{VERSION 6 0 "IBM INTEL LINUX" "6.0" }
{USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 
1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 
0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 256 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal
" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 
-1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" 
-1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 
-1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 
0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Plot" 0 13 1 
{CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 
0 0 0 0 0 -1 0 }}
{SECT 0 {PARA 0 "" 0 "" {TEXT -1 75 "cable_2.mws   9/28/04  (improved \+
version (\"assuming\") cable.mws 6/18/2004)\n" }{TEXT 256 46 "Equilibr
ium shape of (uniform) elastic cable.\n" }{TEXT -1 46 "Use \"B\" for \+
\"D\", which is a predefined symbol." }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 118 "restart:\nDigits:=50;\nxp:=C+1/sqrt(1+((B+epsilon*ss
)/C)^2);\nint(xp,ss=0..s) assuming (epsilon>0,C>0);\nx:=unapply(%,s);
\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'DigitsG\"#]" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>%#xpG,&%\"CG\"\"\"*&F'F'*$,&F'F'*&,&%\"BGF'*&%(epsi
lonGF'%#ssGF'F'\"\"#F&!\"#F'#F'F1!\"\"F'" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#*(%\"CG\"\"\",*-%#lnG6#F$F%-F(6#,&%\"BGF%*$,&*$)F$\"\"#F%F%*$)F-
F2F%F%#F%F2F%!\"\"*&%\"sGF%%(epsilonGF%F%-%(arcsinhG6#*&,&F7F%F-F%F%F$
F6F%F%F9F6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xGj+6#%\"sG6\"6$%)op
eratorG%&arrowGF(*(%\"CG\"\"\",*-%#lnG6#F-F.-F16#,&%\"BGF.*$,&*$)F-\"
\"#F.F.*$)F6F;F.F.#F.F;F.!\"\"*&9$F.%(epsilonGF.F.-%(arcsinhG6#*&,&F@F
.F6F.F.F-F?F.F.FBF?F(F(F(6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 83 "yp:=(B+epsilon*ss)*xp/C;\nint(yp,ss=0..s) assuming (e
psilon>0,C>0);\ny:=unapply(%,s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
#ypG*(,&%\"BG\"\"\"*&%(epsilonGF(%#ssGF(F(F(,&%\"CGF(*&F(F(*$,&F(F(*&F
&\"\"#F-!\"#F(#F(F2!\"\"F(F(F-F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$
*(\"\"#!\"\",**&F%\"\"\",&*$)%\"CGF%F)F)*$)%\"BGF%F)F)#F)F%F&*&F%F),*F
+F)F.F)**F%F)F0F)%(epsilonGF)%\"sGF)F)*&)F5F%F))F6F%F)F)F1F)**F%F)F0F)
F5F)F6F)F)F7F)F)F5F&F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"yGj+6#%
\"sG6\"6$%)operatorG%&arrowGF(,$*&#\"\"\"\"\"#F/*&,**&F0F/,&*$)%\"CGF0
F/F/*$)%\"BGF0F/F/F.!\"\"*&F0F/,*F5F/F8F/**F0F/F:F/%(epsilonGF/9$F/F/*
&)F?F0F/)F@F0F/F/F.F/**F0F/F:F/F?F/F@F/F/FAF/F/F?F;F/F/F(F(F(6#\"\"!" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "bc1:=b=x(1);\nbc2:=d=y(1)
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$bc1G/%\"bG*(%\"CG\"\"\",*-%#ln
G6#F(F)-F,6#,&%\"BGF)*$,&*$)F(\"\"#F)F)*$)F1F6F)F)#F)F6F)!\"\"%(epsilo
nGF)-%(arcsinhG6#*&,&F;F)F1F)F)F(F:F)F)F;F:" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$bc2G/%\"dG,$*(\"\"#!\"\",**&F)\"\"\",&*$)%\"CGF)F-F-
*$)%\"BGF)F-F-#F-F)F**&F)F-,*F/F-F2F-*(F)F-F4F-%(epsilonGF-F-*$)F9F)F-
F-F5F-*(F)F-F4F-F9F-F-F:F-F-F9F*F-" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 475 "Digits:=30;\nepsilon:=0.5;\nns:=50: ds:=evalf(1/ns):
\nfsolve(subs(b=0.5,d=0,\{bc1,bc2\}),\{C=.007,B=-.05\});\nassign(%):\n
[x(s),y(s)];\np1:=plot([x(s),y(s),s=0..1],scaling=constrained,color=re
d):\n#stretched_length:=evalf(Int(diff(x(s),s)^2+diff(y(s),s)^2,s=0..1
));\nE_elastic:=evalf((1/2)\n          *Int( (diff(x(s),s)^2+diff(y(s)
,s)^2-1)^2,s=0..1));\nE_gravitational:=evalf( Int( epsilon*y(s),s=0..1
));\nelastic_to_grav:=E_elastic/E_gravitational;\nunassign('epsilon'):
 unassign('B','C'):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'DigitsG\"#I
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(epsilonG$\"\"&!\"\"" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#<$/%\"BG$!?++++++++++++++D!#I/%\"CG$\"?v+)p&
yn;!)=YkO%\\![!#J" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7$,($\"?+5:2h;*f!
