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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 1 0 316 "restar
t:with(difforms):\ndefform(psi[1]=1,psi[2]=1,psi[3]=1,psi[4]=1,\npsi[5
]=1,rho[1]=1,rho[2]=1,rho[3]=1,rho[4]=1,rho[5]=1,\neta[1]=1,eta[2]=1,e
ta[3]=1,eta[4]=1,eta[5]=1,l[1]=0,l[2]=0,l[3]=0,l[4]=0,l5=0,\na[1]=0,a[
2]=0,a[3]=0,a[4]=0,a5=0,sigma=0,\np[3,1]=0,p[3,2]=0,\np[4,1]=0,p[4,2]=
0,p[5,1]=0,p[5,2]=0,mu=0,m1=0,m3=0):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 281 "This is a Maple file intended to be read in conjunction with \+
\"Exterior differential systems and billiards\", which appeared in \"G
eometry integrability and quantization\", Sophia 2006 -  a .pdf file o
f this article is available on arXiv and Landsberg's homepage www.math
.tamu.edu/~jml" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 194 "This file stud
ies the n=4 case, above I just declared certain variables to be differ
ential forms of degree 0 (functions)\nor 1 (1-forms). Next I introduce
 some shorthand for the signs and cosines" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 137 "s1:=sin(a[1]):s2:=sin(a[2]):s3:=sin(a[3]):s4:=sin(
a[4]):s5:=sin(a5):\nc1:=cos(a[1]):c2:=cos(a[2]):c3:=cos(a[3]):c4:=cos(
a[4]):c5:=cos(a5):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "  Next  rec
ord the derivatives of the various forms in" }}{PARA 0 "" 0 "" {TEXT 
-1 53 "terms of our 8 variables a[1],.,a[4], l[1],..,l[4].  " }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 956 "rhosubs:=\{rho[1]=(-c2*eta[
2]-c1*eta[1])/l[1],rho[2]=(-c3*eta[3]-c2*eta[2])/l[2],\nrho[3]=(-c4*et
a[4]-c3*eta[3])/l[3],rho[4]=(-c1*eta[1]-c4*eta[4])/l[4]\};\ndpsisubs:=
\{d(psi[1])=-(1/2)*(rho[1]+rho[4])&^eta[1],d(psi[2])=-(1/2)*(rho[2]+rh
o[1])&^eta[2],\nd(psi[3])=-(1/2)*(rho[3]+rho[2])&^eta[3],d(psi[4])=-(1
/2)*(rho[4]+rho[3])&^eta[4]\};\netasubs:=\{eta[3]=p[3,1]*eta[1]+p[3,2]
*eta[2],\neta[4]=p[4,1]*eta[1]+p[4,2]*eta[2]\};\ndl1:=s2*eta[2]-s1*eta
[1];dl2:=s3*eta[3]-s2*eta[2];dl3:=s4*eta[4]-s3*eta[3];\ndl4:=s1*eta[1]
-s4*eta[4];\ndlsubs:=\{d(l[1])= dl1 ,d(l[2])= dl2 ,d(l[3])= dl3 ,d(l[4
])= dl4 \};\nda1:=simpform(subs(rhosubs,(1/2)*(rho[1]-rho[4]))):\nda2:
=simpform(subs(rhosubs,(1/2)*(rho[2]-rho[1]))):\nda3:=simpform(subs(rh
osubs,(1/2)*(rho[3]-rho[2]))):\nda4:=simpform(subs(rhosubs,(1/2)*(rho[
4]-rho[3]))):\ndasubs:=\{d(a[1])=da1,d(a[2])=da2,d(a[3])=da3,d(a[4])=d
a4\};\ndetasubs:=\{d(eta[1])=0,d(eta[2])=0,d(eta[3])=0,d(eta[4])=0\};
\ndallsubs:=detasubs union dasubs union dlsubs:" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%(rhosubsG<&/&%$rhoG6#\"\"\"*&,&*&-%$cosG6#&%\"aG6#\"
\"#F*&%$etaGF3F*!\"\"*&-F/6#&F2F)F*&F6F)F*F7F*&%\"lGF)F7/&F(F3*&,&*&-F
/6#&F26#\"\"$F*&F6FGF*F7F-F7F*&F>F3F7/&F(FG*&,&*&-F/6#&F26#\"\"%F*&F6F
SF*F7FCF7F*&F>FGF7/&F(FS*&,&F8F7FOF7F*&F>FSF7" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#>%)dpsisubsG<&/-%\"dG6#&%$psiG6#\"\"\",&*&#F-\"\"#F--%#
&^G6$&%$rhoGF,&%$etaGF,F-!\"\"*&#F-F1F--F36$&F66#\"\"%F7F-F9/-F(6#&F+6
#F1,&*&#F-F1F--F36$&F6FE&F8FEF-F9*&#F-F1F--F36$F5FLF-F9/-F(6#&F+6#\"\"
$,&*&#F-F1F--F36$&F6FU&F8FUF-F9*&#F-F1F--F36$FKFgnF-F9/-F(6#&F+F?,&*&#
F-F1F--F36$F>&F8F?F-F9*&#F-F1F--F36$FfnFeoF-F9" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%(etasubsG<$/&%$etaG6#\"\"$,&*&&%\"pG6$F*\"\"\"F0&F(6#
F0F0F0*&&F.6$F*\"\"#F0&F(6#F6F0F0/&F(6#\"\"%,&*&&F.6$F<F0F0F1F0F0*&&F.
6$F<F6F0F7F0F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$dl1G,&*&-%$sinG6#
&%\"aG6#\"\"#\"\"\"&%$etaGF,F.F.*&-F(6#&F+6#F.F.&F0F5F.!\"\"" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%$dl2G,&*&-%$sinG6#&%\"aG6#\"\"$\"\"\"&%$et
aGF,F.F.*&-F(6#&F+6#\"\"#F.&F0F5F.!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%$dl3G,&*&-%$sinG6#&%\"aG6#\"\"%\"\"\"&%$etaGF,F.F.*&-F(6#&F+6#
\"\"$F.&F0F5F.!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$dl4G,&*&-%$s
inG6#&%\"aG6#\"\"\"F-&%$etaGF,F-F-*&-F(6#&F+6#\"\"%F-&F/F4F-!\"\"" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'dlsubsG<&/-%\"dG6#&%\"lG6#\"\"\",&*
&-%$sinG6#&%\"aG6#\"\"#F-&%$etaGF5F-F-*&-F16#&F4F,F-&F8F,F-!\"\"/-F(6#
&F+F5,&*&-F16#&F46#\"\"$F-&F8FHF-F-F/F>/-F(6#&F+FH,&*&-F16#&F46#\"\"%F
-&F8FTF-F-FDF>/-F(6#&F+FT,&F9F-FPF>" }}{PARA 12 "" 1 "" {XPPMATH 20 "6
#>%'dasubsG<&/-%\"dG6#&%\"aG6#\"\"\",(*&#F-\"\"#F-*,-%$cosGF)F-,&&%\"l
G6#\"\"%!\"\"&F7F,F-F-F;F:F6F:&%$etaGF,F-F-F-*&#F-F1F-*(-F46#&F+6#F1F-
&F=FDF-F;F:F-F:*&F0F-*(-F46#&F+F8F-&F=F8F-F6F:F-F-/-F(FB,(*&F0F-*(F3F-
F<F-F;F:F-F-*&#F-F1F-*,FAF-,&F;F-&F7FDF:F-FUF:F;F:FEF-F-F:*&#F-F1F-*(-
F46#&F+6#\"\"$F-&F=FfnF-FUF:F-F:/-F(FZ,(*&F0F-*(FAF-FEF-FUF:F-F-*&#F-F
1F-*,FYF-,&FUF-&F7FfnF:F-FboF:FUF:FhnF-F-F:*&#F-F1F-*(FHF-FKF-FboF:F-F
:/-F(FI,(*&#F-F1F-*(F3F-F<F-F6F:F-F:*&F0F-*(FYF-FhnF-FboF:F-F-*&#F-F1F
-*,FHF-,&FboF-F6F:F-F6F:FboF:FKF-F-F:" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%)detasubsG<&/-%\"dG6#&%$etaG6#\"\"\"\"\"!/-F(6#&F+6#\"\"#F./-F(6#
&F+6#\"\"$F./-F(6#&F+6#\"\"%F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
115 "Now I calculate the space of integral elements to our orignal sys
tem at a point, it turns out to be one dimensional" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 312 "simpform(subs(etasubs,subs(rhosubs, (subs(
dpsisubs,d(psi[1]))) )));eq1:=scalarpart(%);\nsimpform(subs(etasubs,su
bs(rhosubs, (subs(dpsisubs,d(psi[2]))) ))):eq2:=scalarpart(%);\nsimpfo
rm(subs(etasubs,subs(rhosubs, (subs(dpsisubs,d(psi[3]))) ))):eq3:=scal
arpart(%);\npsols:=solve(\{eq1,eq2,eq3\},\{p[3,1],p[4,2],p[4,1]\});" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"\"\"\"#F&**,&*&-%$cosG6#&%\"
aG6#F'F&&%\"lG6#\"\"%F&F&*(-F,6#&F/F3F&&%\"pG6$F4F'F&&F26#F&F&F&F&F<!
