%Paper: 
%From: laporta@bo.infn.it (Stefano Laporta)
%Date: Fri, 1 Apr 1994 18:10:34 +0200

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\hsize 16.2 true cm             % larghezza della pagina
% \hoffset xyz                   % margine a sinistra
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%\baselineskip = 6truemm        % distanza tra le righe
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% \nopagenumbers
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\def\GC#1{\beta_{#1}}
\def\GCdd{\beta_2^2}
\def\Ztd{\zeta^2(3)}
\def\Qtw{Q_{12}}
\def\r {r}
\def\rr{r}
%\def\rr{\left({m_e\over m_\mu}\right)}
\def\lrr{\ln \rr}
\def\lrrd{\ln^2 \rr}
\def\lrrt{\ln^3 \rr}
\def\lrrq{\ln^4 \rr}
\def\lrrc{\ln^5 \rr}
\def\lr{\ln \r}
\def\lrd{\ln^2 \r}
\def\lrt{\ln^3 \r}
\def\lrq{\ln^4 \r}
\def\lrc{\ln^5 \r}
\def\LGd{\ln 2}
\def\LGdd{\ln^2 2}
\def\LGdt{\ln^3 2}
\def\LGdq{\ln^4 2}
\def\LGdc{\ln^5 2}
\def\A#1{a_{#1}}
\def\Z#1{\zeta(#1)}
\def\parn{\par\noindent}
\def\app{{\left({\alpha\over\pi}\right)}}
\def\PR{{\it Phys.Rev. }}
\def\PRL{{\it Phys.Rev.Lett. }}
\def\ref#1{[#1]}
\def\oo{\infty}
\def\oc{\propto}
\def\Li{{\rm Li}}
\def\LGmM {{\ln  \left({m \over M } \right)} }
%\def\fig#1{[{\rm fig.} #1 ]}
\def\fig#1{{\rm fig. #1} }
\font\fontA=cmbx10 scaled\magstep1
\def\F{\Lambda}
\def\+{\hskip -2pt + \hskip -2pt}
\def\our{\ ({\rm our});\qquad}
\def\refk{\ ({\rm ref.[3]});}
\def\refkk{\ ({\rm ref.[3]}).}
\def\amu{{a_\mu^{(5)}}}
\def\ae{{a_e^{(5)}}}
\def\C#1{{ a_\mu^{(#1)}-a_e^{(#1)} }}
\def\CC#1{{\left(\C#1\right) }}
\def\#{{\hskip -.55 pt}}
\def\riga{\par\noindent
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
----------------------------------------------------------------------------------------------------------------
\par\noindent}
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\def\d${$ \displaystyle }

\rightline{\bf DFUB 94-02}
\rightline{23 March 1994}
\vskip 20truemm
\centerline{\bf Analytical and numerical contributions of some
                tenth-order graphs}
\par
\centerline{\bf containing vacuum polarization insertions }
\par
\centerline{\bf to the muon (g-2) in QED. }
\par
\vskip 20truemm
%%%%%%%%%%%%%%%%%%%%%%% footnote %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\centerline{S.Laporta
\footnote {$^{\S}$}{ {E-mail: {\tt laporta@bo.infn.it} }}  }
%%%%%%%%%%%%%%%%%%%%%%% footnote %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vskip 20truemm
\centerline{\it Dipartimento di Fisica, Universit\`a di Bologna,}
\centerline{\it and INFN, Sezione di Bologna,}
\centerline{\it Via Irnerio 46, I-40126 Bologna, Italy}
\vskip 20truemm
%ABSTRACT-++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
\centerline{\bf Abstract. }  \par
The contributions to the $g-2$ of the muon from some
tenth-order (five-loop) graphs
containing one-loop and two-loop vacuum polarization insertions
have been evaluated analytically in QED perturbation theory,
expanding the results in the
ratio of the electron to muon mass $(m_e / m_\mu)$.
Some results contain also terms known only in numerical form.
Our results agree with the renormalization group results
already existing in the literature.
%ABSTRACT-++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
\vfill\eject
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In this work we have calculated in analytical form
the contributions
to the muon $g$-$2$ from the five-loop graphs
obtained inserting
one-loop and two-loop vacuum
polarizations on a single photon line of the second-order and
fourth-order vertex graphs;
the considered graphs are shown in figs.1 and 2.
Following the method used in previous calculations of the
contributions of
three-loop and four-loop graphs containing vacuum polarizations [1][2],
we found it convenient to express the results
expanding them in the ratio of the electron and muon masses
\d${(m_e/m_\mu)}$.
Due to the length of the expressions of the coefficients of
the high power terms, we will list them only up to \d$(m_e/m_\mu)$.

The analytical expressions of the contributions to the muon anomaly
of the graphs shown in figs.1 and 2,
accounting for the proper multiplicity factors,
are ($r\equiv m_e/m_\mu$):
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%F101=
$$ \eqalign { \amu[{\rm fig.1(a)}]=
& {8\over81} \lrq
 +{200\over243} \lrt
 +\left( {8\over81} \pi^2 +{634\over243} \right) \lrd
+\left( {16\over27}\Z3 +{100\over243} \pi^2
\right.\cr
&\left.
+{8609\over2187} \right) \lr
+{2\over135} \pi^4 +{100\over81}\Z3 +{317\over729} \pi^2
                     +{64613\over26244}
\cr
&
-\r\left[ {18203\over374220} \pi^4 \right]
          +O(\r^2\lrq)\ ,}  \eqno(1) $$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%F102=
$$ \eqalign { \amu[{\rm fig.1(b)}]=
 &\left( {32\over81} \pi^2 -{952\over243} \right) \lrt
 +\left(  {8\over81} \pi^2 -{244\over243} \right) \lrd
+\left( {32\over135} \pi^4  -{104\over81} \pi^2
\right.\cr
&\left.
 -{7627\over729}
                              \right) \lr
 +{16\over27} \pi^2\Z3 +{8\over405} \pi^4 -{476\over81}\Z3
      -{2593\over2187} \pi^2 +{64244\over6561}
\cr
&
          +O(\r^2\lrd)\ ,}  \eqno(2) $$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%F103=
$$ \eqalign { \amu[{\rm fig.1(c)}]=
&\left( {64\over9}\Z3 -{32\over405} \pi^2 -{1886\over243} \right) \lrd
+\left( -{16\over81} \pi^4 +{16\over135}\Z3 +{21532\over6075} \pi^2
\right.\cr
&\left.
       -{57899\over3645} \right) \lr
+{64\over9}\Z5 +{32\over27} \pi^2\Z3
-{106\over6075} \pi^4 +{10732\over2025}\Z3
\cr
&
-{148921\over91125} \pi^2 -{1090561\over109350}
          +O(\r^2\lr) \ ,}  \eqno(3) $$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
$$ \eqalign { \amu[{\rm fig.1(d)}]=
%F104=
&\left(  {16\over135} \pi^4 -{256\over189}\Z3 -{151849\over15309} \right) \lr
 -{16\over9} \pi^2\Z3
 +{124\over8505} \pi^4
\cr
 &
+{92476\over6615}\Z3 +{143\over81} \pi^2 -{46796257\
          +O(\r^2) \ ,}  \eqno(4) $$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
$$ \eqalign { \amu[{\rm fig.1(e)}]=
%F105=
&
 -{1\over3} \lrt
 +\left(  {2\over3}\Z3 -{35\over18} \right) \lrd
 +\left( {25\over9}\Z3 -{1\over6} \pi^2 -{413\over108} \right) \lr
\cr
&
+{1\over9} \pi^2\Z3
+{263\over108}\Z3 -{35\over108} \pi^2
 -{439\over162}
\cr
&
+\r\left[
        -{2516\over945} \pi^3\GC2 -{3401\over2268} \pi^5
         -{11\over54} \pi^4 \LGd   -{1549\over360} \pi^2\Z3
\right.\cr
&
\phantom{+\r[}
\left.
         +{12195383\0} \pi^4+{87998\over8505} \pi^3
         +{970237\over43740} \pi^2
\right]
          +O(\r^2\lrq) \ ,}  \eqno(5) $$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
$$ \eqalign { \amu[{\rm fig.1(f)}]=
%F106=
&
   \left( -{2\over3} \pi^2 +{119\over18} \right) \lrd
  +\left(  {4\over3} \pi^2 \Z3 -{119\over9}\Z3
          +{1\over6} \pi^2   -{13\over8}   \right) \lr
\cr
	   &+{1\over9} \pi^2\Z3  -{2\over15} \pi^4
          -{61\over54}\Z3 +{161\over216} \pi^2 +{3661\over648}
            +O(r^2\lrd)\ ,}  \eqno(6) $$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
$$ \eqalign { \amu[{\rm fig.1(g)}]=
%F107
&
\left( -{4}\Z3 +{2\over45} \pi^2 +{943\over216} \right) \lr
 +{8}\Ztd  -{4\over45} \pi^2\Z3
\cr
&
 +{1\over18} \pi^4
-{3833\over540}\Z3 -{5483\over5400} \pi^2   +{8581\over3240}
            +O(r^2\lr)\ ,}  \eqno(7) $$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
$$ \eqalign { \amu[{\rm fig.1(h)}]=
%F108=
&\left(  -{14\over405} \pi^4 +{128\over9} \A4
        -{16\over27}  \pi^2 \LGdd  +{16\over27} \LGdq
        +{26\over27} \Z3 -{16\over27} \pi^2 \LGd
\right.\cr
&\left.
        -{164\over243} \pi^2  +{673\over81}
	                       \right) \lrd
+\left( -{146\over9} \Z5
      +{256\over9} \A5
      -{76\over27}  \pi^2\Z3
      +{32\over81}  \pi^2 \LGdt
\right.\cr
&\left.
      +{196\over405}  \pi^4 \LGd
      -{32\over135}  \LGdc
      -{8\over27}  \pi^2 \LGdd
      +{16\over3}  \pi^2 \LGd
      -{44\over405}  \pi^4
\right.\cr
&\left.
      +{64\over3} \A4
      +{8\over9}  \LGdq
      -{1213\over81} \Z3
      -{4873\over1458}  \pi^2
      +{33335\over1944}           \right) \lr
      +U_1
      -{463\over27}\Z5
\cr
&
      +{64\over3}\A5
      +{29\over135} \pi^4 \LGd
      -{161\over162} \pi^2\Z3
      +{8\over81} \pi^2 \LGdt
      -{8\over45} \LGdc
\cr
&
      +{301\over7290} \pi^4
      -{1696\over81}\A4
      -{436\over243} \pi^2 \LGdd
      -{212\over243} \LGdq
      -{3211\over324}\Z3
\cr
&
      -{952\over243} \pi^2 \LGd
      +{2635\over648} \pi^2
      +{31603\over1944}
%      +(-3/2)*ESA108
            +O(\r^2\lr) \ ,}  \eqno(8) $$
%                       ???
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
$$ \eqalign { \amu[{\rm fig.1(i)}]=
%F109=
&\left({175\over9}\Z5          +{140\over27} \pi^2\Z3
      -{10\over9} \pi^4 \LGd   +{14\over6075} \pi^4
      +{3328\over135}\A4       -{416\over405} \pi^2 \LGdd
\right.\cr
&\left.
      +{416\over405} \LGdq     +{1561\over675}\Z3
      -{940\over243} \pi^2     +{36653\over1800}
 \right) \lr
%-{3} ESA109
 +U_2
      +{10681\over540}\Z5
\cr
&
      +{3328\over135}\A5
      -{416\over2025} \LGdc
      -{263\over81} \pi^2\Z3
      +{416\over1215} \pi^2 \LGdt
      -{3679\over12150} \pi^4 \LGd
\cr
&
      +{202289\over583200} \pi^4
      +{2176\over2025}\A4
      -{272\over6075} \pi^2 \LGdd
      +{272\over6075} \LGdq
      -{2933101\over60750}\Z3
\cr
&
      +{2984\over243} \pi^2 \LGd
      -{56345\over8748} \pi^2
      +{13416707\over364500}
            +O(\r^2) \ ,}  \eqno(9) $$
%                   ???
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
$$ \eqalign { \amu[{\rm fig.1(j)}]=
%F110=
&{1\over8} \lrd
+\left( -{1\over2}\Z3 +{5\over12} \right) \lr
%+ESA110
+U_3
        -{5\over6}\Z3 +{1\over48} \pi^2 +{409\over1152}
\cr
&
+\r\left[
        {110353\over272160} \pi^6
       +{9725\over756} \pi^5 \LGd
       +{4642\over567} \pi^4 \LGdd
       -{389\over8} \pi^3\Z3
       -{3376\over63} \pi^3 \Qtw
\right.\cr
&
\phantom{+\r[}
\left.
       +{1688\over189} \pi^3 \GC2\LGd
       -{6224\over63} \pi^2\GC4
       -{3376\over189} \pi^2\GCdd
       +{3376\over189} \pi^2\A4
       +{422\over567} \pi^2 \LGdq
\right.\cr
&
\phantom{+\r[}
\left.
       -{1461799\over127008} \pi^5
       -{21247\over756} \pi^4 \LGd
       -{30997\over4410} \pi^3\GC2
       +{557663\over15120} \pi^2\Z3
\right.\cr
&
\phantom{+\r[}
\left.
       +{621830879\0} \pi^4
       +{23060\over189} \pi^3 \LGd
       +{18448\over189} \pi^2\GC2
       -{21563\over13230} \pi^3    -{94319\over22680} \pi^2
\right]
\cr&            +O(r^2\lrt) \ ,}  \eqno(10) $$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
$$ \eqalign { \amu[{\rm fig.1(k)}]=
%F111=
&\left({7\over270} \pi^4 -{32\over3}\A4   +{4\over9} \pi^2 \LGdd
       -{4\over9} \LGdq -{13\over18}\Z3  +{4\over9} \pi^2 \LGd
       +{41\over81} \pi^2
\right.\cr
&\left.
-{673\over108} \right) \lr
-{7\over135} \pi^4\Z3 +{64\over3}\A4\Z3 -{8\over9} \pi^2\Z3\LGdd
+{8\over9}\Z3\LGdq
\cr
&
+{13\over9}\Ztd  -{8\over9} \pi^2\Z3 \LGd
+{73\over12}\Z5 -{32\over3}\A5 -{49\over270} \pi^4 \LGd
+{7\over162} \pi^2\Z3
\cr
&
-{4\over27} \pi^2 \LGdt +{4\over45} \LGdc
+{97\over3240} \pi^4 -{32\over9}\A4 -{2\over27} \pi^2 \LGdd
-{4\over27} \LGdq
\cr
&
+{1985\over108}\Z3 -{59\over27} \pi^2 \LGd
+{1351\over1296} \pi^2 -{6625\over1728}
            +O(r^2\lr) \ ,}  \eqno(11) $$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% L4B(4-{th)-{e-{e-{e-{ w27
%F112=
$$ \eqalign { \amu[{\rm fig.2(a)}]=
&
\left(
      {8\over27} \pi^2 \LGd -{20\over81} \pi^2
       -{4\over9}\Z3 +{31\over27}
\right) \lrt
+\left(
        {11\over162} \pi^4 -{32\over9}\A4
       -{8\over27} \pi^2 \LGdd
\right.\cr
&\left.
       -{4\over27} \LGdq
       -{14\over3}\Z3 +{20\over9} \pi^2 \LGd
       -{158\over81} \pi^2  +{115\over18}
\right) \lrd
\cr
&
+\left(
        {143\over18}\Z5 -{64\over9}\A5
       +{41\over405} \pi^4 \LGd +{2\over9} \pi^2\Z3
       +{16\over81} \pi^2 \LGdt +{8\over135} \LGdc
\right.\cr
&\left.
       +{119\over810} \pi^4  -{160\over9}\A4
       -{40\over27} \pi^2 \LGdd -{20\over27} \LGdq
       -{133\over9}\Z3  +{442\over81} \pi^2 \LGd
\right.\cr
&\left.
       -{1133\over243} \pi^2 +{8719\over648}
\right) \lr
%+ESA112
+U_4
       +{547\over36}\Z5 -{160\over9}\A5 -{1\over27} \pi^2\Z3
\cr
&
       +{40\over81} \pi^2 \LGdt +{41\over162} \pi^4 \LGd
       +{4\over27} \LGdc
       -{485\over5832} \pi^4
       -{1768\over81}\A4
       -{442\over243} \pi^2 \LGdd
\cr
&
       -{221\over243} \LGdq
       -{7199\over486}\Z3
       +{3503\over729} \pi^2 \LGd
       -{34541\over8748} \pi^2
       +{31531\over2916}
\cr
&
+\r\left[
       {101\over3072} \pi^4
\right]
            +O(r^2\lrt) \ ,}  \eqno(12) $$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
$$ \eqalign { \amu[{\rm fig.2(b)}]=
%F113=
&\left(
      {2\over27}  \pi^4 -{35\over6}\Z3 -{16\over9} \pi^2 \LGd
     +{62\over81} \pi^2 +{227\over54}
\right) \lrd
+\left(
      -{40\over27} \pi^2\Z3
\right.\cr
&\left.
      +{409\over1080} \pi^4
      +{104\over9}\A4
      +{35\over27} \pi^2 \LGdd
      +{13\over27} \LGdq +{1475\over162}\Z3
      -{616\over81} \pi^2 \LGd
\right.\cr
&\left.
      -{187\over729} \pi^2
      +{11891\over972}
\right) \lr
%+ESA113
+U_5
        -{1411\over144}\Z5 +{104\over9}\A5 -{26\over27} \pi^2\Z3
\cr
&
	-{35\over81} \pi^2 \LGdt
        -{803\over3240} \pi^4 \LGd
        -{13\over135} \LGdc
	-{24679\over58320} \pi^4
      +{2284\over81}\A4
\cr
&
      +{1277\over486} \pi^2 \LGdd
      +{571\over486} \LGdq
      -{11381\over1944}\Z3 -{1858\over243} \pi^2 \LGd +{58045\over8748} \pi^2
\cr
&
      +{136247\over5832}
            +O(r^2\lr) \ ,}  \eqno(13) $$
%                  ???
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
$$ \eqalign { \amu[{\rm fig.2(c)}]=
%F114=
&\left(
         {385\over18}\Z5 -{11\over9} \pi^4 \LGd +{77\over27} \pi^2\Z3
        +{139\over432} \pi^4 -{1459\over135}\Z3 +{302\over405} \pi^2
\right.\cr
&\left.
        +{3067\over3240}
\right) \lr
%+ESA114
+U_6
        +{929\over32}\Z5 -{683\over216} \pi^2\Z3 -{319\over432} \pi^4 \LGd
\cr
&
        +{4\over27} \pi^2 \LGdd +{102541\over388800} \pi^4
        -{32\over9}\A4 -{4\over27} \LGdq -{128483\over12150}\Z3
        -{58\over27} \pi^2 \LGd
\cr
&
        +{407693\over437400} \pi^2 +{621001\over72900}
            +O(r^2) \ ,}  \eqno(14) $$
%                  ???
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%F115=
$$ \eqalign { \amu[{\rm fig.2(d)}]=
&\left(
         -{2\over3} \pi^2 \LGd +\Z3 +{5\over9} \pi^2 -{31\over12}
\right)	 \lrd
+\left(
         -{2}\Ztd +{4\over3} \pi^2\Z3 \LGd
\right.\cr
&\left.
         -{10\over9} \pi^2\Z3
         -{11\over108} \pi^4 +{16\over3}\A4 +{4\over9} \pi^2 \LGdd
         +{2\over9} \LGdq +{47\over4}\Z3
\right.\cr
&\left.
         -{55\over18} \pi^2 \LGd
	 +{97\over36} \pi^2 -{1225\over144}
\right)  \lr
         +{11\over108} \pi^4\Z3
         -{16\over3}\A4\Z3
\cr
&
         -{4\over9} \pi^2\Z3 \LGdd
         -{2\over9}\Z3 \LGdq
         -{7}\Ztd +{10\over3} \pi^2\Z3 \LGd
         -{143\over24}\Z5
\cr
&
         +{16\over3}\A5
         -{167\over54} \pi^2\Z3 -{4\over27} \pi^2 \LGdt -{41\over540} \pi^4
\LGd
         -{2\over45} \LGdc
         -{1153\over12960} \pi^4
\cr
&
         +{110\over9}\A4 +{55\over54} \pi^2 \LGdd
         +{55\over108} \LGdq
         +{461\over24}\Z3
         -{367\over108} \pi^2 \LGd
\cr
&
	 +{1871\over648} \pi^2 -{3497\over432}
\cr
&
+\r\left[
 {101\over144} \pi^3\GC2 -{101\over144} \pi^4 \LGd +{707\over288} \pi^2\Z3
+{9035\over13824} \pi^4
\right.\cr
&
\phantom{+\r[}
\left.
-{821\over864} \pi^3 -{5081\over1296} \pi^2
\right]
            +O(r^2\lrt) \ ,}  \eqno(15) $$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% L4B(4-{th)-{V4u-{V2e-{   (W.33)
%f116= troppi eptalog
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
$$ \eqalign { \amu[{\rm fig.2(f)}]=
%F117=
&
\left(
-{1\over18} \pi^4
+{35\over8}\Z3
+{4\over3} \pi^2 \LGd
-{31\over54} \pi^2
-{227\over72}
\right) \lr
+{1\over9} \pi^4\Z3
\cr
&
-{35\over4}\Ztd
-{8\over3} \pi^2\Z3 \LGd
+{46\over27} \pi^2\Z3
-{1027\over8640} \pi^4
-{13\over3}\A4
\cr
&
-{35\over72} \pi^2 \LGdd
-{13\over72} \LGdq
+{923\over864}\Z3
+{62\over27} \pi^2 \LGd
+{163\over486} \pi^2
-{4243\over1296}
\cr&            +O(r^2\lr) \ .}  \eqno(16) $$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Here $\zeta(p)$ is the Riemann $\zeta$-function of argument $p$,
\d$ \zeta(p) \equiv \sum_{n=1}^{\infty} {1\over n^p} \ ,$
$a_k$ and $\beta_k$ are the constants defined as
\d$ a_k \equiv \sum_{n=1}^{\infty} {1\over {2^n n^k}}$ and
\d$\beta_k\equiv\sum_{n=0}^\oo {(-1)^n\over (2n+1)^k}$,
whose first values are respectively
$\Z2=\pi^2/6$,
$\Z3=1.202\;056\;903...$,
$\Z4=\pi^4/90$,
$\Z5=1.036\;927\;755...$
$\Z6=\pi^6/945$,
%\d$     a_1 = \LGd$,
%\d$     a_2 = {1\over 2} \Z2-{1\over 2}\LGdd $,
%\d$     a_3 = {7\over 8}\Z3 -{1\over 2}\Z2\LGd +{1\over 6}\LGdt$,
\d$     a_4 = 0.517\;479\;061... \ $,
\d$     a_5 = 0.508\;400\;579... \ $
and
\d$ \beta_1 = \pi/4$,
\d$ \beta_2 = 0.915\;965\;594...$,
\d$ \beta_3 = \pi^3/32$,
\d$ \beta_4 = 0.988\;944\;551...$ .

$\Qtw$ is the transcendentality-three constant defined as
$$ \Qtw\equiv
 \int^{\pi/2}_0 \ln^2\left( 2\cos\left( {\theta/2}\right)\right) d\theta
    = 0.550\;279\;883\;952...\  . \eqno(17) $$

The constants $\beta_2$, $\beta_4$ and $\Qtw$ come from the
values of the polylogarithmic functions
$$ \Li_2(i)   = -{1\over 8}   \zeta(2) + i\beta_2\ , $$
$$ \Li_4(i)   = -{7\over 320} \zeta(4) + i\beta_4\  $$
and
$$ S_{12}(i)  = \int_0^i  {{\ln^2(1-t)}\over 2t} dt
              = {29\over64}\Z3 -{1\over4}\pi\beta_2
               +i\left( {1\over 32}\Z2\pi-{1\over2}\Qtw \right) \ . $$

Eq.(8)-(10) and eq.(12)-(14)
contain also six different combinations $U_i$ of several integrals
with transcendentality six
containing products of logarithms, dilogarithms and  trilogarithms
which we were unable to calculate in analytical form;
an example of such integrals is \d$\int_0^1 {dx \over {x+1}}
\Li_3(x)\Li_2(-x)$.
We have calculated them by using numerical methods to high
precision:
$$ \eqalign{
 U_1=& \phantom{+0} 1.833\;055\;809\;327... \ ,\cr
 U_2=&          \phantom{} -  8.520\;012\;033\;995... \ ,\cr
 U_3=& \phantom{+0}  0.722\;470\;399\;216... \ ,\cr
 U_4=& \phantom{+} 15.183\;055\;006\;269... \ ,\cr
 U_5=& \phantom{+} 18.086\;098\;879\;723... \ ,\cr
 U_6=& \phantom{+} 42.269\;128\;989\;300... \ .\cr
} $$

We expect that these combinations of integrals
can be expressed in analytical form
using the transcendentality-six constants,
$\zeta(6)$, $\Ztd$, $\ln^6 2$, $a_6$, $\zeta(5)\LGd$, etc..
%%%%%%%%%%%%%%%%%%%%%%% footnote %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\footnote {$^1$}{
We found that the numerical value of $U_5$
coincides with that of \d${160\over9}\Z6$ for at least
the first 28 significant figures.},
%%%%%%%%%%%%%%%%%%%%%%% footnote %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
in analogy with similar integrals with transcendentality five.

The calculation of the contribution of the graph of ${\rm Fig.2(e)}$
involves many groups of integrals with transcendentality six and seven,
and we were unable to complete it in analytical form without
the knowledge of the explicit analytical value of these integrals.
The contribution of this graph will be given later
in numerical form.

The expansions in \d$(m_e/m_\mu)$ eqs.(1)-(16)
present some features already found in the expressions
of the three-loop and four-loop contributions
to the muon $g$-$2$ [1][2]:

1) The expression of the contributions of the graphs
containing only electron loops
(eq.(1), (5), (10), (12) and (15))
have a linear $(m_e/m_\mu)$ term
%%%%%%%%%%%%%%%%%%%%%%% footnote %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\footnote {$^2$}{
At present
only the analytical expression
of the contribution of the three-loop light-light graphs
it is known to contain a term proportional to
$(m_e/m_\mu)$ $\ln(m_\mu/m_e)$.}.
%%%%%%%%%%%%%%%%%%%%%%% footnote %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The coefficients of these terms are constant,
being numerically respectively -4.74, -18.00, -20.38, 3.20, -2.81.
As we expected, these numerical values are larger
than those found in four-loop calculations [2].
The $(m_e/m_\mu)$ expansions of the contributions of the other
graphs
containing also muon loops begin with the $(m_e/m_\mu)^2$ term.

2) The coefficients of the expansions
   in $(m_e/m_\mu)$
   of the contributions of the
   graphs containing only electron loops
   contain high powers of $\ln(m_\mu/m_e)$.
   As an example, in table 1 we have listed
   the numerical values
   and the maximum power of $\ln(m_\mu/m_e)$
   of the coefficients of the powers
   of \d$(m_e/m_\mu)$ of eq.(5).
   We found that the coefficients of the even powers
   have the maximum power of $\ln(m_\mu/m_e)$ equal or greater than
   the maximum power of the zero order coefficient;
   the coefficient of $(m_e/m_\mu)^4$
   contains the highest power of $\ln(m_\mu/m_e)$.
   The coefficients of the odd powers (greater than one)
   contain at most a $\ln(m_\mu/m_e)$ term.
   The same behaviour of the coefficients of the expansions was found
   even in all similar sixth- and eighth-order contributions [1][2];
   this observation could be useful in the analytical calculations
   of the contributions
   of other high-order graphs containing only electron loops
   in order to estimate the magnitude of the uncalculated
   terms of the expansions in $(m_e/m_\mu)$.

3) At the five-loop level
   new transcendental constants, $\beta_4$ and $\Qtw$,
   appear  in the
   coefficient of the $(m_e/m_\mu)$ term of the
   expression of the
   contribution of the graph of Fig.1(j),
   containing a double insertion of
   a fourth-order vacuum polarization on a second order vertex graph;
   in analogous way,
   the odd power $\pi^3$
   and
   the constant $\beta_2$ appeared
   at the three-loop level [1] and four-loop level [2] respectively.
   This fact remarks that
   the not leading terms of the contribution
   of the graph of Fig.1(j) cannot be worked out
   from the not leading terms of simpler graphs,
   whereas the leading ones can be easily deduced
   using the renormalization group equations [3].

We have compared our results with
the renormalization-group results of
ref.[3].
The leading terms of our eq.(1) and (5)
agree respectively with eq.(35) and (32) of ref.[3]
The logarithmic and constant terms of our eq.(10)
agree with eq.(25) of ref.[3],
except that our expression contains
the unknown transcendentality-six constant $U_3$, whereas
the expression of ref.[3] contains the analytical term $\Ztd/2$;
these terms are in perfect numerical agreement,
so that the analytical expression of $U_3$ can be inferred.

In table 2 we have listed the numerical values of eqs.(1)-(16)
obtained
using the experimental value [4] $ {(m_\mu/m_e)}=206.768 262(30)$
and taking into account the (not shown) terms up to
$(m_e/m_\mu)^4$.
The value of the contribution of the graph of Fig.2(e)
has been worked out evaluating numerically
 the one-dimensional integral obtained
using the dispersive representation for the vacuum polarization [2].

As expected, the graphs containing muon loops give contributions
substantially
smaller than those containing only electron loops.
The sum of the numerical values of the three graphs of fig.1
calculated using the renormalization group technique
in ref.[3] has a rather large value, about 52.
On the contrary the sum of all contributions of fig.1 and 2
shows a strong cancellation,
the numerical value being
$$ \amu[\fig{1+2}] =-1.261\;574(2) \ . \eqno(18)$$
This fact is due to the cancellation between
the positive contributions of the graphs of fig.1
containing vacuum polarization
insertion on the second-order vertex graph
and the negative contributions of
the graphs of fig.2 containing vacuum
polarization insertions on fourth-order vertex graphs.
We found a similar behaviour at the four-loop level,
even if in that case the cancellation was less marked [2].
Finally, eq.(18) turns out to be much smaller than the estimate
of the total tenth-order contribution [5]
$$ \amu            = 570(140) \ .  \eqno(19) $$
%which is dominated from the contributions of the graphs
%containing a double insertion of a second-order vacuum polarization
%on a light-light vertex.

\par
\vskip 6 truemm
All the algebraic manipulations were carried out through the symbolic
manipulation program ASHMEDAI [6].
\par
\vfill\eject
%\vskip 20 truemm
{\bf References}
\vskip 10 truemm

\item{[1]}  S.Laporta,  {\it Nuovo Cimento   A},{\bf 106}, (1993) 675.
\item{[2]}  S.Laporta,  {\it Phys.Lett.} {\bf B}312, (1993) 495.
\item{[3]}  A.L.Kataev, {\it Phys.Lett.} {\bf B}284, (1992) 401.
\item{[4]} E. R. Cohen and B. N. Taylor, {\it Rev. Mod. Phys.},
           {\bf 59}, (1987) 1121.
\item{[5]}  T.Kinoshita and W.B.Lindquist, \PR D {\bf 41}, (1990) 593;
\par
            T.Kinoshita and W.Marciano in {\it Quantum Electrodynamics},
            edited by T.Kinoshita \par
            (World Scientific, Singapore, 1990), p.419.
\parn
\item{[6]} M. J. Levine, U.S. AEC Report No. CAR-882-25 (1971)
                                                       (unpublished).
\parn
\vfill\eject
{\bf Figure captions}
\vskip 12 truemm
\parn
Fig.1: Tenth-order vertex graph
       obtained with insertions of second- and fourth-order
       vacuum polarization subdiagrams on the second-order vertex graph.
\parn
Fig.2: Examples of tenth-order vertex graph
       obtained with insertions of second- and fourth-order
       vacuum polarization subdiagrams on the fourth-order vertex graphs.
\parn
\vskip 12 truemm
\vfill\eject
\riga
TABLE 1: The numerical values and the maximum power of $\ln(m_\mu/m_e)$
         of the coefficients of the expansion
 in $\left({m_e/m_\mu}\right)$ of the contribution of the
graph of Fig.1(e).
{
\riga
\hskip 9 truemm
 {\it n}  \hskip 5truemm
 Coefficient of
    \hskip 10 truemm
 Maximum power
\parn
\hskip 9 truemm
\phantom {n}
\hskip 8truemm
 $(m_e/m_\mu)^n$
\hskip 15 truemm
 of $\ln(m_\mu/m_e)$
\riga
$$
 \eqalign {
&0 \qquad \cr
&1 \cr
&2 \cr
&3 \cr
&4 \cr
&5 \cr
&6 \cr
&7 \cr
&8 \cr
&9 \cr
&10 \cr
}
 \eqalign {
&27.71         \hskip 30 truemm  {} \cr
-&18.00   \cr
&2493.30   \cr
&154.49   \cr
-&34783.30 \cr
-&3.91    \cr
&452.06   \cr
&11.93    \cr
-&3859.59  \cr
&3.15     \cr
-&1346.48  \cr
}
 \eqalign {
&3 \hskip 86 truemm  {} \cr
&0 \cr
&4 \cr
&0 \cr
&5 \cr
&0 \cr
&4 \cr
&0 \cr
&4 \cr
&0 \cr
&4 \cr
}
 $$
\riga
}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vfill\eject
\riga
TABLE 2: The numerical values of the contributions to $\amu$
of the graphs of figs.1 and 2.
\riga
\hskip 7 truemm
 Figure  \hskip 14truemm
 Contribution to $\amu$
\riga
$$ \eqalign{
&{\rm 1(a)}       \hskip 19 truemm  {}\cr
&{\rm 1(b)} \cr
&{\rm 1(c)} \cr
&{\rm 1(d)} \cr
&{\rm 1(e)} \cr
&{\rm 1(f)} \cr
&{\rm 1(g)} \cr
&{\rm 1(h)} \cr
&{\rm 1(i)} \cr
&{\rm 1(j)} \cr
&{\rm 1(k)} \cr
&{\rm 2(a)} \cr
&{\rm 2(b)} \cr
&{\rm 2(c)} \cr
&{\rm 2(d)} \cr
&{\rm 2(e)} \cr
&{\rm 2(f)} \cr
&{\rm 1+2} \cr
}
 \eqalign{
& 20.142\;811   (3)         \hskip 92 truemm  {}  \cr
&  2.203\;327\;2 (2)      \cr
&  0.206\;959\;08(1)      \cr
&  0.013\;875\;908\;8(3) \cr
& 27.690\;059\;(3)       \cr
&  1.166\;152\;15(6)     \cr
&  0.031\;814\;813\;1(6) \cr
&  1.614\;350\;1(1)      \cr
&  0.164\;714\;747(4)    \cr
&  4.742\;149\;1 (2)     \cr
&  0.399\;245\;484 (8)   \cr
& -28.429\;744\;(2)      \cr
&  -6.792\;291\;9(3)     \cr
&  -0.952\;823\;49 (2)   \cr
& -19.042\;323\;(1)      \cr
&  -3.131\;386\;2(1)     \cr
&  -1.288\;464\;12(2)    \cr
& -1.261\;574(2)
} $$
\riga
\bye
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 @endspecial 1 1518 a Fb(\026)709 b(\026)f(\026)19 1359 y(e)120
1276 y(\026)332 1270 y(\026)769 1288 y(e)279 b(e)409 b(e)1820
1270 y(\026)173 1654 y Fa(\(i\))675 b(\(j\))f(\(k\))495 1477
y @beginspecial @setspecial
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 @endspecial 72 1684 a(\026)g(\026)g(\026)167 1442 y(e)444
1418 y(e)858 1542 y(e)911 1796 y Fa(\(d\))1005 1524 y Fb(e)1401
1507 y(e)1596 1489 y(e)495 1642 y @beginspecial @setspecial
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 @endspecial 72 2168 a(\026)g(\026)g(\026)167 1926 y(e)444
1890 y(\026)858 2026 y(e)911 2280 y Fa(\(e\))982 1997 y Fb(\026)1401
1991 y(e)1596 1961 y(\026)495 2127 y @beginspecial @setspecial
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 @endspecial 72 2652 a(\026)g(\026)g(\026)173 2404 y(\026)444
2387 y(e)858 2499 y(\026)911 2764 y Fa(\(f)5 b(\))1005 2493
y Fb(e)1407 2469 y(\026)1596 2457 y(e)495 2611 y @beginspecial
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 newpath
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 newpath
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 newpath
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 newpath
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 newpath
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 newpath
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 newpath
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 newpath
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 newpath
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 newpath
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 newpath
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 newpath
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 newpath
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 newpath
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 newpath
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 newpath
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 newpath
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 newpath
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 newpath
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 newpath
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 newpath
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 newpath
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 newpath
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 newpath
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 newpath
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 newpath
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 newpath
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 newpath
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 newpath
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 newpath
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 newpath
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 newpath
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 newpath
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 newpath
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 newpath
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 newpath
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 newpath
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 newpath
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 stroke
 newpath
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 stroke
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 stroke
 -099 172 78 0 360 arc
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 newpath
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 newpath
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 stroke
 /xa -200 def /ya 0 def /xb -200 def /yb -100 def /rad 20 def
 /len xb xa sub 2 exp yb ya sub 2 exp add sqrt def
 /nstep len rad 4 mul div 1 add round def
 /dx xb xa sub nstep 1 add div def /dy yb ya sub nstep 1 add div def
 /ph { pop
 /xt xa dy sub dx 2 div add def /yt ya dx add dy 2 div add def
 /xe xa dy add dx 2 div add def /ye ya dx sub dy 2 div add def
 /xa xa dx add def /ya ya dy add def
 xt yt xe ye xa ya curveto } def
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 newpath
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 stroke
 newpath
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 newpath
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 newpath
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 newpath
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 newpath
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 newpath
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 newpath
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 newpath
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 newpath
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 newpath
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 newpath
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 newpath
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 newpath
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 newpath
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 newpath
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 newpath
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 newpath
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 newpath
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 newpath
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 newpath
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 newpath
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 newpath
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 newpath
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 newpath
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 newpath
  531  196 moveto
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  525  190 moveto
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 newpath
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 newpath
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 newpath
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 newpath
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 newpath
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 newpath
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 newpath
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 newpath
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 newpath
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 newpath
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 newpath
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 newpath
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 newpath
  312  30 moveto
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 newpath
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 newpath
  300  21 moveto
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 newpath
  293  16 moveto
  290  11 lineto
 stroke
 newpath
  290  11 moveto
  293  5 lineto
 stroke
 newpath
  293  5 moveto
  300  0 lineto
 stroke
 newpath
 200 0 moveto
 800 0 lineto
 stroke
 450 87 26 0 360 arc
 stroke
 550 87 38 0 360 arc
 stroke
 newpath
 560 50 moveto
 540 124 lineto
 stroke
 newpath
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 513 077 lineto
 stroke
 /xa 500 def /ya 0 def /xb 500 def /yb -100 def /rad 20 def
 /len xb xa sub 2 exp yb ya sub 2 exp add sqrt def
 /nstep len rad 4 mul div 1 add round def
 /dx xb xa sub nstep 1 add div def /dy yb ya sub nstep 1 add div def
 /ph { pop
 /xt xa dy sub dx 2 div add def /yt ya dx add dy 2 div add def
 /xe xa dy add dx 2 div add def /ye ya dx sub dy 2 div add def
 /xa xa dx add def /ya ya dy add def
 xt yt xe ye xa ya curveto } def
 newpath xa ya moveto /xa xa dx 2 div add def /ya ya dy 2 div add def
 xa ya lineto 1 1 nstep { ph } for xb yb lineto stroke
 newpath
  1134  0 moveto
  1140  -3 lineto
 stroke
 newpath
  1140  -3 moveto
  1144  -7 lineto
 stroke
 newpath
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  1144  -11 lineto
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 newpath
  1144  -11 moveto
  1142  -15 lineto
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 newpath
  1142  -15 moveto
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 newpath
  1137  -20 moveto
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 newpath
  1132  -25 moveto
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 newpath
  1129  -30 moveto
  1129  -35 lineto
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 newpath
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 newpath
  1132  -38 moveto
  1138  -41 lineto
 stroke
 newpath
  1138  -41 moveto
  1144  -43 lineto
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 newpath
  1144  -43 moveto
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 newpath
  1151  -45 moveto
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 newpath
  1154  -48 moveto
  1155  -52 lineto
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 newpath
  1155  -52 moveto
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 newpath
  1153  -57 moveto
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 newpath
  1149  -64 moveto
  1147  -70 lineto
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 newpath
  1147  -70 moveto
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 newpath
  1147  -75 moveto
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 newpath
  1149  -79 moveto
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 newpath
  1155  -81 moveto
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 newpath
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 newpath
  1169  -80 moveto
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 newpath
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 newpath
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 newpath
  1178  -88 moveto
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 newpath
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 newpath
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 newpath
  1177  -108 moveto
  1179  -112 lineto
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 newpath
  1179  -112 moveto
  1184  -113 lineto
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 newpath
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 newpath
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 newpath
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 newpath
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 newpath
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 newpath
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 newpath
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 newpath
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 newpath
  1216  -129 moveto
  1219  -133 lineto
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 newpath
  1219  -133 moveto
  1223  -135 lineto
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 newpath
  1223  -135 moveto
  1228  -133 lineto
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 newpath
  1228  -133 moveto
  1234  -128 lineto
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 newpath
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  1239  -124 lineto
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 newpath
  1239  -124 moveto
  1244  -121 lineto
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 newpath
  1244  -121 moveto
  1248  -121 lineto
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 newpath
  1248  -121 moveto
  1251  -124 lineto
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 newpath
  1251  -124 moveto
  1255  -129 lineto
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 newpath
  1255  -129 moveto
  1258  -135 lineto
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 newpath
  1258  -135 moveto
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 newpath
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 newpath
  1267  -142 moveto
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 newpath
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 newpath
  1276  -135 moveto
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 newpath
  1279  -129 moveto
  1283  -124 lineto
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 newpath
  1283  -124 moveto
  1286  -121 lineto
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 newpath
  1286  -121 moveto
  1290  -121 lineto
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 newpath
  1290  -121 moveto
  1295  -124 lineto
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 newpath
  1295  -124 moveto
  1300  -128 lineto
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 newpath
  1300  -128 moveto
  1306  -133 lineto
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 newpath
  1306  -133 moveto
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 newpath
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 newpath
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 newpath
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 newpath
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 newpath
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 newpath
  1323  -110 moveto
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 newpath
  1326  -107 moveto
  1331  -107 lineto
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 newpath
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 newpath
  1337  -109 moveto
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 newpath
  1344  -112 moveto
  1350  -113 lineto
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 newpath
  1350  -113 moveto
  1355  -112 lineto
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 newpath
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 newpath
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 newpath
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 newpath
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 newpath
  1356  -88 moveto
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 newpath
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 newpath
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 newpath
  1365  -80 moveto
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 newpath
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 newpath
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 newpath
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 newpath
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 newpath
  1387  -70 moveto
  1385  -64 lineto
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 newpath
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  1381  -57 lineto
 stroke
 newpath
  1381  -57 moveto
  1379  -52 lineto
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 newpath
  1379  -52 moveto
  1380  -48 lineto
 stroke
 newpath
  1380  -48 moveto
  1383  -45 lineto
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 newpath
  1383  -45 moveto
  1390  -43 lineto
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 newpath
  1390  -43 moveto
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 newpath
  1396  -41 moveto
  1402  -38 lineto
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 newpath
  1402  -38 moveto
  1405  -35 lineto
 stroke
 newpath
  1405  -35 moveto
  1405  -30 lineto
 stroke
 newpath
  1405  -30 moveto
  1402  -25 lineto
 stroke
 newpath
  1402  -25 moveto
  1397  -20 lineto
 stroke
 newpath
  1397  -20 moveto
  1392  -15 lineto
 stroke
 newpath
  1392  -15 moveto
  1390  -11 lineto
 stroke
 newpath
  1390  -11 moveto
  1390  -7 lineto
 stroke
 newpath
  1390  -7 moveto
  1394  -3 lineto
 stroke
 newpath
  1394  -3 moveto
  1400  0 lineto
 stroke
 newpath
 900 0 moveto
 1500 0 lineto
 stroke
 1201 114 52 0 360 arc
 stroke
 1066 115 35 0 360 arc
 stroke
 newpath
 1213 66 moveto
 1186 164 lineto
 stroke
 newpath
 1249 128 moveto
 1150 102 lineto
 stroke
 /xa 1067 def /ya 0 def /xb 1067 def /yb -100 def /rad 20 def
 /len xb xa sub 2 exp yb ya sub 2 exp add sqrt def
 /nstep len rad 4 mul div 1 add round def
 /dx xb xa sub nstep 1 add div def /dy yb ya sub nstep 1 add div def
 /ph { pop
 /xt xa dy sub dx 2 div add def /yt ya dx add dy 2 div add def
 /xe xa dy add dx 2 div add def /ye ya dx sub dy 2 div add def
 /xa xa dx add def /ya ya dy add def
 xt yt xe ye xa ya curveto } def
 newpath xa ya moveto /xa xa dx 2 div add def /ya ya dy 2 div add def
 xa ya lineto 1 1 nstep { ph } for xb yb lineto stroke
 newpath
 125 0 moveto
 175 0 lineto
 stroke
 newpath
 150 -25 moveto
 150 +25 lineto
 stroke
 newpath
 825 0 moveto
 875 0 lineto
 stroke
 newpath
 850 -25 moveto
 850 +25 lineto
 stroke
%%EndDocument
 @endspecial 836 3125 a Fa(Figure)19 b(2)p eop
%%Trailer
end
userdict /end-hook known{end-hook}if
%%EOF
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%% End of the 2' postscript file: Figure #1      %%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