px;GvfA!#I\"\"\"*&$\"?v+)p&yn;!)=YkO%\\![!#JF(%\"sGF(F(*&$\"?],'RrbL.w
B*Gt))4'*F,F(-%(arcsinhG6#,&*&$\"?\\j9%[9)[ZPjE\\fS5!#GF(F-F(F($\"?X<t
?C2WP(oJjuH?&!#H!\"\"F(F(,*$\"?+df)\\%fB7d0\"[7:4&F'F<*&$\"?++++++++++
++++?F;F(,($\"?*zPK*zDhBl=O[(3['F,F(*&$\"?++++++++++++++DF'F(F-F(F<*&$
\"#D!\"#F()F-\"\"#F(F(#F(FNF(*&$\"?++++++++++++++DF'F(F-F(F<*&$\"?++++
++++++++++DF'F(FMF(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*E_elasticG
$\"?5%)><J$e^\"fq2KLXb!#J" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%0E_grav
itationalG$!?Z'>7\"z(o0/&z.;js8!#I" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%0elastic_to_gravG$!?SZJ9,+J!=.2EG*RS!#I" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 408 "Digits:=30;\nepsilon:=0.1;\nns:=50: ds:=evalf(1/ns
):\nfsolve(subs(b=0.5,d=0,\{bc1,bc2\}),\{C=.007,B=-.05\});\nassign(%):
\n[x(s),y(s)];\np2:=plot([x(s),y(s),s=0..1],scaling=constrained,color=
blue):\nE_elastic:=evalf((1/2)\n          *Int( (diff(x(s),s)^2+diff(y
(s),s)^2-1)^2,s=0..1));\nE_gravitational:=evalf( Int( epsilon*y(s),s=0
..1));\nelastic_to_grav:=E_elastic/E_gravitational;\nunassign('epsilon
'): unassign('B','C'):\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'DigitsG
\"#I" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(epsilonG$\"\"\"!\"\"" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/%\"CG$\"?^k4A^'**z\"yj)3QI5\"!#J/%
\"BG$!\"&!\"#" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7$,($\"?x^*Qu,+\"4\"o
b4[[W#!#I\"\"\"*&$\"?^k4A^'**z\"yj)3QI5\"!#JF(%\"sGF(F(*&$F+F'F(-%(arc
sinhG6#,&*&$\"?>+%*)*[ISeSdv+(e1*!#HF(F-F(F($\"?5+Z\\C:?HqyP]$H`%F7!\"
\"F(F(,*$\"?lyEZ=rl?YKBRA?^F'F:*&$\"?++++++++++++++5!#GF(,($\"?)zmMh!Q
3x&)\\-$p;i#!#KF(*&$\"#5!\"$F(F-F(F:*&$F(!\"#F()F-\"\"#F(F(#F(FNF(*&$
\"?++++++++++++++]F,F(F-F(F:*&$\"?++++++++++++++]F,F(FMF(F(" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%*E_elasticG$\"?6CTrpsg@T))37-$)>!#K" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%0E_gravitationalG$!?$\\;r+<;!=6'zBpP
P#!#J" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%0elastic_to_gravG$!?4X:fXVE
'3W>5#*QN)!#J" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 467 "epsilon:=
0.02;\nns:=50: ds:=evalf(1/ns):\nfsolve(subs(b=0.5,d=0,\{bc1,bc2\}),\{
C=.007,B=-.05\});\nassign(%):\n[x(s),y(s)];\np3:=plot([x(s),y(s),s=0..
1],scaling=constrained,color=black):\n#stretched_length:=evalf(Int(dif
f(x(s),s)^2+diff(y(s),s)^2,s=0..1));\nE_elastic:=evalf((1/2)\n        \+
  *Int( (diff(x(s),s)^2+diff(y(s),s)^2-1)^2,s=0..1));\nE_gravitational
:=evalf( Int( epsilon*y(s),s=0..1));\nelastic_to_grav:=E_elastic/E_gra
vitational;\nunassign('epsilon'): unassign('B','C'):\n" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%(epsilonG$\"\"#!\"#" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#<$/%\"BG$!?++++++++++++++5!#J/%\"CG$\"?\">`YJE\\`Dzk4Dv
F#!#K" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7$,($\"?cnUo`Ks.w^uBh)[#!#I\"
\"\"*&$\"?#>`YJE\\`Dzk4DvF#!#KF(%\"sGF(F(*&$\"?'fEt:juwiR#[DwQ6F'F(-%(
arcsinhG6#,&*&$\"?3e&*Qb$3pc>5'>Y\"y)!#HF(F-F(F($\"?/zZpxTX$y40)4t!R%F
8!\"\"F(F(,*$\"?IG9Rp?J%ot^iQ!G^F'F;*&$\"?++++++++++++++]!#GF(,($\"?e$
3y!pQTG4l07(=0\"!#LF(*&$\"?++++++++++++++SFFF(F-F(F;*&$\"\"%!\"%F()F-
\"\"#F(F(#F(FOF(*&$\"?++++++++++++++5!#JF(F-F(F;*&$\"?++++++++++++++5F
TF(FNF(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*E_elasticG$\"?qx6glqcB
OtR2eix!#M" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%0E_gravitationalG$!?(G
.#R!yG-vpYC%e%f%!#K" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%0elastic_to_g
ravG$!?LTp'\\$Gk,yrsm]*o\"!#J" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "plots[display](p1,p2,p3);" }
}{PARA 13 "" 1 "" {GLPLOT2D 489 489 489 {PLOTDATA 2 "6(-%'CURVESG6$7S7
$$\"3m9PZ1$Q2k'!#O$\"\"!F,7$$\"3%y0#>!=T.D&!#?$!3RMt'G8a=n#!#>7$$\"3hP
;xR>Er**F0$!32_***[5HU(\\F37$$\"3)eO'fo*yka\"F3$!3IFPPVn8QvF37$$\"3&z!
Q$*R_&47#F3$!3?'[1%*pM\"45!#=7$$\"3%yi`@a'=;FF3$!3/S<o(>6/E\"FC7$$\"3w
0N@N<0\"H$F3$!3o\"fkB]x2\\\"FC7$$\"3sOCXR\"4?\"RF3$!3x.<^F1eE<FC7$$\"3
LJ7kAYn%e%F3$!3d!Gwt83u'>FC7$$\"3Zr]8B0c*G&F3$!3`^Bh::B/AFC7$$\"33+.?V
S&[0'F3$!3dZ\\z=&fUW#FC7$$\"3J)fCLLdpw'F3$!3-m1X[,Y_EFC7$$\"3lq$[$[&>p
h(F3$!3#Gjx\"f$HH)GFC7$$\"3nv7CZ$y!H&)F3$!3N<T*3NR(4JFC7$$\"3ZSO%\\SLA
Z*F3$!3\"\\'=h*G(QBLFC7$$\"3UF:'GKa\"R5FC$!3!yzYOQ*f7NFC7$$\"3;LPaVZ^d
6FC$!3!>r_qj(eIPFC7$$\"3Uu$3f=9nE\"FC$!3:O7*QF'43RFC7$$\"38K35*3IeS\"F
C$!3/%4'p)))fU5%FC7$$\"3()>E.Y/bT:FC$!3zYd<%f=hE%FC7$$\"3;/Is*f_gq\"FC
$!3+K#Q^2dpU%FC7$$\"3>U\")Hl?Sz=FC$!3,I)*)e>6&eXFC7$$\"3UD+'QGuz2#FC$!
3*HyNs*f'em%FC7$$\"3%>`'Rl*zTF#FC$!3J;$oTZ1,t%FC7$$\"3g)za)ep#[\\#FC$!