\"\"F1F>-%#&^G6$&%$etaGF0&FCF=F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%$eq1G,$*&#\"\"\"\"\"#F(*(,&*&-%$cosG6#&%\"aG6#F)F(&%\"lG6#\"\"%F(F
(*(-F.6#&F1F5F(&%\"pG6$F6F)F(&F46#F(F(F(F(F>!\"\"F3F@F(F(" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%$eq2G,$*&#\"\"\"\"\"#F(*(,&*(-%$cosG6#&%\"aG6
#\"\"$F(&%\"pG6$F3F(F(&%\"lG6#F(F(F(*&-F.6#&F1F9F(&F86#F)F(F(F(F>!\"\"
F7F@F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq3G,$*&#\"\"\"\"\"#F(*
(,(**-%$cosG6#&%\"aG6#\"\"%F(&%\"lG6#F)F(&%\"pG6$\"\"$F)F(&F86$F3F(F(!
\"\"**F-F(F4F(&F86$F:F(F(&F86$F3F)F(F(*(F?F(-F.6#&F1F6F(&F56#F:F(F(F(F
GF=F4F=F(F=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&psolsG<%/&%\"pG6$\"
\"$\"\"\",$**-%$cosG6#&%\"aG6#F+F+&%\"lG6#\"\"#F+-F/6#&F26#F*!\"\"&F5F
3F<F</&F(6$\"\"%F+*0F.F+-F/6#&F2F6F+,&*&F4F+&F56#FAF+F+*&&F5F;F+F=F+F<
F+-F/6#&F2FIF<&F(6$F*F7F<F8F<F=!\"#/&F(6$FAF7,$**FCF+FHF+FLF<F=F<F<" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 131 "Here is the prolonged ideal, or
 more precisely, the two new forms added to the prolonged ideal. I som
etimes call the forms that are" }}{PARA 0 "" 0 "" {TEXT -1 34 "zero ph
i3 and phi4 in what follows" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
102 "\nI2:=\{eta[3]=subs(psols,p[3,1]*eta[1]+p[3,2]*eta[2]),\neta[4]=s
ubs(psols,p[4,1]*eta[1]+p[4,2]*eta[2])\};" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#I2G<$/&%$etaG6#\"\"$,&*,-%$cosG6#&%\"aG6#\"\"\"F3&%
\"lG6#\"\"#F3-F.6#&F1F)!\"\"&F5F2F;&F(F2F3F;*&&%\"pG6$F*F7F3&F(F6F3F3/
&F(6#\"\"%,&*2F-F3-F.6#&F1F6F3,&*&F4F3&F5FEF3F3*&&F5F)F3F<F3F;F3-F.6#&
F1FEF;F?F;F8F;F<!\"#F=F3F3*,FIF3FNF3FQF;F<F;FBF3F;" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 105 "Now I calculate the space of integral elements o
f our prolonged system, I need to introduce new variables" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "defform(p[3,2,1]=0,p[3,2,2]=0):\ndp
subs:=\{d(p[3,2])=p[3,2,1]*eta[1]+p[3,2,2]*eta[2]\};" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>%'dpsubsG<#/-%\"dG6#&%\"pG6$\"\"$\"\"#,&*&&F+6%F-F.
\"\"\"F3&%$etaG6#F3F3F3*&&F+6%F-F.F.F3&F56#F.F3F3" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
133 "dphi3:=simpform( subs(dpsubs,subs(I2,subs(dallsubs,d(subs(I2, (et
a[3]))))))):\ndphi3:=simpform(dphi3);ephi3:=numer(scalarpart(dphi3));
" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&dphi3G,$*&#\"\"\"\"\"#F(*,,2*.F
)F(&%\"lG6#F)F(-%$sinG6#&%\"aG6#F(F(-%$cosG6#&F4F/F(-F76#&F46#\"\"$F(&
F.F=F(F(*0F>F(-F7F2F(F:F(&F.F5F(F?F(-F1F;F(&%\"pG6$F>F)F(F(*.F)F(FAF(F
:F(FBF(F?F(-F1F8F(!\"\"*,FAF(FCF(F6F(F?F(FBF(F(*.FAF(FCF(FDF(F:F(FBF(F
-F(FI*,FAF(FCF(F6F(F-F(&F.6#\"\"%F(F(*.F)F(FAF(F-F(FHF(F:F(F?F(FI*,F)F
(&FE6%F>F)F(F()F:F)F()FBF)F(F?F(F(F(F:!\"#FBFVF?FI-%#&^G6$&%$etaGF/&Fe
nF5F(F(FI" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&ephi3G,2*.\"\"#\"\"\"&
%\"lG6#F'F(-%$sinG6#&%\"aG6#F(F(-%$cosG6#&F0F+F(-F36#&F06#\"\"$F(&F*F9
F(!\"\"*0F:F(-F3F.F(F6F(&F*F1F(F;F(-F-F7F(&%\"pG6$F:F'F(F<*.F'F(F>F(F6
F(F?F(F;F(-F-F4F(F(*,F>F(F@F(F2F(F;F(F?F(F<*.F>F(F@F(FAF(F6F(F?F(F)F(F
(*,F>F(F@F(F2F(F)F(&F*6#\"\"%F(F<*.F'F(F>F(F)F(FEF(F6F(F;F(F(*,F'F(&FB
6%F:F'F(F()F6F'F()F?F'F(F;F(F<" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 143 "dphi4:=simpform( subs(dpsubs,subs(I2,subs(dallsubs,d(subs(I2,
 (eta[4]))))))):\ndphi4:=simpform(dphi4):ephi4:=simplify(numer(scalarp
art(dphi4)));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&ephi4G,L*6\"\"#\"
\"\"-%$cosG6#&%\"aG6#F'F(&%\"lG6#\"\"%F(-%$sinG6#&F-6#F(F(-F*6#&F-F1F(
)&%\"pG6$\"\"$F'F'F()-F*6#&F-6#F?F'F(&F0F7F(&F0F.F(&F0FDF(F(*4F/F(-F4F
+F(-F*F5F(F8F(F;F(F@F(FEF(FFF(FGF(!\"\"*0FJF()F)F'F(-F4FBF(F8F(F<F()FG
F'F()FEF'F(FK*0FJF(FMF(FNF(F8F(F<F()FFF'F()F/F'F(F(*0FJF(FIF(F8F(F;F(F
@F(FOF(FPF(FK*4F'F(FMF(F3F(F8F(F<F(FAF(FFF(FOF(FEF(FK*4F'F(FMF(F3F(F8F
(F<F(FAF(FRF(FGF(F/F(F(*6\"\"&F(FJF(FIF(F8F(F<F(FAF(FGF(FRF(F/F(F)F(FK
*6F?F(FJF(FIF(F8F(F<F(FAF(FOF(FEF(F)F(FFF(F(*2FJF(FIF(F8F(F<F(FAF(FOF(
FPF(F)F(FK*6F?F(FJF(F)F(FNF(F8F(F;F(FGF(FPF(FAF(FFF(F(*4FJF(F)F(FNF(F8
F(F;F(FRF(F/F(FAF(FEF(FK*2FJF(F)F(FNF(F8F(F;F(FOF(FPF(FAF(FK*6F'F(FJF(
FMF(-F4F9F(F<F(FAF(FFF(F/F(FGF(FEF(F(*6F'F(FJF(F)F(&F=6%F?F'F'F(F8F(FA
F(FEF(FRF(FGF(F/F(FK*4F'F(FJF(FMF(FinF(F<F(FAF(FFF(FOF(FEF(F(*4F'F(FJF
(F)F(FinF(F;F(F@F(FFF(FPF(FGF(FK*4F'F(F)F(F;F(F@F(FFF(FGF(F3F(F8F(FPF(
F(*4F'F(FJF(F)F(F[oF(F8F(FAF(FPF(FFF(FOF(F(*8F?F(FJF(F)F(FNF(F8F(F;F(F
FF(F/F(FAF(FEF(FGF(F(*6FJF(FIF(F8F(F<F(FAF(FGF(FFF(F/F(F)F(FEF(FK" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 148 "zephi4:=collect(simplify(su
bs(\{cos(a[1])=0,sin(a[1])=1\},ephi4)),\{p[3,2]\});zephi3:=collect(sim
plify(subs(\{cos(a[1])=0,sin(a[1])=1\},ephi3)),\{p[3,2]\});" }}{PARA 
12 "" 1 "" {XPPMATH 20 "6#>%'zephi4G,&*2\"\"#\"\"\"-%$cosG6#&%\"aG6#F'
F(-F*6#&F-6#\"\"%F(-F*6#&F-6#\"\"$F(&%\"lGF.F(&F:F7F(,&*(&F:F2F(F4F(&F
:6#F(F(!\"\"*&F4F()F?F'F(FAF()&%\"pG6$F8F'F'F(FA*2F'F(F)F(F/F(F4F(F9F(
F;F(,&*(F)F(F;F(F?F(F(*(F)F(F9F(F>F(FAF(FEF(FA" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%'zephi3G,$**\"\"#\"\"\"-%$cosG6#&%\"aG6#\"\"$F(&%\"lG
F.F(,&*&-F*6#&F-6#F'F(&F1F7F(F(*(&%\"pG6%F/F'F(F(F)F()&F16#F(F'F(F(F(!
\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 176 "ppsolns are the solution
s - there is a unique integral element at each point - note below if o
ne collects the terms the expressions\nfor these derivatives are not s
o complicated" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "zppsolns:=
simplify(solve(\{zephi3,zephi4\},\{p[3,2,1],p[3,2]\}));" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%)zppsolnsG<$/&%\"pG6$\"\"$\"\"#\"\"!/&F(6%F*F+
\"\"\",$**-%$cosG6#&%\"aG6#F+F0&%\"lGF8F0-F46#&F76#F*!\"\"&F:6#F0!\"#F
?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "solve(collect(zephi4/p
[3,2],p[3,2]),p[3,2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*,-%$cosG6
#&%\"aG6#\"\"#\"\"\",&*&&%\"lGF*F,&F06#\"\"%F,F,*&&F06#\"\"$F,&F06#F,F
,!\"\"F,-F&6#&F)F6F:F8F:,&F8F,F1F,F:F:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 " " }{TEXT -1 78 "thus the two values of p[3,2] over a \+
point where \\a_1=Pi/2 are 0 and the above" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 197 "ppsolns:=simplify(solve(\{ephi3,ephi4\},\{p[3,2,1]
,p[3,2,2]\}));\nEF:=collect( simplify(subs(ppsolns,p[3,2,1])) ,\{c1,s3
,c3,c2,s2,p[3,2]\});EE:=coeff(EF,p[3,2]);FF:=collect(EF-EE*p[3,2],\{c1
,s3,c3,c2,s2\});" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(ppsolnsG<$/&%\"
pG6%\"\"$\"\"#\"\"\",$*&#F,F+F,**&%\"lG6#F,!\"#&F26#F*!\"\",0*.F+F,&F2
6#F+F,-%$sinG6#&%\"aGF3F,-%$cosG6#&F@F;F,-FB6#&F@F6F,F5F,F,*0F*F,-FBF>
F,FEF,F1F,F5F,-F=FFF,&F(6$F*F+F,F,*.F+F,FIF,FEF,F1F,F5F,-F=FCF,F7*,FIF
,FJF,FAF,F5F,F1F,F,*.FIF,FJF,FKF,FEF,F1F,F:F,F7*,FIF,FJF,FAF,F:F,&F26#
\"\"%F,F,*.F+F,FIF,F:F,FNF,FEF,F5F,F7F,FEF4F,F7/&F(6%F*F+F+,$*&#F,F+F,
*6FKF,,H*6F+F,FAF,FRF,F<F,-FB6#&F@FSF,FKF,)FEF+F,F1F,F:F,F5F,F7*4FRF,F
NF,FIF,FinF,FKF,F\\oF,F1F,F:F,F5F,F,*.FIF,)FAF+F,FJF,FinF,)F5F+F,)F1F+
F,F,*.FIF,F_oF,FJF,FinF,)F:F+F,)FRF+F,F7*0FIF,FNF,FinF,FKF,F\\oF,F`oF,
FaoF,F,*2F+F,F_oF,F<F,FinF,FEF,F:F,F`oF,F1F,F,*2F+F,F_oF,F<F,FinF,FEF,
FcoF,F5F,FRF,F7*4\"\"&F,FIF,FNF,FinF,FEF,F5F,FcoF,FRF,FAF,F,*4F*F,FIF,
FNF,FinF,FEF,F`oF,F1F,FAF,F:F,F7*0FIF,FNF,FinF,FEF,F`oF,FaoF,FAF,F,*6F
*F,FIF,FAF,FJF,FinF,FKF,F5F,FaoF,FEF,F:F,F7*4FIF,FAF,FJF,FinF,FKF,FcoF
,FRF,FEF,F1F,F,*2FIF,FAF,FJF,FinF,FKF,F`oF,FaoF,FEF,F,*4F+F,FIF,F_oF,-
F=FjnF,FEF,F:F,FRF,F5F,F1F,F7*2F+F,FIF,F_oF,F`pF,FEF,F:F,F`oF,F1F,F7*4
F+F,FIF,FAF,F`pF,FKF,F\\oF,F:F,FaoF,F5F,F,*4F+F,FAF,FKF,F\\oF,F:F,F5F,
F<F,FinF,FaoF,F7*8F*F,FIF,FAF,FJF,FinF,FKF,F:F,FRF,FEF,F1F,F5F,F7*4FIF
,FNF,FinF,FEF,F5F,F:F,FRF,FAF,F1F,F,F,F:F7F5F7F1F7FinF7FEF7FAF7FIF7,&*
&F:F,FRF,F7*&F5F,F1F,F,F7F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#EF
G,(*&#\"\"\"\"\"#F(*0&%\"lG6#F(!\"#&F,6#\"\"$!\"\",&*&F+F(&F,6#F)F(F2*
(F1F(F/F(F+F(F(F(-%$sinG6#&%\"aGF0F(-%$cosGF:F2-F>6#&F<F-F(&%\"pG6$F1F
)F(F(F2*&,&*&#F(F)F(*.F+F.F/F2,&*&F5F(&F,6#\"\"%F(F(*&F/F(F+F(F(F(F8F(
F=F.-F>6#&F<F6F(F(F2*&#F(F)F(*,F+F.F/F2,&*(F)F(F/F(F+F(F2*(F)F(F5F(F/F
(F2F(-F9FQF(F=F2F(F2F(F?F(F(*,F5F(F=F2F+F.-F9F@F(FPF(F2" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%#EEG,$*&#\"\"\"\"\"#F(*.&%\"lG6#F(!\"#&F,6#\"
\"$!\"\",&*&F+F(&F,6#F)F(F2*(F1F(F/F(F+F(F(F(-%$sinG6#&%\"aGF0F(-%$cos
GF:F2-F>6#&F<F-F(F(F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#FFG,&*&,&*
&#\"\"\"\"\"#F**.&%\"lG6#F*!\"#&F.6#\"\"$!\"\",&*&&F.6#F+F*&F.6#\"\"%F
*F**&F1F*F-F*F*F*-%$sinG6#&%\"aGF2F*-%$cosGF?F0-FC6#&FAF8F*F*F4*&#F*F+
F**,F-F0F1F4,&*(F+F*F1F*F-F*F4*(F+F*F7F*F1F*F4F*-F>FEF*FBF4F*F4F*-FC6#
&FAF/F*F**,F7F*FBF4F-F0-F>FOF*FDF*F4" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 114 "GH:=collect(simplify(subs(ppsolns,p[3,2,2])),\{c1,s3
,c3,c2,s2,p[3,2]\});GG:=coeff(GH,p[3,2]^2);HH:=coeff(GH,p[3,2]);" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#GHG,&*&,**,-%$sinG6#&%\"aG6#\"\"%\"
\"\"&%\"lG6#F0F0-%$cosGF+!\"\",&*&&F26#\"\"#F0&F2F.F0F6*&&F26#\"\"$F0F
1F0F0F6-F56#&F-F?F0F0*&#F0F;F0*0,***F4F0)F9F;F0F<F0F1F0F0*(F4F0)F>F;F0
)F1F;F0F0*,F@F0F4F0F>F0FLF0F9F0F6*.F@F0F4F0F9F0F<F0F1F0F>F0F6F0F9F6F>F
6F1F6F4F6F7F6-F*FBF0F0F0*&FEF0*4,&*,F4F0F9F0F<F0F1F0F>F0F0FJF0F0F9F6F>
F6F1F6F4F6F7F6-F*6#&F-F:F0FAF0-F5FUF6F0F0*&FEF0*2,&*0F;F0F<F0-F*6#&F-F
3F0F4F0F1F0F9F0F>F0F6*.F;F0F9F0F>F0FfnF0F4F0FLF0F6F0F9F6F>F6F1F6F4F6F7
F6FAF0-F5FgnF6F0F0F0)&%\"pG6$F@F;F;F0F0*&,(*&,&*&FEF0*.,&*.F;F0F)F0F9F
0F<F0F>F0F1F0F6*,F;F0F)F0F9F0FKF0F1F0F6F0F9F6F>F6F1F6F4F6F7F6F0F0*&FEF
0*2,&*(F4F0FIF0)F<F;F0F6FJF0F0F9F6F>F6F1F6F4F6F7F6FOF0FAF6F0F0F0FWF0F0
*&FEF0*0,*FJF0*,\"\"&F0F4F0F>F0FIF0F<F0F0*,F@F0F4F0FKF0F1F0F9F0F6FSF0F
0F9F6F>F6F1F6F4F6F7F6FTF0F0F0*&FEF0*2,&*.F;F0FfnF0F4F0F9F0FKF0F1F0F0*.
F;F0FfnF0F4F0FIF0F>F0F<F0F6F0F9F6F>F6F1F6F4F6F7F6FWF0FjnF6F0F0F0F\\oF0
F0" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#GGG,**,-%$sinG6#&%\"aG6#\"\"%
\"\"\"&%\"lG6#F.F.-%$cosGF)!\"\",&*&&F06#\"\"#F.&F0F,F.F4*&&F06#\"\"$F
.F/F.F.F4-F36#&F+F=F.F.*&#F.F9F.*0,***F2F.)F7F9F.F:F.F/F.F.*(F2F.)F<F9
F.)F/F9F.F.*,F>F.F2F.F<F.FJF.F7F.F4*.F>F.F2F.F7F.F:F.F/F.F<F.F4F.F7F4F
<F4F/F4F2F4F5F4-F(F@F.F.F.*&FCF.*4,&*,F2F.F7F.F:F.F/F.F<F.F.FHF.F.F7F4
F<F4F/F4F2F4F5F4-F(6#&F+F8F.F?F.-F3FSF4F.F.*&FCF.*2,&*0F9F.F:F.-F(6#&F
+F1F.F2F.F/F.F7F.F<F.F4*.F9F.F7F.F<F.FZF.F2F.FJF.F4F.F7F4F<F4F/F4F2F4F
5F4F?F.-F3FenF4F.F." }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#HHG,(*&,&*&#
\"\"\"\"\"#F**.,&*.F+F*-%$sinG6#&%\"aG6#\"\"%F*&%\"lG6#F+F*&F7F4F*&F76
#\"\"$F*&F76#F*F*!\"\"*,F+F*F/F*F6F*)F:F+F*F=F*F?F*F6F?F:F?F=F?-%$cosG
F1F?,&*&F6F*F9F*F?*&F:F*F=F*F*F?F*F**&F)F**2,&*(FBF*)F6F+F*)F9F+F*F?*(
FBF*FAF*)F=F+F*F*F*F6F?F:F?F=F?FBF?FDF?-F06#&F3F;F*-FCFPF?F*F*F*-FC6#&
F3F8F*F**&F)F**0,*FMF**,\"\"&F*FBF*F:F*FKF*F9F*F**,F<F*FBF*FAF*F=F*F6F
*F?*,FBF*F6F*F9F*F=F*F:F*F*F*F6F?F:F?F=F?FBF?FDF?-F0FTF*F*F**&F)F**2,&
*.F+F*-F06#&F3F>F*FBF*F6F*FAF*F=F*F**.F+F*F\\oF*FBF*FKF*F:F*F9F*F?F*F6
F?F:F?F=F?FBF?FDF?FSF*-FCF]oF?F*F*" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 34 "GG:=simplify(GG):HH:=simplify(HH):" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 258 "Note that p32 appears linearly in the equation
 for p[3,2,1] and quadratically in the eqn for p[3,2,2] but there is n
o deg 0 term in the\np[3,2,2] eqn. Next I calculate 0=d^2(p[3,2])= d(p
[3,2,1]eta[1] + p[3,2,2]eta[2]) to obtain a new scalar equation involv
ing" }}{PARA 0 "" 0 "" {TEXT -1 119 "our base variables and p[3,2] whi
ch is quadratic in p[3,2]. Thus over any point in the original manifol
d, there are two" }}{PARA 0 "" 0 "" {TEXT -1 112 "integral elements. A
fterwards, letting  xp[3,2] denote the solved for p[3,2], we need to s
ee if we actually have" }}{PARA 0 "" 0 "" {TEXT -1 44 "d(xp[3,2])= p[3
,2,1]eta[1] + p[3,2,2]eta[2]." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 181 "dmueqn:=simpform( subs(dpsubs,subs(I2,subs(dallsubs,d(subs( pps
olns,p[3,2,1]*eta[1]+p[3,2,2]*eta[2])))))):\ndmueqn:=simpform(dmueqn):
ddmueqn:=subs(ppsolns,numer(scalarpart(dmueqn))):" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 126 "since the equation is complicated, I look at each
 coeffiecient. I write the equations in two different ways - collect w
rt the " }}{PARA 0 "" 0 "" {TEXT -1 54 "sines and cosines, and collect
ed wrt the side lengths." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
210 "dmueqn2:=simplify(coeff(ddmueqn,p[3,2]^2)):\n\ndmueqn1:=simplify(
coeff(ddmueqn,p[3,2])):\n\ndmueqn0:=simplify(ddmueqn-dmueqn2*p[3,2]^2-
dmueqn1*p[3,2]):\n\nsimplify(ddmueqn- dmueqn2*p[3,2]^2-dmueqn1*p[3,2]-
dmueqn0);\n\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "collect(dmueqn2,\{c1,c2,c3,c4,s1,s2
,s3,s4\});  " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,(*&,(*&,(*,\"\"#\"\"
\",&*(&%\"lG6#F*F*)&F.6#F)\"\"$F*)&F.6#\"\"%F)F*!\"\"*()F-F3F*)&F.6#F3
F)F*F1F*F*F*F<F*)F-F)F*)-%$cosG6#&%\"aGF=F7F*F**2F)F*,&*,F)F*F>F*F;F*F
5F*F1F*F**,F)F*F>F*F<F*F5F*)F1F)F*F*F*F<F*F>F*-%$sinG6#&FDF6F*-FKFBF*-
FAFLF*)F@F3F*F**.F)F*,.F,F8*,F)F*F>F*F;F*F5F*F1F*F8*(F>F*)F<F3F*F5F*F8
**F>F*F<F*F5F*FIF*F8**F-F*F;F*F4F*F1F*F8*,F)F*F-F*F<F*F4F*FIF*F8F*F<F*
F>F*)FOF)F*)F@F)F*F*F*)-FA6#&FDF2F)F*F**4F)F*,&*,F)F*F-F*F<F*F4F*FIF*F
**,F)F*F-F*F;F*F4F*F1F*F*F*F<F*F>F*FNF*-FKFgnF*FYF*FPF*FfnF*F**.F)F*,&
FTF8*(F<F*)F5F3F*FIF*F*F*F<F*F>F*FYF*F?F*F*F*)-FA6#&FDF/F)F*F**&,&*4F)
F*,&*,F)F*F>F*F<F*F5F*FIF*F8*,F)F*F-F*F<F*F4F*FIF*F8F*F<F*F>F*FJF*-FKF
doF*FOF*F?F*FenF*F**4F)F*,&*,F)F*F-F*F;F*F4F*F1F*F8*,F)F*F>F*F;F*F5F*F
1F*F8F*F<F*F>F*F]oF*F\\pF*FYF*F?F*FfnF*F*F*FcoF*F**0F)F*,.*,F)F*F-F*F<
F*F4F*FIF*F*FWF*F9F**,F)F*F>F*F;F*F5F*F1F*F*F`oF*FVF*F*F<F*F>F*FYF*F?F
*FenF*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "collect(dmueqn1
,\{c1,c2,c3,c4,s1,s2,s3,s4\});" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,(*&
,(*&,(*&,&*,\"\"%\"\"\",2*,\"\"$F,)&%\"lG6#F,\"\"#F,&F26#F/F,)&F26#F+F
4F,)&F26#F4F4F,!\"\"*,F/F,F1F,F7F,)F5F4F,F:F,F=*,F/F,F0F,F?F,F:F,F8F,F
,*,F/F,F0F,)F5F/F,F8F,F;F,F,*,F/F,F1F,F5F,)F;F/F,F7F,F=*,F/F,F0F,F?F,F
7F,F;F,F,*,F/F,F1F,F5F,)F8F/F,F:F,F=*,F/F,)F1F/F,F?F,F;F,F8F,F,F,F5F,F
1F,)-%$cosG6#&%\"aGF9F4F,F,**F+F,,**(F1F,FDF,FGF,F,**FIF,F?F,F;F,F8F,F
=**F1F,F5F,FDF,F7F,F,*(FIF,FBF,F;F,F=F,F5F,F1F,F,F,)-FL6#&FOF6F/F,F,*2
F+F,,2*,F+F,F0F,F5F,F7F,F:F,F=**F0F,F?F,F7F,F;F,F,*,F/F,FIF,F?F,F;F,F8
F,F,**F0F,F?F,F:F,F8F,F,*,F4F,F1F,F5F,FGF,F:F,F=**F0F,FBF,F8F,F;F,F,*,
F4F,F1F,F5F,FDF,F7F,F=*,F4F,F1F,F7F,F?F,F:F,F=F,F5F,F1F,-%$sinGFMF,-F_
oFXF,FKF,)FWF4F,F,*.F+F,,***F1F,F5F,FGF,F:F,F,FRF,FSF=*(FIF,FBF,F8F,F=
F,F5F,F1F,FJF,FWF,F,F,)-FL6#&FOF<F/F,F,*&,&*2F+F,,2*,F4F,F1F,F5F,FDF,F
7F,F,*,F4F,F0F,F?F,F:F,F8F,F=*,F4F,F0F,FBF,F8F,F;F,F=*,F4F,FIF,F?F,F;F
,F8F,F=*,F4F,F1F,F5F,FGF,F:F,F,*,F4F,F0F,F5F,F7F,F:F,F,*,F4F,F0F,F?F,F
7F,F;F,F=*,F4F,F1F,F7F,F?F,F:F,F,F,F5F,F1F,F^oF,-F_oFhoF,FKF,FVF,F,*2F
+F,,2*,F/F,F1F,F5F,FGF,F:F,F=FTF=*,F4F,F0F,F?F,F:F,F8F,F,**F1F,F7F,F?F
,F:F,F=*,F4F,FIF,F?F,F;F,F8F,F,**F0F,F5F,F7F,F:F,F=*,F+F,F0F,F?F,F7F,F
;F,F,*,F4F,F0F,FBF,F8F,F;F,F,F,F5F,F1F,F`oF,FfpF,FJF,FaoF,F,F,)FgoF4F,
F,*0F+F,,*F[oF=FeoF=*(F5F,FGF,FDF,F,FdoF,F,F5F,F1F,FJF,FVF,FgoF,F,F,)-
FL6#&FOF3F4F,F,*&,&*&,&*2F+F,,2*,F/F,F1F,F5F,FDF,F7F,F=FdoF=F[qF=F]qF=
*,F4F,FIF,F?F,F;F,F8F,F,*,F4F,F0F,F?F,F7F,F;F,F,*,F+F,F0F,F?F,F:F,F8F,
F,*,F4F,F0F,FBF,F8F,F;F,F,F,F5F,F1F,F^oF,-F_oFfqF,FKF,FVF,F,*2F+F,F]pF
,F5F,F1F,F`oF,FcrF,FJF,FaoF,F,F,FfoF,F,*4F+F,,2FSF,FgnF,*,F4F,F1F,F5F,
FGF,F:F,F=*,F4F,F0F,F5F,F7F,F:F,F=*,F+F,F1F,F7F,F?F,F:F,F=*,F4F,F1F,F5
F,FDF,F7F,F=FinF,*,F/F,F0F,FBF,F8F,F;F,F,F,F5F,F1F,FfpF,FcrF,FJF,FVF,F
`qF,F,F,FeqF,F,*0F+F,,*F[oF=FcqF,FTF,FUF=F,F5F,F1F,FJF,FVF,FfoF,F," }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "collect(dmueqn0,\{c1,c2,c3,
c4,s1,s2,s3,s4\});" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,(*&,(*&,(**\"\"
#\"\"\",.**&%\"lG6#F*F*)&F.6#\"\"$F)F*)&F.6#F)F3F*)&F.6#\"\"%F)F*!\"\"
*()F-F3F*)F1F:F*F5F*F**(F-F*F4F*)F8F:F*F;*,F)F*F-F*F1F*F4F*)F8F3F*F;*,
F)F*F=F*)F1F3F*F5F*F8F*F***F=F*F0F*F5F*F7F*F*F*F1F*)-%$cosG6#&%\"aGF2F
)F*F**0F)F*,**,F)F*)F-F)F*F1F*)F5F)F*FBF*F**,F)F*F=F*F0F*F5F*F7F*F;*,F
)F*FOF*F0F*FPF*F7F*F**,F)F*F=F*FDF*F5F*F8F*F;F*F1F*-%$sinG6#&FKF9F*-FU
FIF*-FHFVF*FGF*F***F)F*,**()F-F:F*FDF*F8F*F;F?F;**FOF*F1F*FPF*FBF*F*FE
F*F*F1F*)FYF)F*F*F*)-FH6#&FKF6F:F*F**2F)F*,**,F)F*FOF*F0F*FPF*F7F*F**,
F)F*F-F*F1F*F4F*FBF*F;*,F)F*F=F*F0F*F5F*F7F*F**,F)F*FOF*F1F*FPF*FBF*F;
F*F1F*FXF*-FUF\\oF*FinF*FGF*)F[oF3F*F**.F)F*,.**FOF*FDF*F8F*FPF*F;*,F)
F*F=F*FDF*F5F*F8F*F;*,F)F*F-F*F1F*F4F*FBF*F**(F1F*FBF*)F5F:F*F*FfnF;Fh
nF*F*F1F*FinF*FFF*)F[oF)F*F*F*)-FH6#&FKF/F)F*F**&,&*2F)F*,**,F)F*F-F*F
0F*F4F*F7F*F**,F)F*FOF*F0F*FPF*F7F*F;*,F)F*FOF*FDF*F8F*FPF*F;*,F)F*F-F
*F1F*F4F*FBF*F*F*F1F*FTF*-FUF`pF*FYF*FFF*FjnF*F**2F)F*,**,F)F*F-F*F0F*
F4F*F7F*F;*,F)F*F=F*FDF*F5F*F8F*F**,F)F*FOF*FDF*F8F*FPF*F**,F)F*FOF*F0
F*FPF*F7F*F;F*F1F*FdoF*FjpF*FinF*FFF*FeoF*F*F*F_pF*F**.F)F*,*F<F*F,F;F
[pF*FhoF;F*F1F*FinF*FFF*FjnF*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 421 "ddmueqn0:=simpform(simpform(subs(I2,subs(dallsubs,d(dmueqn0))
))):\nddmueqn1:=simpform(simpform(subs(I2,subs(dallsubs,d(dmueqn1)))))
:\nddmueqn2:=simpform(simpform(subs(I2,subs(dallsubs,d(dmueqn2))))):yy
y;\np32sub:=(-dmueqn1+DDs)/(4*dmueqn2):\nddmueqn0:=simpform(simpform(s
ubs(p[3,2]=p32sub,ddmueqn0))):\nddmueqn1:=simpform(simpform(subs(p[3,2
]=p32sub,ddmueqn1))):\nddmueqn2:=simpform(simpform(subs(p[3,2]=p32sub,
ddmueqn2))):yyz;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%$yyyG" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 213 "ddmueqn01:=collect(coeff(ddmueqn0,
eta[1]),\{c1,c2,c3,c4,s1,s2,s3,s4\}):\nddmueqn11:=collect(coeff(ddmueq
n1,eta[1]),\{c1,c2,c3,c4,s1,s2,s3,s4\}):\nddmueqn21:=collect(coeff(ddm
ueqn2,eta[1]),\{c1,c2,c3,c4,s1,s2,s3,s4\}):zzz;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%$zzzG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 148 "d
peta1coeff:= -(1/2)*dmueqn2*ddmueqn11+(-dmueqn1/2+DDs/2)*ddmueqn21+(dm
ueqn2*(2*dmueqn1*ddmueqn11-4*ddmueqn21*dmueqn0-4*dmueqn2*ddmueqn01))/(
4*DDs):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "dpeta1coeffnDDS:
= -(1/2)*dmueqn2*ddmueqn11+(-dmueqn1/2 )*ddmueqn21:" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 61 "dpeta1coeff:=collect(dpeta1coeff,\{c1,c2,
c3,c4,s1,s2,s3,s4\}):\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 114 
"dpeta1coeffb:=simplify(EE*(-dmueqn1+DDs)/(2*dmueqn2)+FF):dpeta1coeffb
nDDs:=simplify(EE*(-dmueqn1)/(2*dmueqn2)+FF):" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 53 "%collect(dpeta1coeffb,\{c1,c2,c3,c4,s1,s2,s3,s4,
DDs\}):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "firsteqn:=collec
t((dpeta1coeff-dpeta1coeffb),\{c1,c2,c3,c4,s1,s2,s3,s4,DDs\}):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "feqnDDs:=collect(coeff(first
eqn,DDs),\{c1,c2,c3,c4,s1,s2,s3,s4\}):" }}{PARA 8 "" 1 "" {TEXT -1 31 
"Error, unable to compute coeff\n" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 78 "feqnmDDs:=collect(dpeta1coeffnDDS-dpeta1coeffbnDDs,\{
c1,c2,c3,c4,s1,s2,s3,s4\});" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%)feqn
mDDsG,(*,,R*.&%\"lG6#\"\"\"F,)-%$cosG6#&%\"aGF+\"\"#F,)-F/6#&F26#\"\"$
F3F,)&F*6#F3F9F,)-F/6#&F2F<F3F,)&F*6#\"\"%F3F,!\"\"*.)F)F3F,)&F*F8F9F,
FBF,F-F,)-F/6#&F2FCF3F,F4F,FE*.)F)F9F,)FIF3F,F4F,F;F,FJF,F=F,F,*2F3F,F
GF,FPF,F4F,F;F,FBF,F=F,FJF,F,*2F3F,F)F,FIF,)F;F3F,FAF,F4F,F=F,FJF,F,*.
FIF,)FBF9F,F-F,FJF,F4F,FSF,F,*.FOF,FPF,F-F,F4F,F;F,F=F,F,*0FGF,FIF,FSF
,FBF,F4F,FJF,F=F,F,*8F3F,FGF,FPF,F-F,-%$sinGF6F,F5F,FKF,FBF,F=F,-FZFLF
,F;F,F,*0F)F,FPF,F-F,FJF,FAF,F=F,F;F,FE*2F3F,F)F,FIF,F-F,FJF,FAF,F=F,F
SF,FE*2F3F,FGF,FPF,F-F,FJF,FBF,F=F,F;F,FE*.FGF,FHF,F-F,FJF,FBF,F=F,FE*
0FGF,FIF,FJF,F-F,FBF,F=F,FSF,FE*.F)F,F:F,FAF,FJF,F-F,F=F,FE*8F3F,FGF,F
PF,F4F,F;F,FBF,F>F,F.F,-FZF?F,FJF,-FZF0F,FE*8F3F,FGF,FIF,F4F,FSF,FBF,F
=F,F.F,FenF,F^oF,FKF,FE*8F3F,FGF,FIF,FKF,F-F,FBF,F=F,FYF,F5F,FSF,FenF,
F,*8F3F,F)F,FPF,F-F,F>F,FYF,FJF,FAF,F5F,F]oF,F;F,F,*8F3F,F)F,FPF,F>F,F
AF,F^oF,FJF,F4F,F;F,F]oF,F.F,FE*8F3F,F)F,FIF,F=F,FAF,F^oF,FKF,F4F,FSF,
F.F,FenF,FE*8F3F,F)F,FIF,F-F,FYF,FSF,FJF,F>F,FAF,F]oF,F5F,F,*0F)F,FPF,
F4F,F;F,FJF,F=F,FAF,F,*.FIF,F4F,FSF,FJF,F=F,FUF,F,F,F4F,FIF,FGF,,&*(F3
F,,ddm*6\"#BF,F=F,FHF,)F.F9F,)F5F9F,FJF,FGF,FAF,FSF,F]oF,FE*:FDF,F=F,F
\\pF,FAF,FKF,F4F,FPF,FOF,FYF,FSF,F]oF,FenF,FE*8FDF,F4F,FHF,F\\pF,FUF,F
JF,F>F,FYF,FSF,)F]oF3F,F)F,F,*4\"\"'F,F>F,)FIFDF,F-F,F]pF,FJF,FOF,FAF,
F^oF,F,*8\"\")F,F]pF,FHF,F\\pF,FGF,FUF,FKF,F>F,FenF,F;F,F`pF,F,*8FDF,)
F>F9F,F\\pF,)FBFDF,FKF,)FYF3F,FenF,F:F,F5F,FIF,F)F,FE*8FDF,F]pF,FHF,F-
F,FGF,FAF,FJF,F>F,FSF,F`pF,F^oF,FE*6FbpF,FHF,F\\pF,FOF,FAF,FJF,F=F,F;F
,F]oF,F5F,F,*:FepF,FgpF,FPF,F-F,F4F,FKF,FGF,FYF,FAF,F^oF,F:F,FenF,F,*4
FgpF,F\\pF,F4F,FGF,FPF,FSF,FUF,FYF,)FenF3F,FE*6\"\"(F,FgpF,FcpF,F\\pF,
F]pF,FKF,FGF,F;F,FAF,FenF,FE*6F9F,F=F,F\\pF,FUF,FJF,F]pF,)F;FDF,F]oF,F
IF,F)F,FE*6F9F,FgpF,FPF,F-F,F]pF,FJF,FUF,F^oF,F:F,F)F,F,*2F3F,F]pF,FPF
,F\\pF,FhpF,FJF,F]oF,F:F,FE*8FDF,F]pF,FHF,F-F,FOF,FAF,FJF,F>F,F;F,F`pF
,F^oF,FE*6F3F,FgpF,F\\pF,FKF,FGF,FUF,F:F,FenF,F5F,FIF,F,*8\"\"*F,FgpF,
FPF,F.F,F]pF,FKF,FGF,F:F,FAF,FenF,)F^oF3F,FE*8FepF,FgpF,F]pF,FHF,FiqF,
FOF,FAF,FKF,F;F,F.F,FenF,FE*4F3F,FgpF,F.F,F]pF,FKF,)F)FDF,FcpF,F;F,Fen
F,FE*:F3F,F=F,FHF,F-F,F]pF,FKF,FUF,F^oF,FSF,F]oF,FenF,F)F,F,*8\"#7F,Fg
pF,F-F,FUF,FJF,FPF,FGF,F^oF,FSF,FipF,F5F,FE*8\"#5F,FgpF,FHF,F.F,F4F,FJ
F,FGF,FUF,FiqF,F;F,FYF,F,*6\"\"&F,F>F,F\\pF,FAF,FJF,F4F,FHF,FOF,FYF,F;
F,FE*4FDF,FgpF,FPF,FAF,F]pF,FGF,F:F,FJF,F^oF,F,*0FgpF,F\\pF,FhpF,F4F,F
bqF,FYF,F)F,FE*4F3F,FgpF,F-F,F]pF,F:F,FhpF,F^oF,FIF,F)F,FE*4F3F,FgpF,F
]pF,FPF,F^oF,FUF,FJF,F:F,F)F,F,*6FhqF,FgpF,FHF,F\\pF,F]pF,FKF,FGF,FAF,
FSF,FenF,FE*4F3F,FgpF,FYF,F\\pF,F:F,FhpF,FJF,FIF,F)F,FE*0FgpF,F\\pF,FA
F,FJF,FcpF,FOF,FYF,FE*:\"#=F,FgpF,FHF,F-F,F4F,FKF,FOF,FAF,FYF,F;F,F^oF
,FenF,FE*8FepF,F]pF,FcpF,F\\pF,FGF,FAF,FKF,F>F,FenF,F;F,F`pF,F,*2F>F,F
\\pF,FhpF,FJF,F4F,FbqF,FYF,FIF,F,*6F`qF,FgpF,FPF,F\\pF,F]pF,FKF,FhpF,F
enF,FSF,F)F,F,*6F3F,FgpF,F\\pF,F4F,FYF,FhpF,F^qF,F:F,FIF,F)F,FE*:F3F,F
=F,F-F,FhpF,FJF,F4F,FYF,F:F,F]oF,F^oF,FIF,F)F,FE*:\"#9F,F=F,FHF,F-F,F]
pF,FKF,FGF,FAF,F^oF,FSF,F]oF,FenF,FE*6F3F,F=F,F\\pF,F]pF,FUF,F^qF,FbqF
,F]oF,FIF,F)F,F,*6F3F,FgpF,F-F,FJF,FhpF,F:F,F^oF,F5F,FIF,F)F,FE*:F3F,F
gpF,FHF,F-F,F4F,FKF,FGF,FAF,F^oF,FSF,FYF,FenF,FE*8FbpF,FgpF,FPF,F.F,F4
F,FJF,FYF,FUF,FiqF,F:F,F)F,FE*6FcrF,F>F,F\\pF,FAF,FJF,F4F,FHF,FGF,FYF,
FSF,FE*8FDF,F4F,FPF,F\\pF,FUF,FJF,F>F,FYF,F:F,F`pF,F)F,F,*8F_rF,F=F,FH
F,F.F,F]pF,FJF,FUF,FiqF,FSF,F]oF,F)F,F,*:\"#;F,F=F,FPF,F-F,F4F,FJF,FUF
,FYF,F:F,F]oF,F^oF,F)F,FE*8FDF,FgpF,F\\pF,FAF,FKF,FPF,FGF,FipF,F:F,Fen
F,F5F,F,*8FepF,F]pF,FPF,F-F,FGF,FUF,FJF,F>F,F^oF,FSF,F`pF,F,*8F9F,FgpF
,FPF,F.F,F]pF,FKF,FOF,FSF,FAF,FiqF,FenF,FE*8FDF,FgpF,F-F,FUF,FJF,FGF,F
^oF,F:F,FipF,F5F,FIF,F,*4FgpF,F.F,FAF,FJF,F4F,FcpF,FGF,FYF,F;F,FE*6F[p
F,F=F,FcpF,F\\pF,F]pF,FJF,FGF,FAF,F;F,F]oF,FE*4F3F,F]pF,FPF,F\\pF,FhpF
,FJF,F]oF,FSF,F)F,FE*8FDF,FgpF,F\\pF,FhpF,FKF,FPF,FipF,FenF,FSF,F5F,F)
F,F,*4FgpF,F\\pF,FUF,FJF,F4F,FYF,FbqF,FIF,F)F,FE*:FDF,F=F,F-F,F]pF,FKF
,FbqF,FUF,FenF,F^oF,F]oF,FIF,F)F,FE*2FgpF,F\\pF,F4F,FcpF,FOF,F;F,FYF,F
BF,FE*6F3F,F=F,F\\pF,F]pF,FOF,FHF,F^qF,FSF,F]oF,FBF,FE*6F3F,FgpF,F.F,F
]pF,FKF,FOF,FcpF,F;F,FenF,FBF,FE*2FgpF,F\\pF,FAF,F4F,FHF,FOF,FYF,F;F,F
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JF,FOF,F;F,F^oF,FBF,FE*:FDF,F=F,F\\pF,F4F,FKF,FOF,FHF,FYF,FSF,FenF,F]o
F,FBF,F,*6F3F,F>F,F\\pF,F]pF,FKF,FOF,FcpF,F;F,FenF,FBF,FE*6F3F,FgpF,FP
F,F-F,FJF,FhpF,FSF,F^oF,F5F,F)F,FE*6FcrF,FgpF,F-F,FUF,FJF,F]pF,F^oF,Fb
qF,FIF,F)F,F,*:F3F,FgpF,FPF,F-F,F4F,FKF,FOF,FYF,FAF,FenF,FSF,F^oF,FE*4
FbpF,FgpF,F]pF,FcpF,F^oF,FOF,FJF,F;F,FBF,F,*6F3F,F=F,F\\pF,F]pF,FhpF,F
^qF,F:F,F]oF,FIF,F)F,F,*8FDF,F4F,FPF,F\\pF,FGF,FUF,FJF,F>F,FYF,FSF,F`p
F,F,*:FDF,F=F,F\\pF,F4F,FKF,F\\rF,FHF,FYF,F;F,F]oF,FenF,FBF,F,*6F=F,FH
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rF,F;F,F^oF,FBF,FE*6F3F,FgpF,F.F,F]pF,FKF,FOF,FHF,FSF,FenF,FBF,FE*4F3F
,F>F,F\\pF,F]pF,FKF,F\\rF,FcpF,FenF,FBF,FE*6F3F,FgpF,F-F,F]pF,FOF,F;F,
FcpF,F^oF,F^qF,FBF,F,*6F3F,F>F,F\\pF,F]pF,FKF,FOF,FHF,FAF,FenF,F;F,FE*
6F3F,F=F,F\\pF,F]pF,F\\rF,FHF,F^qF,F;F,F]oF,FBF,FE*6FDF,FgpF,F-F,F]pF,
FOF,FHF,FSF,F^qF,F^oF,FBF,F,*6FcrF,F=F,FcpF,F\\pF,F]pF,FJF,FOF,F;F,F]o
F,FBF,F,*:F\\sF,F4F,FPF,F\\pF,FGF,FUF,FKF,F=F,FYF,FSF,F]oF,FenF,F,*4F3
F,F>F,F\\pF,F]pF,FKF,FhpF,FbqF,FenF,FIF,F,*6FcrF,FgpF,F.F,FJF,F4F,FHF,
FOF,FSF,FYF,FBF,FE*6F9F,FgpF,FcpF,F\\pF,F4F,FJF,FOF,FYF,F;F,FBF,FE*6FD
F,FgpF,FHF,F-F,F]pF,FJF,FOF,FSF,F^oF,FBF,FE*6\"#<F,FgpF,FHF,F-F,F]pF,F
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,F]pF,FKF,F\\rF,F;F,FiqF,FenF,FBF,F,*6F3F,F=F,F\\pF,F]pF,FOF,FcpF,F^qF
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,F\\pF,F4F,FOF,FYF,FcpF,F^qF,F;F,FBF,F,*8F3F,FgpF,FcpF,F.F,F]pF,FKF,FO
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,F;F,FYF,FBF,FE*6FbpF,F]pF,FcpF,F.F,FOF,FJF,F=F,F;F,F]oF,FBF,F,*4FcrF,
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enF,F5F,FBF,FE*0FgpF,F\\pF,FhpF,FJF,FYF,FbqF,F)F,F,*8FDF,FgpF,FHF,F.F,
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,F;F,F]oF,F^oF,FBF,F,*8F3F,FgpF,FcpF,F.F,F4F,FJF,FOF,FYF,F;F,FiqF,FBF,
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F,F]pF,FKF,FOF,F;F,FenF,FBF,F,*8F3F,FgpF,FHF,F.F,F4F,FJF,FOF,FYF,FiqF,
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pF,FKF,FOF,FSF,FenF,FBF,F,*6FcrF,FgpF,FHF,F\\pF,F4F,FJF,F\\rF,FYF,F;F,
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\\rF,F;F,F]oF,FBF,F,*4FbpF,FgpF,FcpF,F-F,F]pF,FOF,F;F,F^oF,FBF,F,*4Fbp
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,F]pF,FJF,FGF,FAF,F;F,F^oF,FE*4FgpF,F-F,F]pF,F:F,FUF,FPF,F^oF,F^qF,F)F
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F,FJF,F4F,FE*0F.F,FYF,FgpF,F:F,FhpF,F4F,FJF,FE*6FDF,F.F,FYF,FgpF,FSF,F
AF,FGF,FIF,F4F,FJF,FE*:FbpF,FYF,F\\pF,F=F,FGF,FIF,F4F,FKF,FBF,FenF,F:F
,F]oF,F,*:FbpF,FYF,F\\pF,F=F,FOF,FIF,F4F,FKF,FBF,FenF,FSF,F]oF,F,*6F3F
,F.F,FYF,FgpF,FSF,FBF,FOF,FIF,F4F,FJF,FE*4F3F,FYF,FgpF,F;F,FAF,FGF,FPF
,F\\pF,FJF,F,*:F\\tF,FYF,F-F,F=F,F)F,FPF,F4F,FJF,FAF,F^oF,FSF,F]oF,FE*
4F3F,F.F,FYF,FgpF,F:F,FUF,F)F,F4F,FJF,FE*:FepF,FYF,F\\pF,F=F,FGF,FIF,F
AF,FKF,F4F,FenF,FSF,F]oF,F,*4FDF,FYF,FgpF,FSF,FAF,FGF,FIF,F\\pF,FJF,F,
*2F.F,FYF,FgpF,F:F,FAF,FGF,F4F,FJF,FE*2F3F,FYF,FgpF,F:F,FAF,FGF,FJF,F
\\pF,F,*:F3F,FYF,F\\pF,F=F,FOF,FPF,F4F,FKF,FBF,FenF,F;F,F]oF,FE*6FhqF,
FYF,F\\pF,FgpF,FOF,FPF,FJF,FBF,F4F,F;F,F,*6FcrF,FYF,F>F,F;F,FAF,FGF,FP
F,F\\pF,FJF,F4F,F,*:FbpF,FYF,F-F,FgpF,FGF,F4F,FSF,FBF,FPF,F^oF,FKF,Fen
F,F,*6F\\tF,FYF,F\\pF,FgpF,FGF,FIF,FJF,FAF,F4F,FSF,FE*2F.F,FYF,FgpF,FP
F,F\\rF,F4F,F;F,FJF,FE*0FYF,FgpF,FGF,F\\pF,F4F,F:F,FAF,F,*4F3F,F.F,FPF
,F]oF,F]pF,F\\rF,F;F,FJF,F=F,F,*6F3F,F.F,FYF,FgpF,FPF,FOF,F4F,F;F,FBF,
FJF,FE*6F3F,F.F,FYF,FgpF,FSF,FUF,F)F,FIF,F4F,FJF,FE*6F9F,FYF,F\\pF,Fgp
F,FGF,FIF,F4F,F:F,FJF,FBF,FE*2F3F,FHF,F]oF,F]pF,FOF,FBF,F\\pF,FJF,FE*6
FbpF,FYF,F\\pF,FgpF,FGF,FJF,FBF,F4F,FPF,FSF,F,*2F3F,F]oF,F]pF,FGF,F\\p
F,F:F,F=F,FAF,FE*8F3F,FYF,F-F,F=F,FGF,F4F,FJF,FAF,F^oF,F:F,F]oF,F,*0FY
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F=F,F;F,F,*4F3F,FHF,F]oF,F5F,FOF,F\\pF,FJF,FBF,F=F,FE*6FDF,F.F,FPF,F]o
F,F]pF,FOF,F;F,FBF,F=F,FJF,F,*2F3F,FPF,F]oF,F]pF,F\\rF,F\\pF,F;F,F=F,F
,*4F.F,FYF,FgpF,FPF,FGF,F4F,F;F,FJF,FAF,FE*2F3F,FPF,F]oF,F]pF,FSF,FOF,
F\\pF,F=F,F,*0FYF,F\\pF,FgpF,FOF,FPF,F4F,FSF,F,*6FDF,F.F,FIF,F]oF,F]pF
,FGF,FSF,FAF,F=F,FJF,F,*4F3F,FIF,F]oF,F]pF,F)F,FUF,F\\pF,FJF,FSF,F,*6F
DF,FIF,F]oF,F5F,FOF,FJF,F\\pF,FBF,F=F,FSF,FE*2F]oF,F5F,FGF,F:F,FAF,FJF
,F\\pF,F=F,FE*:FbpF,FYF,F\\pF,F=F,F)F,FPF,F4F,FKF,FAF,FenF,FSF,F]oF,F,
*6F3F,F.F,FIF,F]oF,F]pF,FOF,FSF,FBF,FJF,F=F,F,*8F3F,FYF,F\\pF,F=F,FGF,
F4F,FKF,FAF,FenF,F:F,F]oF,FE*6FbdlF,FIF,F]oF,F5F,FGF,F\\pF,FJF,FAF,F=F
,FSF,FE*4F9F,FYF,F\\pF,FgpF,F)F,FJF,FUF,F4F,F:F,F,*6FepF,FPF,F]oF,F5F,
FGF,F\\pF,FJF,FAF,F=F,F;F,F,*2F]oF,F5F,FbqF,F)F,FAF,FJF,F\\pF,F=F,FE*6
F3F,F.F,FPF,F]oF,F]pF,FGF,F;F,FJF,F=F,FAF,F,*:FepF,FYF,F-F,FgpF,FGF,FI
F,F4F,F:F,FBF,FenF,F^oF,FKF,FE*:FDF,FIF,F]oF,F]pF,FGF,F=F,FAF,F^oF,FKF
,FSF,F-F,FenF,FE*4F3F,FYF,F\\pF,FgpF,FOF,FIF,FJF,FBF,FSF,F,*:FDF,FIF,F
]oF,F]pF,FOF,FSF,FBF,F=F,F-F,FenF,F^oF,FKF,FE*6FcrF,FPF,F]oF,F5F,FSF,F
)F,F\\pF,FJF,FAF,F=F,FE*:F9F,FYF,F-F,FgpF,F)F,F4F,FSF,FAF,FPF,F^oF,FKF
,FenF,FE*6FbdlF,FYF,F\\pF,FgpF,F)F,FJF,FAF,F4F,FPF,FSF,FE*4F3F,FbqF,F^
oF,FgpF,F]pF,F)F,FAF,FJF,F-F,F,*4F3F,F\\pF,FgpF,F)F,FKF,FAF,F5F,FbqF,F
enF,F,F,F)!\"#F5FgepF'FEF,FE" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "N
ow here are the coefficients collected with respect to side lengths" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "collect(dmueqn2,\{l[1],l[2
],l[3],l[4]\}):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "collect(
dmueqn1,\{l[1],l[2],l[3],l[4]\}):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 39 "collect(dmueqn0,\{l[1],l[2],l[3],l[4]\}):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "disc:=simplify(dmueqn1^2-4*dmueqn2*
dmueqn0):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 126 "disc is the express
ion for the discriminant in the quadratic equation. It takes up about \+
2 pages. Note that it has the factor:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 104 "disfactor:=l[3]^3*l[1]^3*l[2]*l[4]*16*cos(a[2])^3*co
s(a[3])^3*cos(a[1])*cos(a[4])*(l[1]+l[4]+l[2]+l[3]);" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>%*disfactorG,$*6\"#;\"\"\")%#l3G\"\"$F()%#l1GF+F(%#
l2GF(%#l4GF()-%$cosG6#%#a2GF+F()-F26#%#a3GF+F(-F26#%#a1GF(-F26#%#a4GF(
,*F-F(F/F(F.F(F*F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 305 "which \+
pulls out of it. What is left is of deg 9 in side lengths and degree 8
 in trigonometric functions. It is this expression that I'm hoping ya'
ll can simplify.\nIn dealing with the equation for p[3,2] I now work o
nly with dmueqn2,dmueqn1 and disc using the quadratic formula. I print
 out the expressions" }}{PARA 0 "" 0 "" {TEXT -1 47 "for disc in the m
aple worksheet disc040505.mws." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "defform(A=0,B=0,
C=0,DD=0):d((-B+sqrt(DD))/2*A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*
&#\"\"\"\"\"#F&*&%\"AGF&-%\"dG6#%\"BGF&F&!\"\"*&,&*&F'F.F-F&F.*&F'F.%#
DDG#F&F'F&F&-F+6#F)F&F&*&#F&\"\"%F&*(F)F&F3#F.F'-F+6#F3F&F&F&" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "sqdisc:=sqrt(disc):" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 214 "dmuplus is an expression for dp[3
,2] without explicitly solving for p[3,2] obtained by taking the deriv
ative of\nthe solution in the quadratic formula (with the positive roo
t), I then get an expression involving our" }}{PARA 0 "" 0 "" {TEXT 
-1 113 "base variables and p[3,2] which must be equal to the previousl
y solved for d(p[3,2])=p[3,2,1]eta[1]+p322eta[2].\n " }}}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 161 "dmuplus:=-(1/2)*dmueqn2*d(dmueqn1)+ (dmueqn2*d(
disc))/(4*sqdisc)\n+(-dmueqn1/2+ sqdisc/2)*d(dmueqn2):xxx;\ndmuplus:=s
impform(subs(I2,subs(dallsubs,dmuplus))):yyy;\n" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%$xxxG" }}{PARA 0 "" 0 "" {XPPMATH 20 "6#%$yyyG" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "1 0 0" 281 }
{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }