(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 3.0, MathReader 3.0, or any compatible application. The data for the notebook starts with the line of stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. ***********************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 261627, 8000]*) (*NotebookOutlinePosition[ 277315, 8568]*) (* CellTagsIndexPosition[ 277094, 8558]*) (*WindowFrame->Normal*) Notebook[{ Cell["Creeping flow caused by a rotating sphere", "Title"], Cell[TextData[{ "This notebook has been written in ", StyleBox["Mathematica ", FontSlant->"Italic"], "by \n\nMark J. McCready\nProfessor and Chair of Chemical Engineering\n\ University of Notre Dame\nNotre Dame IN 46556\nUSA\n\n\ Mark.J.McCready.1@nd.edu\n", ButtonBox["http://www.nd.edu/~mjm/", ButtonData:>{ URL[ "http://www.nd.edu/~mjm/"], None}, ButtonStyle->"Hyperlink"], "\n\n\nIt is copyrighted to the extent allowed by whatever laws pertain to \ the World Wide Web and the Internet.\n\nI would hope that as a professional \ courtesy, if you use it, this notice remain visible to other users. \nThere \ is no charge for copying and dissemination. \n\nVersion: 6/8/99\nMore \ recent versions of this notebook should be available at the web site:\n", ButtonBox["http://www.nd.edu/~mjm/rotatingsphere.nb", ButtonData:>{ URL[ "http://www.nd.edu/~mjm/rotatingsphere.nb"], None}, ButtonStyle->"Hyperlink"] }], "Text"], Cell[CellGroupData[{ Cell[TextData[StyleBox["Summary", "Subtitle"]], "Subtitle"], Cell[TextData[{ StyleBox[ "This notebook shows how to solve the flow field for a rotating sphere at \ low Reynolds number showing the 0 order and first order solutions. A regular \ perturbation expansion is used. At the lowest order, inertial terms are \ neglected and the flow is pure rotation that decays away from the sphere. \ The perturbation improvement takes into account weak inertia and this causes \ a secondary flow in the ", FontSize->13, FontWeight->"Plain"], StyleBox["r", FontSize->13, FontWeight->"Plain", FontSlant->"Italic"], StyleBox[ " and \[Theta] directions. \n\nThis notebook is intended as a supplement to \ ", FontSize->13, FontWeight->"Plain"], StyleBox["\nL. G. Leal (1992) ", FontSize->13], StyleBox["Laminar flow and Convective Transport Processes", FontSize->13, FontWeight->"Plain", FontSlant->"Italic"], StyleBox[", Butterworth", FontSize->13, FontWeight->"Plain"], StyleBox[".", FontSize->13] }], "Text", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell["Keywords", "Subtitle"], Cell["\<\ Creeping flow, Rotating Sphere, Regular Perturbation, Low Reynolds \ number fluid dynamics\ \>", "Text"] }, Closed]], Cell[CellGroupData[{ Cell["Problem formulation", "Subtitle"], Cell[CellGroupData[{ Cell[TextData[{ "Consider an infinite quiescent fluid. A sphere of radius ", Cell[BoxData[ StyleBox["a", FontSlant->"Italic"]]], " is placed in the region and caused to rotate with angular velocity, \ \[CapitalOmega], such that the Reynolds number, ", StyleBox["R", FontSlant->"Italic"], " = ", Cell[BoxData[ RowBox[{ SuperscriptBox[ StyleBox["a", FontSlant->"Italic"], "2"], "\[CapitalOmega]", " ", \(\[Rho]/\[Mu]\)}]]], " is small compared to 1. The 0 order governing equation is just the ", Cell[BoxData[ \(v\_\[Phi]\)]], " equation without any inertia terms. The answer will have the same \ angular dependence as the velocity on the surface of the sphere (See Leal's \ discussion starting on Page 135) and decay with distance. Since inertia is \ weak, any radial flow associated with centrifugal effects will enter at \ higher order.\n\nWe solve the 0 order problem first. Then we look to find \ out how the 0 order solution produces weak inertia to cause a flow at first \ order. 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0`00Oomoo`1EOol005Moo`03001oogoo05Eoo`00Egoo00<007ooOol0EGoo001GOol00`00Oomoo`1E Ool005Moo`03001oogoo05Eoo`00Egoo00<007ooOol0EGoo001GOol00`00Oomoo`1EOol005Moo`03 001oogoo05Eoo`00EGoo1P00E7oo001COol20002Ool01000Oomoogoo1@00Cgoo001BOol00`00Oomo o`02Ool00`00Oomoo`06Ool00`00Oomoo`1"], "Graphics", CellMargins->{{20, Inherited}, {Inherited, Inherited}}, Evaluatable->False, ImageSize->{175, 311}, ImageMargins->{{0, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "The 0 order problem. Flow is only in ", StyleBox["\[Phi]", FontFamily->"Symbol"], " direction." }], "Subtitle", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ " A sphere is rotating in an otherwise quiescent medium. The equation for \ ", Cell[BoxData[ \(v\_\[Phi]\)]], " assuming creeping flow is:" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(v\[Phi]eq = \(\[PartialD]\_r\((r\^2\ \[PartialD]\_r v\[Phi][r, \ \[Theta]])\)\/r\^2 + \ \[PartialD]\_\[Theta]\(\[PartialD]\_\[Theta]\((v\[Phi][r, \[Theta]]\ Sin[\ \[Theta]])\)\/Sin[\[Theta]]\)\/r\^2\)\/R\)], "Input", AspectRatioFixed->True], Cell[BoxData[ FormBox[ RowBox[{\(1\/R\), RowBox[{"(", RowBox[{ RowBox[{\(1\/r\^2\), RowBox[{"(", RowBox[{ RowBox[{\(csc(\[Theta])\), " ", RowBox[{"(", RowBox[{\(\(-\(sin(\[Theta])\)\)\ \(v\[Phi]( r, \[Theta])\)\), "+", RowBox[{"2", " ", \(cos(\[Theta])\), " ", RowBox[{ SuperscriptBox["v\[Phi]", TagBox[\((0, 1)\), Derivative], MultilineFunction->None], "(", \(r, \[Theta]\), ")"}]}], "+", RowBox[{\(sin(\[Theta])\), " ", RowBox[{ SuperscriptBox["v\[Phi]", TagBox[\((0, 2)\), Derivative], MultilineFunction->None], "(", \(r, \[Theta]\), ")"}]}]}], ")"}]}], "-", RowBox[{\(cot(\[Theta])\), " ", \(csc(\[Theta])\), " ", RowBox[{"(", RowBox[{\(\(cos(\[Theta])\)\ \(v\[Phi](r, \[Theta])\)\), "+", RowBox[{\(sin(\[Theta])\), " ", RowBox[{ SuperscriptBox["v\[Phi]", TagBox[\((0, 1)\), Derivative], MultilineFunction->None], "(", \(r, \[Theta]\), ")"}]}]}], ")"}]}]}], ")"}]}], "+", FractionBox[ RowBox[{ RowBox[{ RowBox[{ SuperscriptBox["v\[Phi]", TagBox[\((2, 0)\), Derivative], MultilineFunction->None], "(", \(r, \[Theta]\), ")"}], " ", \(r\^2\)}], "+", RowBox[{"2", " ", RowBox[{ SuperscriptBox["v\[Phi]", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "(", \(r, \[Theta]\), ")"}], " ", "r"}]}], \(r\^2\)]}], ")"}]}], TraditionalForm]], "Output"] }, Open ]], Cell["\<\ We can assume a form of the solution from the boundary conditions. \ Note that this is simply saying the velocity is highest at the equator and 0 \ at the \"poles\".\ \>", "Text", Evaluatable->False, AspectRatioFixed->True, CellTags->"velocityboundary"], Cell[TextData[{ Cell[BoxData[ FormBox[ ButtonBox["return", ButtonData:>"conclusions", ButtonStyle->"Hyperlink"], TraditionalForm]]], " to conclusions" }], "Text"], Cell[BoxData[ \(\(vfi = f[r]\ Sin[\[Theta]];\)\)], "Input", AspectRatioFixed->True], Cell[TextData["Now substitute and see what happens"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"v\[Phi]eqO01", "=", RowBox[{"v\[Phi]eq", "/.", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ SuperscriptBox["v\[Phi]", TagBox[\((a1_, a2_)\), Derivative], MultilineFunction->None], "[", \(r, \[Theta]\), "]"}], "\[RuleDelayed]", \(\[PartialD]\_\({r, a1}, {\[Theta], a2}\)vfi\ \)}], ",", \(v\[Phi][r, \[Theta]] \[Rule] vfi\)}], "}"}]}]}]], "Input", AspectRatioFixed->True], Cell[BoxData[ FormBox[ FractionBox[ RowBox[{\(\(\(csc(\[Theta])\)\ \((2\ \(\(cos\^2\)(\[Theta])\)\ \(f( r)\) - 2\ \(f(r)\)\ \(\(sin\^2\)(\[Theta])\))\) - 2\ \(cos(\[Theta])\)\ \(cot(\[Theta])\)\ \(f(r)\)\)\/r\^2\), "+", FractionBox[ RowBox[{ RowBox[{\(sin(\[Theta])\), " ", RowBox[{ SuperscriptBox["f", "\[Prime]\[Prime]", MultilineFunction->None], "(", "r", ")"}], " ", \(r\^2\)}], "+", RowBox[{"2", " ", \(sin(\[Theta])\), " ", RowBox[{ SuperscriptBox["f", "\[Prime]", MultilineFunction->None], "(", "r", ")"}], " ", "r"}]}], \(r\^2\)]}], "R"], TraditionalForm]], "Output"] }, Open ]], Cell["\<\ The substitution has worked and the angular dependence of the \ equation has simplified.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(v\[Phi]eqO02 = Expand[v\[Phi]eqO01]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ FormBox[ RowBox[{\(-\(\(2\ \(f(r)\)\ \(sin(\[Theta])\)\)\/\(r\^2\ R\)\)\), "+", FractionBox[ RowBox[{"2", " ", RowBox[{ SuperscriptBox["f", "\[Prime]", MultilineFunction->None], "(", "r", ")"}], " ", \(sin(\[Theta])\)}], \(r\ R\)], "+", FractionBox[ RowBox[{ RowBox[{ SuperscriptBox["f", "\[Prime]\[Prime]", MultilineFunction->None], "(", "r", ")"}], " ", \(sin(\[Theta])\)}], "R"]}], TraditionalForm]], "Output"] }, Open ]], Cell[TextData[{ "We can put this into a standard form, an ", StyleBox["Euler", FontSlant->"Italic"], " equation, by multiplying by the appropriate factor" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(v\[Phi]eqO03 = Expand[\(v\[Phi]eqO02\ r\^2\ R\)\/Sin[\[Theta]]]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{ SuperscriptBox["f", "\[Prime]\[Prime]", MultilineFunction->None], "(", "r", ")"}], " ", \(r\^2\)}], "+", RowBox[{"2", " ", RowBox[{ SuperscriptBox["f", "\[Prime]", MultilineFunction->None], "(", "r", ")"}], " ", "r"}], "-", \(2\ \(f(r)\)\)}], TraditionalForm]], "Output"] }, Open ]], Cell["This is readily solved.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(ansO01 = DSolve[{v\[Phi]eqO03 == 0}, f[r], r]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ FormBox[ RowBox[{"{", RowBox[{"{", RowBox[{\(f(r)\), "\[Rule]", RowBox[{ RowBox[{"r", " ", SubscriptBox[ TagBox["c", C], "1"]}], "+", FractionBox[ SubscriptBox[ TagBox["c", C], "2"], \(r\^2\)]}]}], "}"}], "}"}], TraditionalForm]], "Output"] }, Open ]], Cell[TextData[{ "Because we need the solution to decay away from the sphere and be 0 \ sufficiently far away, C[1] = 0. On the surface of the sphere, the velocity \ is ", StyleBox[" ", FontSlant->"Italic"], Cell[BoxData[ StyleBox["a", FontSlant->"Italic"]]], " \[CapitalOmega] Sin[\[Theta]]. The Sin[\[Theta]] is already built in so \ we just have that C[2] = ", Cell[BoxData[ SuperscriptBox[ StyleBox["a", FontSlant->"Italic"], "3"]]], " \[CapitalOmega]" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(ansO02 = f[r] /. ansO01\[LeftDoubleBracket]1\[RightDoubleBracket]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"r", " ", SubscriptBox[ TagBox["c", C], "1"]}], "+", FractionBox[ SubscriptBox[ TagBox["c", C], "2"], \(r\^2\)]}], TraditionalForm]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(ansO03 = ansO02 /. {C[1] \[Rule] 0, C[2] \[Rule] a^3 \[CapitalOmega]}\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(TraditionalForm\`\(a\^3\ \[CapitalOmega]\)\/r\^2\)], "Output"] }, Open ]], Cell[TextData[{ "This gives a solution where the velocity is only in the \[Phi] direction \ and decay as 1/", Cell[BoxData[ \(TraditionalForm\`r\^2\)]], ". The \[Theta] dependence is the same as the velocity on the sphere. " }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(O0ans = ansO03\ Sin[\[Theta]]\)], "Input", AspectRatioFixed->True, CellTags->"zerosolution"], Cell[BoxData[ \(TraditionalForm\`\(a\^3\ \[CapitalOmega]\ \(sin(\[Theta])\)\)\/r\^2\)], \ "Output"] }, Open ]], Cell[TextData[{ Cell[BoxData[ FormBox[ ButtonBox["return", ButtonData:>"conclusions", ButtonStyle->"Hyperlink"], TraditionalForm]]], " to conclusions" }], "Text"], Cell[TextData[{ "The 1/", Cell[BoxData[ \(TraditionalForm\`r\^2\)]], " dependence is fast enough that a regular peturbation method can be used \ to capture the first effects of inertia. " }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell["The first order problem which comes from perturbation theory", \ "Subtitle", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "\nNow suppose that we want to improve this by using a perturbation \ solution. If we examine the inertia (nonlinear) terms in the ", Cell[BoxData[ \(v\_\[Phi]\)]], " equation, we see that the 0 order solution for ", Cell[BoxData[ \(v\_\[Phi]\)]], " produces no correction terms at ", Cell[BoxData[ \(R\^1\)]], " order. The pure rotation solution that was obtained is an exact solution \ to the ", Cell[BoxData[ \(v\_\[Phi]\)]], "equation in the absence of ", Cell[BoxData[ \(v\_\(\(r\)\(\ \)\)\)]], "or ", Cell[BoxData[ \(v\_\[Theta]\)]], ". Thus we need to examine the ", Cell[BoxData[ \(v\_r\)]], " and ", Cell[BoxData[ \(v\_\[Theta]\)]], " equations to find the first improvement term. In the ", Cell[BoxData[ \(v\_r\)]], " equation, there is a -", Cell[BoxData[ \(v\_\[Phi]\)]], Cell[BoxData[ \(\^2\)]], "/", StyleBox["r", FontSlant->"Italic"], " term. In the \[Theta] equation, there is a similar term. We are seeing \ that rotation with finite ", StyleBox["R", FontSlant->"Italic"], " causes a weak centrifugal force that leads to a flow in the ", Cell[BoxData[ \(r\)]], " direction -- we suspect outward at the equator. To conserve mass, there \ is necessarily an inflow someplace, presumably at the point of weakest \ inertia, the poles. This flow at ", Cell[BoxData[ \(R\^1\)]], "requires both the ", Cell[BoxData[ \(v\_r\)]], " and ", Cell[BoxData[ \(v\_\[Theta]\)]], " equations to describe it. Let's see what we can do. \n\n\nHere is the \ axisymmetric (about the \[Phi] axis) ", Cell[BoxData[ \(v\_r\)]], " equation with the additional inertia term. This is produced by the \ expansion of the equations at ", Cell[BoxData[ \(R\^1\)]], " order. " }], "Text", Evaluatable->False, AspectRatioFixed->True, CellTags->"phicorrection"], Cell[TextData[{ Cell[BoxData[ FormBox[ ButtonBox["return", ButtonData:>"conclusions", ButtonStyle->"Hyperlink"], TraditionalForm]]], " to conclusions" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(vr1eq = \(-\(v\[Phi][r, \[Theta]]\^2\/r\)\) - \((\[PartialD]\_r\(\ \[PartialD]\_r\((vr[ r, \[Theta]]\ r\^2)\)\/r\^2\) + \[PartialD]\_\ \[Theta]\((Sin[\[Theta]]\ \[PartialD]\_\[Theta] vr[r, \[Theta]])\)\/\(r\^2\ \ Sin[\[Theta]]\) - \(2\ \[PartialD]\_\[Theta]\((v\[Theta][r, \[Theta]]\ Sin[\ \[Theta]])\)\)\/\(r\^2\ Sin[\[Theta]]\) - \[PartialD]\_r p[ r, \[Theta]])\)\)], "Input", AspectRatioFixed->True], Cell[BoxData[ FormBox[ RowBox[{\(General::"spell1"\), \(\(:\)\(\ \)\), "\<\"Possible spelling \ error: new symbol name \\\"\\!\\(TraditionalForm\\`v\[Theta]\\)\\\" is \ similar to existing symbol \\\"\\!\\(TraditionalForm\\`v\[Phi]\\)\\\".\"\>"}], TraditionalForm]], "Message"], Cell[BoxData[ FormBox[ RowBox[{\(-\(\(v\[Phi](r, \[Theta])\)\^2\/r\)\), "+", FractionBox[ RowBox[{"2", " ", \(csc(\[Theta])\), " ", RowBox[{"(", RowBox[{\(\(cos(\[Theta])\)\ \(v\[Theta](r, \[Theta])\)\), "+", RowBox[{\(sin(\[Theta])\), " ", RowBox[{ SuperscriptBox["v\[Theta]", TagBox[\((0, 1)\), Derivative], MultilineFunction->None], "(", \(r, \[Theta]\), ")"}]}]}], ")"}]}], \(r\^2\)], "-", FractionBox[ RowBox[{\(csc(\[Theta])\), " ", RowBox[{"(", RowBox[{ RowBox[{\(cos(\[Theta])\), " ", RowBox[{ SuperscriptBox["vr", TagBox[\((0, 1)\), Derivative], MultilineFunction->None], "(", \(r, \[Theta]\), ")"}]}], "+", RowBox[{\(sin(\[Theta])\), " ", RowBox[{ SuperscriptBox["vr", TagBox[\((0, 2)\), Derivative], MultilineFunction->None], "(", \(r, \[Theta]\), ")"}]}]}], ")"}]}], \(r\^2\)], "+", RowBox[{ SuperscriptBox["p", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "(", \(r, \[Theta]\), ")"}], "+", FractionBox[ RowBox[{"2", " ", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{ SuperscriptBox["vr", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "(", \(r, \[Theta]\), ")"}], " ", \(r\^2\)}], "+", \(2\ \(vr(r, \[Theta])\)\ r\)}], ")"}]}], \(r\^3\)], "-", FractionBox[ RowBox[{ RowBox[{ RowBox[{ SuperscriptBox["vr", TagBox[\((2, 0)\), Derivative], MultilineFunction->None], "(", \(r, \[Theta]\), ")"}], " ", \(r\^2\)}], "+", RowBox[{"4", " ", RowBox[{ SuperscriptBox["vr", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "(", \(r, \[Theta]\), ")"}], " ", "r"}], "+", \(2\ \(vr(r, \[Theta])\)\)}], \(r\^2\)]}], TraditionalForm]], "Output"] }, Open ]], Cell[TextData[{ "Here is the corresponding ", Cell[BoxData[ \(v\_\[Theta]\)]], " equation." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(v\[Theta]1eq = \(-\(\(v\[Phi][r, \[Theta]]\^2\ Cot[\[Theta]]\)\/r\)\) - \ \((\[PartialD]\_r\((r\^2\ \[PartialD]\_r v\[Theta][r, \[Theta]])\)\/r\^2 + \ \[PartialD]\_\[Theta]\(\[PartialD]\_\[Theta]\((v\[Theta][r, \[Theta]]\ Sin[\ \[Theta]])\)\/Sin[\[Theta]]\)\/r\^2 + \(2\ \[PartialD]\_\[Theta] vr[r, \ \[Theta]]\)\/r\^2 - \[PartialD]\_\[Theta] p[r, \[Theta]]\/r)\)\)], "Input", AspectRatioFixed->True], Cell[BoxData[ FormBox[ RowBox[{\(-\(\(\(cot(\[Theta])\)\ \(v\[Phi](r, \[Theta])\)\^2\)\/r\)\), "+", FractionBox[ RowBox[{ SuperscriptBox["p", TagBox[\((0, 1)\), Derivative], MultilineFunction->None], "(", \(r, \[Theta]\), ")"}], "r"], "-", FractionBox[ RowBox[{"2", " ", RowBox[{ SuperscriptBox["vr", TagBox[\((0, 1)\), Derivative], MultilineFunction->None], "(", \(r, \[Theta]\), ")"}]}], \(r\^2\)], "-", RowBox[{\(1\/r\^2\), RowBox[{"(", RowBox[{ RowBox[{\(csc(\[Theta])\), " ", RowBox[{"(", RowBox[{\(\(-\(sin(\[Theta])\)\)\ \(v\[Theta]( r, \[Theta])\)\), "+", RowBox[{"2", " ", \(cos(\[Theta])\), " ", RowBox[{ SuperscriptBox["v\[Theta]", TagBox[\((0, 1)\), Derivative], MultilineFunction->None], "(", \(r, \[Theta]\), ")"}]}], "+", RowBox[{\(sin(\[Theta])\), " ", RowBox[{ SuperscriptBox["v\[Theta]", TagBox[\((0, 2)\), Derivative], MultilineFunction->None], "(", \(r, \[Theta]\), ")"}]}]}], ")"}]}], "-", RowBox[{\(cot(\[Theta])\), " ", \(csc(\[Theta])\), " ", RowBox[{"(", RowBox[{\(\(cos(\[Theta])\)\ \(v\[Theta](r, \[Theta])\)\), "+", RowBox[{\(sin(\[Theta])\), " ", RowBox[{ SuperscriptBox["v\[Theta]", TagBox[\((0, 1)\), Derivative], MultilineFunction->None], "(", \(r, \[Theta]\), ")"}]}]}], ")"}]}]}], ")"}]}], "-", FractionBox[ RowBox[{ RowBox[{ RowBox[{ SuperscriptBox["v\[Theta]", TagBox[\((2, 0)\), Derivative], MultilineFunction->None], "(", \(r, \[Theta]\), ")"}], " ", \(r\^2\)}], "+", RowBox[{"2", " ", RowBox[{ SuperscriptBox["v\[Theta]", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "(", \(r, \[Theta]\), ")"}], " ", "r"}]}], \(r\^2\)]}], TraditionalForm]], "Output"] }, Open ]], Cell["\<\ Now we can derive the equation for the stream function easily \ enough, or could we do something else? (See Leal for a general method). Let's get the stream function \ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(eqO11 = Expand[\[PartialD]\_\[Theta] vr1eq - \[PartialD]\_r\((r\ v\[Theta]1eq)\)]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ FormBox[ RowBox[{\(-\(\(\(v\[Theta]( r, \[Theta])\)\ \(\(cot\^2\)(\[Theta])\)\)\/r\^2\)\), "+", FractionBox[ RowBox[{ RowBox[{ SuperscriptBox["vr", TagBox[\((0, 1)\), Derivative], MultilineFunction->None], "(", \(r, \[Theta]\), ")"}], " ", \(\(cot\^2\)(\[Theta])\)}], \(r\^2\)], "-", FractionBox[ RowBox[{ RowBox[{ SuperscriptBox["v\[Theta]", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "(", \(r, \[Theta]\), ")"}], " ", \(\(cot\^2\)(\[Theta])\)}], "r"], "+", FractionBox[ RowBox[{ RowBox[{ SuperscriptBox["v\[Theta]", TagBox[\((0, 1)\), Derivative], MultilineFunction->None], "(", \(r, \[Theta]\), ")"}], " ", \(cot(\[Theta])\)}], \(r\^2\)], "-", FractionBox[ RowBox[{ RowBox[{ SuperscriptBox["vr", TagBox[\((0, 2)\), 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SuperscriptBox["\[Psi]", TagBox[\((2, 0)\), Derivative], MultilineFunction->None], "(", \(r, \[Theta]\), ")"}], \(r\^2\)], "+", FractionBox[ RowBox[{"2", " ", \(cot(\[Theta])\), " ", RowBox[{ SuperscriptBox["\[Psi]", TagBox[\((2, 1)\), Derivative], MultilineFunction->None], "(", \(r, \[Theta]\), ")"}]}], \(r\^2\)], "-", FractionBox[ RowBox[{"2", " ", RowBox[{ SuperscriptBox["\[Psi]", TagBox[\((2, 2)\), Derivative], MultilineFunction->None], "(", \(r, \[Theta]\), ")"}]}], \(r\^2\)], "-", RowBox[{ SuperscriptBox["\[Psi]", TagBox[\((4, 0)\), Derivative], MultilineFunction->None], "(", \(r, \[Theta]\), ")"}]}], TraditionalForm]], "Output"] }, Open ]], Cell[TextData[{ "Now we have a PDE that we need to solve. The problem is, how?\nOne way \ is to guess the function form for the \[Theta] dependence. The boundary \ relation for the 0 order problem is clear but does readily not give insight \ into the first order problem. Of course we could just guess and try. But \ what is this equation really?\n\nIt should be an axisymmetric creeping flow \ problem with an inhomogeneous term\n\n", Cell[BoxData[ \(E\^4\)]], " \[Psi] = 6 ", Cell[BoxData[ \(a\^6\)]], " ", Cell[BoxData[ \(\[CapitalOmega]\^2\)]], " Cos[\[Theta]] ", Cell[BoxData[ \(Sin[\[Theta]]\^2\)]], "/", Cell[BoxData[ \(r\^5\)]], ".\n\nWe know the general solutions to this equation. We would need a \ particular equation. The boundary conditions for the rotating problem are \ all fit at 0 order so the homogenous solution and the particular solution \ must cancel on the boundary for this perturbation correction to work. \n\n\ Now that we have some familiarity, we could translate using \ \[Eta]\[Congruent]cos(\[Theta]) (see page 160 of Leal) and look at the \ general solutions. A second way would be to look at the \[Eta] solutions in \ terms of Gegenbauer polynomials for the expected form of the \[Theta] \ dependence(see page 163). What are the choices:\nCos[\[Theta]] + 1, which is \ not in the general solution." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["The next one is", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Simplify[\((Cos[\[Theta]]^2 - 1)\)/2]\)], "Input"], Cell[BoxData[ \(TraditionalForm\`\(-\(1\/2\)\)\ \(\(sin\^2\)(\[Theta])\)\)], "Output"] }, Open ]], Cell["It does not look promising. Continue on with the third,", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Simplify[Cos[\[Theta]]^3 - Cos[\[Theta]]]\)], "Input"], Cell[BoxData[ \(TraditionalForm\`\(-\(cos(\[Theta])\)\)\ \(\(sin\^2\)(\[Theta])\)\)], \ "Output"] }, Open ]], Cell[TextData[ "This looks great!! It has the same form as the inhomogeneous term. Try this \ form for the \[Theta] dependence."], "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"eqO15", "=", RowBox[{"Expand", "[", RowBox[{"eqO14", "/.", RowBox[{"{", RowBox[{ RowBox[{ SuperscriptBox["\[Psi]", TagBox[\((a1_, a2_)\), Derivative], MultilineFunction->None], "[", \(r, \[Theta]\), "]"}], "\[RuleDelayed]", \(\[PartialD]\_\({r, a1}, {\[Theta], \ a2}\)\((Sin[\[Theta]]\^2\ Cos[\[Theta]]\ f1[r])\)\)}], "}"}]}], "]"}]}]], "Input", AspectRatioFixed->True], Cell[BoxData[ FormBox[ RowBox[{\(-\(\(6\ \[CapitalOmega]\^2\ \(cos(\[Theta])\)\ \(\(sin\^2\)(\ \[Theta])\)\ a\^6\)\/r\^5\)\), "-", \(\(17\ \(cos(\[Theta])\)\ \(f1( r)\)\ \(\(sin\^2\)(\[Theta])\)\)\/r\^4\), "-", \(\(19\ \(\(cos\^3\)(\[Theta])\)\ \(f1(r)\)\)\/r\^4\), "-", \(\(2\ \(\(cos\^3\)(\[Theta])\)\ \(\(cot\^2\)(\[Theta])\)\ \(f1( r)\)\)\/r\^4\), "+", \(\(2\ \(cos(\[Theta])\)\ \(\(cot\^2\)(\[Theta])\)\ \(f1( r)\)\)\/r\^4\), "+", \(\(17\ \(cos(\[Theta])\)\ \(f1(r)\)\)\/r\^4\), "-", FractionBox[ RowBox[{ "24", " ", \(cos(\[Theta])\), " ", \(\(sin\^2\)(\[Theta])\), " ", RowBox[{ SuperscriptBox["f1", "\[Prime]", MultilineFunction->None], "(", "r", ")"}]}], \(r\^3\)], "+", FractionBox[ RowBox[{\(\(cos\^3\)(\[Theta])\), " ", RowBox[{ SuperscriptBox["f1", "\[Prime]\[Prime]", MultilineFunction->None], "(", "r", ")"}]}], \(r\^2\)], "+", FractionBox[ RowBox[{ "13", " ", \(cos(\[Theta])\), " ", \(\(sin\^2\)(\[Theta])\), " ", RowBox[{ SuperscriptBox["f1", "\[Prime]\[Prime]", MultilineFunction->None], "(", "r", ")"}]}], \(r\^2\)], "-", FractionBox[ RowBox[{\(cos(\[Theta])\), " ", RowBox[{ SuperscriptBox["f1", "\[Prime]\[Prime]", MultilineFunction->None], "(", "r", ")"}]}], \(r\^2\)], "-", RowBox[{\(cos(\[Theta])\), " ", \(\(sin\^2\)(\[Theta])\), " ", RowBox[{ SuperscriptBox["f1", TagBox[\((4)\), Derivative], MultilineFunction->None], "(", "r", ")"}]}]}], TraditionalForm]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(eqO16 = \ Simplify[eqO15]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ FormBox[ RowBox[{"-", FractionBox[ RowBox[{\(cos(\[Theta])\), " ", \(\(sin\^2\)(\[Theta])\), " ", RowBox[{"(", RowBox[{\(6\ \[CapitalOmega]\^2\ a\^6\), "+", RowBox[{"24", " ", \(r\^2\), " ", RowBox[{ SuperscriptBox["f1", "\[Prime]", MultilineFunction->None], "(", "r", ")"}]}], "-", RowBox[{"12", " ", \(r\^3\), " ", RowBox[{ SuperscriptBox["f1", "\[Prime]\[Prime]", MultilineFunction->None], "(", "r", ")"}]}], "+", RowBox[{\(r\^5\), " ", RowBox[{ SuperscriptBox["f1", TagBox[\((4)\), Derivative], MultilineFunction->None], "(", "r", ")"}]}]}], ")"}]}], \(r\^5\)]}], TraditionalForm]], "Output"] }, Open ]], Cell["By knowing what we want we can simplify the equation, ", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(eqO17 = \ Expand[\(r\^4\ eqO16\)\/\(Cos[\[Theta]]\ Sin[\[Theta]]\^2\)]\)], "Input",\ AspectRatioFixed->True], Cell[BoxData[ FormBox[ RowBox[{\(-\(\(6\ \[CapitalOmega]\^2\ a\^6\)\/r\)\), "-", RowBox[{"24", " ", "r", " ", RowBox[{ SuperscriptBox["f1", "\[Prime]", MultilineFunction->None], "(", "r", ")"}]}], "+", RowBox[{"12", " ", \(r\^2\), " ", RowBox[{ SuperscriptBox["f1", "\[Prime]\[Prime]", MultilineFunction->None], "(", "r", ")"}]}], "-", RowBox[{\(r\^4\), " ", RowBox[{ SuperscriptBox["f1", TagBox[\((4)\), Derivative], MultilineFunction->None], "(", "r", ")"}]}]}], TraditionalForm]], "Output"] }, Open ]], Cell["The equation is Euler's equation which is easily solved.", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(ans011 = DSolve[eqO17 == 0, f1[r], r]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ FormBox[ RowBox[{"{", RowBox[{"{", RowBox[{\(f1(r)\), "\[Rule]", RowBox[{\(\(\[CapitalOmega]\^2\ a\^6\)\/\(4\ r\)\), "+", FractionBox[ SubscriptBox[ TagBox["c", C], "1"], \(r\^2\)], "+", SubscriptBox[ TagBox["c", C], "2"], "+", RowBox[{\(r\^3\), " ", SubscriptBox[ TagBox["c", C], "3"]}], "+", RowBox[{\(r\^5\), " ", SubscriptBox[ TagBox["c", C], "4"]}]}]}], "}"}], "}"}], TraditionalForm]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(ans012 = f1[r] /. ans011\[LeftDoubleBracket]1\[RightDoubleBracket]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ FormBox[ RowBox[{\(\(\[CapitalOmega]\^2\ a\^6\)\/\(4\ r\)\), "+", FractionBox[ SubscriptBox[ TagBox["c", C], "1"], \(r\^2\)], "+", SubscriptBox[ TagBox["c", C], "2"], "+", RowBox[{\(r\^3\), " ", SubscriptBox[ TagBox["c", C], "3"]}], "+", RowBox[{\(r\^5\), " ", SubscriptBox[ TagBox["c", C], "4"]}]}], TraditionalForm]], "Output"] }, Open ]], Cell[TextData[{ "As mentioned earlier, we need the ", Cell[BoxData[ \(R\^1\)]], " solution to be 0 on the boundaries. As r -->\[Infinity] this requires \ that C[3], C[4] =0. " }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(ans013 = ans012 /. {C[3] \[Rule] 0, C[4] \[Rule] 0}\)], "Input", AspectRatioFixed->True], Cell[BoxData[ FormBox[ RowBox[{\(\(\[CapitalOmega]\^2\ a\^6\)\/\(4\ r\)\), "+", FractionBox[ SubscriptBox[ TagBox["c", C], "1"], \(r\^2\)], "+", SubscriptBox[ TagBox["c", C], "2"]}], TraditionalForm]], "Output"] }, Open ]], Cell[TextData["Now fit the boundary conditions on the sphere"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(bc1 = ans013\/r /. r \[Rule] a\)], "Input", AspectRatioFixed->True], Cell[BoxData[ FormBox[ FractionBox[ RowBox[{\(\(\[CapitalOmega]\^2\ a\^5\)\/4\), "+", SubscriptBox[ TagBox["c", C], "2"], "+", FractionBox[ SubscriptBox[ TagBox["c", C], "1"], \(a\^2\)]}], "a"], TraditionalForm]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(bc2 = \(-\(\[PartialD]\_r ans013\/r\^2\)\) /. r \[Rule] a\)], "Input", AspectRatioFixed->True], Cell[BoxData[ FormBox[ RowBox[{"-", FractionBox[ RowBox[{\(\(-\(1\/4\)\)\ \[CapitalOmega]\^2\ a\^4\), "-", FractionBox[ RowBox[{"2", " ", SubscriptBox[ TagBox["c", C], "1"]}], \(a\^3\)]}], \(a\^2\)]}], TraditionalForm]], "Output"] }, Open ]], Cell[TextData["Everything is 0 on the boundaries"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(constsO1 = Solve[{bc1 == 0, bc2 == 0}, {C[1], C[2]}]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ FormBox[ RowBox[{"{", RowBox[{"{", RowBox[{ RowBox[{ SubscriptBox[ TagBox["c", C], "2"], "\[Rule]", \(\(-\(1\/8\)\)\ a\^5\ \[CapitalOmega]\^2\)}], ",", RowBox[{ SubscriptBox[ TagBox["c", C], "1"], "\[Rule]", \(\(-\(1\/8\)\)\ a\^7\ \[CapitalOmega]\^2\)}]}], "}"}], "}"}], TraditionalForm]], "Output"] }, Open ]], Cell[TextData[{ "Thus, the stream function for ", Cell[BoxData[ \(TraditionalForm\`R\^1\)]], " order is" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(ansO1last = \((ans013 /. constsO1\[LeftDoubleBracket]1\[RightDoubleBracket])\)\ Sin[\ \[Theta]]\^2\ Cos[\[Theta]]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(TraditionalForm\`\((\(-\(\(\[CapitalOmega]\^2\ a\^7\)\/\(8\ r\^2\)\)\) \ + \(\[CapitalOmega]\^2\ a\^6\)\/\(4\ r\) - \(\[CapitalOmega]\^2\ a\^5\)\/8)\)\ \ \(cos(\[Theta])\)\ \(\(sin\^2\)(\[Theta])\)\)], "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Examine the first order solution graphically. ", "Subtitle"], Cell["Make the requisite polar to Cartesian transformation,", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(ansO1Cartesian = ansO1last /. {r \[Rule] \@\(x\^2 + y\^2\), \[Theta] \[Rule] ArcTan[y\/x]}\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(TraditionalForm\`\(y\^2\ \((\(-\(\(\[CapitalOmega]\^2\ a\^7\)\/\(8\ \ \((x\^2 + y\^2)\)\)\)\) + \(\[CapitalOmega]\^2\ a\^6\)\/\(4\ \@\(x\^2 + \ y\^2\)\) - \(\[CapitalOmega]\^2\ a\^5\)\/8)\)\)\/\(x\^2\ \((y\^2\/x\^2 + 1)\)\ \^\(3/2\)\)\)], "Output"] }, Open ]], Cell["Substitute in some numbers,", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(anstoplot = ansO1Cartesian /. {\[CapitalOmega] \[Rule] 5, a \[Rule] 1}\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(TraditionalForm\`\(y\^2\ \((\(-\(25\/8\)\) + 25\/\(4\ \@\(x\^2 + y\^2\)\ \) - 25\/\(8\ \((x\^2 + y\^2)\)\))\)\)\/\(x\^2\ \((y\^2\/x\^2 + 1)\)\^\(3/2\)\ \)\)], "Output"], 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L .45561 .9219 L .45409 .94152 L Mistroke .45255 .96115 L .45101 .98077 L Mfstroke .8 g .01923 .29811 m .03885 .30529 L .05848 .31245 L .06157 .31358 L .0781 .3196 L .09772 .32673 L .11557 .3332 L .11735 .33385 L .13697 .34093 L .15659 .348 L .17005 .35283 L .17622 .35503 L .19584 .36201 L .21546 .36895 L .22541 .37245 L .23509 .37582 L .25471 .38261 L .27433 .38929 L .28262 .39207 L .29396 .39582 L .31358 .40213 L .3332 .40814 L .34551 .4117 L .35283 .41367 L .37245 .41842 L .39207 .42184 L .4117 .4227 L .43132 .41842 L .4441 .4117 L .45094 .4016 L .45507 .39207 L .45924 .37245 L .46038 .35283 L .46021 .3332 L .45936 .31358 L .45812 .29396 L .45662 .27433 L .45495 .25471 L .45316 .23509 L .45128 .21546 L .45094 .21202 L .44934 .19584 L .44735 .17622 L .44532 .15659 L .44325 .13697 L .44116 .11735 L .43904 .09772 L .43691 .0781 L .43476 .05848 L .43259 .03885 L .43132 .02734 L .43042 .01923 L .01923 .01923 L F 0 g .01923 .29811 m .03885 .30529 L .05848 .31245 L .06157 .31358 L .0781 .3196 L .09772 .32673 L .11557 .3332 L .11735 .33385 L .13697 .34093 L .15659 .348 L .17005 .35283 L .17622 .35503 L .19584 .36201 L .21546 .36895 L .22541 .37245 L .23509 .37582 L .25471 .38261 L .27433 .38929 L .28262 .39207 L .29396 .39582 L .31358 .40213 L .3332 .40814 L .34551 .4117 L .35283 .41367 L .37245 .41842 L .39207 .42184 L .4117 .4227 L .43132 .41842 L .4441 .4117 L .45094 .4016 L .45507 .39207 L .45924 .37245 L .46038 .35283 L .46021 .3332 L .45936 .31358 L .45812 .29396 L .45662 .27433 L .45495 .25471 L .45316 .23509 L .45128 .21546 L .45094 .21202 L .44934 .19584 L .44735 .17622 L .44532 .15659 L .44325 .13697 L .44116 .11735 L .43904 .09772 L .43691 .0781 L .43476 .05848 L .43259 .03885 L Mistroke .43132 .02734 L .43042 .01923 L Mfstroke .8 g .01923 .70189 m .03885 .69471 L .05848 .68755 L .06157 .68642 L .0781 .6804 L .09772 .67327 L .11557 .6668 L .11735 .66615 L .13697 .65907 L .15659 .652 L .17005 .64717 L .17622 .64497 L .19584 .63799 L .21546 .63105 L .22541 .62755 L .23509 .62418 L .25471 .61739 L .27433 .61071 L .28262 .60793 L .29396 .60418 L .31358 .59787 L .3332 .59186 L .34551 .5883 L .35283 .58633 L .37245 .58158 L .39207 .57816 L .4117 .5773 L .43132 .58158 L .4441 .5883 L .45094 .5984 L .45507 .60793 L .45924 .62755 L .46038 .64717 L .46021 .6668 L .45936 .68642 L .45812 .70604 L .45662 .72567 L .45495 .74529 L .45316 .76491 L .45128 .78454 L .45094 .78798 L .44934 .80416 L .44735 .82378 L .44532 .84341 L .44325 .86303 L .44116 .88265 L .43904 .90228 L .43691 .9219 L .43476 .94152 L .43259 .96115 L .43132 .97266 L .43042 .98077 L .01923 .98077 L F 0 g .01923 .70189 m .03885 .69471 L .05848 .68755 L .06157 .68642 L .0781 .6804 L .09772 .67327 L .11557 .6668 L .11735 .66615 L .13697 .65907 L .15659 .652 L .17005 .64717 L .17622 .64497 L .19584 .63799 L .21546 .63105 L .22541 .62755 L .23509 .62418 L .25471 .61739 L .27433 .61071 L .28262 .60793 L .29396 .60418 L .31358 .59787 L .3332 .59186 L .34551 .5883 L .35283 .58633 L .37245 .58158 L .39207 .57816 L .4117 .5773 L .43132 .58158 L .4441 .5883 L .45094 .5984 L .45507 .60793 L .45924 .62755 L .46038 .64717 L .46021 .6668 L .45936 .68642 L .45812 .70604 L .45662 .72567 L .45495 .74529 L .45316 .76491 L .45128 .78454 L .45094 .78798 L .44934 .80416 L .44735 .82378 L .44532 .84341 L .44325 .86303 L .44116 .88265 L .43904 .90228 L .43691 .9219 L .43476 .94152 L .43259 .96115 L Mistroke .43132 .97266 L .43042 .98077 L Mfstroke .75 g .01923 .73828 m .03885 .72988 L .04872 .72567 L .05848 .7215 L .0781 .71314 L .0948 .70604 L .09772 .7048 L .11735 .69649 L .13697 .68821 L .14122 .68642 L .15659 .67996 L .17622 .67176 L .18815 .6668 L .19584 .66361 L .21546 .65553 L .23509 .64754 L .23599 .64717 L .25471 .63966 L .27433 .63192 L .28565 .62755 L .29396 .62439 L .31358 .61714 L .3332 .61031 L .34036 .60793 L .35283 .60412 L .37245 .59895 L .39207 .59565 L .4117 .59598 L .43132 .60542 L .43391 .60793 L .44347 .62755 L .446 .64717 L .44633 .6668 L .44556 .68642 L .44415 .70604 L .44235 .72567 L .44028 .74529 L .43802 .76491 L .43563 .78454 L .43314 .80416 L .43132 .8182 L .43059 .82378 L .42797 .84341 L .42531 .86303 L .42261 .88265 L .41987 .90228 L .4171 .9219 L .41432 .94152 L .4117 .95986 L .41151 .96115 L .40869 .98077 L .01923 .98077 L F 0 g .01923 .73828 m .03885 .72988 L .04872 .72567 L .05848 .7215 L .0781 .71314 L .0948 .70604 L .09772 .7048 L .11735 .69649 L .13697 .68821 L .14122 .68642 L .15659 .67996 L .17622 .67176 L .18815 .6668 L .19584 .66361 L .21546 .65553 L .23509 .64754 L .23599 .64717 L .25471 .63966 L .27433 .63192 L .28565 .62755 L .29396 .62439 L .31358 .61714 L .3332 .61031 L .34036 .60793 L .35283 .60412 L .37245 .59895 L .39207 .59565 L .4117 .59598 L .43132 .60542 L .43391 .60793 L .44347 .62755 L .446 .64717 L .44633 .6668 L .44556 .68642 L .44415 .70604 L .44235 .72567 L .44028 .74529 L .43802 .76491 L .43563 .78454 L .43314 .80416 L .43132 .8182 L .43059 .82378 L .42797 .84341 L .42531 .86303 L .42261 .88265 L .41987 .90228 L .4171 .9219 L .41432 .94152 L .4117 .95986 L .41151 .96115 L Mistroke .40869 .98077 L Mfstroke .75 g .01923 .26172 m .03885 .27012 L .04872 .27433 L .05848 .2785 L .0781 .28686 L .0948 .29396 L .09772 .2952 L .11735 .30351 L .13697 .31179 L .14122 .31358 L .15659 .32004 L .17622 .32824 L .18815 .3332 L .19584 .33639 L .21546 .34447 L .23509 .35246 L .23599 .35283 L .25471 .36034 L .27433 .36808 L .28565 .37245 L .29396 .37561 L .31358 .38286 L .3332 .38969 L .34036 .39207 L .35283 .39588 L .37245 .40105 L .39207 .40435 L .4117 .40402 L .43132 .39458 L .43391 .39207 L .44347 .37245 L .446 .35283 L .44633 .3332 L .44556 .31358 L .44415 .29396 L .44235 .27433 L .44028 .25471 L .43802 .23509 L .43563 .21546 L .43314 .19584 L .43132 .1818 L .43059 .17622 L .42797 .15659 L .42531 .13697 L .42261 .11735 L .41987 .09772 L .4171 .0781 L .41432 .05848 L .4117 .04014 L .41151 .03885 L .40869 .01923 L .01923 .01923 L F 0 g .01923 .26172 m .03885 .27012 L .04872 .27433 L .05848 .2785 L .0781 .28686 L .0948 .29396 L .09772 .2952 L .11735 .30351 L .13697 .31179 L .14122 .31358 L .15659 .32004 L .17622 .32824 L .18815 .3332 L .19584 .33639 L .21546 .34447 L .23509 .35246 L .23599 .35283 L .25471 .36034 L .27433 .36808 L .28565 .37245 L .29396 .37561 L .31358 .38286 L .3332 .38969 L .34036 .39207 L .35283 .39588 L .37245 .40105 L .39207 .40435 L .4117 .40402 L .43132 .39458 L .43391 .39207 L .44347 .37245 L .446 .35283 L .44633 .3332 L .44556 .31358 L .44415 .29396 L .44235 .27433 L .44028 .25471 L .43802 .23509 L .43563 .21546 L .43314 .19584 L .43132 .1818 L .43059 .17622 L .42797 .15659 L .42531 .13697 L .42261 .11735 L .41987 .09772 L .4171 .0781 L .41432 .05848 L .4117 .04014 L .41151 .03885 L Mistroke .40869 .01923 L Mfstroke .7 g .01923 .77563 m .03885 .76603 L .04113 .76491 L .05848 .75645 L .0781 .74689 L .08139 .74529 L .09772 .73736 L .11735 .72787 L .12192 .72567 L .13697 .71842 L .15659 .70902 L .16283 .70604 L .17622 .69968 L .19584 .69041 L .20435 .68642 L .21546 .68123 L .23509 .67217 L .24689 .6668 L .25471 .66327 L .27433 .65456 L .29153 .64717 L .29396 .64615 L .31358 .63811 L .3332 .63068 L .34185 .62755 L .35283 .62419 L .37245 .61922 L .39207 .61725 L .4117 .62248 L .41902 .62755 L .42731 .64717 L .42922 .6668 L .42914 .68642 L .42795 .70604 L .42606 .72567 L .42373 .74529 L .4211 .76491 L .41826 .78454 L .41527 .80416 L .41216 .82378 L .4117 .82668 L .40898 .84341 L .40573 .86303 L .40241 .88265 L .39905 .90228 L .39564 .9219 L .3922 .94152 L .39207 .94226 L .38874 .96115 L .38526 .98077 L .01923 .98077 L F 0 g .01923 .77563 m .03885 .76603 L .04113 .76491 L .05848 .75645 L .0781 .74689 L .08139 .74529 L .09772 .73736 L .11735 .72787 L .12192 .72567 L .13697 .71842 L .15659 .70902 L .16283 .70604 L .17622 .69968 L .19584 .69041 L .20435 .68642 L .21546 .68123 L .23509 .67217 L .24689 .6668 L .25471 .66327 L .27433 .65456 L .29153 .64717 L .29396 .64615 L .31358 .63811 L .3332 .63068 L .34185 .62755 L .35283 .62419 L .37245 .61922 L .39207 .61725 L .4117 .62248 L .41902 .62755 L .42731 .64717 L .42922 .6668 L .42914 .68642 L .42795 .70604 L .42606 .72567 L .42373 .74529 L .4211 .76491 L .41826 .78454 L .41527 .80416 L .41216 .82378 L .4117 .82668 L .40898 .84341 L .40573 .86303 L .40241 .88265 L .39905 .90228 L .39564 .9219 L .3922 .94152 L .39207 .94226 L .38874 .96115 L .38526 .98077 L Mistroke Mfstroke .7 g .01923 .22437 m .03885 .23397 L .04113 .23509 L .05848 .24355 L .0781 .25311 L .08139 .25471 L .09772 .26264 L .11735 .27213 L .12192 .27433 L .13697 .28158 L .15659 .29098 L .16283 .29396 L .17622 .30032 L .19584 .30959 L .20435 .31358 L .21546 .31877 L .23509 .32783 L .24689 .3332 L .25471 .33673 L .27433 .34544 L .29153 .35283 L .29396 .35385 L .31358 .36189 L .3332 .36932 L .34185 .37245 L .35283 .37581 L .37245 .38078 L .39207 .38275 L .4117 .37752 L .41902 .37245 L .42731 .35283 L .42922 .3332 L .42914 .31358 L .42795 .29396 L .42606 .27433 L .42373 .25471 L .4211 .23509 L .41826 .21546 L .41527 .19584 L .41216 .17622 L .4117 .17332 L .40898 .15659 L .40573 .13697 L .40241 .11735 L .39905 .09772 L .39564 .0781 L .3922 .05848 L .39207 .05774 L .38874 .03885 L .38526 .01923 L .01923 .01923 L F 0 g .01923 .22437 m .03885 .23397 L .04113 .23509 L .05848 .24355 L .0781 .25311 L .08139 .25471 L .09772 .26264 L .11735 .27213 L .12192 .27433 L .13697 .28158 L .15659 .29098 L .16283 .29396 L .17622 .30032 L .19584 .30959 L .20435 .31358 L .21546 .31877 L .23509 .32783 L .24689 .3332 L .25471 .33673 L .27433 .34544 L .29153 .35283 L .29396 .35385 L .31358 .36189 L .3332 .36932 L .34185 .37245 L .35283 .37581 L .37245 .38078 L .39207 .38275 L .4117 .37752 L .41902 .37245 L .42731 .35283 L .42922 .3332 L .42914 .31358 L .42795 .29396 L .42606 .27433 L .42373 .25471 L .4211 .23509 L .41826 .21546 L .41527 .19584 L .41216 .17622 L .4117 .17332 L .40898 .15659 L .40573 .13697 L .40241 .11735 L .39905 .09772 L .39564 .0781 L .3922 .05848 L .39207 .05774 L .38874 .03885 L .38526 .01923 L Mistroke Mfstroke .65 g .01923 .18426 m .03885 .19509 L .04022 .19584 L .05848 .20589 L .07592 .21546 L .0781 .21666 L .09772 .22738 L .11187 .23509 L .11735 .23806 L .13697 .24868 L .14817 .25471 L .15659 .25923 L .17622 .2697 L .18498 .27433 L .19584 .28006 L .21546 .29028 L .2226 .29396 L .23509 .30034 L .25471 .31016 L .26166 .31358 L .27433 .31969 L .29396 .32879 L .30387 .3332 L .31358 .33726 L .3332 .34481 L .35283 .35069 L .36421 .35283 L .37245 .35348 L .38128 .35283 L .39207 .34912 L .40478 .3332 L .40746 .31358 L .40748 .29396 L .40611 .27433 L .40389 .25471 L .40113 .23509 L .398 .21546 L .39461 .19584 L .39207 .18179 L .39105 .17622 L .38737 .15659 L .38355 .13697 L .37964 .11735 L .37565 .09772 L .37245 .08217 L .37161 .0781 L .36753 .05848 L .3634 .03885 L .35923 .01923 L .01923 .01923 L F 0 g .01923 .18426 m .03885 .19509 L .04022 .19584 L .05848 .20589 L .07592 .21546 L .0781 .21666 L .09772 .22738 L .11187 .23509 L .11735 .23806 L .13697 .24868 L .14817 .25471 L .15659 .25923 L .17622 .2697 L .18498 .27433 L .19584 .28006 L .21546 .29028 L .2226 .29396 L .23509 .30034 L .25471 .31016 L .26166 .31358 L .27433 .31969 L .29396 .32879 L .30387 .3332 L .31358 .33726 L .3332 .34481 L .35283 .35069 L .36421 .35283 L .37245 .35348 L .38128 .35283 L .39207 .34912 L .40478 .3332 L .40746 .31358 L .40748 .29396 L .40611 .27433 L .40389 .25471 L .40113 .23509 L .398 .21546 L .39461 .19584 L .39207 .18179 L .39105 .17622 L .38737 .15659 L .38355 .13697 L .37964 .11735 L .37565 .09772 L .37245 .08217 L .37161 .0781 L .36753 .05848 L .3634 .03885 L .35923 .01923 L s .65 g .01923 .81574 m .03885 .80491 L .04022 .80416 L .05848 .79411 L .07592 .78454 L .0781 .78334 L .09772 .77262 L .11187 .76491 L .11735 .76194 L .13697 .75132 L .14817 .74529 L .15659 .74077 L .17622 .7303 L .18498 .72567 L .19584 .71994 L .21546 .70972 L .2226 .70604 L .23509 .69966 L .25471 .68984 L .26166 .68642 L .27433 .68031 L .29396 .67121 L .30387 .6668 L .31358 .66274 L .3332 .65519 L .35283 .64931 L .36421 .64717 L .37245 .64652 L .38128 .64717 L .39207 .65088 L .40478 .6668 L .40746 .68642 L .40748 .70604 L .40611 .72567 L .40389 .74529 L .40113 .76491 L .398 .78454 L .39461 .80416 L .39207 .81821 L .39105 .82378 L .38737 .84341 L .38355 .86303 L .37964 .88265 L .37565 .90228 L .37245 .91783 L .37161 .9219 L .36753 .94152 L .3634 .96115 L .35923 .98077 L .01923 .98077 L F 0 g .01923 .81574 m .03885 .80491 L .04022 .80416 L .05848 .79411 L .07592 .78454 L .0781 .78334 L .09772 .77262 L .11187 .76491 L .11735 .76194 L .13697 .75132 L .14817 .74529 L .15659 .74077 L .17622 .7303 L .18498 .72567 L .19584 .71994 L .21546 .70972 L .2226 .70604 L .23509 .69966 L .25471 .68984 L .26166 .68642 L .27433 .68031 L .29396 .67121 L .30387 .6668 L .31358 .66274 L .3332 .65519 L .35283 .64931 L .36421 .64717 L .37245 .64652 L .38128 .64717 L .39207 .65088 L .40478 .6668 L .40746 .68642 L .40748 .70604 L .40611 .72567 L .40389 .74529 L .40113 .76491 L .398 .78454 L .39461 .80416 L .39207 .81821 L .39105 .82378 L .38737 .84341 L .38355 .86303 L .37964 .88265 L .37565 .90228 L .37245 .91783 L .37161 .9219 L .36753 .94152 L .3634 .96115 L .35923 .98077 L s .6 g .01923 .13895 m .03885 .15106 L .04785 .15659 L .05848 .16312 L .0781 .17513 L .07989 .17622 L .09772 .18707 L .11221 .19584 L .11735 .19894 L .13697 .21072 L .14492 .21546 L .15659 .2224 L .17622 .23393 L .1782 .23509 L .19584 .2453 L .21239 .25471 L .21546 .25644 L .23509 .26732 L .24813 .27433 L .25471 .27778 L .27433 .28773 L .28733 .29396 L .29396 .29685 L .31358 .30478 L .3332 .31048 L .35283 .31163 L .37245 .29622 L .37329 .29396 L .37697 .27433 L .37671 .25471 L .37484 .23509 L .37245 .21741 L .37209 .21546 L .36884 .19584 L .36513 .17622 L .36111 .15659 L .35686 .13697 L .35283 .11898 L .35246 .11735 L .34796 .09772 L .34334 .0781 L .33862 .05848 L .33383 .03885 L .3332 .0363 L .329 .01923 L .01923 .01923 L F 0 g .01923 .13895 m .03885 .15106 L .04785 .15659 L .05848 .16312 L .0781 .17513 L .07989 .17622 L .09772 .18707 L .11221 .19584 L .11735 .19894 L .13697 .21072 L .14492 .21546 L .15659 .2224 L .17622 .23393 L .1782 .23509 L .19584 .2453 L .21239 .25471 L .21546 .25644 L .23509 .26732 L .24813 .27433 L .25471 .27778 L .27433 .28773 L .28733 .29396 L .29396 .29685 L .31358 .30478 L .3332 .31048 L .35283 .31163 L .37245 .29622 L .37329 .29396 L .37697 .27433 L .37671 .25471 L .37484 .23509 L .37245 .21741 L .37209 .21546 L .36884 .19584 L .36513 .17622 L .36111 .15659 L .35686 .13697 L .35283 .11898 L .35246 .11735 L .34796 .09772 L .34334 .0781 L .33862 .05848 L .33383 .03885 L .3332 .0363 L .329 .01923 L s .6 g .01923 .86105 m .03885 .84894 L .04785 .84341 L .05848 .83688 L .0781 .82487 L .07989 .82378 L .09772 .81293 L .11221 .80416 L .11735 .80106 L .13697 .78928 L .14492 .78454 L .15659 .7776 L .17622 .76607 L .1782 .76491 L .19584 .7547 L .21239 .74529 L .21546 .74356 L .23509 .73268 L .24813 .72567 L .25471 .72222 L .27433 .71227 L .28733 .70604 L .29396 .70315 L .31358 .69522 L .3332 .68952 L .35283 .68837 L .37245 .70378 L .37329 .70604 L .37697 .72567 L .37671 .74529 L .37484 .76491 L .37245 .78259 L .37209 .78454 L .36884 .80416 L .36513 .82378 L .36111 .84341 L .35686 .86303 L .35283 .88102 L .35246 .88265 L .34796 .90228 L .34334 .9219 L .33862 .94152 L .33383 .96115 L .3332 .9637 L .329 .98077 L .01923 .98077 L F 0 g .01923 .86105 m .03885 .84894 L .04785 .84341 L .05848 .83688 L .0781 .82487 L .07989 .82378 L .09772 .81293 L .11221 .80416 L .11735 .80106 L .13697 .78928 L .14492 .78454 L .15659 .7776 L .17622 .76607 L .1782 .76491 L .19584 .7547 L .21239 .74529 L .21546 .74356 L .23509 .73268 L .24813 .72567 L .25471 .72222 L .27433 .71227 L .28733 .70604 L .29396 .70315 L .31358 .69522 L .3332 .68952 L .35283 .68837 L .37245 .70378 L .37329 .70604 L .37697 .72567 L .37671 .74529 L .37484 .76491 L .37245 .78259 L .37209 .78454 L .36884 .80416 L .36513 .82378 L .36111 .84341 L .35686 .86303 L .35283 .88102 L .35246 .88265 L .34796 .90228 L .34334 .9219 L .33862 .94152 L .33383 .96115 L .3332 .9637 L .329 .98077 L s .55 g .01923 .08407 m .03885 .09748 L .03922 .09772 L .05848 .11081 L .06816 .11735 L .0781 .12404 L .09746 .13697 L .09772 .13715 L .11735 .15013 L .12721 .15659 L .13697 .16294 L .15659 .17553 L .15768 .17622 L .17622 .18786 L .18921 .19584 L .19584 .19983 L .21546 .21132 L .22266 .21546 L .23509 .22215 L .25471 .23199 L .25907 .23509 L .27433 .24027 L .29396 .24581 L .31358 .24488 L .32402 .23509 L .33014 .21546 L .3297 .19584 L .32761 .17622 L .32445 .15659 L .32058 .13697 L .31625 .11735 L .31358 .10593 L .31162 .09772 L .30676 .0781 L .30167 .05848 L .2964 .03885 L .29396 .02995 L .29103 .01923 L .01923 .01923 L F 0 g .01923 .08407 m .03885 .09748 L .03922 .09772 L .05848 .11081 L .06816 .11735 L .0781 .12404 L .09746 .13697 L .09772 .13715 L .11735 .15013 L .12721 .15659 L .13697 .16294 L .15659 .17553 L .15768 .17622 L .17622 .18786 L .18921 .19584 L .19584 .19983 L .21546 .21132 L .22266 .21546 L .23509 .22215 L .25471 .23199 L .25907 .23509 L .27433 .24027 L .29396 .24581 L .31358 .24488 L .32402 .23509 L .33014 .21546 L .3297 .19584 L .32761 .17622 L .32445 .15659 L .32058 .13697 L .31625 .11735 L .31358 .10593 L .31162 .09772 L .30676 .0781 L .30167 .05848 L .2964 .03885 L .29396 .02995 L .29103 .01923 L s .55 g .01923 .91593 m .03885 .90252 L .03922 .90228 L .05848 .88919 L .06816 .88265 L .0781 .87596 L .09746 .86303 L .09772 .86285 L .11735 .84987 L .12721 .84341 L .13697 .83706 L .15659 .82447 L .15768 .82378 L .17622 .81214 L .18921 .80416 L .19584 .80017 L .21546 .78868 L .22266 .78454 L .23509 .77785 L .25471 .76801 L .25907 .76491 L .27433 .75973 L .29396 .75419 L .31358 .75512 L .32402 .76491 L .33014 .78454 L .3297 .80416 L .32761 .82378 L .32445 .84341 L .32058 .86303 L .31625 .88265 L .31358 .89407 L .31162 .90228 L .30676 .9219 L .30167 .94152 L .2964 .96115 L .29396 .97005 L .29103 .98077 L .01923 .98077 L F 0 g .01923 .91593 m .03885 .90252 L .03922 .90228 L .05848 .88919 L .06816 .88265 L .0781 .87596 L .09746 .86303 L .09772 .86285 L .11735 .84987 L .12721 .84341 L .13697 .83706 L .15659 .82447 L .15768 .82378 L .17622 .81214 L .18921 .80416 L .19584 .80017 L .21546 .78868 L .22266 .78454 L .23509 .77785 L .25471 .76801 L .25907 .76491 L .27433 .75973 L .29396 .75419 L .31358 .75512 L .32402 .76491 L .33014 .78454 L .3297 .80416 L .32761 .82378 L .32445 .84341 L .32058 .86303 L .31625 .88265 L .31358 .89407 L .31162 .90228 L .30676 .9219 L .30167 .94152 L .2964 .96115 L .29396 .97005 L .29103 .98077 L s .5 g .23271 .01923 m .23509 .03721 L .23709 .03885 L .24083 .05848 L .24341 .0781 L .24396 .09772 L .23777 .11735 L .23509 .11976 L .21546 .12564 L .19584 .12081 L .18716 .11735 L .17622 .11236 L .15659 .10187 L .14921 .09772 L .13697 .09018 L .11805 .0781 L .11735 .07762 L .09772 .06451 L .08891 .05848 L .0781 .05093 L .06106 .03885 L .05848 .03699 L .03885 .02278 L .03401 .01923 L F 0 g .23271 .01923 m .23509 .03721 L .23709 .03885 L .24083 .05848 L .24341 .0781 L .24396 .09772 L .23777 .11735 L .23509 .11976 L .21546 .12564 L .19584 .12081 L .18716 .11735 L .17622 .11236 L .15659 .10187 L .14921 .09772 L .13697 .09018 L .11805 .0781 L .11735 .07762 L .09772 .06451 L .08891 .05848 L .0781 .05093 L .06106 .03885 L .05848 .03699 L .03885 .02278 L .03401 .01923 L s .5 g .23271 .98077 m .23509 .96279 L .23709 .96115 L .24083 .94152 L .24341 .9219 L .24396 .90228 L .23777 .88265 L .23509 .88024 L .21546 .87436 L .19584 .87919 L .18716 .88265 L .17622 .88764 L .15659 .89813 L .14921 .90228 L .13697 .90982 L .11805 .9219 L .11735 .92238 L .09772 .93549 L .08891 .94152 L .0781 .94907 L .06106 .96115 L .05848 .96301 L .03885 .97722 L .03401 .98077 L F 0 g .23271 .98077 m .23509 .96279 L .23709 .96115 L .24083 .94152 L .24341 .9219 L .24396 .90228 L .23777 .88265 L .23509 .88024 L .21546 .87436 L .19584 .87919 L .18716 .88265 L .17622 .88764 L .15659 .89813 L .14921 .90228 L .13697 .90982 L .11805 .9219 L .11735 .92238 L .09772 .93549 L .08891 .94152 L .0781 .94907 L .06106 .96115 L .05848 .96301 L .03885 .97722 L .03401 .98077 L s .95 g .49019 .46849 m .50981 .46849 L .52723 .47057 L .52943 .47837 L .52978 .49019 L .52978 .50981 L .52943 .52163 L .52723 .52943 L .50981 .53151 L .49019 .53151 L .47277 .52943 L .47057 .52163 L .47022 .50981 L .47022 .49019 L .47057 .47837 L .47277 .47057 L F 0 g .49019 .46849 m .50981 .46849 L .52723 .47057 L .52943 .47837 L .52978 .49019 L .52978 .50981 L .52943 .52163 L .52723 .52943 L .50981 .53151 L .49019 .53151 L .47277 .52943 L .47057 .52163 L .47022 .50981 L .47022 .49019 L .47057 .47837 L .47277 .47057 L .49019 .46849 L s .9 g .49019 .46954 m .50981 .46954 L .51929 .47057 L .52921 .49019 L .52921 .50981 L .51929 .52943 L .50981 .53046 L .49019 .53046 L .48071 .52943 L .47079 .50981 L .47079 .49019 L .48071 .47057 L F 0 g .49019 .46954 m .50981 .46954 L .51929 .47057 L .52921 .49019 L .52921 .50981 L .51929 .52943 L .50981 .53046 L .49019 .53046 L .48071 .52943 L .47079 .50981 L .47079 .49019 L .48071 .47057 L .49019 .46954 L s .85 g .49019 .47045 m .50981 .47045 L .51116 .47057 L .52886 .49019 L .52886 .50981 L .51116 .52943 L .50981 .52955 L .49019 .52955 L .48884 .52943 L .47114 .50981 L .47114 .49019 L .48884 .47057 L F 0 g .49019 .47045 m .50981 .47045 L .51116 .47057 L .52886 .49019 L .52886 .50981 L .51116 .52943 L .50981 .52955 L .49019 .52955 L .48884 .52943 L .47114 .50981 L .47114 .49019 L .48884 .47057 L .49019 .47045 L s .8 g .49019 .47088 m .50981 .47088 L .52852 .49019 L .52852 .50981 L .50981 .52912 L .49019 .52912 L .47148 .50981 L .47148 .49019 L F 0 g .49019 .47088 m .50981 .47088 L .52852 .49019 L .52852 .50981 L .50981 .52912 L .49019 .52912 L .47148 .50981 L .47148 .49019 L .49019 .47088 L s .75 g .49019 .47123 m .50981 .47123 L .52818 .49019 L .52818 .50981 L .50981 .52877 L .49019 .52877 L .47182 .50981 L .47182 .49019 L F 0 g .49019 .47123 m .50981 .47123 L .52818 .49019 L .52818 .50981 L .50981 .52877 L .49019 .52877 L .47182 .50981 L .47182 .49019 L .49019 .47123 L s .7 g .49019 .47158 m .50981 .47158 L .52785 .49019 L .52785 .50981 L .50981 .52842 L .49019 .52842 L .47215 .50981 L .47215 .49019 L F 0 g .49019 .47158 m .50981 .47158 L .52785 .49019 L .52785 .50981 L .50981 .52842 L .49019 .52842 L .47215 .50981 L .47215 .49019 L .49019 .47158 L s .65 g .49019 .47193 m .50981 .47193 L .52752 .49019 L .52752 .50981 L .50981 .52807 L .49019 .52807 L .47248 .50981 L .47248 .49019 L F 0 g .49019 .47193 m .50981 .47193 L .52752 .49019 L .52752 .50981 L .50981 .52807 L .49019 .52807 L .47248 .50981 L .47248 .49019 L .49019 .47193 L s .6 g .49019 .47226 m .50981 .47226 L .52719 .49019 L .52719 .50981 L .50981 .52774 L .49019 .52774 L .47281 .50981 L .47281 .49019 L F 0 g .49019 .47226 m .50981 .47226 L .52719 .49019 L .52719 .50981 L .50981 .52774 L .49019 .52774 L .47281 .50981 L .47281 .49019 L .49019 .47226 L s .55 g .49019 .4726 m .50981 .4726 L .52687 .49019 L .52687 .50981 L .50981 .5274 L .49019 .5274 L .47313 .50981 L .47313 .49019 L F 0 g .49019 .4726 m .50981 .4726 L .52687 .49019 L .52687 .50981 L .50981 .5274 L .49019 .5274 L .47313 .50981 L .47313 .49019 L .49019 .4726 L s .5 g .49019 .47293 m .50981 .47293 L .52656 .49019 L .52656 .50981 L .50981 .52707 L .49019 .52707 L .47344 .50981 L .47344 .49019 L F 0 g .49019 .47293 m .50981 .47293 L .52656 .49019 L .52656 .50981 L .50981 .52707 L .49019 .52707 L .47344 .50981 L .47344 .49019 L .49019 .47293 L s .45 g .49019 .47325 m .50981 .47325 L .52625 .49019 L .52625 .50981 L .50981 .52675 L .49019 .52675 L .47375 .50981 L .47375 .49019 L F 0 g .49019 .47325 m .50981 .47325 L .52625 .49019 L .52625 .50981 L .50981 .52675 L .49019 .52675 L .47375 .50981 L .47375 .49019 L .49019 .47325 L s .4 g .49019 .47357 m .50981 .47357 L .52594 .49019 L .52594 .50981 L .50981 .52643 L .49019 .52643 L .47406 .50981 L .47406 .49019 L F 0 g .49019 .47357 m .50981 .47357 L .52594 .49019 L .52594 .50981 L .50981 .52643 L .49019 .52643 L .47406 .50981 L .47406 .49019 L .49019 .47357 L s .35 g .49019 .47389 m .50981 .47389 L .52563 .49019 L .52563 .50981 L .50981 .52611 L .49019 .52611 L .47437 .50981 L .47437 .49019 L F 0 g .49019 .47389 m .50981 .47389 L .52563 .49019 L .52563 .50981 L .50981 .52611 L .49019 .52611 L .47437 .50981 L .47437 .49019 L .49019 .47389 L s .3 g .49019 .4742 m .50981 .4742 L .52533 .49019 L .52533 .50981 L .50981 .5258 L .49019 .5258 L .47467 .50981 L .47467 .49019 L F 0 g .49019 .4742 m .50981 .4742 L .52533 .49019 L .52533 .50981 L .50981 .5258 L .49019 .5258 L .47467 .50981 L .47467 .49019 L .49019 .4742 L s .25 g .49019 .47451 m .50981 .47451 L .52503 .49019 L .52503 .50981 L .50981 .52549 L .49019 .52549 L .47497 .50981 L .47497 .49019 L F 0 g .49019 .47451 m .50981 .47451 L .52503 .49019 L .52503 .50981 L .50981 .52549 L .49019 .52549 L .47497 .50981 L .47497 .49019 L .49019 .47451 L s .2 g .49019 .47482 m .50981 .47482 L .52473 .49019 L .52473 .50981 L .50981 .52518 L .49019 .52518 L .47527 .50981 L .47527 .49019 L F 0 g .49019 .47482 m .50981 .47482 L .52473 .49019 L .52473 .50981 L .50981 .52518 L .49019 .52518 L .47527 .50981 L .47527 .49019 L .49019 .47482 L s .15 g .49019 .47513 m .50981 .47513 L .52444 .49019 L .52444 .50981 L .50981 .52487 L .49019 .52487 L .47556 .50981 L .47556 .49019 L F 0 g .49019 .47513 m .50981 .47513 L .52444 .49019 L .52444 .50981 L .50981 .52487 L .49019 .52487 L .47556 .50981 L .47556 .49019 L .49019 .47513 L s .1 g .49019 .47543 m .50981 .47543 L .52415 .49019 L .52415 .50981 L .50981 .52457 L .49019 .52457 L .47585 .50981 L .47585 .49019 L F 0 g .49019 .47543 m .50981 .47543 L .52415 .49019 L .52415 .50981 L .50981 .52457 L .49019 .52457 L .47585 .50981 L .47585 .49019 L .49019 .47543 L s .05 g .49019 .47573 m .50981 .47573 L .52385 .49019 L .52385 .50981 L .50981 .52427 L .49019 .52427 L .47615 .50981 L .47615 .49019 L F 0 g .49019 .47573 m .50981 .47573 L .52385 .49019 L .52385 .50981 L .50981 .52427 L .49019 .52427 L .47615 .50981 L .47615 .49019 L .49019 .47573 L s .49019 .47603 m .50981 .47603 L .52356 .49019 L .52356 .50981 L .50981 .52397 L .49019 .52397 L .47644 .50981 L .47644 .49019 L F .49019 .47603 m .50981 .47603 L .52356 .49019 L .52356 .50981 L .50981 .52397 L .49019 .52397 L .47644 .50981 L .47644 .49019 L .49019 .47603 L s .9 g .98077 .62339 m .96115 .61896 L .94152 .61452 L .9219 .61008 L .91239 .60793 L .90228 .60567 L .88265 .60128 L .86303 .59688 L .84341 .59248 L .82473 .5883 L .82378 .5881 L .80416 .5838 L .78454 .57951 L .76491 .57521 L .74529 .57093 L .73487 .56868 L .72567 .56679 L .70604 .56278 L .68642 .55881 L .6668 .55494 L .64717 .55129 L .63407 .54906 L .62755 .54824 L .60793 .54618 L .5883 .54556 L .56868 .54822 L .56358 .54906 L .54906 .55736 L .53331 .56868 L .52943 .57257 L .52226 .5883 L .51879 .60793 L .5176 .62755 L .51734 .64717 L .51754 .6668 L .51798 .68642 L .51855 .70604 L .5192 .72567 L .5199 .74529 L .52063 .76491 L .52138 .78454 L .52215 .80416 L .52292 .82378 L .52369 .84341 L .52447 .86303 L .52525 .88265 L .52603 .90228 L .52681 .9219 L .52759 .94152 L .52838 .96115 L .52917 .98077 L .98077 .98077 L F 0 g .98077 .62339 m .96115 .61896 L .94152 .61452 L .9219 .61008 L .91239 .60793 L .90228 .60567 L .88265 .60128 L .86303 .59688 L .84341 .59248 L .82473 .5883 L .82378 .5881 L .80416 .5838 L .78454 .57951 L .76491 .57521 L .74529 .57093 L .73487 .56868 L .72567 .56679 L .70604 .56278 L .68642 .55881 L .6668 .55494 L .64717 .55129 L .63407 .54906 L .62755 .54824 L .60793 .54618 L .5883 .54556 L .56868 .54822 L .56358 .54906 L .54906 .55736 L .53331 .56868 L .52943 .57257 L .52226 .5883 L .51879 .60793 L .5176 .62755 L .51734 .64717 L .51754 .6668 L .51798 .68642 L .51855 .70604 L .5192 .72567 L .5199 .74529 L .52063 .76491 L .52138 .78454 L .52215 .80416 L .52292 .82378 L .52369 .84341 L .52447 .86303 L .52525 .88265 L .52603 .90228 L .52681 .9219 L .52759 .94152 L .52838 .96115 L Mistroke .52917 .98077 L Mfstroke .9 g .98077 .37661 m .96115 .38104 L .94152 .38548 L .9219 .38992 L .91239 .39207 L .90228 .39433 L .88265 .39872 L .86303 .40312 L .84341 .40752 L .82473 .4117 L .82378 .4119 L .80416 .4162 L .78454 .42049 L .76491 .42479 L .74529 .42907 L .73487 .43132 L .72567 .43321 L .70604 .43722 L .68642 .44119 L .6668 .44506 L .64717 .44871 L .63407 .45094 L .62755 .45176 L .60793 .45382 L .5883 .45444 L .56868 .45178 L .56358 .45094 L .54906 .44264 L .53331 .43132 L .52943 .42743 L .52226 .4117 L .51879 .39207 L .5176 .37245 L .51734 .35283 L .51754 .3332 L .51798 .31358 L .51855 .29396 L .5192 .27433 L .5199 .25471 L .52063 .23509 L .52138 .21546 L .52215 .19584 L .52292 .17622 L .52369 .15659 L .52447 .13697 L .52525 .11735 L .52603 .09772 L .52681 .0781 L .52759 .05848 L .52838 .03885 L .52917 .01923 L .98077 .01923 L F 0 g .98077 .37661 m .96115 .38104 L .94152 .38548 L .9219 .38992 L .91239 .39207 L .90228 .39433 L .88265 .39872 L .86303 .40312 L .84341 .40752 L .82473 .4117 L .82378 .4119 L .80416 .4162 L .78454 .42049 L .76491 .42479 L .74529 .42907 L .73487 .43132 L .72567 .43321 L .70604 .43722 L .68642 .44119 L .6668 .44506 L .64717 .44871 L .63407 .45094 L .62755 .45176 L .60793 .45382 L .5883 .45444 L .56868 .45178 L .56358 .45094 L .54906 .44264 L .53331 .43132 L .52943 .42743 L .52226 .4117 L .51879 .39207 L .5176 .37245 L .51734 .35283 L .51754 .3332 L .51798 .31358 L .51855 .29396 L .5192 .27433 L .5199 .25471 L .52063 .23509 L .52138 .21546 L .52215 .19584 L .52292 .17622 L .52369 .15659 L .52447 .13697 L .52525 .11735 L .52603 .09772 L .52681 .0781 L .52759 .05848 L .52838 .03885 L Mistroke .52917 .01923 L Mfstroke .85 g .98077 .33541 m .96115 .3413 L .94152 .34718 L .92267 .35283 L .9219 .35306 L .90228 .35891 L .88265 .36475 L .86303 .37059 L .85675 .37245 L .84341 .37639 L .82378 .38216 L .80416 .38792 L .78995 .39207 L .78454 .39364 L .76491 .39929 L .74529 .40489 L .72567 .41043 L .72112 .4117 L .70604 .41581 L .68642 .42105 L .6668 .42608 L .64717 .43076 L .64476 .43132 L .62755 .43477 L .60793 .43777 L .5883 .43876 L .56868 .43574 L .55446 .43132 L .54906 .4259 L .53652 .4117 L .53011 .39207 L .52943 .38796 L .52813 .37245 L .52767 .35283 L .52788 .3332 L .52845 .31358 L .52924 .29396 L .52943 .28966 L .5303 .27433 L .53151 .25471 L .53279 .23509 L .53414 .21546 L .53553 .19584 L .53695 .17622 L .5384 .15659 L .53988 .13697 L .54137 .11735 L .54287 .09772 L .54439 .0781 L .54591 .05848 L .54745 .03885 L .54899 .01923 L .98077 .01923 L F 0 g .98077 .33541 m .96115 .3413 L .94152 .34718 L .92267 .35283 L .9219 .35306 L .90228 .35891 L .88265 .36475 L .86303 .37059 L .85675 .37245 L .84341 .37639 L .82378 .38216 L .80416 .38792 L .78995 .39207 L .78454 .39364 L .76491 .39929 L .74529 .40489 L .72567 .41043 L .72112 .4117 L .70604 .41581 L .68642 .42105 L .6668 .42608 L .64717 .43076 L .64476 .43132 L .62755 .43477 L .60793 .43777 L .5883 .43876 L .56868 .43574 L .55446 .43132 L .54906 .4259 L .53652 .4117 L .53011 .39207 L .52943 .38796 L .52813 .37245 L .52767 .35283 L .52788 .3332 L .52845 .31358 L .52924 .29396 L .52943 .28966 L .5303 .27433 L .53151 .25471 L .53279 .23509 L .53414 .21546 L .53553 .19584 L .53695 .17622 L .5384 .15659 L .53988 .13697 L .54137 .11735 L .54287 .09772 L .54439 .0781 L .54591 .05848 L Mistroke .54745 .03885 L .54899 .01923 L Mfstroke .85 g .98077 .66459 m .96115 .6587 L .94152 .65282 L .92267 .64717 L .9219 .64694 L .90228 .64109 L .88265 .63525 L .86303 .62941 L .85675 .62755 L .84341 .62361 L .82378 .61784 L .80416 .61208 L .78995 .60793 L .78454 .60636 L .76491 .60071 L .74529 .59511 L .72567 .58957 L .72112 .5883 L .70604 .58419 L .68642 .57895 L .6668 .57392 L .64717 .56924 L .64476 .56868 L .62755 .56523 L .60793 .56223 L .5883 .56124 L .56868 .56426 L .55446 .56868 L .54906 .5741 L .53652 .5883 L .53011 .60793 L .52943 .61204 L .52813 .62755 L .52767 .64717 L .52788 .6668 L .52845 .68642 L .52924 .70604 L .52943 .71034 L .5303 .72567 L .53151 .74529 L .53279 .76491 L .53414 .78454 L .53553 .80416 L .53695 .82378 L .5384 .84341 L .53988 .86303 L .54137 .88265 L .54287 .90228 L .54439 .9219 L .54591 .94152 L .54745 .96115 L .54899 .98077 L .98077 .98077 L F 0 g .98077 .66459 m .96115 .6587 L .94152 .65282 L .92267 .64717 L .9219 .64694 L .90228 .64109 L .88265 .63525 L .86303 .62941 L .85675 .62755 L .84341 .62361 L .82378 .61784 L .80416 .61208 L .78995 .60793 L .78454 .60636 L .76491 .60071 L .74529 .59511 L .72567 .58957 L .72112 .5883 L .70604 .58419 L .68642 .57895 L .6668 .57392 L .64717 .56924 L .64476 .56868 L .62755 .56523 L .60793 .56223 L .5883 .56124 L .56868 .56426 L .55446 .56868 L .54906 .5741 L .53652 .5883 L .53011 .60793 L .52943 .61204 L .52813 .62755 L .52767 .64717 L .52788 .6668 L .52845 .68642 L .52924 .70604 L .52943 .71034 L .5303 .72567 L .53151 .74529 L .53279 .76491 L .53414 .78454 L .53553 .80416 L .53695 .82378 L .5384 .84341 L .53988 .86303 L .54137 .88265 L .54287 .90228 L .54439 .9219 L .54591 .94152 L Mistroke .54745 .96115 L .54899 .98077 L Mfstroke .8 g .98077 .29811 m .96115 .30529 L .94152 .31245 L .93843 .31358 L .9219 .3196 L .90228 .32673 L .88443 .3332 L .88265 .33385 L .86303 .34093 L .84341 .348 L .82995 .35283 L .82378 .35503 L .80416 .36201 L .78454 .36895 L .77459 .37245 L .76491 .37582 L .74529 .38261 L .72567 .38929 L .71738 .39207 L .70604 .39582 L .68642 .40213 L .6668 .40814 L .65449 .4117 L .64717 .41367 L .62755 .41842 L .60793 .42184 L .5883 .4227 L .56868 .41842 L .5559 .4117 L .54906 .4016 L .54493 .39207 L .54076 .37245 L .53962 .35283 L .53979 .3332 L .54064 .31358 L .54188 .29396 L .54338 .27433 L .54505 .25471 L .54684 .23509 L .54872 .21546 L .54906 .21202 L .55066 .19584 L .55265 .17622 L .55468 .15659 L .55675 .13697 L .55884 .11735 L .56096 .09772 L .56309 .0781 L .56524 .05848 L .56741 .03885 L .56868 .02734 L .56958 .01923 L .98077 .01923 L F 0 g .98077 .29811 m .96115 .30529 L .94152 .31245 L .93843 .31358 L .9219 .3196 L .90228 .32673 L .88443 .3332 L .88265 .33385 L .86303 .34093 L .84341 .348 L .82995 .35283 L .82378 .35503 L .80416 .36201 L .78454 .36895 L .77459 .37245 L .76491 .37582 L .74529 .38261 L .72567 .38929 L .71738 .39207 L .70604 .39582 L .68642 .40213 L .6668 .40814 L .65449 .4117 L .64717 .41367 L .62755 .41842 L .60793 .42184 L .5883 .4227 L .56868 .41842 L .5559 .4117 L .54906 .4016 L .54493 .39207 L .54076 .37245 L .53962 .35283 L .53979 .3332 L .54064 .31358 L .54188 .29396 L .54338 .27433 L .54505 .25471 L .54684 .23509 L .54872 .21546 L .54906 .21202 L .55066 .19584 L .55265 .17622 L .55468 .15659 L .55675 .13697 L .55884 .11735 L .56096 .09772 L .56309 .0781 L .56524 .05848 L .56741 .03885 L Mistroke .56868 .02734 L .56958 .01923 L Mfstroke .8 g .98077 .70189 m .96115 .69471 L .94152 .68755 L .93843 .68642 L .9219 .6804 L .90228 .67327 L .88443 .6668 L .88265 .66615 L .86303 .65907 L .84341 .652 L .82995 .64717 L .82378 .64497 L .80416 .63799 L .78454 .63105 L .77459 .62755 L .76491 .62418 L .74529 .61739 L .72567 .61071 L .71738 .60793 L .70604 .60418 L .68642 .59787 L .6668 .59186 L .65449 .5883 L .64717 .58633 L .62755 .58158 L .60793 .57816 L .5883 .5773 L .56868 .58158 L .5559 .5883 L .54906 .5984 L .54493 .60793 L .54076 .62755 L .53962 .64717 L .53979 .6668 L .54064 .68642 L .54188 .70604 L .54338 .72567 L .54505 .74529 L .54684 .76491 L .54872 .78454 L .54906 .78798 L .55066 .80416 L .55265 .82378 L .55468 .84341 L .55675 .86303 L .55884 .88265 L .56096 .90228 L .56309 .9219 L .56524 .94152 L .56741 .96115 L .56868 .97266 L .56958 .98077 L .98077 .98077 L F 0 g .98077 .70189 m .96115 .69471 L .94152 .68755 L .93843 .68642 L .9219 .6804 L .90228 .67327 L .88443 .6668 L .88265 .66615 L .86303 .65907 L .84341 .652 L .82995 .64717 L .82378 .64497 L .80416 .63799 L .78454 .63105 L .77459 .62755 L .76491 .62418 L .74529 .61739 L .72567 .61071 L .71738 .60793 L .70604 .60418 L .68642 .59787 L .6668 .59186 L .65449 .5883 L .64717 .58633 L .62755 .58158 L .60793 .57816 L .5883 .5773 L .56868 .58158 L .5559 .5883 L .54906 .5984 L .54493 .60793 L .54076 .62755 L .53962 .64717 L .53979 .6668 L .54064 .68642 L .54188 .70604 L .54338 .72567 L .54505 .74529 L .54684 .76491 L .54872 .78454 L .54906 .78798 L .55066 .80416 L .55265 .82378 L .55468 .84341 L .55675 .86303 L .55884 .88265 L .56096 .90228 L .56309 .9219 L .56524 .94152 L .56741 .96115 L Mistroke .56868 .97266 L .56958 .98077 L Mfstroke .75 g .98077 .26172 m .96115 .27012 L .95128 .27433 L .94152 .2785 L .9219 .28686 L .9052 .29396 L .90228 .2952 L .88265 .30351 L .86303 .31179 L .85878 .31358 L .84341 .32004 L .82378 .32824 L .81185 .3332 L .80416 .33639 L .78454 .34447 L .76491 .35246 L .76401 .35283 L .74529 .36034 L .72567 .36808 L .71435 .37245 L .70604 .37561 L .68642 .38286 L .6668 .38969 L .65964 .39207 L .64717 .39588 L .62755 .40105 L .60793 .40435 L .5883 .40402 L .56868 .39458 L .56609 .39207 L .55653 .37245 L .554 .35283 L .55367 .3332 L .55444 .31358 L .55585 .29396 L .55765 .27433 L .55972 .25471 L .56198 .23509 L .56437 .21546 L .56686 .19584 L .56868 .1818 L .56941 .17622 L .57203 .15659 L .57469 .13697 L .57739 .11735 L .58013 .09772 L .5829 .0781 L .58568 .05848 L .5883 .04014 L .58849 .03885 L .59131 .01923 L .98077 .01923 L F 0 g .98077 .26172 m .96115 .27012 L .95128 .27433 L .94152 .2785 L .9219 .28686 L .9052 .29396 L .90228 .2952 L .88265 .30351 L .86303 .31179 L .85878 .31358 L .84341 .32004 L .82378 .32824 L .81185 .3332 L .80416 .33639 L .78454 .34447 L .76491 .35246 L .76401 .35283 L .74529 .36034 L .72567 .36808 L .71435 .37245 L .70604 .37561 L .68642 .38286 L .6668 .38969 L .65964 .39207 L .64717 .39588 L .62755 .40105 L .60793 .40435 L .5883 .40402 L .56868 .39458 L .56609 .39207 L .55653 .37245 L .554 .35283 L .55367 .3332 L .55444 .31358 L .55585 .29396 L .55765 .27433 L .55972 .25471 L .56198 .23509 L .56437 .21546 L .56686 .19584 L .56868 .1818 L .56941 .17622 L .57203 .15659 L .57469 .13697 L .57739 .11735 L .58013 .09772 L .5829 .0781 L .58568 .05848 L .5883 .04014 L .58849 .03885 L Mistroke .59131 .01923 L Mfstroke .75 g .98077 .73828 m .96115 .72988 L .95128 .72567 L .94152 .7215 L .9219 .71314 L .9052 .70604 L .90228 .7048 L .88265 .69649 L .86303 .68821 L .85878 .68642 L .84341 .67996 L .82378 .67176 L .81185 .6668 L .80416 .66361 L .78454 .65553 L .76491 .64754 L .76401 .64717 L .74529 .63966 L .72567 .63192 L .71435 .62755 L .70604 .62439 L .68642 .61714 L .6668 .61031 L .65964 .60793 L .64717 .60412 L .62755 .59895 L .60793 .59565 L .5883 .59598 L .56868 .60542 L .56609 .60793 L .55653 .62755 L .554 .64717 L .55367 .6668 L .55444 .68642 L .55585 .70604 L .55765 .72567 L .55972 .74529 L .56198 .76491 L .56437 .78454 L .56686 .80416 L .56868 .8182 L .56941 .82378 L .57203 .84341 L .57469 .86303 L .57739 .88265 L .58013 .90228 L .5829 .9219 L .58568 .94152 L .5883 .95986 L .58849 .96115 L .59131 .98077 L .98077 .98077 L F 0 g .98077 .73828 m .96115 .72988 L .95128 .72567 L .94152 .7215 L .9219 .71314 L .9052 .70604 L .90228 .7048 L .88265 .69649 L .86303 .68821 L .85878 .68642 L .84341 .67996 L .82378 .67176 L .81185 .6668 L .80416 .66361 L .78454 .65553 L .76491 .64754 L .76401 .64717 L .74529 .63966 L .72567 .63192 L .71435 .62755 L .70604 .62439 L .68642 .61714 L .6668 .61031 L .65964 .60793 L .64717 .60412 L .62755 .59895 L .60793 .59565 L .5883 .59598 L .56868 .60542 L .56609 .60793 L .55653 .62755 L .554 .64717 L .55367 .6668 L .55444 .68642 L .55585 .70604 L .55765 .72567 L .55972 .74529 L .56198 .76491 L .56437 .78454 L .56686 .80416 L .56868 .8182 L .56941 .82378 L .57203 .84341 L .57469 .86303 L .57739 .88265 L .58013 .90228 L .5829 .9219 L .58568 .94152 L .5883 .95986 L .58849 .96115 L Mistroke .59131 .98077 L Mfstroke .7 g .98077 .77563 m .96115 .76603 L .95887 .76491 L .94152 .75645 L .9219 .74689 L .91861 .74529 L .90228 .73736 L .88265 .72787 L .87808 .72567 L .86303 .71842 L .84341 .70902 L .83717 .70604 L .82378 .69968 L .80416 .69041 L .79565 .68642 L .78454 .68123 L .76491 .67217 L .75311 .6668 L .74529 .66327 L .72567 .65456 L .70847 .64717 L .70604 .64615 L .68642 .63811 L .6668 .63068 L .65815 .62755 L .64717 .62419 L .62755 .61922 L .60793 .61725 L .5883 .62248 L .58098 .62755 L .57269 .64717 L .57078 .6668 L .57086 .68642 L .57205 .70604 L .57394 .72567 L .57627 .74529 L .5789 .76491 L .58174 .78454 L .58473 .80416 L .58784 .82378 L .5883 .82668 L .59102 .84341 L .59427 .86303 L .59759 .88265 L .60095 .90228 L .60436 .9219 L .6078 .94152 L .60793 .94226 L .61126 .96115 L .61474 .98077 L .98077 .98077 L F 0 g .98077 .77563 m .96115 .76603 L .95887 .76491 L .94152 .75645 L .9219 .74689 L .91861 .74529 L .90228 .73736 L .88265 .72787 L .87808 .72567 L .86303 .71842 L .84341 .70902 L .83717 .70604 L .82378 .69968 L .80416 .69041 L .79565 .68642 L .78454 .68123 L .76491 .67217 L .75311 .6668 L .74529 .66327 L .72567 .65456 L .70847 .64717 L .70604 .64615 L .68642 .63811 L .6668 .63068 L .65815 .62755 L .64717 .62419 L .62755 .61922 L .60793 .61725 L .5883 .62248 L .58098 .62755 L .57269 .64717 L .57078 .6668 L .57086 .68642 L .57205 .70604 L .57394 .72567 L .57627 .74529 L .5789 .76491 L .58174 .78454 L .58473 .80416 L .58784 .82378 L .5883 .82668 L .59102 .84341 L .59427 .86303 L .59759 .88265 L .60095 .90228 L .60436 .9219 L .6078 .94152 L .60793 .94226 L .61126 .96115 L .61474 .98077 L Mistroke Mfstroke .7 g .98077 .22437 m .96115 .23397 L .95887 .23509 L .94152 .24355 L .9219 .25311 L .91861 .25471 L .90228 .26264 L .88265 .27213 L .87808 .27433 L .86303 .28158 L .84341 .29098 L .83717 .29396 L .82378 .30032 L .80416 .30959 L .79565 .31358 L .78454 .31877 L .76491 .32783 L .75311 .3332 L .74529 .33673 L .72567 .34544 L .70847 .35283 L .70604 .35385 L .68642 .36189 L .6668 .36932 L .65815 .37245 L .64717 .37581 L .62755 .38078 L .60793 .38275 L .5883 .37752 L .58098 .37245 L .57269 .35283 L .57078 .3332 L .57086 .31358 L .57205 .29396 L .57394 .27433 L .57627 .25471 L .5789 .23509 L .58174 .21546 L .58473 .19584 L .58784 .17622 L .5883 .17332 L .59102 .15659 L .59427 .13697 L .59759 .11735 L .60095 .09772 L .60436 .0781 L .6078 .05848 L .60793 .05774 L .61126 .03885 L .61474 .01923 L .98077 .01923 L F 0 g .98077 .22437 m .96115 .23397 L .95887 .23509 L .94152 .24355 L .9219 .25311 L .91861 .25471 L .90228 .26264 L .88265 .27213 L .87808 .27433 L .86303 .28158 L .84341 .29098 L .83717 .29396 L .82378 .30032 L .80416 .30959 L .79565 .31358 L .78454 .31877 L .76491 .32783 L .75311 .3332 L .74529 .33673 L .72567 .34544 L .70847 .35283 L .70604 .35385 L .68642 .36189 L .6668 .36932 L .65815 .37245 L .64717 .37581 L .62755 .38078 L .60793 .38275 L .5883 .37752 L .58098 .37245 L .57269 .35283 L .57078 .3332 L .57086 .31358 L .57205 .29396 L .57394 .27433 L .57627 .25471 L .5789 .23509 L .58174 .21546 L .58473 .19584 L .58784 .17622 L .5883 .17332 L .59102 .15659 L .59427 .13697 L .59759 .11735 L .60095 .09772 L .60436 .0781 L .6078 .05848 L .60793 .05774 L .61126 .03885 L .61474 .01923 L Mistroke Mfstroke .65 g .98077 .81574 m .96115 .80491 L .95978 .80416 L .94152 .79411 L .92408 .78454 L .9219 .78334 L .90228 .77262 L .88813 .76491 L .88265 .76194 L .86303 .75132 L .85183 .74529 L .84341 .74077 L .82378 .7303 L .81502 .72567 L .80416 .71994 L .78454 .70972 L .7774 .70604 L .76491 .69966 L .74529 .68984 L .73834 .68642 L .72567 .68031 L .70604 .67121 L .69613 .6668 L .68642 .66274 L .6668 .65519 L .64717 .64931 L .63579 .64717 L .62755 .64652 L .61872 .64717 L .60793 .65088 L .59522 .6668 L .59254 .68642 L .59252 .70604 L .59389 .72567 L .59611 .74529 L .59887 .76491 L .602 .78454 L .60539 .80416 L .60793 .81821 L .60895 .82378 L .61263 .84341 L .61645 .86303 L .62036 .88265 L .62435 .90228 L .62755 .91783 L .62839 .9219 L .63247 .94152 L .6366 .96115 L .64077 .98077 L .98077 .98077 L F 0 g .98077 .81574 m .96115 .80491 L .95978 .80416 L .94152 .79411 L .92408 .78454 L .9219 .78334 L .90228 .77262 L .88813 .76491 L .88265 .76194 L .86303 .75132 L .85183 .74529 L .84341 .74077 L .82378 .7303 L .81502 .72567 L .80416 .71994 L .78454 .70972 L .7774 .70604 L .76491 .69966 L .74529 .68984 L .73834 .68642 L .72567 .68031 L .70604 .67121 L .69613 .6668 L .68642 .66274 L .6668 .65519 L .64717 .64931 L .63579 .64717 L .62755 .64652 L .61872 .64717 L .60793 .65088 L .59522 .6668 L .59254 .68642 L .59252 .70604 L .59389 .72567 L .59611 .74529 L .59887 .76491 L .602 .78454 L .60539 .80416 L .60793 .81821 L .60895 .82378 L .61263 .84341 L .61645 .86303 L .62036 .88265 L .62435 .90228 L .62755 .91783 L .62839 .9219 L .63247 .94152 L .6366 .96115 L .64077 .98077 L s .65 g .98077 .18426 m .96115 .19509 L .95978 .19584 L .94152 .20589 L .92408 .21546 L .9219 .21666 L .90228 .22738 L .88813 .23509 L .88265 .23806 L .86303 .24868 L .85183 .25471 L .84341 .25923 L .82378 .2697 L .81502 .27433 L .80416 .28006 L .78454 .29028 L .7774 .29396 L .76491 .30034 L .74529 .31016 L .73834 .31358 L .72567 .31969 L .70604 .32879 L .69613 .3332 L .68642 .33726 L .6668 .34481 L .64717 .35069 L .63579 .35283 L .62755 .35348 L .61872 .35283 L .60793 .34912 L .59522 .3332 L .59254 .31358 L .59252 .29396 L .59389 .27433 L .59611 .25471 L .59887 .23509 L .602 .21546 L .60539 .19584 L .60793 .18179 L .60895 .17622 L .61263 .15659 L .61645 .13697 L .62036 .11735 L .62435 .09772 L .62755 .08217 L .62839 .0781 L .63247 .05848 L .6366 .03885 L .64077 .01923 L .98077 .01923 L F 0 g .98077 .18426 m .96115 .19509 L .95978 .19584 L .94152 .20589 L .92408 .21546 L .9219 .21666 L .90228 .22738 L .88813 .23509 L .88265 .23806 L .86303 .24868 L .85183 .25471 L .84341 .25923 L .82378 .2697 L .81502 .27433 L .80416 .28006 L .78454 .29028 L .7774 .29396 L .76491 .30034 L .74529 .31016 L .73834 .31358 L .72567 .31969 L .70604 .32879 L .69613 .3332 L .68642 .33726 L .6668 .34481 L .64717 .35069 L .63579 .35283 L .62755 .35348 L .61872 .35283 L .60793 .34912 L .59522 .3332 L .59254 .31358 L .59252 .29396 L .59389 .27433 L .59611 .25471 L .59887 .23509 L .602 .21546 L .60539 .19584 L .60793 .18179 L .60895 .17622 L .61263 .15659 L .61645 .13697 L .62036 .11735 L .62435 .09772 L .62755 .08217 L .62839 .0781 L .63247 .05848 L .6366 .03885 L .64077 .01923 L s .6 g .98077 .13895 m .96115 .15106 L .95215 .15659 L .94152 .16312 L .9219 .17513 L .92011 .17622 L .90228 .18707 L .88779 .19584 L .88265 .19894 L .86303 .21072 L .85508 .21546 L .84341 .2224 L .82378 .23393 L .8218 .23509 L .80416 .2453 L .78761 .25471 L .78454 .25644 L .76491 .26732 L .75187 .27433 L .74529 .27778 L .72567 .28773 L .71267 .29396 L .70604 .29685 L .68642 .30478 L .6668 .31048 L .64717 .31163 L .62755 .29622 L .62671 .29396 L .62303 .27433 L .62329 .25471 L .62516 .23509 L .62755 .21741 L .62791 .21546 L .63116 .19584 L .63487 .17622 L .63889 .15659 L .64314 .13697 L .64717 .11898 L .64754 .11735 L .65204 .09772 L .65666 .0781 L .66138 .05848 L .66617 .03885 L .6668 .0363 L .671 .01923 L .98077 .01923 L F 0 g .98077 .13895 m .96115 .15106 L .95215 .15659 L .94152 .16312 L .9219 .17513 L .92011 .17622 L .90228 .18707 L .88779 .19584 L .88265 .19894 L .86303 .21072 L .85508 .21546 L .84341 .2224 L .82378 .23393 L .8218 .23509 L .80416 .2453 L .78761 .25471 L .78454 .25644 L .76491 .26732 L .75187 .27433 L .74529 .27778 L .72567 .28773 L .71267 .29396 L .70604 .29685 L .68642 .30478 L .6668 .31048 L .64717 .31163 L .62755 .29622 L .62671 .29396 L .62303 .27433 L .62329 .25471 L .62516 .23509 L .62755 .21741 L .62791 .21546 L .63116 .19584 L .63487 .17622 L .63889 .15659 L .64314 .13697 L .64717 .11898 L .64754 .11735 L .65204 .09772 L .65666 .0781 L .66138 .05848 L .66617 .03885 L .6668 .0363 L .671 .01923 L s .6 g .98077 .86105 m .96115 .84894 L .95215 .84341 L .94152 .83688 L .9219 .82487 L .92011 .82378 L .90228 .81293 L .88779 .80416 L .88265 .80106 L .86303 .78928 L .85508 .78454 L .84341 .7776 L .82378 .76607 L .8218 .76491 L .80416 .7547 L .78761 .74529 L .78454 .74356 L .76491 .73268 L .75187 .72567 L .74529 .72222 L .72567 .71227 L .71267 .70604 L .70604 .70315 L .68642 .69522 L .6668 .68952 L .64717 .68837 L .62755 .70378 L .62671 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