3!\\$z7j/^bZFC7$$\"3P@$yR]FTs#FC$!3:O&RZa([IZFC7$$\"3o]r1d[s:HFC$!3M>P
4>*e&oYFC7$$\"3)e*[U&H*f3JFC$!3')*[e*=5FmXFC7$$\"3cCKYH\"H,H$FC$!381$R
***)f-V%FC7$$\"3\\YbX)zC0X$FC$!3!*yt80msuUFC7$$\"33R]iMC+\"f$FC$!3iSo/
!\\e$3TFC7$$\"3rl!3Mo-At$FC$!3T?dz$z\\(4RFC7$$\"3M*R$*>i#pZQFC$!3Ah\">
Z&Gd@PFC7$$\"3xYI[ZR(3'RFC$!3$f;p@&Ra7NFC7$$\"3+Q<%fssc0%FC$!3L#p)f(QG
rJ$FC7$$\"34Cae)o[?:%FC$!3%*Hpz#fPz4$FC7$$\"3>LsSMNrOUFC$!3UX9rKE2()GF
C7$$\"3`X<s-2y>VFC$!3&[&yr+rRiEFC7$$\"3'\\Rfl')ziR%FC$!3ZB)pAJ;*QCFC7$
$\"3[Er'\\Wm>Z%FC$!3,`im'\\J7?#FC7$$\"3+1$p[:'3TXFC$!3%)oG*R3d*o>FC7$$
\"3*QP(*ztc$3YFC$!3N]5-8K@G<FC7$$\"3_(=(3&Q7?n%FC$!3\\#4a8K6k[\"FC7$$
\"3_)=!QF/3GZFC$!38NnuJ\"[;E\"FC7$$\"3G[mr7Hv*y%FC$!35N&\\15=6+\"FC7$$
\"3dvV3`\"zF%[FC$!3Ln/696DbwF37$$\"3E-(eoT!H(*[FC$!3A%>GRJjs6&F37$$\"3
yP&ePImw%\\FC$!38$eTF@3Mm#F37$$\"3++++++++]FCF+-%'COLOURG6&%$RGBG$\"*+
+++\"!\")F+F+-F$6$7S7$$!3qt;;ag/pU!#N$!3oX9y!yyxQ\"!#M7$$\"3lzgdPYYO]F
0$!3z'Q?u*))*GB#F37$$\"3!>#*y'\\*p1f*F0$!3d6*Q^u.z;%F37$$\"3S-2%)zz(>
\\\"F3$!3N@QO\\lsMjF37$$\"3x/CjJwa_?F3$!3h/w\\tLB0&)F37$$\"3#p^g-2snj#
F3$!3ZzP,)z'Ql5FC7$$\"3%)**Gvnh9/KF3$!3+QrVwmYj7FC7$$\"3+73zQ1I?QF3$!3
`pj)*4*3tY\"FC7$$\"3GgBu@%*Q\"\\%F3$!3^\"=m4*\\iw;FC7$$\"3?&*zrPCQ)>&F
3$!3reVC,6c$)=FC7$$\"3cO\"=_TU*pfF3$!3kFQ8@TQ%4#FC7$$\"3Sh!\\yH<7p'F3$
!34Z3V\"[/\"yAFC7$$\"3<P*=!*y,eb(F3$!3^Q(z%z?M#[#FC7$$\"3boJFo]O([)F3$
!3l+ig^99%o#FC7$$\"3;4H-Pxp`%*F3$!3oR;w[d%[(GFC7$$\"3<J(z>Wm(R5FC$!3(R
I/'=-6WIFC7$$\"3B*3m0PP9;\"FC$!3:@u'3!oLRKFC7$$\"338(zS#)[OF\"FC$!3%*>
GvAMD)R$FC7$$\"3*ox1^$fD;9FC$!3*3F(*41`Md$FC7$$\"3kfy&GV%pa:FC$!3:yF5(
>/tr$FC7$$\"3Wa*)[y*e6s\"FC$!3*3#*o_ZG\"fQFC7$$\"3!G^$\\tkg%*=FC$!3[\"
*eO\"RRQ(RFC7$$\"3+dgI^#414#FC$!3dqB:RV?mSFC7$$\"38ZI()\\>z\"G#FC$!33i
tt4/x?TFC7$$\"3w%>Gen4]\\#FC$!3>QkIpY<UTFC7$$\"3R0$\\&psc;FFC$!3BS)\\]
D#4@TFC7$$\"3hK207j@.HFC$!3Zp(*4%R.&oSFC7$$\"3)3DwB6pM4$FC$!3(p9&3$4e0
)RFC7$$\"3Koj$fm(*\\F$FC$!3.>-Wm[-iQFC7$$\"3/$*oj\"o\\sV$FC$!3d4f=jx#
\\s$FC7$$\"31WeCWT]!e$FC$!3!*fQs7b5xNFC7$$\"3v4R`$**Q_s$FC$!3+Q_QTEt*R
$FC7$$\"3[sa+qh\"R%QFC$!3w\"=#zyCEJKFC7$$\"3WC&)e2EEgRFC$!3$R#Q[%fgS/$
FC7$$\"3h\\:9(=,w0%FC$!3$=g'Q`=DpGFC7$$\"3VFQ->RLcTFC$!3Ts&)zVKitEFC7$
$\"3te%4]X%zUUFC$!3S66/L;-'[#FC7$$\"3-z$[$\\;IFVFC$!3*G!GY5@*oG#FC7$$
\"3$Rnm%Q)*y/WFC$!3eg+z7%z'*3#FC7$$\"3s4zTg$*3\"[%FC$!3?j;%e&H$4)=FC7$
$\"3IhN*)=XT]XFC$!3BDq)e$\\(zn\"FC7$$\"3W7!Q,$)Hvh%FC$!3=FdsTSso9FC7$$
\"3ev9@B7p!o%FC$!38B\"*3gEqf7FC7$$\"3S!HFceEgt%FC$!3l^5YovWm5FC7$$\"3+
,!pm-alz%FC$!3Pfj(y?*)oV)F37$$\"3T*\\D5j)H[[FC$!3aQZ&Hj2SV'F37$$\"3\\*
*R-aT>,\\FC$!3(GZIys]%)G%F37$$\"3N@&)*Hk)z\\\\FC$!3!Gje*f#>eA#F37$FfzF
f[l-Fiz6&F[[lF+F+F\\[l-F$6$7S7$$!32=O#R&zCt>Fh[l$!3>9O&>&pWpMFe[l7$$\"
339J.aR8$*\\F0$!3oprd2/BW@F37$$\"3+b+mKyK9&*F0$!3w;RIAx$\\+%F37$$\"3+F
Np'o(=\"[\"F3$!37\"R_!)zU84'F37$$\"3)3aCUL:#R?F3$!3fT$*eo?@%=)F37$$\"3
LM`g!*oe@EF3$!3P2H0EG)e-\"FC7$$\"3H<%z$3x)y=$F3$!3$QCEjt%Q<7FC7$$\"3)*
)Qc$Qwi.QF3$!3(=z6i?)p99FC7$$\"3Scr')[43vWF3$!3)>1o7kcvh\"FC7$$\"3l)[v
n<*G$=&F3$!3n3\\y#=W$==FC7$$\"3]m?'[*e.dfF3$!3u/xKU38B?FC7$$\"3Jw]v()z
0\"o'F3$!3rt)=oek<?#FC7$$\"3*f@Qgi!o\\vF3$!3`<It@y^+CFC7$$\"3id9HRnI'[
)F3$!354tA:y/(f#FC7$$\"3<?[$GNJ%e%*F3$!3awvL'*f(Gy#FC7$$\"3&*=&[]9M3/
\"FC$!3+*f9:\\iy%HFC7$$\"3)>x>I2tK;\"FC$!3IQ]EB&e\"QJFC7$$\"3\"*yleaK:
w7FC$!3#>n>2\"z+$H$FC7$$\"3_tZR<7\\>9FC$!3]nO-TPejMFC7$$\"3zpDNy[We:FC
$!3,ytn;QW.OFC7$$\"3*G/qAY\">D<FC$!3pyK1S!e5u$FC7$$\"3W$[,j\\r%)*=FC$!
3HL-KZ/2_QFC7$$\"3OZ!otD)p$4#FC$!3b.)>()*H>TRFC7$$\"3VV\"e'f\"4OG#FC$!
33^/=B=q$*RFC7$$\"3+1`*z(G0&\\#FC$!3RGOXb`E9SFC7$$\"3fgptsEw9FFC$!3%4M
B`<6S*RFC7$$\"3m!oq!G7;+HFC$!3z9i#\\p2M%RFC7$$\"3\\SB,$yL'*3$FC$!3])H<
8<i&eQFC7$$\"3tSfw/O'4F$FC$!3MA%3#>Z'Qu$FC7$$\"3P*4d,]wMV$FC$!3\\G5;j!
\\3h$FC7$$\"3$>t1NNasd$FC$!3%>C4LXPrY$FC7$$\"3+7))zl#GFs$FC$!3w48+P)[W
H$FC7$$\"3e@@_rO6UQFC$!3z.G@&**)GIJFC7$$\"3#4+P#*3&>fRFC$!3!)Q<_=T\"y%
HFC7$$\"3@,ny.i9dSFC$!3&G&prqSUxFFC7$$\"3.34VS)ok:%FC$!3(*H]^G6!oe#FC7
$$\"3/#\\<!4$)RVUFC$!3G(ye01+TS#FC7$$\"3hhQh\"Q-$GVFC$!3YJ9q2FJ5AFC7$$
\"3bP&)zqk31WFC$!3gzF?^%e&=?FC7$$\"3E9\\Q#z+E[%FC$!3WKhS1Fz:=FC7$$\"3h
-%\\l&[/_XFC$!3!o*H,Hd')=;FC7$$\"3<.7$*fr>>YFC$!3<)Rt2&)ogT\"FC7$$\"3v
e&*fGcJ#o%FC$!3-\\&H:bVP@\"FC7$$\"3W/O=XeaPZFC$!3-o?IYx!p-\"FC7$$\"3=(
32+D!)yz%FC$!3?MBvw$o#=\")F37$$\"3N`Zkz1R\\[FC$!3D<$H*4#4q='F37$$\"3h4
G!>nw>!\\FC$!3!p%)HO&Q$47%F37$$\"3Pwrdi1B]\\FC$!3\\h7[FpUP@F37$Ffz$\"3
0u')o%osmK)Fh[l-Fiz6&F[[lF,F,F,-%+AXESLABELSG6%Q!6\"Fbjm-%%FONTG6#%(DE
FAULTG-%(SCALINGG6#%,CONSTRAINEDG-%%VIEWG6$FgjmFgjm" 1 2 0 1 10 0 2 9 
1 4 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" }
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 627 "# LOOK AT CONVERGENCE AS
 EPSILON GOES TO ZERO:\nBB:=-.05: CC:=.007; #STARTING GUESSES\nepsilon
:=.1;\nfac:=0.8;\nn:=20;\nfor i from 1 to n do\n   sol:=convert(fsolve
(subs(b=0.5,d=0,\{bc1,bc2\}),\{C=CC,B=BB\}),list);\n   BB:=eval(B,sol)
; CC:=eval(C,sol);\n   Barr[i]:=[epsilon,BB]:\n   Carr[i]:=[epsilon,CC
]:\n   ymid[i]:=[epsilon,eval(y(0.5),sol)];\n   epsilon:=epsilon*fac;
\nod: unassign('i'):\nBdat:=seq(Barr[i],i=1..n);\nCdat:=seq(Carr[i],i=
1..n);\nymiddat:=seq(ymid[i],i=1..n);\nplot([Bdat,Cdat],style=point);
\nplot([ymiddat],style=point,labels=[\"epsilon\",\" \"],title=\"height
 at midpoint\",\n           view=[0..0.1,ymiddat[1][2]..ymiddat[n][2]]
);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#CCG$\"\"(!\"$" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%(epsilonG$\"\"\"!\"\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$facG$\"\")!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%\"nG\"#?" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%BdatG667$$\"\"\"!\"\"$
!\"&!\"#7$$\"\")F,$!S++++++++++++++++++++++++S!#^7$$\"#k!\"$$!S+++++++
+++++++++++++++++KF27$$\"$7&!\"%$!S+++++++++++++++++++++++gDF27$$\"%'4
%F+$!S+++++++++++++++++++++++[?F27$$\"&oF$!\"'$!S+++++++++++++++++++++
+SQ;F27$$\"'W@E!\"($!S++++++++++++++++++++++s58F27$$\"(_r4#!\")$!S++++
+++++++++++++++++gd[5F27$$\");sx;!\"*$!S+++++++++++++++++++++!3')Q)!#_
7$$\"*Gx@M\"!#5$!S+++++++++++++++++++++k)3r'Ffn7$$\"+C=ut5!#6$!S++++++
++++++++++++++?\"4(o`Ffn7$$\"+#fM**e)!#7$!S++++++++++++++++++++'Hn\\H%
Ffn7$$\",OnZ>(o!#8$!S+++++++++++++++++++!o$Q(fV$Ffn7$$\"-))Q\"ev\\&!#9
$!S+++++++++++++++++++Wp!z([FFfn7$$\"./6^Y!)R%!#:$!S++++++++++++++++++
?bbK-*>#Ffn7$$\"/K))3sV=N!#;$!S++++++++++++++++++;W/'=#f<Ffn7$$\"0c1rw
\\Z\"G!#<$!S+++++++++++++++++!G`N)[P29Ffn7$$\"1[_o8)*z^A!#=$!S++++++++
+++++++++CE%o!***e7\"Ffn7$$\"2%)>[4&)R9!=!#>$!S+++++++++++++++++#*4ua#
*>2!*!#`7$$\"3se&e2)=:T9!#?$!S++++++++++++++++g$z#z.%fd?(Fcr" }}{PARA 
12 "" 1 "" {XPPMATH 20 "6#>%%CdatG667$$\"\"\"!\"\"$\"S5`hywBO-Vl]k4A^'
**z\"yj)3QI5\"!#^7$$\"\")!\"#$\"S&3lT<M--$phW9dd0()*GH!fwQW$*))!#_7$$
\"#k!\"$$\"S?]L`a&zL\\]UY.m6M7)e\"f7xl)frF37$$\"$7&!\"%$\"S#f3pEodBCi@
:PE(yi!R7Q$\\'=sv&F37$$\"%'4%!\"&$\"SI#*>#=#)\\D%y!3)f)z5&\\Oy4DaEyCYF
37$$\"&oF$!\"'$\"SXeu1[[k#4ei'[4Z](4x?Hu-;@r$F37$$\"'W@E!\"($\"SSX'fQC
ch0bFgiW%p&z0W-*4MixHF37$$\"(_r4#!\")$\"S$fa7;;,([cNA2MK&4$z`w+f!3sQ#F
37$$\");sx;!\"*$\"S.mLYa3**G<:&4&z[aJL/q.*)Q08>F37$$\"*Gx@M\"!#5$\"Svw
&e3&yNgi14Fg![\\,BQGl4cD`\"F37$$\"+C=ut5!#6$\"S5vx?R*f'3E5>#pB>KA&)eR'
R=SF7F37$$\"+#fM**e)!#7$\"SAc<$Q$)G$=2;W!>.K@,D**[uABz#)*!#`7$$\",OnZ>
(o!#8$\"SAmC&>))Q)*)zk;cZ&QKvB+<lMCz'yFjo7$$\"-))Q\"ev\\&!#9$\"SY29TMh
pMyqo*3T5Z_B`Dgp?zH'Fjo7$$\"./6^Y!)R%!#:$\"SZk1:vnNm')\\1s#Gd+-oZ2pyJ1
/&Fjo7$$\"/K))3sV=N!#;$\"S.\"[&e$ov/3xQQ\"R.#[*Q+!\\Y/wR.%Fjo7$$\"0c1r
w\\Z\"G!#<$\"Sp*H$*)Qw&QrBJw/9cUY_+D\\)G7GKFjo7$$\"1[_o8)*z^A!#=$\"Si!
R'ou$z0'[&H<p[a\\zz&*4\"e;5$e#Fjo7$$\"2%)>[4&)R9!=!#>$\"SlIGak3pp-.cZo
7._%35mapno1#Fjo7$$\"3se&e2)=:T9!#?$\"S_VaAuLS^q_e^3S(yUB<hxaTPl\"Fjo
" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(ymiddatG667$$\"\"\"!\"\"$!SCn%
\\AJpsuqITr^sYdE!ooMe=UT!#]7$$\"\")!\"#$!S3_i$>3H,:-GRZBJ0r_GpH?7/6%F,
7$$\"#k!\"$$!Sw`ZRRjDCP/<Xn%46_x*e%32)*[3%F,7$$\"$7&!\"%$!SI^%o#QOBX>S
B4CByurQ^y8SUkSF,7$$\"%'4%!\"&$!SVIaxm6]De*[H\"pDeQ?'zmVk.![SF,7$$\"&o
F$!\"'$!Sa:n%Qv&)[yId\\_(Q)3t?G1-_S[.%F,7$$\"'W@E!\"($!S9Uc\"4#=wue'G$
zDoC>JTvt#e#HCSF,7$$\"(_r4#!\")$!SlDC8zcjlu-U**4e[.XyeEKH%e,%F,7$$\");
sx;!\"*$!Sa/D\">4y'y;/T!\\.)o%)>$))*f1f24SF,7$$\"*Gx@M\"!#5$!S/\"\\)>1
Me2pWBX\"pCWVHx*zsvl.SF,7$$\"+C=ut5!#6$!S`c,S!RWAv(GPR&ok!Hb4alp)>$**R
F,7$$\"+#fM**e)!#7$!SL([#y_5&=h&3!z3jFL$3v4!RvZe*RF,7$$\",OnZ>(o!#8$!S
m(H*Gw!3Sbo4O%)=qh]NV**f!)oI*RF,7$$\"-))Q\"ev\\&!#9$!Sf1$GYL&3F(3G3?b7
p13!ofT[%3*RF,7$$\"./6^Y!)R%!#:$!SH&\\*H9G#)4^Fjt(>2.(Q'f)z_^1*)RF,7$$
\"/K))3sV=N!#;$!S@'yWL\"y&3[46dyJ(pY[&4#**p5k()RF,7$$\"0c1rw\\Z\"G!#<$
!S;k1<gP)y?9Ft$H$>IfD=]5f,l)RF,7$$\"1[_o8)*z^A!#=$!S+96Ocv=7'H;;\")*oy
.R8Ler)*e&)RF,7$$\"2%)>[4&)R9!=!#>$!SeW3D%y!4f5AoK_<Gi%z7t'3/'[)RF,7$$
\"3se&e2)=:T9!#?$!SjG1#fzq?HB;jGu^(>b#>+CywU)RF," }}{PARA 13 "" 1 "" 
{GLPLOT2D 489 489 489 {PLOTDATA 2 "6&-%'CURVESG6$7J7$$\"3/+++++++5!#=$
!3G+++++++]!#>7$$\"3=+++++++!)F-$!33+++++++SF-7$$\"37+++++++kF-$!33+++
++++KF-7$$\"3C++++++?^F-$!37++++++gDF-7$$\"3M++++++'4%F-$!3<++++++[?F-
7$$\"3&)**********zwKF-$!3$***********RQ;F-7$$\"3))*********R9i#F-$!3%
**********>2J\"F-7$$\"3/++++?:(4#F-$!3-++++gd[5F-7$$\"35++++;sx;F-$!3_
++++!3')Q)!#?7$$\"30+++!Gx@M\"F-$!3D++++k)3r'FV7$$\"3%******R#=ut5F-$!
3o******>\"4(o`FV7$$\"3$)*****>fM**e)FV$!3#******fHn\\H%FV7$$\"3@++gtw
%>(oFV$!35++!o$Q(fV$FV7$$\"3+++))Q\"ev\\&FV$!3+++Wp!z([FFV7$$\"3M+S56l
/)R%FV$!3;+?bbK-*>#FV7$$\"3$**>$))3sV=NFV$!3'**fTWg=#f<FV7$$\"3?gl5n(
\\Z\"GFV$!35!G`N)[P29FV7$$\"3;[_o8)*z^AFV$!33CE%o!***e7\"FV7$$\"3O)>[4
&)R9!=FV$!3y\"*4ua#*>2!*!#@7$$\"3ke&e2)=:T9FV$!3?$z#z.%fd?(Fcq7$F($\"3
.=yj)3QI5\"F-7$F/$\"3c#H!fwQW$*))FV7$F4$\"3ge\"f7xl)frFV7$F9$\"3yB\"Q$
\\'=sv&FV7$F>$\"32y4DaEyCYFV7$FC$\"3*y?Hu-;@r$FV7$FH$\"3`SC!*4MixHFV7$
FM$\"3$Ql2!f!3sQ#FV7$FR$\"3B/q.*)Q08>FV7$FX$\"3K#QGl4cD`\"FV7$Fgn$\"3Z
)eR'R=SF7FV7$F\\o$\"3$H**[uABz#)*Fcq7$Fao$\"3!H+<lMCz'yFcq7$Ffo$\"3;Kb
-'p?zH'Fcq7$F[p$\"3_wu!pyJ1/&Fcq7$F`p$\"3j+!\\Y/wR.%Fcq7$Fep$\"3<0]#\\
)G7GKFcq7$Fjp$\"3yd*4\"e;5$e#Fcq7$F_q$\"3*35mapno1#Fcq7$Feq$\"3@s6wZ:u
`;Fcq-%'COLOURG6&%$RGBG$\"#5!\"\"$\"\"!F]vF\\v-%+AXESLABELSG6$Q!6\"Fav
-%&STYLEG6#%&POINTG-%%VIEWG6$%(DEFAULTGFjv" 1 5 0 1 10 0 2 9 1 4 2 
1.000000 45.000000 45.000000 0 0 "Curve 1" }}}{PARA 13 "" 1 "" 
{GLPLOT2D 489 489 489 {PLOTDATA 2 "6'-%'CURVESG6$767$$\"3/+++++++5!#=$
!3!H!ooMe=UTF*7$$\"3=+++++++!)!#>$!3A&GpH?7/6%F*7$$\"37+++++++kF0$!3j(
*e%32)*[3%F*7$$\"3C++++++?^F0$!3'*Q^y8SUkSF*7$$\"3M++++++'4%F0$!30'zmV
k.![SF*7$$\"3&)**********zwKF0$!3.#G1-_S[.%F*7$$\"3))*********R9i#F0$!
3UTvt#e#HCSF*7$$\"3/++++?:(4#F0$!3TyeEKH%e,%F*7$$\"35++++;sx;F0$!3A$))
*f1f24SF*7$$\"30+++!Gx@M\"F0$!33t(*zsvl.SF*7$$\"3%******R#=ut5F0$!3W4a
lp)>$**RF*7$$\"3$)*****>fM**e)!#?$!39v4!RvZe*RF*7$$\"3@++gtw%>(oF]o$!3
[L%**f!)oI*RF*7$$\"3+++))Q\"ev\\&F]o$!3k+ofT[%3*RF*7$$\"3M+S56l/)R%F]o
$!3`'f)z_^1*)RF*7$$\"3$**>$))3sV=NF]o$!3c&4#**p5k()RF*7$$\"3?gl5n(\\Z
\"GF]o$!3w#=]5f,l)RF*7$$\"3;[_o8)*z^AF]o$!3C8Ler)*e&)RF*7$$\"3O)>[4&)R
9!=F]o$!3>GJn3/'[)RF*7$$\"3ke&e2)=:T9F]o$!3N#>+CywU)RF*-%'COLOURG6&%$R
GBG$\"#5!\"\"$\"\"!F`rF_r-%+AXESLABELSG6$Q(epsilon6\"Q\"~Fer-%&STYLEG6
#%&POINTG-%&TITLEG6#Q3height~at~midpointFer-%%VIEWG6$;F_r$\"\"\"F^r;$!
0!ooMe=UT!#:$!0>+CywU)RFhs" 1 5 0 1 10 0 2 9 1 4 2 1.000000 45.000000 
45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "?
eval" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "9 0 0" 
0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }