq
\ninej{3/2}{3/2}{2}{2}{1/2}{3/2}{3/2}{1}{1/2}
\eeq
which has $XF=0$, $YF=1$, $ZF=2$, implying that it is a polynomial
zero of degree ($XF+YF+ZF=$)~3. However, since $p1=3$, $p2=0$,
$p3=-1$, the constraints on the summation indices~:
\beq
y+z\geq p1\hbox{\ \ and if\ \ }p2,p3<0,\hbox{\ \  then\ \ }
x+y\geq |p2|,\  x+z\geq|p3|,
\eeq
restrict $z$ to $1\leq z\leq 2$. The number of terms which occur in
the triple sum series is thus~4, 
but by definition it is a polynomial
zero of degree~3~. Therefore, we may call this 
 a ``4-term'' degree~3 zero.\\

\section{Tabulation of the degree~2 zeros}

The first 355 polynomial zeros of degree~2 of the 
9-$j$ coefficient,
which correspond to $12\leq\sigma\leq 22$ 
are presented in Table~1. We
restrict $a,b,d,e$ to $\leq 5$ and the total 
number of degree~2 zeros
generated is about 1600. Simple  Fortran programs were written to
check for the conditions (11), (14) and (17). 
As stated in section~2,
there are 24 conditions and hence 24 Fortran 
programs were written to
generate the zeros. 
The programs consist essentially of four {\tt DO}
loops for $a,b,d$ and $e$ and five more loops for $c,f,g,h$ and $i$
being generated using the triangle 
inequalities that must be satisfied
by the triads~:
\beq
(abc),\, (adg),\, (def),\, (beh),\, (cfi),\,\hbox{ and } (ghi).
\eeq
The quantities $N_1, N_2, D_1, D_2$ are 
computed for the given values
$a,b,\ldots,i$ and if the condition (11), (14) or (17) is satisfied
then the set of values $a,b,\ldots,i,$ 
along with their sum $\sigma$
are stored in a file, until the upper limits for $a,b,d,e$ set as 5
are reached. 
The 24 output files contain degree~2 zeros with several
repetitions.
A sort program is used to sort on $\sigma  $
 and the entries in each of the output
files are ordered for increasing $\sigma$. 
An edit program then weeds
out the identical entries, i.e.~entries in which $a,b,\ldots,i$ are
all the same string of characters.
 The 24 files containing the repetitions
weeded out are now merged to give the Table~1 presented here. The
cases when any one (or more) of the nine angular 
momenta is a zero are
avoided, since it is well-known that a 9-$j$ coefficient in such a
case reduces to a 6-$j$ coefficient.\\

In Table~2 are presented the first 102 polynomial zeros of 
the $9-j$ coefficient, wherein symmetries of the coefficient
are taken into account (and each zero appears only once). In 
addition, additional useful information is also provided here.
The tabulated results are part of the list of 829 zeros 
generated for values of $a,b,\cdots i$ such that their 
sum $\sigma $ is $\leq 30$. For the sake of easy readability,
in this table, we have given twice the values for all
angular momenta : the first nine columns of the table give 
the values of $2a, 2b, \cdots 2i $, and the remaining five 
columns provide the values of :  $2\sigma $ ($\sigma = $
$a + b + \cdots + i$); ntr gives the minimum number of terms 
in the triple sum series representation for the coefficient;
n6j gives the minimum number of terms that occur in the 
single sum over the product of three $6-j$ coefficients 
formula for the $9-j$ coefficient; str represents the number 
of strechings in the $9-j$ coefficient and the final column 
gives the type of stretching (viz. one, two, three or four
of the nine angular meomenta being stretched). At the end of 
this table the types A, B, T and Q are explicitly given. 
We note that for more than 4 stretchings, closed form expressions
exist, which implies that they will not give rise to a zero 
of the $9-j$ coefficient. 
%Also, the A's T's and Q's correspond 
%to zeros of the $3-j$ coefficient. 
The tables provided 
here will be very useful in  further analysis of the nature of 
the polynomial zeros of the $9-j$ coefficient.

\section{Concluding remarks}

In this article we have studied the polynomial zeros 
of degree~2 of the
9-$j$ coefficient, arising from the triple-sum series of
Jucys-Bandzaitis. The first few hundred of these are generated from
the conditions (11), (14) and (17) 
to be satisfied by the nine angular
momenta $a,b,\ldots,i,$ for the 9-$j$ coefficient to be a degree~2
zero. It is noted that the triple sum 
series reduces to a double series
in these cases of interest which can be 
considered as special cases of
such series studied by Kamp\'e de F\'eriet$^{(15)}$
 and expressed in a compact
notation by Burchnall and Chaundy$^{(16)}$.\\

An interesting observation made is that due to the inherent lack of
symmetry of the triple sum series, the 72 
symmetries of a given 9-$j$
coefficient, which is a polynomial zero of 
degree $n$, will not be all
of the same degree $n$. 
It is perhaps worthwhile to study the relation
between the degree of the zero, 
defined as the sum of the upper limits
$XF$, $YF$, $ZF$, and the number of terms which add up to give the
degree $n$ zero. 
The number of terms is always $n+1$, for polynomial
zeros of degree $n$, if it is a 3-$j$ or a 6-$j$ angular momentum
coefficient, since in these cases, 
the series is a single sum series.\\

Another point to be noted is that the Howell$^{(18)}$ procedure for
ordering the nine angular momenta cannot be resorted to, 
to reduce the
Table~1, by retaining only the ``generic'' zeros and discarding its
symmetries. This is again due to the inherent 
lack of symmetry of the
triple sum series for the 9-$j$ coefficient.
It is our sincere hope that the analysis of the polynomial 
zeros presented here will be the source for further studies 
to understand this complicated problem of their nature.\\


\section*{Acknowledgements}

It is a pleasure to thank Professors G. Vanden Berghe and 
Hans de Meyer for stimulating discussions. This work was partly 
supported by EEC (contract no. CI1*-CT92-0101).\\

\section*{References}

\begin{enumerate}
%1
\item S.H.~Koozekanani and L.C.~Biedenharn (1974) Rev.\ Mex.\ Fis.\
 {\bf 23}, 327. See also Ch.~V, Topic~11 in L.C.Biedenharn and 
J.D.Louck, {\it The Racah-Wigner Algebra in Quantum Theory},
Encyclopedia of Maths. and Applns. {\bf 9}, Addison-Wesley
 Pub. Co. (1981).
%2
\item D.A.~Varshalovich, A.N.~Moskalev and V.K.~Khersonskii (1975)
{\it Quantum Theory of Angular Momentum}, Nauka, 
Leningrad (in Russian);
English Edition 1988, World Scientific.
%3
\item T.~Regge (1958) Nuo.\ Cim.\ {\bf 10}, 544; (1959), ibid~{\bf
11}, 116.
%4
\item
M.J.~Bowick (1976) {\it Regge symmetries and null 3-$j$ and 6-$j$
symbols}, Thesis, Univ.\ of Canterbury, Christchurch, New Zealand.
%5
\item
see, for instance~: S.~Kastin and J.L.~McGregor (1961) Scripta
Math.\ {\bf 26}, 33.
%6
\item
Ya.A.~Smorodinskii and S.K.~Suslov (1982) Sov.\ J.\ Nucl.\ Phys.\
{\bf 35}, 108; ibid {\bf 36}, 623.
%7
\item
J.~Wilson (1980) SIAM J.\ Math.\ Anal.\ {\bf 11}, 690.
%8
\item
R.~Askey and J.~Wilson (1985) Memoirs of Am.\ Math.\ Soc.~{\bf 54},
No.~319.
%9
\item
K.~Srinivasa Rao and V.~Rajeswari (1988) J.\ Phys.~A~{\bf 21}, 4255.
%10
\item
A.P.~Jucys and A.A.~Bandzaitis (1977) {\it Angular Momentum 
Theory in
Quantum Physics}, Mokslas, Vilnius. See also, S.J.~Alisauskas and
A.P.~Jucys (1971) J.\ Math.\ Phys.~{\bf 12}, 594.
%11
\item
G.~Lauricella (1883) Rend.\ Circ.\ Mat.\ Palermo {\bf 7}, 111;
S.~Saran (1954) Ganita~{\bf 5}, 77; H.M.~Srivastava (1967) Proc.\
Camb.\ Phil.\ Soc.~{\bf 63}, 425. Also refer Harold Exton, 
{\it }  
%12
\item
E.T.~Bell (1933) Am.\ J.\ Math.~{\bf 55}, 50; Morgan Ward (1933)
ibid.\ p.~68; K.\ Srinivasa Rao, T.S.~Santhanam and V.~Rajeswari
(1992) Ind.\ J.\ Pure Appl\ Math.~{\bf 23}, 171.
%13
\item
K.~Srinivasa Rao and V.~Rajeswari (1993) {\it Quantum Theory of 
Angular Momentum : Selected Topics}, Springer-Verlag.
%14
\item
A.R.~Edmonds (1957) {\it Angular Momentum in Quantum Mechanics},
Princeton Univ.\ Press, Princeton.

%15
\item
J.\ Kamp\'e de F\'eriet (1921) C.R.\ Acad.\ Sci.\ Paris {\bf 173},
489.
%16
\item
J.L.~Burchnall and T.W.~Chaundy (1941) Quart.\ J.\ Math.\ Oxford
Ser.~{\bf 12}, 112.
%17
\item E.W.~Barnes (1907) Proc.\ Lon.\ Math.\ Soc.~{\bf 5}, 59;
ibid.~{\bf 6}, 141; Trans.\ Camb.\ Phil.\ Soc.~{\bf 20}, 253.
%18
\item K.M.~Howell (1959) {\it Tables of 9-$j$ symbols}, University 
of Southampton (England), Research Report 59-2.
\end{enumerate} 
\newpage
\noindent Table~1. Degree~2 zeros of the 9-$j$ coefficient
$\ninej{a}{b}{c}{d}{e}{f}{g}{h}{i}$.\\
 The values of $a,b,\ldots,i$ and
the total sum $\sigma=a+b+\cdots +i$ are given.
\begin{center}
\begin{verbatim}
-----------------------------------------------------------
  a     b     c     d     e     f     g     h     i       
----------------------------------------------------------- 
 1.5   2.0   1.5   1.5   0.5   1.0   2.0   1.5   0.5    12
 1.5   1.0   0.5   2.5   1.5   1.0   2.0   1.5   1.5    13
 2.0   2.5   1.5   1.5   1.0   0.5   1.5   1.5   1.0    13
 2.0   1.5   1.5   1.5   0.5   1.0   2.5   1.0   1.5    13
 2.0   1.5   1.5   1.5   1.0   0.5   2.5   1.5   1.0    13
 1.5   2.0   1.5   0.5   1.5   1.0   1.0   2.5   1.5    13
 1.5   2.0   1.5   1.0   1.5   0.5   1.5   2.5   1.0    13
 1.5   1.0   2.5   1.0   0.5   1.5   1.5   1.5   2.0    13
 0.5   1.5   1.0   1.0   2.5   1.5   1.5   2.0   1.5    13
 1.0   1.5   0.5   2.5   2.0   1.5   1.5   1.5   1.0    13
 2.0   1.5   0.5   2.5   1.0   1.5   1.5   1.5   2.0    14
 2.5   2.0   0.5   2.0   1.0   1.0   1.5   2.0   1.5    14
 1.5   2.0   1.5   2.0   1.0   1.0   2.5   2.0   0.5    14
 1.5   2.5   2.0   1.5   0.5   1.0   2.0   2.0   1.0    14
 1.5   2.0   1.5   2.0   0.5   1.5   2.5   1.5   1.0    14
 1.0   1.5   2.5   1.5   0.5   2.0   1.5   2.0   1.5    14
 1.0   1.5   0.5   2.0   1.5   2.5   1.0   2.0   2.0    14
 1.5   2.0   0.5   2.5   1.5   2.0   1.0   1.5   1.5    14
 0.5   1.5   1.0   2.5   1.5   2.0   2.0   2.0   1.0    14
 2.0   3.0   2.0   1.5   1.0   0.5   1.5   2.0   1.5    15
 2.0   1.5   1.5   2.0   0.5   1.5   3.0   1.0   2.0    15
 2.0   1.5   1.5   2.0   1.5   0.5   3.0   2.0   1.0    15
 1.5   2.0   1.5   0.5   2.0   1.5   1.0   3.0   2.0    15
 1.5   2.0   1.5   1.5   2.0   0.5   2.0   3.0   1.0    15
 3.0   2.5   0.5   2.5   1.5   1.0   1.5   2.0   1.5    16
 1.5   2.0   1.5   2.5   1.5   1.0   3.0   2.5   0.5    16
 1.5   3.0   2.5   1.5   0.5   1.0   2.0   2.5   1.5    16
 1.5   2.0   1.5   2.5   0.5   2.0   3.0   1.5   1.5    16
 1.5   2.0   1.5   2.5   1.0   1.5   3.0   2.0   1.0    16
 1.0   1.5   0.5   2.5   1.5   3.0   1.5   2.0   2.5    16
 0.5   1.5   1.0   3.0   1.5   2.5   2.5   2.0   1.5    16
----------------------------------------------------------- 
\end{verbatim}
\newpage
\centerline{\bf Table~1 (contd.)}
\begin{verbatim}
-----------------------------------------------------------
  a     b     c     d     e     f     g     h     i
-----------------------------------------------------------
 1.5   1.0   0.5   3.5   2.0   1.5   3.0   2.0   2.0    17
 2.0   3.0   2.0   2.0   0.5   1.5   3.0   2.5   0.5    17
 2.0   3.5   2.5   1.5   1.0   0.5   1.5   2.5   2.0    17
 3.0   3.5   1.5   2.0   1.5   0.5   2.0   2.0   1.0    17
 2.0   1.5   1.5   2.5   0.5   2.0   3.5   1.0   2.5    17
 2.0   1.5   1.5   2.5   1.0   1.5   3.5   1.5   2.0    17
 2.0   1.5   1.5   2.5   1.5   1.0   3.5   2.0   1.5    17
 2.0   1.5   1.5   2.5   2.0   0.5   3.5   2.5   1.0    17
 3.0   2.0   2.0   1.5   0.5   1.0   3.5   1.5   2.0    17
 3.0   2.0   2.0   1.5   1.0   0.5   3.5   2.0   1.5    17
 1.5   2.0   1.5   0.5   2.5   2.0   1.0   3.5   2.5    17
 1.5   2.0   1.5   1.0   2.5   1.5   1.5   3.5   2.0    17
 1.5   2.0   1.5   1.5   2.5   1.0   2.0   3.5   1.5    17
 1.5   2.0   1.5   2.0   2.5   0.5   2.5   3.5   1.0    17
 2.0   3.0   2.0   0.5   1.5   1.0   1.5   3.5   2.0    17
 2.0   3.0   2.0   1.0   1.5   0.5   2.0   3.5   1.5    17
 3.0   3.5   1.5   1.0   1.5   0.5   2.0   3.0   1.0    17
 2.0   1.5   3.5   1.0   0.5   1.5   2.0   2.0   3.0    17
 0.5   1.5   1.0   1.5   3.5   2.0   2.0   3.0   2.0    17
 1.5   2.0   0.5   3.5   3.0   1.5   2.0   2.0   1.0    17
 1.5   2.5   1.0   2.0   3.5   1.5   1.5   2.0   1.5    17
 2.0   1.5   0.5   2.0   1.5   2.5   3.0   2.0   3.0    18
 2.5   2.0   0.5   1.5   1.5   2.0   3.0   2.5   2.5    18
 2.5   2.0   0.5   3.0   1.5   1.5   2.5   2.5   2.0    18
 1.5   2.0   1.5   2.5   3.0   0.5   2.0   3.0   2.0    18
 1.5   2.5   2.0   2.0   2.5   0.5   1.5   3.0   2.5    18
 3.5   3.0   0.5   3.0   2.0   1.0   1.5   2.0   1.5    18
 1.5   1.5   2.0   2.0   2.5   0.5   2.5   3.0   2.5    18
 1.5   1.5   2.0   2.5   2.0   0.5   3.0   2.5   2.5    18
 1.5   2.0   2.5   1.5   2.0   0.5   2.0   3.0   3.0    18
 2.0   1.5   2.5   2.0   1.5   0.5   3.0   2.0   3.0    18
 1.5   2.0   1.5   3.0   2.0   1.0   3.5   3.0   0.5    18
 1.5   3.5   3.0   1.5   0.5   1.0   2.0   3.0   2.0    18
 3.0   3.5   1.5   2.0   1.0   1.0   3.0   2.5   0.5    18
 1.5   2.0   1.5   3.0   0.5   2.5   3.5   1.5   2.0    18
-----------------------------------------------------------
\end{verbatim}
\newpage
\centerline{\bf Table~1 (contd.)}
\begin{verbatim}
-----------------------------------------------------------
  a     b     c     d     e     f     g     h     i
----------------------------------------------------------- 
 1.5   2.0   1.5   3.0   1.0   2.0   3.5   2.0   1.5    18
 1.5   2.0   1.5   3.0   1.5   1.5   3.5   2.5   1.0    18
 3.0   3.0   2.0   1.5   0.5   1.0   3.5   2.5   1.0    18
 1.5   2.0   1.5   2.0   3.0   2.0   0.5   3.0   2.5    18
 2.0   2.5   1.5   2.5   3.0   1.5   0.5   2.5   2.0    18
 2.0   3.0   2.0   1.5   2.0   1.5   0.5   3.0   2.5    18
 2.5   3.0   1.5   2.0   2.5   1.5   0.5   2.5   2.0    18
 1.5   2.0   1.5   2.0   0.5   2.5   2.5   2.5   3.0    18
 1.5   2.5   2.0   1.5   0.5   2.0   2.0   3.0   3.0    18
 1.5   0.5   2.0   1.5   2.5   2.0   2.0   3.0   3.0    18
 2.0   0.5   2.5   1.5   2.0   1.5   2.5   2.5   3.0    18
 1.5   1.5   3.0   2.0   0.5   2.5   2.5   2.0   2.5    18
 1.5   2.0   0.5   3.0   2.5   2.5   1.5   2.5   2.0    18
 0.5   2.0   1.5   2.5   2.0   1.5   3.0   3.0   2.0    18
 0.5   2.5   2.0   2.0   1.5   1.5   2.5   3.0   2.5    18
 1.0   1.5   0.5   3.0   1.5   3.5   2.0   2.0   3.0    18
 0.5   1.5   1.0   3.5   1.5   3.0   3.0   2.0   2.0    18
 0.5   1.5   2.0   2.5   1.5   2.0   3.0   2.0   3.0    18
 0.5   2.0   2.5   2.0   1.5   1.5   2.5   2.5   3.0    18
 2.0   1.5   1.5   0.5   2.5   2.0   2.5   3.0   2.5    18
 2.5   2.0   1.5   0.5   2.0   1.5   3.0   3.0   2.0    18
 2.0   1.5   0.5   3.5   2.5   1.0   3.5   3.0   1.5    19
 2.5   2.0   0.5   3.5   1.0   2.5   2.0   2.0   3.0    19
 3.5   3.0   0.5   2.5   1.0   1.5   2.0   3.0   2.0    19
 2.0   3.0   2.0   2.5   1.0   1.5   3.5   3.0   0.5    19
 2.0   3.5   2.5   2.0   0.5   1.5   3.0   3.0   1.0    19
 2.0   4.0   3.0   1.5   1.0   0.5   1.5   3.0   2.5    19
 3.0   4.0   2.0   2.0   1.5   0.5   2.0   2.5   1.5    19
 2.0   3.0   2.0   2.5   0.5   2.0   3.5   2.5   1.0    19
 2.0   1.5   1.5   3.0   0.5   2.5   4.0   1.0   3.0    19
 2.0   1.5   1.5   3.0   1.0   2.0   4.0   1.5   2.5    19
 2.0   1.5   1.5   3.0   2.0   1.0   4.0   2.5   1.5    19
 2.0   1.5   1.5   3.0   2.5   0.5   4.0   3.0   1.0    19
 3.0   2.0   2.0   2.0   0.5   1.5   4.0   1.5   2.5    19
 3.0   2.0   2.0   2.0   1.5   0.5   4.0   2.5   1.5    19
-----------------------------------------------------------
\end{verbatim}
\newpage 
\centerline{\bf Table~1 (contd.)}
\begin{verbatim} 
-----------------------------------------------------------
  a     b     c     d     e     f     g     h     i   
-----------------------------------------------------------
 1.5   2.0   1.5   0.5   3.0   2.5   1.0   4.0   3.0    19
 1.5   2.0   1.5   1.0   3.0   2.0   1.5   4.0   2.5    19
 1.5   2.0   1.5   2.0   3.0   1.0   2.5   4.0   1.5    19
 1.5   2.0   1.5   2.5   3.0   0.5   3.0   4.0   1.0    19
 2.0   3.0   2.0   0.5   2.0   1.5   1.5   4.0   2.5    19
 2.0   3.0   2.0   1.5   2.0   0.5   2.5   4.0   1.5    19
 3.0   3.5   1.5   1.5   2.0   0.5   2.5   3.5   1.0    19
 3.0   3.5   1.5   1.0   2.0   1.0   2.0   3.5   1.5    19
 1.0   2.5   3.5   2.0   0.5   2.5   2.0   3.0   2.0    19
 2.5   1.0   3.5   1.5   0.5   2.0   3.0   1.5   3.5    19
 0.5   2.0   1.5   1.0   3.5   2.5   1.5   3.5   3.0    19
 1.0   1.5   0.5   3.5   3.5   2.0   2.5   3.0   1.5    19
 1.0   2.0   1.0   2.0   3.5   1.5   3.0   3.5   1.5    19
 1.5   2.0   0.5   2.5   2.0   3.5   1.0   3.0   3.0    19
 2.0   3.0   1.0   2.5   4.0   1.5   1.5   2.0   1.5    19
 2.0   3.5   1.5   1.0   2.0   1.0   3.0   3.5   1.5    19
 2.5   3.0   0.5   3.5   2.0   2.5   1.0   2.0   2.0    19
 0.5   2.0   1.5   3.5   2.0   2.5   3.0   3.0   1.0    19
 1.5   2.5   1.0   3.5   2.0   2.5   2.0   2.5   1.5    19
 1.0   2.0   2.0   2.5   3.0   0.5   2.5   4.0   2.5    20
 1.0   2.5   2.5   2.0   2.5   0.5   2.0   4.0   3.0    20
 2.0   1.5   0.5   4.0   1.5   2.5   3.0   2.0   3.0    20
 2.5   2.0   0.5   2.5   1.0   2.5   4.0   2.0   3.0    20
 2.5   3.0   1.5   2.0   1.5   1.5   1.5   3.5   3.0    20
 3.0   2.5   0.5   2.0   1.0   2.0   4.0   2.5   2.5    20
 3.5   3.0   0.5   2.5   1.5   1.0   3.0   3.5   1.5    20
 4.0   3.5   0.5   3.5   2.5   1.0   1.5   2.0   1.5    20
 2.0   1.0   2.0   3.0   2.5   0.5   4.0   2.5   2.5    20
 2.5   1.0   2.5   2.5   2.0   0.5   4.0   2.0   3.0    20
 1.5   2.0   1.5   3.5   2.5   1.0   4.0   3.5   0.5    20
 1.5   4.0   3.5   1.5   0.5   1.0   2.0   3.5   2.5    20
 3.0   3.5   1.5   2.5   1.5   1.0   3.5   3.0   0.5    20
 3.0   4.0   2.0   2.0   1.0   1.0   3.0   3.0   1.0    20
 1.5   2.0   1.5   3.5   0.5   3.0   4.0   1.5   2.5    20
 1.5   2.0   1.5   3.5   1.0   2.5   4.0   2.0   2.0    20
-----------------------------------------------------------
\end{verbatim}
\newpage
\centerline{\bf Table~1 (contd.)}
\begin{verbatim}
-----------------------------------------------------------
  a     b     c     d     e     f     g     h     i
-----------------------------------------------------------
 1.5   2.0   1.5   3.5   1.5   2.0   4.0   2.5   1.5    20
 1.5   2.0   1.5   3.5   2.0   1.5   4.0   3.0   1.0    20
 1.5   2.5   2.0   3.0   1.5   1.5   3.5   3.0   1.5    20
 3.0   3.0   2.0   2.0   0.5   1.5   4.0   2.5   1.5    20
 3.0   3.0   2.0   2.0   1.0   1.0   4.0   3.0   1.0    20
 3.0   3.5   2.5   1.5   0.5   1.0   3.5   3.0   1.5    20
 2.5   4.0   2.5   2.0   2.0   1.0   0.5   3.0   2.5    20
 3.0   3.5   1.5   2.5   1.5   2.0   1.5   3.0   1.5    20
 3.0   4.0   2.0   2.5   2.5   1.0   0.5   2.5   2.0    20
 2.0   2.0   1.0   2.5   4.0   2.5   0.5   3.0   2.5    20
 2.5   2.5   1.0   3.0   4.0   2.0   0.5   2.5   2.0    20
 1.0   2.0   2.0   2.5   0.5   3.0   2.5   2.5   4.0    20
 1.0   2.5   2.5   2.0   0.5   2.5   2.0   3.0   4.0    20
 2.0   0.5   2.5   1.0   2.5   2.5   2.0   3.0   4.0    20
 2.5   0.5   3.0   1.0   2.0   2.0   2.5   2.5   4.0    20
 1.5   2.5   4.0   1.5   0.5   2.0   2.0   3.0   3.0    20
 0.5   2.5   2.0   2.5   2.5   1.0   3.0   4.0   2.0    20
 0.5   3.0   2.5   2.0   2.0   1.0   2.5   4.0   2.5    20
 1.0   1.5   0.5   2.5   3.0   3.5   1.5   3.5   3.0    20
 1.5   3.0   1.5   2.0   1.5   2.5   1.5   3.5   3.0    20
 2.5   3.0   0.5   4.0   3.0   2.0   1.5   2.0   1.5    20
 1.0   1.5   0.5   3.5   1.5   4.0   2.5   2.0   3.5    20
 1.0   2.0   1.0   4.0   3.0   2.0   3.0   3.0   1.0    20
 1.5   3.0   1.5   2.5   1.5   2.0   3.0   3.5   1.5    20
 0.5   1.5   1.0   4.0   1.5   3.5   3.5   2.0   2.5    20
 2.0   2.0   1.0   0.5   3.0   2.5   2.5   4.0   2.5    20
 2.5   2.5   1.0   0.5   2.5   2.0   3.0   4.0   2.0    20
 0.5   2.0   2.5   2.5   1.0   2.5   3.0   2.0   4.0    20
 0.5   2.5   3.0   2.0   1.0   2.0   2.5   2.5   4.0    20
 1.5   1.0   0.5   4.5   2.5   2.0   4.0   2.5   2.5    21
 4.0   3.5   0.5   3.0   1.5   1.5   2.0   3.0   2.0    21
 2.0   3.0   2.0   3.0   1.5   1.5   4.0   3.5   0.5    21
 2.0   4.0   3.0   2.0   0.5   1.5   3.0   3.5   1.5    21
 2.0   4.5   3.5   1.5   1.0   0.5   1.5   3.5   3.0    21
 3.0   4.5   2.5   2.0   1.5   0.5   2.0   3.0   2.0    21
-----------------------------------------------------------
\end{verbatim}
\newpage
\centerline{\bf Table~1 (contd.)}
\begin{verbatim}
-----------------------------------------------------------
  a     b     c     d     e     f     g     h     i
-----------------------------------------------------------
 4.0   4.5   1.5   2.5   2.0   0.5   2.5   2.5   1.0    21
 2.0   3.0   2.0   3.0   0.5   2.5   4.0   2.5   1.5    21
 2.0   3.0   2.0   3.0   1.0   2.0   4.0   3.0   1.0    21
 2.0   1.5   1.5   3.5   0.5   3.0   4.5   1.0   3.5    21
 2.0   1.5   1.5   3.5   1.0   2.5   4.5   1.5   3.0    21
 2.0   1.5   1.5   3.5   1.5   2.0   4.5   2.0   2.5    21
 2.0   1.5   1.5   3.5   2.0   1.5   4.5   2.5   2.0    21
 2.0   1.5   1.5   3.5   2.5   1.0   4.5   3.0   1.5    21
 2.0   1.5   1.5   3.5   3.0   0.5   4.5   3.5   1.0    21
 3.0   2.0   2.0   2.5   0.5   2.0   4.5   1.5   3.0    21
 3.0   2.0   2.0   2.5   1.0   1.5   4.5   2.0   2.5    21
 3.0   2.0   2.0   2.5   1.5   1.0   4.5   2.5   2.0    21
 3.0   2.0   2.0   2.5   2.0   0.5   4.5   3.0   1.5    21
 4.0   2.5   2.5   1.5   0.5   1.0   4.5   2.0   2.5    21
 4.0   2.5   2.5   1.5   1.0   0.5   4.5   2.5   2.0    21
 1.5   2.0   1.5   0.5   3.5   3.0   1.0   4.5   3.5    21
 1.5   2.0   1.5   1.0   3.5   2.5   1.5   4.5   3.0    21
 1.5   2.0   1.5   1.5   3.5   2.0   2.0   4.5   2.5    21
 1.5   2.0   1.5   2.0   3.5   1.5   2.5   4.5   2.0    21
 1.5   2.0   1.5   2.5   3.5   1.0   3.0   4.5   1.5    21
 1.5   2.0   1.5   3.0   3.5   0.5   3.5   4.5   1.0    21
 2.0   3.0   2.0   0.5   2.5   2.0   1.5   4.5   3.0    21
 2.0   3.0   2.0   1.0   2.5   1.5   2.0   4.5   2.5    21
 2.0   3.0   2.0   1.5   2.5   1.0   2.5   4.5   2.0    21
 2.0   3.0   2.0   2.0   2.5   0.5   3.0   4.5   1.5    21
 2.5   4.0   2.5   0.5   1.5   1.0   2.0   4.5   2.5    21
 2.5   4.0   2.5   1.0   1.5   0.5   2.5   4.5   2.0    21
 3.0   3.5   1.5   2.0   2.5   0.5   3.0   4.0   1.0    21
 3.0   3.5   1.5   1.0   2.5   1.5   2.0   4.0   2.0    21
 3.0   3.5   1.5   1.5   2.5   1.0   2.5   4.0   1.5    21
 2.5   2.0   4.5   1.0   0.5   1.5   2.5   2.5   4.0    21
 0.5   1.5   1.0   2.0   4.5   2.5   2.5   4.0   2.5    21
 1.0   2.5   1.5   1.5   4.0   2.5   1.5   3.5   3.0    21
 1.0   2.5   1.5   1.5   3.5   2.0   2.5   4.0   2.5    21
 1.5   2.5   1.0   2.5   4.5   2.0   2.0   3.0   2.0    21
------------------------------------------------------------
\end{verbatim}
\newpage
\centerline{\bf Table~1 (contd.)}
\begin{verbatim}
-----------------------------------------------------------
  a     b     c     d     e     f     g     h     i
-----------------------------------------------------------
 1.5   2.5   1.0   2.5   4.0   1.5   3.0   3.5   1.5    21
 1.5   3.5   2.0   1.0   2.5   1.5   2.5   4.0   2.5    21
 2.0   2.5   0.5   4.5   4.0   1.5   2.5   2.5   1.0    21
 2.5   3.5   1.0   3.0   4.5   1.5   1.5   2.0   1.5    21
 2.5   4.0   1.5   1.5   2.5   1.0   3.0   3.5   1.5    21
 1.5   2.0   0.5   3.0   2.0   4.0   1.5   3.0   3.5    21
 0.5   2.0   1.5   4.0   2.0   3.0   3.5   3.0   1.5    21
 1.0   2.0   2.0   3.5   3.0   0.5   3.5   4.0   2.5    22
 1.0   2.5   2.5   3.0   2.5   0.5   3.0   4.0   3.0    22
 1.5   1.0   0.5   2.5   2.5   4.0   3.0   2.5   4.5    22
 1.5   1.0   0.5   3.0   2.0   4.0   3.5   2.0   4.5    22
 2.0   2.0   1.0   2.0   2.0   3.0   3.0   3.0   4.0    22
 2.5   2.0   0.5   4.0   2.0   2.0   3.5   3.0   2.5    22
 3.0   2.5   0.5   1.0   2.5   2.5   3.0   4.0   3.0    22
 3.0   2.5   0.5   3.5   2.0   1.5   3.5   3.5   2.0    22
 3.0   3.0   1.0   1.5   1.5   2.0   3.5   3.5   3.0    22
 3.5   3.0   0.5   1.0   2.0   2.0   3.5   4.0   2.5    22
 3.5   3.0   0.5   3.5   1.0   2.5   2.0   3.0   3.0    22
 1.5   2.0   1.5   3.0   4.0   1.0   2.5   4.0   2.5    22
 1.5   3.0   2.5   2.0   3.0   1.0   1.5   4.0   3.5    22
 2.0   2.0   1.0   4.0   4.5   0.5   3.0   3.5   1.5    22
 2.0   2.5   1.5   3.0   4.0   1.0   2.0   3.5   2.5    22
 2.0   3.0   2.0   2.5   3.5   1.0   1.5   3.5   3.0    22
 2.0   4.0   3.0   2.0   2.5   0.5   1.0   3.5   3.5    22
 2.5   2.5   1.0   4.0   4.5   0.5   2.5   3.0   1.5    22
 2.5   4.0   2.5   2.5   3.0   0.5   1.0   3.0   3.0    22
 3.0   3.5   1.5   2.5   2.0   1.5   1.5   3.5   3.0    22
 4.0   3.5   0.5   3.0   2.0   1.0   3.0   3.5   1.5    22
 4.5   4.0   0.5   4.0   3.0   1.0   1.5   2.0   1.5    22
 1.5   1.5   2.0   2.5   3.5   1.0   3.0   4.0   3.0    22
 1.5   1.5   2.0   3.0   3.0   1.0   3.5   3.5   3.0    22
 1.5   1.5   2.0   3.5   2.5   1.0   4.0   3.0   3.0    22
 1.5   2.0   2.5   3.0   2.0   1.0   3.5   3.0   3.5    22
 1.5   2.5   3.0   1.5   2.5   1.0   2.0   4.0   4.0    22
 2.0   1.0   2.0   3.0   3.5   0.5   4.0   3.5   2.5    22
-----------------------------------------------------------
\end{verbatim}
\newpage
\centerline{\bf Table~1 (contd.)}
\begin{verbatim}
-----------------------------------------------------------
  a     b     c     d     e     f     g     h     i  
-----------------------------------------------------------
 2.0   1.5   2.5   2.0   3.0   1.0   3.0   3.5   3.5    22
 2.0   2.0   3.0   1.5   2.5   1.0   2.5   3.5   4.0    22
 2.0   2.0   3.0   2.0   2.0   1.0   3.0   3.0   4.0    22
 2.0   2.0   3.0   2.5   1.5   1.0   3.5   2.5   4.0    22
 2.0   3.0   4.0   1.0   1.5   0.5   2.0   3.5   4.5    22
 2.5   1.0   2.5   2.5   3.0   0.5   4.0   3.0   3.0    22
 2.5   1.5   1.0   2.0   2.0   3.0   3.5   2.5   4.0    22
 2.5   1.5   1.0   2.5   1.5   3.0   4.0   2.0   4.0    22
 2.5   1.5   3.0   2.5   1.5   1.0   4.0   2.0   4.0    22
 2.5   2.5   4.0   1.0   1.5   0.5   2.5   3.0   4.5    22
 2.5   2.5   4.0   1.5   1.0   0.5   3.0   2.5   4.5    22
 3.0   2.0   1.0   1.5   2.0   2.5   3.5   3.0   3.5    22
 3.0   2.0   4.0   1.5   1.0   0.5   3.5   2.0   4.5    22
 3.5   2.5   1.0   1.5   1.5   2.0   4.0   3.0   3.0    22
 1.5   2.0   1.5   4.0   3.0   1.0   4.5   4.0   0.5    22
 1.5   4.5   4.0   1.5   0.5   1.0   2.0   4.0   3.0    22
 2.5   4.0   2.5   2.5   0.5   2.0   4.0   3.5   0.5    22
 3.0   3.5   1.5   3.0   2.0   1.0   4.0   3.5   0.5    22
 3.0   4.5   2.5   1.5   0.5   1.0   2.5   4.0   2.5    22
 3.0   4.5   2.5   2.0   1.0   1.0   3.0   3.5   1.5    22
 3.5   4.5   2.0   1.5   0.5   1.0   3.0   4.0   2.0    22
 1.5   2.0   1.5   4.0   0.5   3.5   4.5   1.5   3.0    22
 1.5   2.0   1.5   4.0   1.0   3.0   4.5   2.0   2.5    22
 1.5   2.0   1.5   4.0   1.5   2.5   4.5   2.5   2.0    22
 1.5   2.0   1.5   4.0   2.0   2.0   4.5   3.0   1.5    22
 1.5   2.0   1.5   4.0   2.5   1.5   4.5   3.5   1.0    22
 1.5   3.0   2.5   3.0   1.5   1.5   3.5   3.5   2.0    22
 3.0   3.0   1.0   3.0   0.5   2.5   4.0   2.5   2.5    22
 3.5   3.5   1.0   2.5   0.5   2.0   4.0   3.0   2.0    22
 3.0   3.0   2.0   2.5   0.5   2.0   4.5   2.5   2.0    22
 3.0   3.0   2.0   2.5   1.0   1.5   4.5   3.0   1.5    22
 3.0   3.0   2.0   2.5   1.5   1.0   4.5   3.5   1.0    22
 3.0   4.0   3.0   1.5   0.5   1.0   3.5   3.5   2.0    22
 2.5   3.0   1.5   2.5   2.5   1.0   4.0   4.5   0.5    22
 2.5   4.0   2.5   3.0   3.0   1.0   0.5   3.0   2.5    22
-----------------------------------------------------------
\end{verbatim}
\newpage
\centerline{\bf Table~1 (contd.)}
\begin{verbatim}
-----------------------------------------------------------
  a     b     c     d     e     f     g     h     i 
-----------------------------------------------------------
 3.0   3.5   1.5   2.0   2.0   1.0   4.0   4.5   0.5    22
 3.0   4.0   2.0   3.5   3.5   1.0   0.5   2.5   2.0    22
 1.0   2.0   2.0   1.5   3.5   3.0   0.5   4.5   4.0    22
 1.0   2.5   2.5   1.5   3.0   2.5   0.5   4.5   4.0    22
 1.5   2.0   1.5   2.5   4.0   2.5   1.0   4.0   3.0    22
 1.5   2.5   2.0   2.5   3.5   2.0   1.0   4.0   3.0    22
 1.5   3.0   2.5   1.0   2.5   2.5   0.5   4.5   4.0    22
 1.5   3.5   3.0   1.0   2.0   2.0   0.5   4.5   4.0    22
 2.0   3.0   2.0   2.0   3.0   2.0   1.0   4.0   3.0    22
 2.0   3.0   2.0   3.0   3.5   1.5   1.0   3.5   2.5    22
 2.5   3.0   1.5   3.5   4.0   1.5   1.0   3.0   2.0    22
 2.5   3.5   2.0   1.5   2.5   2.0   1.0   4.0   3.0    22
 2.5   4.0   2.5   1.5   2.0   1.5   1.0   4.0   3.0    22
 3.0   3.0   1.0   2.5   4.0   2.5   0.5   3.0   2.5    22
 3.0   3.5   1.5   2.0   3.0   2.0   1.0   3.5   2.5    22
 3.0   3.5   1.5   3.0   3.5   1.5   1.0   3.0   2.0    22
 3.5   3.5   1.0   3.0   4.0   2.0   0.5   2.5   2.0    22
 3.5   4.0   1.5   2.5   3.0   1.5   1.0   3.0   2.0    22
 1.5   2.0   1.5   2.5   1.0   3.5   3.0   3.0   4.0    22
 1.5   2.0   1.5   3.0   1.0   3.0   3.5   3.0   3.5    22
 1.5   3.0   2.5   1.5   1.0   2.5   2.0   4.0   4.0    22
 2.0   2.0   1.0   3.0   0.5   3.5   4.0   2.5   3.5    22
 2.0   2.5   1.5   2.0   1.0   3.0   3.0   3.5   3.5    22
 2.0   3.0   2.0   1.5   1.0   2.5   2.5   4.0   3.5    22
 2.0   3.0   2.0   2.0   1.0   2.0   3.0   4.0   3.0    22
 2.0   4.0   3.0   1.0   0.5   1.5   2.0   4.5   3.5    22
 2.5   2.5   1.0   2.5   0.5   3.0   4.0   3.0   3.0    22
 2.5   4.0   2.5   1.0   0.5   1.5   2.5   4.5   3.0    22
 1.0   0.5   1.5   2.0   4.0   3.0   2.0   4.5   3.5    22
 1.0   0.5   1.5   2.5   4.0   2.5   2.5   4.5   3.0    22
 1.5   1.0   2.5   1.5   3.0   2.5   2.0   4.0   4.0    22
 1.5   1.0   2.5   2.0   3.0   2.0   2.5   4.0   3.5    22
 2.0   1.0   2.0   2.0   3.0   2.0   3.0   4.0   3.0    22
 2.0   1.0   3.0   2.0   2.5   1.5   3.0   3.5   3.5    22
 2.5   0.5   3.0   2.5   2.5   1.0   4.0   3.0   3.0    22
-----------------------------------------------------------
\end{verbatim}
\newpage
\centerline{\bf Table~1 (contd.)}
\begin{verbatim}
-----------------------------------------------------------
  a     b     c     d     e     f     g     h     i
-----------------------------------------------------------
 2.5   1.0   3.5   1.5   2.0   1.5   3.0   3.0   4.0    22
 3.0   0.5   3.5   2.0   2.0   1.0   4.0   2.5   3.5    22
 3.0   1.0   3.0   1.5   2.0   1.5   3.5   3.0   3.5    22
 1.0   2.5   3.5   3.0   0.5   3.5   3.0   3.0   2.0    22
 2.0   1.5   3.5   2.5   0.5   3.0   3.5   2.0   3.5    22
 2.0   2.0   4.0   2.0   0.5   2.5   3.0   2.5   3.5    22
 0.5   2.5   2.0   3.5   3.5   1.0   3.0   4.0   2.0    22
 0.5   3.0   2.5   3.0   3.0   1.0   2.5   4.0   2.5    22
 1.5   2.0   0.5   3.5   3.5   3.0   2.0   3.5   2.5    22
 2.0   2.5   0.5   4.0   3.5   2.5   2.0   3.0   2.0    22
 2.5   3.0   0.5   3.5   2.0   3.5   1.0   3.0   3.0    22
 0.5   1.5   1.0   4.0   2.5   2.5   4.5   3.0   2.5    22
 0.5   1.5   1.0   4.0   3.0   2.0   4.5   3.5   2.0    22
 0.5   3.0   2.5   2.5   1.0   2.5   3.0   3.0   4.0    22
 0.5   3.5   3.0   2.0   1.0   2.0   2.5   3.5   4.0    22
 1.0   1.5   0.5   3.0   3.0   4.0   2.0   3.5   3.5    22
 1.0   1.5   0.5   4.0   1.5   4.5   3.0   2.0   4.0    22
 1.0   2.5   1.5   3.0   2.0   2.0   4.0   3.5   2.5    22
 1.0   2.5   1.5   3.0   2.5   1.5   4.0   4.0   2.0    22
 1.0   3.0   2.0   2.5   1.5   2.0   3.5   3.5   3.0    22
 1.0   3.5   2.5   2.0   1.5   1.5   3.0   4.0   3.0    22
 1.5   3.0   1.5   2.5   1.5   3.0   2.0   3.5   3.5    22
 0.5   1.5   1.0   4.5   1.5   4.0   4.0   2.0   3.0    22
 1.5   3.0   1.5   3.0   1.5   2.5   3.5   3.5   2.0    22
 0.5   1.0   1.5   4.0   2.0   3.0   4.5   2.0   3.5    22
 0.5   1.0   1.5   4.0   2.5   2.5   4.5   2.5   3.0    22
 0.5   2.5   3.0   2.5   2.5   1.0   3.0   4.0   3.0    22
 0.5   3.0   3.5   2.0   2.0   1.0   2.5   4.0   3.5    22
 1.0   1.5   2.5   3.0   1.5   2.5   4.0   2.0   4.0    22
 1.0   1.5   2.5   3.0   2.0   2.0   4.0   2.5   3.5    22
 1.0   2.0   2.0   3.0   2.0   2.0   4.0   3.0   3.0    22
 1.0   2.0   3.0   2.5   2.0   1.5   3.5   3.0   3.5    22
 1.0   2.5   3.5   2.0   1.5   1.5   3.0   3.0   4.0    22
 1.0   3.0   3.0   2.0   1.5   1.5   3.0   3.5   3.5    22
 2.0   1.0   2.0   0.5   3.5   3.0   2.5   3.5   4.0    22
-----------------------------------------------------------
\end{verbatim}
\newpage
\centerline{\bf Table~1 (contd.)}
\begin{verbatim}
-----------------------------------------------------------
  a     b     c     d     e     f     g     h     i
----------------------------------------------------------- 
 2.0   1.5   1.5   1.0   3.0   3.0   3.0   3.5   3.5    22
 2.0   1.5   1.5   1.0   3.5   2.5   3.0   4.0   3.0    22
 2.5   1.0   2.5   0.5   3.0   2.5   3.0   3.0   4.0    22
 2.5   1.5   2.0   1.0   3.0   2.0   3.5   3.5   3.0    22
 3.0   2.0   2.0   1.0   2.0   2.0   4.0   3.0   3.0    22
 3.0   2.0   2.0   1.0   2.5   1.5   4.0   3.5   2.5    22
 3.0   2.5   1.5   1.0   2.5   1.5   4.0   4.0   2.0    22
 4.0   2.5   2.5   0.5   1.5   1.0   4.5   3.0   2.5    22
 4.0   3.0   2.0   0.5   1.5   1.0   4.5   3.5   2.0    22
-----------------------------------------------------------
\end{verbatim}
\end{center}
\newpage

%\input gt.tex
\end{document}
 1.5   1.5   1.0   3.5   3.0   2.5   3.0   3.5   3.5    23
 1.5   2.0   1.5   2.5   3.0   2.5   2.0   4.0   4.0    23
 1.5   2.0   1.5   4.0   2.0   2.0   3.5   3.0   3.5    23
 1.5   3.0   2.5   3.0   1.5   1.5   2.5   3.5   4.0    23
 2.0   3.0   2.0   3.0   2.0   1.0   3.0   4.0   3.0    23
 3.0   2.5   0.5   4.0   1.5   2.5   3.0   3.0   3.0    23
 3.5   3.0   0.5   3.5   1.5   2.0   3.0   3.5   2.5    23
 2.5   2.0   0.5   3.0   4.5   2.5   1.5   3.5   3.0    23
 2.5   2.0   0.5   3.0   4.0   2.0   2.5   4.0   2.5    23
 3.0   2.5   0.5   3.5   4.5   2.0   1.5   3.0   2.5    23
 3.0   2.5   0.5   3.5   4.0   1.5   2.5   3.5   2.0    23
 3.0   4.0   2.0   2.5   2.0   0.5   2.5   4.0   2.5    23
 3.0   4.5   2.5   2.5   2.0   0.5   1.5   3.5   3.0    23
 3.5   4.0   1.5   3.0   2.5   0.5   2.5   3.5   2.0    23
 3.5   4.5   2.0   3.0   2.5   0.5   1.5   3.0   2.5    23
 4.5   4.0   0.5   3.5   2.0   1.5   2.0   3.0   2.0    23
 1.5   3.0   2.5   3.0   2.5   0.5   3.5   4.5   2.0    23
 1.5   3.5   3.0   2.5   2.0   0.5   3.0   4.5   2.5    23
 2.0   3.0   2.0   3.5   2.0   1.5   4.5   4.0   0.5    23
 2.0   4.5   3.5   2.0   0.5   1.5   3.0   4.0   2.0    23
 2.0   5.0   4.0   1.5   1.0   0.5   1.5   4.0   3.5    23
 2.5   3.5   2.0   3.0   2.5   0.5   3.5   4.0   1.5    23
 2.5   4.0   2.5   2.5   2.0   0.5   3.0   4.0   2.0    23
 3.0   5.0   3.0   2.0   1.5   0.5   2.0   3.5   2.5    23
 4.0   5.0   2.0   2.5   2.0   0.5   2.5   3.0   1.5    23
 2.0   3.0   2.0   3.5   0.5   3.0   4.5   2.5   2.0    23
 2.0   3.0   2.0   3.5   1.0   2.5   4.5   3.0   1.5    23
 2.0   3.0   2.0   3.5   1.5   2.0   4.5   3.5   1.0    23
 1.5   2.5   3.0   3.0   0.5   2.5   3.5   2.0   4.5    23
 1.5   3.0   3.5   2.5   0.5   2.0   3.0   2.5   4.5    23
 2.0   1.5   1.5   4.0   0.5   3.5   5.0   1.0   4.0    23
 2.0   1.5   1.5   4.0   1.0   3.0   5.0   1.5   3.5    23
 2.0   1.5   1.5   4.0   1.5   2.5   5.0   2.0   3.0    23
 2.0   1.5   1.5   4.0   2.5   1.5   5.0   3.0   2.0    23
 2.0   1.5   1.5   4.0   3.0   1.0   5.0   3.5   1.5    23
 2.0   1.5   1.5   4.0   3.5   0.5   5.0   4.0   1.0    23
 2.5   2.5   3.0   2.5   0.5   2.0   4.0   2.0   4.0    23
 2.5   3.0   2.5   2.5   2.0   0.5   4.0   4.0   2.0    23
 2.5   3.0   3.5   2.0   0.5   1.5   3.5   2.5   4.0    23
 2.5   3.5   3.0   2.0   1.5   0.5   3.5   4.0   2.5    23
 3.0   2.0   2.0   3.0   0.5   2.5   5.0   1.5   3.5    23
 3.0   2.0   2.0   3.0   1.0   2.0   5.0   2.0   3.0    23
 3.0   2.0   2.0   3.0   2.0   1.0   5.0   3.0   2.0    23
 3.0   2.0   2.0   3.0   2.5   0.5   5.0   3.5   1.5    23
 4.0   2.5   2.5   2.0   0.5   1.5   5.0   2.0   3.0    23
 4.0   2.5   2.5   2.0   1.5   0.5   5.0   3.0   2.0    23
 1.5   2.0   1.5   0.5   4.0   3.5   1.0   5.0   4.0    23
 1.5   2.0   1.5   1.0   4.0   3.0   1.5   5.0   3.5    23
 1.5   2.0   1.5   1.5   4.0   2.5   2.0   5.0   3.0    23
 1.5   2.0   1.5   2.5   4.0   1.5   3.0   5.0   2.0    23
 1.5   2.0   1.5   3.0   4.0   1.0   3.5   5.0   1.5    23
 1.5   2.0   1.5   3.5   4.0   0.5   4.0   5.0   1.0    23
 2.0   3.0   2.0   0.5   3.0   2.5   1.5   5.0   3.5    23
 2.0   3.0   2.0   1.0   3.0   2.0   2.0   5.0   3.0    23
 2.0   3.0   2.0   2.0   3.0   1.0   3.0   5.0   2.0    23
 2.0   3.0   2.0   2.5   3.0   0.5   3.5   5.0   1.5    23
 2.5   4.0   2.5   0.5   2.0   1.5   2.0   5.0   3.0    23
 2.5   4.0   2.5   1.5   2.0   0.5   3.0   5.0   2.0    23
 1.5   3.5   3.0   3.0   2.5   1.5   1.5   4.0   2.5    23
 2.0   3.0   2.0   4.0   3.5   1.5   2.0   3.5   1.5    23
 3.0   3.5   1.5   2.5   3.0   0.5   3.5   4.5   1.0    23
 3.0   3.5   1.5   3.5   3.0   1.5   2.5   3.5   1.0    23
 3.0   4.0   2.0   2.5   2.0   1.5   2.5   4.0   1.5    23
 4.0   5.0   2.0   1.0   1.5   0.5   3.0   4.5   1.5    23
 4.5   5.0   1.5   1.5   2.0   0.5   3.0   4.0   1.0    23
 3.0   3.5   1.5   1.0   3.0   2.0   2.0   4.5   2.5    23
 3.0   3.5   1.5   1.5   3.0   1.5   2.5   4.5   2.0    23
 3.0   3.5   1.5   2.0   3.0   1.0   3.0   4.5   1.5    23
 1.5   1.5   3.0   3.0   2.5   1.5   3.5   4.0   2.5    23
 1.5   2.0   3.5   3.0   0.5   3.5   3.5   2.5   3.0    23
 1.5   2.5   4.0   2.5   0.5   3.0   3.0   3.0   3.0    23
 2.0   1.0   3.0   3.0   2.0   2.0   4.0   3.0   3.0    23
 2.0   2.0   4.0   2.0   1.5   1.5   3.0   3.5   3.5    23
 3.0   2.5   2.5   2.0   1.5   1.5   4.0   4.0   2.0    23
 3.0   2.5   3.5   1.5   1.0   1.5   3.5   3.5   3.0    23
 1.0   3.0   2.0   3.0   3.0   2.0   2.0   4.0   3.0    23
 1.5   3.0   1.5   2.0   4.5   2.5   1.5   3.5   3.0    23
 1.5   4.0   2.5   3.0   2.5   1.5   1.5   3.5   3.0    23
 2.0   2.5   0.5   3.5   3.0   3.5   1.5   3.5   3.0    23
 2.0   3.0   1.0   3.0   5.0   2.0   2.0   3.0   2.0    23
 2.0   3.0   1.0   3.0   4.5   1.5   3.0   3.5   1.5    23
 2.0   3.5   1.5   4.0   3.5   1.5   2.0   3.0   2.0    23
 2.5   3.0   0.5   4.0   3.0   3.0   1.5   3.0   2.5    23
 2.5   3.5   1.0   3.5   3.0   1.5   3.0   3.5   1.5    23
 2.5   4.0   1.5   2.5   2.0   1.5   3.0   4.0   2.0    23
 3.0   4.0   1.0   3.5   5.0   1.5   1.5   2.0   1.5    23
 3.0   4.5   1.5   2.0   3.0   1.0   3.0   3.5   1.5    23
 0.5   2.0   1.5   3.0   2.5   3.5   2.5   3.5   4.0    23
 0.5   2.5   2.0   2.5   2.5   3.0   2.0   4.0   4.0    23
 0.5   2.5   2.0   3.0   1.5   3.5   2.5   3.0   4.5    23
 0.5   3.0   2.5   2.5   1.5   3.0   2.0   3.5   4.5    23
 1.5   2.0   0.5   3.5   2.0   4.5   2.0   3.0   4.0    23
 1.5   2.0   0.5   3.5   2.5   3.0   4.0   3.5   2.5    23
 2.0   2.5   0.5   3.0   2.5   2.5   4.0   4.0   2.0    23
 2.0   2.5   0.5   3.5   1.5   3.0   4.5   3.0   2.5    23
 2.5   3.0   0.5   3.0   1.5   2.5   4.5   3.5   2.0    23
 1.0   3.0   2.0   4.0   1.5   3.5   3.0   3.5   1.5    23
 2.0   3.0   1.0   4.5   3.0   2.5   2.5   3.0   1.5    23
 2.5   3.5   1.0   4.0   1.5   3.5   1.5   3.0   2.5    23
 0.5   2.0   1.5   4.5   2.0   3.5   4.0   3.0   2.0    23
 1.0   1.5   1.5   4.0   4.5   0.5   4.0   5.0   2.0    24
 1.0   2.0   2.0   3.0   4.0   1.0   3.0   5.0   3.0    24
 1.0   3.0   3.0   2.0   3.0   1.0   2.0   5.0   4.0    24
 1.0   4.0   4.0   1.5   2.0   0.5   1.5   5.0   4.5    24
 2.0   1.5   0.5   4.0   1.0   4.0   5.0   1.5   4.5    24
 2.0   2.0   1.0   1.5   3.0   3.5   2.5   4.0   4.5    24
 2.0   2.0   1.0   3.0   1.5   3.5   4.0   2.5   4.5    24
 2.0   2.0   1.0   4.0   2.0   3.0   3.0   3.0   4.0    24
 2.0   3.0   2.0   2.5   2.0   2.5   1.5   4.0   4.5    24
 3.0   2.5   0.5   4.5   1.0   3.5   2.5   2.5   4.0    24
 4.5   4.0   0.5   1.5   1.0   1.5   5.0   4.0   2.0    24
 4.5   4.0   0.5   3.0   1.0   2.0   2.5   4.0   2.5    24
 3.0   2.5   0.5   3.0   4.0   3.0   1.0   3.5   3.5    24
 3.5   3.0   0.5   3.0   3.0   2.0   2.5   4.0   2.5    24
 3.5   3.0   0.5   3.5   4.0   2.5   1.0   3.0   3.0    24
 3.5   4.0   1.5   3.0   2.5   1.5   1.5   3.5   3.0    24
 4.5   4.0   0.5   3.5   2.5   1.0   3.0   3.5   1.5    24
 5.0   4.5   0.5   4.5   3.5   1.0   1.5   2.0   1.5    24
 1.5   1.0   1.5   4.5   4.0   0.5   5.0   4.0   2.0    24
 1.5   3.0   3.5   2.0   2.0   1.0   2.5   4.0   4.5    24
 2.0   1.0   2.0   4.0   3.0   1.0   5.0   3.0   3.0    24
 3.0   1.0   3.0   3.0   2.0   1.0   5.0   2.0   4.0    24
 3.0   1.5   3.5   2.0   2.0   1.0   4.0   2.5   4.5    24
 3.0   2.0   1.0   3.0   1.0   3.0   5.0   2.0   4.0    24
 4.0   1.0   4.0   2.0   1.5   0.5   5.0   1.5   4.5    24
 4.0   3.0   1.0   2.0   1.0   2.0   5.0   3.0   3.0    24
 1.5   2.0   1.5   4.5   3.5   1.0   5.0   4.5   0.5    24
 1.5   5.0   4.5   1.5   0.5   1.0   2.0   4.5   3.5    24
 2.5   4.0   2.5   3.0   1.0   2.0   4.5   4.0   0.5    24
 2.5   4.5   3.0   2.5   0.5   2.0   4.0   4.0   1.0    24
 3.0   3.5   1.5   3.5   2.5   1.0   4.5   4.0   0.5    24
 3.0   5.0   3.0   2.0   1.0   1.0   3.0   4.0   2.0    24
 4.5   5.0   1.5   2.5   1.5   1.0   4.0   3.5   0.5    24
 1.0   3.0   3.0   3.5   0.5   3.0   3.5   2.5   4.0    24
 1.0   3.5   3.5   3.0   0.5   2.5   3.0   3.0   4.0    24
 1.5   2.0   1.5   4.5   0.5   4.0   5.0   1.5   3.5    24
 1.5   2.0   1.5   4.5   1.0   3.5   5.0   2.0   3.0    24
 1.5   2.0   1.5   4.5   1.5   3.0   5.0   2.5   2.5    24
 1.5   2.0   1.5   4.5   2.0   2.5   5.0   3.0   2.0    24
 1.5   2.0   1.5   4.5   2.5   2.0   5.0   3.5   1.5    24
 1.5   2.0   1.5   4.5   3.0   1.5   5.0   4.0   1.0    24
 1.5   3.5   3.0   3.0   1.5   1.5   3.5   4.0   2.5    24
 2.0   3.0   2.0   3.5   1.0   2.5   3.5   3.0   3.5    24
 2.5   4.0   2.5   2.5   1.0   1.5   3.0   4.0   3.0    24
 2.5   4.0   2.5   3.0   0.5   2.5   4.5   3.5   1.0    24
 2.5   3.5   3.0   2.5   0.5   2.0   4.0   3.0   3.0    24
 3.0   3.0   2.0   3.0   0.5   2.5   5.0   2.5   2.5    24
 3.0   3.0   2.0   3.0   1.0   2.0   5.0   3.0   2.0    24
 3.0   3.0   2.0   3.0   1.5   1.5   5.0   3.5   1.5    24
 3.0   3.0   2.0   3.0   2.0   1.0   5.0   4.0   1.0    24
 3.0   4.5   3.5   1.5   0.5   1.0   3.5   4.0   2.5    24
 4.5   4.0   2.5   1.5   0.5   1.0   5.0   3.5   1.5    24
 2.0   3.0   2.0   4.0   3.0   2.0   3.0   4.0   1.0    24
 2.0   4.0   3.0   2.5   1.5   2.0   2.5   4.5   2.0    24
 2.0   5.0   4.0   1.5   1.5   1.0   0.5   4.5   4.0    24
 2.5   4.0   2.5   3.5   2.5   2.0   1.0   3.5   2.5    24
 3.0   5.0   3.0   2.0   2.0   1.0   1.0   4.0   3.0    24
 4.0   5.0   2.0   3.0   3.0   1.0   1.0   3.0   2.0    24
 4.5   5.0   1.5   4.0   4.0   1.0   0.5   2.0   1.5    24
 1.5   1.5   1.0   2.0   5.0   4.0   0.5   4.5   4.0    24
 2.0   2.0   1.0   3.0   5.0   3.0   1.0   4.0   3.0    24
 2.0   2.5   1.5   2.0   4.0   3.0   1.0   4.5   3.5    24
 2.0   4.0   3.0   2.0   2.5   1.5   1.0   4.5   3.5    24
 3.0   3.0   1.0   4.0   5.0   2.0   1.0   3.0   2.0    24
 4.0   4.0   1.0   4.5   5.0   1.5   0.5   2.0   1.5    24
 1.0   1.5   1.5   4.0   0.5   4.5   4.0   2.0   5.0    24
 1.0   2.0   2.0   3.0   1.0   4.0   3.0   3.0   5.0    24
 1.0   3.0   3.0   2.0   1.0   3.0   2.0   4.0   5.0    24
 1.0   4.0   4.0   1.5   0.5   2.0   1.5   4.5   5.0    24
 1.5   3.5   3.0   2.0   1.0   2.0   2.5   4.5   4.0    24
 3.0   3.5   1.5   2.0   1.0   2.0   4.0   4.5   2.5    24
 1.5   0.5   2.0   1.0   4.0   4.0   1.5   4.5   5.0    24
 2.0   1.0   2.0   1.5   3.5   3.0   2.5   4.5   4.0    24
 2.0   1.0   2.0   3.0   3.5   1.5   4.0   4.5   2.5    24
 2.0   1.0   3.0   1.0   3.0   3.0   2.0   4.0   5.0    24
 3.0   1.0   4.0   1.0   2.0   2.0   3.0   3.0   5.0    24
 4.0   0.5   4.5   1.0   1.5   1.5   4.0   2.0   5.0    24
 1.0   3.5   4.5   2.5   0.5   3.0   2.5   4.0   2.5    24
 2.0   2.5   2.5   3.0   2.0   2.0   4.0   4.5   1.5    24
 2.0   3.0   4.0   2.0   1.0   2.0   3.0   4.0   3.0    24
 0.5   2.0   1.5   4.0   4.0   1.0   4.5   5.0   1.5    24
 0.5   4.5   4.0   1.5   1.5   1.0   2.0   5.0   4.0    24
 1.0   3.0   2.0   3.0   3.0   1.0   4.0   5.0   2.0    24
 1.0   4.0   3.0   2.0   2.0   1.0   3.0   5.0   3.0    24
 2.0   2.5   0.5   3.0   2.5   4.5   1.0   4.0   4.0    24
 2.5   4.5   2.0   2.5   1.5   2.0   2.0   4.0   3.0    24
 3.0   4.0   1.0   4.0   3.0   2.0   2.0   3.0   2.0    24
 3.5   4.0   0.5   4.5   2.5   3.0   1.0   2.5   2.5    24
 1.0   1.5   0.5   3.5   3.0   4.5   2.5   3.5   4.0    24
 1.0   1.5   0.5   4.5   1.5   5.0   3.5   2.0   4.5    24
 1.5   3.0   1.5   3.0   1.5   3.5   2.5   3.5   4.0    24
 2.0   2.5   0.5   3.0   2.5   3.5   3.0   4.0   3.0    24
 2.5   3.0   0.5   3.5   1.0   3.5   4.0   3.0   3.0    24
 3.0   3.5   0.5   3.0   1.0   3.0   4.0   3.5   2.5    24
 0.5   2.5   2.0   4.5   2.5   3.0   4.0   4.0   1.0    24
 1.0   3.5   2.5   3.5   2.5   2.0   2.5   4.0   2.5    24
 1.5   3.0   1.5   4.5   2.5   3.0   3.0   3.5   1.5    24
 2.5   3.5   1.0   4.5   2.5   3.0   2.0   3.0   2.0    24
 0.5   1.5   1.0   5.0   1.5   4.5   4.5   2.0   3.5    24
 0.5   3.0   2.5   3.5   1.0   3.5   3.0   3.0   4.0    24
 0.5   3.5   3.0   3.0   1.0   3.0   2.5   3.5   4.0    24
 1.5   3.0   1.5   3.5   1.5   3.0   4.0   3.5   2.5    24
 1.5   1.5   1.0   0.5   4.5   4.0   2.0   5.0   4.0    24
 2.0   2.0   1.0   1.0   4.0   3.0   3.0   5.0   3.0    24
 3.0   3.0   1.0   1.0   3.0   2.0   4.0   5.0   2.0    24
 4.0   4.0   1.0   0.5   2.0   1.5   4.5   5.0   1.5    24
 0.5   1.5   2.0   4.0   1.0   4.0   4.5   1.5   5.0    24
 0.5   4.0   4.5   1.5   1.0   1.5   2.0   4.0   5.0    24
 1.0   2.0   2.0   3.5   1.5   3.0   4.5   2.5   4.0    24
 1.0   2.0   2.0   3.5   3.0   1.5   4.5   4.0   2.5    24
 1.0   2.0   3.0   3.0   1.0   3.0   4.0   2.0   5.0    24
 1.0   3.0   4.0   2.0   1.0   2.0   3.0   3.0   5.0    24
 3.5   1.5   3.0   1.0   2.0   2.0   4.5   2.5   4.0    24
 3.5   3.0   1.5   1.0   2.0   2.0   4.5   4.0   2.5    24
 2.0   1.5   0.5   5.0   3.5   1.5   5.0   4.0   2.0    25
 2.5   2.0   0.5   4.5   3.5   1.0   5.0   4.5   1.5    25
 2.5   3.0   1.5   3.5   1.5   3.0   2.0   3.5   4.5    25
 3.5   3.0   0.5   3.0   3.0   3.0   1.5   4.0   3.5    25
 4.0   3.5   0.5   3.5   3.0   2.5   1.5   3.5   3.0    25
 5.0   4.5   0.5   4.0   2.5   1.5   2.0   3.0   2.0    25
 2.0   3.0   2.0   4.0   2.5   1.5   5.0   4.5   0.5    25
 2.0   5.0   4.0   2.0   0.5   1.5   3.0   4.5   2.5    25
 4.0   5.0   2.0   2.5   1.0   1.5   4.5   4.0   0.5    25
 1.5   3.5   3.0   3.5   0.5   3.0   4.0   3.0   3.0    25
 1.5   4.0   3.5   3.0   0.5   2.5   3.5   3.5   3.0    25
 2.0   3.0   2.0   4.0   0.5   3.5   5.0   2.5   2.5    25
 2.0   3.0   2.0   4.0   1.0   3.0   5.0   3.0   2.0    25
 2.0   3.0   2.0   4.0   1.5   2.5   5.0   3.5   1.5    25
 2.0   3.0   2.0   4.0   2.0   2.0   5.0   4.0   1.0    25
 2.0   1.5   1.5   4.5   0.5   4.0   5.5   1.0   4.5    25
 2.0   1.5   1.5   4.5   1.0   3.5   5.5   1.5   4.0    25
 2.0   1.5   1.5   4.5   1.5   3.0   5.5   2.0   3.5    25
 2.0   1.5   1.5   4.5   2.0   2.5   5.5   2.5   3.0    25
 2.0   1.5   1.5   4.5   2.5   2.0   5.5   3.0   2.5    25
 2.0   1.5   1.5   4.5   3.0   1.5   5.5   3.5   2.0    25
 2.0   1.5   1.5   4.5   3.5   1.0   5.5   4.0   1.5    25
 2.0   1.5   1.5   4.5   4.0   0.5   5.5   4.5   1.0    25
 3.0   2.0   2.0   3.5   0.5   3.0   5.5   1.5   4.0    25
 3.0   2.0   2.0   3.5   1.0   2.5   5.5   2.0   3.5    25
 3.0   2.0   2.0   3.5   1.5   2.0   5.5   2.5   3.0    25
 3.0   2.0   2.0   3.5   2.0   1.5   5.5   3.0   2.5    25
 3.0   2.0   2.0   3.5   2.5   1.0   5.5   3.5   2.0    25
 3.0   2.0   2.0   3.5   3.0   0.5   5.5   4.0   1.5    25
 4.0   2.5   2.5   2.5   0.5   2.0   5.5   2.0   3.5    25
 4.0   2.5   2.5   2.5   1.0   1.5   5.5   2.5   3.0    25
 4.0   2.5   2.5   2.5   1.5   1.0   5.5   3.0   2.5    25
 4.0   2.5   2.5   2.5   2.0   0.5   5.5   3.5   2.0    25
 4.0   4.5   2.5   2.0   0.5   1.5   5.0   4.0   1.0    25
 5.0   3.0   3.0   1.5   0.5   1.0   5.5   2.5   3.0    25
 5.0   3.0   3.0   1.5   1.0   0.5   5.5   3.0   2.5    25
 1.5   2.0   1.5   0.5   4.5   4.0   1.0   5.5   4.5    25
 1.5   2.0   1.5   1.0   4.5   3.5   1.5   5.5   4.0    25
 1.5   2.0   1.5   1.5   4.5   3.0   2.0   5.5   3.5    25
 1.5   2.0   1.5   2.0   4.5   2.5   2.5   5.5   3.0    25
 1.5   2.0   1.5   2.5   4.5   2.0   3.0   5.5   2.5    25
 1.5   2.0   1.5   3.0   4.5   1.5   3.5   5.5   2.0    25
 1.5   2.0   1.5   3.5   4.5   1.0   4.0   5.5   1.5    25
 1.5   2.0   1.5   4.0   4.5   0.5   4.5   5.5   1.0    25
 2.0   3.0   2.0   0.5   3.5   3.0   1.5   5.5   4.0    25
 2.0   3.0   2.0   1.0   3.5   2.5   2.0   5.5   3.5    25
 2.0   3.0   2.0   1.5   3.5   2.0   2.5   5.5   3.0    25
 2.0   3.0   2.0   2.0   3.5   1.5   3.0   5.5   2.5    25
 2.0   3.0   2.0   2.5   3.5   1.0   3.5   5.5   2.0    25
 2.0   3.0   2.0   3.0   3.5   0.5   4.0   5.5   1.5    25
 2.5   4.0   2.5   0.5   2.5   2.0   2.0   5.5   3.5    25
 2.5   4.0   2.5   1.0   2.5   1.5   2.5   5.5   3.0    25
 2.5   4.0   2.5   1.5   2.5   1.0   3.0   5.5   2.5    25
 2.5   4.0   2.5   2.0   2.5   0.5   3.5   5.5   2.0    25
 3.0   5.0   3.0   0.5   1.5   1.0   2.5   5.5   3.0    25
 3.0   5.0   3.0   1.0   1.5   0.5   3.0   5.5   2.5    25
 1.5   3.5   3.0   3.5   2.0   2.5   3.0   4.5   1.5    25
 3.0   3.5   1.5   3.0   3.5   0.5   4.0   5.0   1.0    25
 4.0   5.0   2.0   1.5   2.0   0.5   3.5   5.0   1.5    25
 4.5   5.0   1.5   2.0   2.5   0.5   3.5   4.5   1.0    25
 3.0   3.5   1.5   1.0   3.5   2.5   2.0   5.0   3.0    25
 3.0   3.5   1.5   1.5   3.5   2.0   2.5   5.0   2.5    25
 3.0   3.5   1.5   2.0   3.5   1.5   3.0   5.0   2.0    25
 3.0   3.5   1.5   2.5   3.5   1.0   3.5   5.0   1.5    25
 4.0   5.0   2.0   1.0   2.0   1.0   3.0   5.0   2.0    25
 4.5   5.0   1.5   1.5   2.5   1.0   3.0   4.5   1.5    25
 1.5   3.0   3.5   3.0   1.5   2.5   3.5   4.5   2.0    25
 3.0   2.5   5.5   1.0   0.5   1.5   3.0   3.0   5.0    25
 3.5   1.0   4.5   2.0   0.5   2.5   4.5   1.5   5.0    25
 3.5   1.5   5.0   1.5   0.5   2.0   4.0   2.0   5.0    25
 0.5   2.0   1.5   1.5   5.0   3.5   2.0   5.0   4.0    25
 0.5   2.5   2.0   1.0   4.5   3.5   1.5   5.0   4.5    25
 1.0   1.5   0.5   4.5   5.0   2.5   3.5   4.5   2.0    25
 1.0   2.0   1.0   3.0   5.0   2.0   4.0   5.0   2.0    25
 1.0   2.5   1.5   2.0   4.5   2.5   3.0   5.0   3.0    25
 1.5   2.0   0.5   5.0   5.0   2.0   3.5   4.0   1.5    25
 1.5   2.5   1.0   3.0   4.5   1.5   4.5   5.0   1.5    25
 2.0   3.5   1.5   2.5   5.0   2.5   1.5   3.5   3.0    25
 2.0   3.5   1.5   2.5   4.5   2.0   2.5   4.0   2.5    25
 2.0   4.5   2.5   1.0   2.5   1.5   3.0   5.0   3.0    25
 2.5   3.5   1.0   3.5   5.0   1.5   3.0   3.5   1.5    25
 2.5   4.5   2.0   2.0   3.5   1.5   2.5   4.0   2.5    25
 3.0   4.5   1.5   1.5   2.5   1.0   4.5   5.0   1.5    25
 3.0   4.5   1.5   3.5   2.0   2.5   1.5   3.5   3.0    25
 3.0   5.0   2.0   1.0   2.0   1.0   4.0   5.0   2.0    25
 3.5   5.0   1.5   2.5   3.5   1.0   3.0   3.5   1.5    25
 1.5   2.0   0.5   4.0   2.0   5.0   2.5   3.0   4.5    25
 2.5   3.0   0.5   3.5   1.5   4.0   3.0   3.5   3.5    25
 3.0   3.5   0.5   3.0   1.5   3.5   3.0   4.0   3.0    25
 2.0   3.5   1.5   4.5   2.0   3.5   2.5   3.5   2.0    25
 0.5   2.0   1.5   5.0   2.0   4.0   4.5   3.0   2.5    25
 0.5   3.0   2.5   4.0   1.5   3.5   3.5   3.5   3.0    25
 0.5   3.5   3.0   3.5   1.5   3.0   3.0   4.0   3.0    25
 1.0   3.0   2.0   4.5   1.5   4.0   3.5   3.5   2.0    25

\end{verbatim}  
\end{document}

