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Notebook[{

Cell[CellGroupData[{
Cell["Integrating Equations of Motion in Mathematica", "Title"],

Cell[TextData[StyleBox[
"Gary L. Gray\nAssistant Professor\nEngineering Science and Mechanics\nThe \
Pennsylvania State University\n227 Hammond Building\nUniversity Park, PA \
16802\ngray@engr.psu.edu", "Subtitle",
  FontSize->14]], "Subtitle"],

Cell["\<\
Double-click the cell brackets on the right to open and close them.\
\
\>", "Text"],

Cell[CellGroupData[{

Cell["Introduction", "Section"],

Cell[TextData[{
  "There is really only one thing you need to know about ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " \[LongDash] it can do almost ",
  StyleBox["anything",
    FontSlant->"Italic"],
  " you would want to do mathematically. Of course, getting it to do ",
  StyleBox["anything",
    FontSlant->"Italic"],
  " is quite another matter. What you will learn here is probably much less \
than 1% of what ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " is capable of."
}], "Text"],

Cell[TextData[{
  "This is a quick tutorial on how to use ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " for many of problems you will be solving this semester. For example, you \
will numerically solve ordinary differential equations (equations of motion), \
solve systems of algebraic equations, and plot many types of functions. If \
you have suggestions, comments, or corrections, please send them to me at the \
above email address."
}], "Text"],

Cell[CellGroupData[{

Cell[TextData[{
  "Executing Commands in ",
  StyleBox["Mathematica",
    FontSlant->"Italic"]
}], "Subsection"],

Cell[TextData[{
  "Commands are entered in \"cells\". The vertical lines you see on the right \
are cell brackets and they define boundaries between groups of objects. Each \
cell has associated with it a certain set of attributes, but you only need be \
concerned with a couple of them. The cell containing this paragraph is a text \
cell as you can see in the toolbar at the top of the window. A text cell is \
used for adding comments and explanation to a notebook and is not ",
  StyleBox["executable",
    FontSlant->"Italic"],
  ". You can only do mathematics in ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " withing executable cells. Any time you start a new cell, it is an \"Input \
cell\" by default and ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " commands can be executed there since it is executable. Below you see an \
example of an input cell with a ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " command and the resulting output cell. The command is executed by placing \
the cursor ",
  StyleBox["anywhere within the cell",
    FontSlant->"Italic"],
  " and pressing either ",
  StyleBox["Shift-Return",
    FontWeight->"Bold"],
  " or ",
  StyleBox["Enter",
    FontWeight->"Bold"],
  "."
}], "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(2 + 2\ Cos[3.1]\)], "Input"],

Cell[BoxData[
    \(0.00172969945344103592`\)], "Output"]
}, Open  ]],

Cell[TextData[{
  "When you execute the very first command in a ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " notebook (this is a ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " notebook and it runs in an application called the \"front end\"), ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " must start a program called the \"kernel\" which is the application that \
does all the computation. Therefore, you may experience a small delay the \
first time you execute an input cell as you wait for the kernel to start up.\n\
\nNote that ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " knows about constants such as \[Pi] and E (the ",
  StyleBox["N",
    FontFamily->"Courier",
    FontWeight->"Bold"],
  " command tells ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " to return a numerical value). You can get the \"pi\" from the palettes on \
the right or using some special keystrokes."
}], "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(N[\[Pi]]\)], "Input"],

Cell[BoxData[
    \(3.14159265358979311`\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(N[E]\)], "Input"],

Cell[BoxData[
    \(2.71828182845904509`\)], "Output"]
}, Open  ]],

Cell[TextData[{
  "Without the ",
  StyleBox["N",
    FontFamily->"Courier",
    FontWeight->"Bold"],
  ", you simply get"
}], "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(\[Pi] - E\)], "Input"],

Cell[BoxData[
    \(\(-E\) + \[Pi]\)], "Output"]
}, Open  ]],

Cell[TextData[{
  "since ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " thinks of these as ",
  StyleBox["exact",
    FontSlant->"Italic"],
  " constants. This is an important thing to keep in mind about ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  "."
}], "Text"]
}, Open  ]]
}, Open  ]],

Cell[CellGroupData[{

Cell[TextData[{
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " Syntax"
}], "Section"],

Cell[TextData[{
  "You can use any name you like as a variable name, but it cannot have \
spaces, dashes, or begin with a number. The equals sign \"=\" assigns one \
thing to another. For example, this command sets ",
  StyleBox["q",
    FontSlant->"Italic"],
  " equal to 4. A semicolon suppresses the output."
}], "Text"],

Cell[BoxData[
    \(\(q = 4; \)\)], "Input"],

Cell[TextData[{
  "We now set ",
  StyleBox["p",
    FontSlant->"Italic"],
  " equal to 5."
}], "Text"],

Cell[BoxData[
    \(\(p = 5; \)\)], "Input"],

Cell[TextData[{
  "If you ask ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " what ",
  StyleBox["p",
    FontSlant->"Italic"],
  " is, it returns what you would expect."
}], "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(p\)], "Input"],

Cell[BoxData[
    \(5\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(p + q\)], "Input"],

Cell[BoxData[
    \(9\)], "Output"]
}, Open  ]],

Cell[TextData[{
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " never forgets these values for ",
  StyleBox["p",
    FontSlant->"Italic"],
  " and ",
  StyleBox["q",
    FontSlant->"Italic"],
  " until you quit and restart the program. (Actually, there are ways to get \
it to forget them, but we won't get into them here.) In addition, I highly \
recommend that you start all variable names with a lower case letter since ",
  StyleBox["all",
    FontWeight->"Bold",
    FontSlant->"Italic"],
  " ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " commands begin with capital letters and then there is no was you can use \
a variable name that is also a ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " command.\n\nMultiplication is represented by a space or an asterisk."
}], "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(p\ q\)], "Input"],

Cell[BoxData[
    \(20\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(p*q\)], "Input"],

Cell[BoxData[
    \(20\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Fancy Formatting", "Subsubsection"],

Cell[TextData[{
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " is capable of some pretty fancy formatting. Note that all ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " commands start with capital letters (",
  StyleBox["e.g.",
    FontSlant->"Italic"],
  ", cosine) and also note that the argument of each ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " command is contained in square brackets (more on this later)."
}], "Text"],

Cell[BoxData[
    \(z = 
      \[Integral]\_\(-1
            \)\%\[Infinity]\( Cos[a\ x]\/\(x\^2 + \@\(y - z\^3\)\)\) 
          \[DifferentialD]x\)], "Input",
  Evaluatable->False],

Cell["\<\
You can use the Palettes submenu of the File menu to get a whole \
bunch of palettes for formatting.\
\>", "Text"]
}, Open  ]],

Cell[CellGroupData[{

Cell["The Double Equals Sign", "Subsection"],

Cell[TextData[{
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " uses the double equals sign \"==\" to equate one thing to another. This \
is not the same as an assignment given by the single equals sign. For \
example, say way want to solve the simultaneous equations ",
  StyleBox["x",
    FontSlant->"Italic"],
  " + ",
  StyleBox["y",
    FontSlant->"Italic"],
  " = 3 and ",
  StyleBox["x",
    FontSlant->"Italic"],
  " \[Dash] ",
  StyleBox["y",
    FontSlant->"Italic"],
  " = 4. You would enter the equations in the following way (notice that I \
have put more than one command in a single cell bracket):"
}], "Text"],

Cell[CellGroupData[{

Cell[BoxData[{
    \(eq1 = x + y == 1\), 
    \(eq2 = x - y == 2\)}], "Input"],

Cell[BoxData[
    \(x + y == 1\)], "Output"],

Cell[BoxData[
    \(x - y == 2\)], "Output"]
}, Open  ]],

Cell[TextData[{
  "I then solve the equations using the ",
  StyleBox["Solve",
    FontFamily->"Courier",
    FontWeight->"Bold"],
  " command:"
}], "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(xysoln = Solve[{eq1, eq2}, {x, y}]\)], "Input"],

Cell[BoxData[
    \({{x \[Rule] 3\/2, y \[Rule] \(-\(1\/2\)\)}}\)], "Output"]
}, Open  ]],

Cell[TextData[{
  "The output from the solution is a list of ",
  StyleBox["replacement rules",
    FontSlant->"Italic"],
  ". A replacement rule has the form:\n\t",
  StyleBox["left -> right",
    FontFamily->"Courier",
    FontWeight->"Bold"]
}], "Text"],

Cell[TextData[{
  "That is, anywhere ",
  StyleBox["left",
    FontFamily->"Courier",
    FontWeight->"Bold"],
  " appears, replace it with ",
  StyleBox["right",
    FontFamily->"Courier",
    FontWeight->"Bold"],
  ". Here is how you use them."
}], "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(x /. xysoln\)], "Input"],

Cell[BoxData[
    \({3\/2}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(y /. xysoln\)], "Input"],

Cell[BoxData[
    \({\(-\(1\/2\)\)}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(x\ y /. xysoln\)], "Input"],

Cell[BoxData[
    \({\(-\(3\/4\)\)}\)], "Output"]
}, Open  ]],

Cell[TextData[{
  "This says to multiply ",
  StyleBox["x",
    FontSlant->"Italic"],
  " by ",
  StyleBox["y. ",
    FontSlant->"Italic"],
  "Then the ",
  StyleBox["/.",
    FontFamily->"Courier",
    FontWeight->"Bold"],
  " command tells ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " to substitute the replacement rule ",
  StyleBox["xysoln",
    FontFamily->"Courier",
    FontWeight->"Bold"],
  " into ",
  StyleBox["x y. ",
    FontSlant->"Italic"],
  "For now, don't worry about the extra square brackets that are hanging \
around."
}], "Text"],

Cell["Let's have a look at one more example.", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(list = {d -> 4, e -> 3.2, f -> 1.2}\)], "Input"],

Cell[BoxData[
    RowBox[{"{", 
      RowBox[{\(d \[Rule] 4\), ",", 
        RowBox[{"e", "\[Rule]", 
          StyleBox["3.20000000000000017`",
            StyleBoxAutoDelete->True,
            PrintPrecision->2]}], ",", 
        RowBox[{"f", "\[Rule]", 
          StyleBox["1.19999999999999996`",
            StyleBoxAutoDelete->True,
            PrintPrecision->2]}]}], "}"}]], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(d /. list\)], "Input"],

Cell[BoxData[
    \(4\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(d + e + f + g /. list\)], "Input"],

Cell[BoxData[
    \(\(8.40000000000000035`\[InvisibleSpace]\) + g\)], "Output"]
}, Open  ]],

Cell[TextData[{
  "Notice it didn't substitute anything for ",
  StyleBox["g",
    FontFamily->"Courier",
    FontWeight->"Bold"],
  " since it was not in our list."
}], "Text"]
}, Open  ]],

Cell[CellGroupData[{

Cell["\<\
You can get information about any command by entering a question \
mark followed by the command.\
\>", "Subsection"],

Cell[TextData[{
  "Here is information on the ",
  StyleBox["Solve",
    FontFamily->"Courier",
    FontWeight->"Bold"],
  " command we used above."
}], "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(?Solve\)\)], "Input"],

Cell[BoxData[
    \("Solve[eqns, vars] attempts to solve an equation or set of equations \
for the variables vars. Solve[eqns, vars, elims] attempts to solve the \
equations for vars, eliminating the variables elims."\)], "Print"]
}, Open  ]],

Cell["\<\
Even more information can be obtained by using the double question \
mark.\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(?? Solve\)], "Input"],

Cell[BoxData[
    \("Solve[eqns, vars] attempts to solve an equation or set of equations \
for the variables vars. Solve[eqns, vars, elims] attempts to solve the \
equations for vars, eliminating the variables elims."\)], "Print"],

Cell[BoxData[
    InterpretationBox[
      StyleBox[\(Attributes[Solve]\  = \ {Protected}\n\ \n
        Options[Solve]\  = 
          \ {InverseFunctions\  -> \ Automatic, \ MakeRules\  -> \ False, \ 
            Method\  -> \ 3, \ Mode\  -> \ Generic, \ Sort\  -> \ True, \ 
            VerifySolutions\  -> \ Automatic, \ 
            WorkingPrecision\  -> \ Infinity}\),
        ShowStringCharacters->True,
        NumberMarks->True],
      InputForm[ 
        Definition[ Solve]],
      Editable->True,
      AutoDelete->True]], "Print"]
}, Open  ]]
}, Open  ]]
}, Open  ]],

Cell[CellGroupData[{

Cell[TextData[{
  "Turn off annoying ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " warnings."
}], "Section"],

Cell[CellGroupData[{

Cell[BoxData[
    \(fred = 12\)], "Input"],

Cell[BoxData[
    \(12\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(freda = 13\)], "Input"],

Cell[BoxData[
    \(General::"spell1" \( : \ \) 
      "Possible spelling error: new symbol name \"\!\(freda\)\" is similar to \
existing symbol \"\!\(fred\)\"."\)], "Message"],

Cell[BoxData[
    \(13\)], "Output"]
}, Open  ]],

Cell[TextData[{
  "These keep ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " from complaining when you name a variable ",
  StyleBox["freda",
    FontFamily->"Courier",
    FontWeight->"Bold"],
  " when you already have a variable named ",
  StyleBox["fred",
    FontFamily->"Courier",
    FontWeight->"Bold"],
  ". I execute these commands in almost every ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " notebook I create."
}], "Text"],

Cell[BoxData[{
    \(Off[General::spell]\), 
    \(Off[General::spell1]\)}], "Input"],

Cell["Now watch what happens:", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(joe = 3\)], "Input"],

Cell[BoxData[
    \(3\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(joey = 1\)], "Input"],

Cell[BoxData[
    \(1\)], "Output"]
}, Open  ]]
}, Open  ]],

Cell[CellGroupData[{

Cell["Doing Vector Operations", "Section"],

Cell[TextData[{
  "To do the type of operations we need to do in this class, you first need \
to load some ",
  StyleBox["external packages",
    FontSlant->"Italic"],
  ". These are packages that ship with ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  ", but are not loaded automatically."
}], "Text"],

Cell[BoxData[
    \(Needs["\<Calculus`VectorAnalysis`\>"]\)], "Input"],

Cell[CellGroupData[{

Cell["Defining a Vector", "Subsection"],

Cell[TextData[{
  "Vectors are defines as ",
  StyleBox["lists",
    FontSlant->"Italic"],
  " enclosed in curly brackets."
}], "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(f = {1, 2, 3}\)], "Input"],

Cell[BoxData[
    \({1, 2, 3}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(g = {4, 5, 6}\)], "Input"],

Cell[BoxData[
    \({4, 5, 6}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(dot = DotProduct[f, g]\)], "Input"],

Cell[BoxData[
    \(32\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(cross = CrossProduct[f, g]\)], "Input"],

Cell[BoxData[
    \({\(-3\), 6, \(-3\)}\)], "Output"]
}, Open  ]],

Cell["These operations also work on more than numbers.", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(DotProduct[{l, m, n}, {r, s, t}]\)], "Input"],

Cell[BoxData[
    \(l\ r + m\ s + n\ t\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(CrossProduct[{l, m, n}, {r, s, t}]\)], "Input"],

Cell[BoxData[
    \({\(-n\)\ s + m\ t, n\ r - l\ t, \(-m\)\ r + l\ s}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Picking out components of a vector or list.", "Subsubsection"],

Cell["\<\
You can pick out one of the components of the vector (or, \
equivalently, a part of a list) by using the following notation.\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(cross[\([1]\)]\)], "Input"],

Cell[BoxData[
    \(\(-3\)\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(cross[\([2]\)]\)], "Input"],

Cell[BoxData[
    \(6\)], "Output"]
}, Open  ]],

Cell["You get the idea.", "Text"],

Cell[TextData[{
  "Now, recall ",
  StyleBox["xysoln",
    FontFamily->"Courier",
    FontWeight->"Bold"],
  "."
}], "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(xysoln\)], "Input"],

Cell[BoxData[
    \({{x \[Rule] 3\/2, y \[Rule] \(-\(1\/2\)\)}}\)], "Output"]
}, Open  ]],

Cell[TextData[{
  "It is really a ",
  StyleBox["list of lists",
    FontSlant->"Italic"],
  ". Therefore:"
}], "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(xysoln[\([1]\)]\)], "Input"],

Cell[BoxData[
    \({x \[Rule] 3\/2, y \[Rule] \(-\(1\/2\)\)}\)], "Output"]
}, Open  ]],

Cell["Therefore we need to do the following:", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(xysoln[\([1, 1]\)]\)], "Input"],

Cell[BoxData[
    \(x \[Rule] 3\/2\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(xysoln[\([1, 2]\)]\)], "Input"],

Cell[BoxData[
    \(y \[Rule] \(-\(1\/2\)\)\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(xysoln[\([2, 1]\)]\)], "Input"],

Cell[BoxData[
    \(Part::"partw" \( : \ \) 
      "Part \!\(2\) of \!\({{\(x \\[Rule] 3\/2\), \(y \\[Rule] \
\(-\(1\/2\)\)\)}}\) does not exist."\)], "Message"],

Cell[BoxData[
    \({{x \[Rule] 3\/2, y \[Rule] \(-\(1\/2\)\)}}\[LeftDoubleBracket]2, 1
      \[RightDoubleBracket]\)], "Output"]
}, Open  ]],

Cell["\<\
Get it? One more example. Form a another type of list (actually, \
this is a 2 by 2 matrix):\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(mat = {{1, 2}, {3, 4}}\)], "Input"],

Cell[BoxData[
    \({{1, 2}, {3, 4}}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(mat[\([1]\)]\)], "Input"],

Cell[BoxData[
    \({1, 2}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(mat[\([1, 2]\)]\)], "Input"],

Cell[BoxData[
    \(2\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(mat[\([2]\)]\)], "Input"],

Cell[BoxData[
    \({3, 4}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(mat[\([2, 1]\)]\)], "Input"],

Cell[BoxData[
    \(3\)], "Output"]
}, Open  ]]
}, Open  ]]
}, Open  ]]
}, Open  ]],

Cell[CellGroupData[{

Cell["Solving Systems of Algebraic Equations", "Section"],

Cell[TextData[{
  "We have already seen this, but ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " is ",
  StyleBox["very",
    FontSlant->"Italic"],
  " powerful!!!"
}], "Text"],

Cell[CellGroupData[{

Cell["Define 8 equations.", "Subsection"],

Cell["\<\
These eight equations are the equations of motion of the \
slider-crank mechanism inside an IC engine.\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[{
    \(eq1 = N + C\_x == 0\), 
    \(eq2 = C\_y - P == \(m\_C\) a\_C\), 
    \(eq3 = B\_y - C\_y == \(m\_BC\) a\_\(D\_y\)\), 
    \(eq4 = B\_x - C\_x == \(m\_BC\) a\_\(D\_x\)\), 
    \(eq5 = 
      \(C\_x\) L\ Cos[\[Phi]] - \(C\_y\) L\ Sin[\[Phi]] == 
        \(I\_BC\) \[Alpha]\_BC + \(m\_BC\) a\_\(D\_y\)\ a\ Sin[\[Phi]] - 
          \(m\_BC\) a\_\(D\_x\)\ a\ Cos[\[Phi]]\), 
    \(eq6 = A\_x - B\_x == 0\), 
    \(eq7 = A\_y - B\_y == 0\), 
    \(eq8 = 
      \(B\_x\) r\ Cos[\[Theta]] + \(B\_y\) r\ Sin[\[Theta]] + T == 0\)}], 
  "Input"],

Cell[BoxData[
    \(N + C\_x == 0\)], "Output"],

Cell[BoxData[
    \(\(-P\) + C\_y == a\_C\ m\_C\)], "Output"],

Cell[BoxData[
    \(B\_y - C\_y == a\_\(D\_y\)\ m\_BC\)], "Output"],

Cell[BoxData[
    \(B\_x - C\_x == a\_\(D\_x\)\ m\_BC\)], "Output"],

Cell[BoxData[
    \(L\ Cos[\[Phi]]\ C\_x - L\ Sin[\[Phi]]\ C\_y == 
      \(-a\)\ Cos[\[Phi]]\ a\_\(D\_x\)\ m\_BC + 
        a\ Sin[\[Phi]]\ a\_\(D\_y\)\ m\_BC + I\_BC\ \[Alpha]\_BC\)], "Output"],

Cell[BoxData[
    \(A\_x - B\_x == 0\)], "Output"],

Cell[BoxData[
    \(A\_y - B\_y == 0\)], "Output"],

Cell[BoxData[
    \(T + r\ Cos[\[Theta]]\ B\_x + r\ Sin[\[Theta]]\ B\_y == 0\)], "Output"]
}, Open  ]]
}, Open  ]],

Cell[CellGroupData[{

Cell["Solve the 8 equations of motion.", "Subsection"],

Cell["These are the various forces in the problem.", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(soln = 
      Solve[{eq1, eq2, eq3, eq4, eq5, eq6, eq7, eq8}, {A\_x, B\_x, A\_y, 
          B\_y, C\_x, C\_y, N, T}]\)], "Input"],

Cell[BoxData[
    \({{T \[Rule] 
          r\ Sin[\[Theta]]\ \((\(-P\) - a\_\(D\_y\)\ m\_BC - a\_C\ m\_C)\) + 
            \(1\/L\((
              r\ Cos[\[Theta]]\ 
                \((a\ a\_\(D\_x\)\ m\_BC - L\ a\_\(D\_x\)\ m\_BC - 
                    Sec[\[Phi]]\ I\_BC\ \[Alpha]\_BC - L\ P\ Tan[\[Phi]] - 
                    a\ a\_\(D\_y\)\ m\_BC\ Tan[\[Phi]] - 
                    L\ a\_C\ m\_C\ Tan[\[Phi]])\))\)\), 
        A\_x \[Rule] 
          \(-\(1\/L\((
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                Sec[\[Phi]]\ I\_BC\ \[Alpha]\_BC - L\ P\ Tan[\[Phi]] - 
                a\ a\_\(D\_y\)\ m\_BC\ Tan[\[Phi]] - 
                L\ a\_C\ m\_C\ Tan[\[Phi]])\)\)\), 
        A\_y \[Rule] P + a\_\(D\_y\)\ m\_BC + a\_C\ m\_C, 
        N \[Rule] 
          \(-\(\(\(-a\)\ a\_\(D\_x\)\ m\_BC + 
                  Sec[\[Phi]]\ I\_BC\ \[Alpha]\_BC + L\ P\ Tan[\[Phi]] + 
                  a\ a\_\(D\_y\)\ m\_BC\ Tan[\[Phi]] + 
                  L\ a\_C\ m\_C\ Tan[\[Phi]]\)\/L\)\), 
        B\_x \[Rule] 
          \(-\(1\/L\((
              a\ a\_\(D\_x\)\ m\_BC - L\ a\_\(D\_x\)\ m\_BC - 
                Sec[\[Phi]]\ I\_BC\ \[Alpha]\_BC - L\ P\ Tan[\[Phi]] - 
                a\ a\_\(D\_y\)\ m\_BC\ Tan[\[Phi]] - 
                L\ a\_C\ m\_C\ Tan[\[Phi]])\)\)\), 
        C\_x \[Rule] 
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                  L\ P\ Tan[\[Phi]] - a\ a\_\(D\_y\)\ m\_BC\ Tan[\[Phi]] - 
                  L\ a\_C\ m\_C\ Tan[\[Phi]]\)\/L\)\), 
        B\_y \[Rule] P + a\_\(D\_y\)\ m\_BC + a\_C\ m\_C, 
        C\_y \[Rule] P + a\_C\ m\_C}}\)], "Output"]
}, Open  ]],

Cell["\<\
If I wanted to assign the one of the eight solutions to a variable, \
I would do it as before (get used to this, because you will be doing it a \
lot).\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(bX = B\_x /. soln[\([1]\)]\)], "Input"],

Cell[BoxData[
    \(\(-\(\(a\ a\_\(D\_x\)\ m\_BC - L\ a\_\(D\_x\)\ m\_BC - 
            Sec[\[Phi]]\ I\_BC\ \[Alpha]\_BC - L\ P\ Tan[\[Phi]] - 
            a\ a\_\(D\_y\)\ m\_BC\ Tan[\[Phi]] - 
            L\ a\_C\ m\_C\ Tan[\[Phi]]\)\/L\)\)\)], "Output"]
}, Open  ]],

Cell["Notice I didn't get the curly braces this time.", "Text"]
}, Open  ]]
}, Open  ]],

Cell[CellGroupData[{

Cell["\<\
Solving a Single Second-Order Ordinary Differential Equation.\
\>", 
  "Section"],

Cell[CellGroupData[{

Cell["You begin by defining the equation to be solved.", "Subsection"],

Cell[TextData[{
  "For a single second-order ODE, you simply define the equation in the \
manner shown below. Notice, that you can include the initial conditions in \
the list (for numerical solutions, you ",
  StyleBox["must",
    FontSlant->"Italic"],
  " include the initial conditions). This equation happens to be a form of an \
equation called ",
  StyleBox["Duffing's equation",
    FontSlant->"Italic"],
  "."
}], "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(duffing = {
        \(\(x'\)'\)[t] + \[Gamma]\ \(x'\)[t] - x[t] + x[t]\^3 == A\ Cos[t], 
        x[0] == 0.6, \(x'\)[0] == 1.25}\)], "Input"],

Cell[BoxData[
    RowBox[{"{", 
      RowBox[{
        RowBox[{
          RowBox[{\(-x[t]\), "+", \(x[t]\^3\), "+", 
            RowBox[{"\[Gamma]", " ", 
              RowBox[{
                SuperscriptBox["x", "\[Prime]",
                  MultilineFunction->None], "[", "t", "]"}]}], "+", 
            RowBox[{
              SuperscriptBox["x", "\[DoublePrime]",
                MultilineFunction->None], "[", "t", "]"}]}], "==", 
          \(A\ Cos[t]\)}], ",", 
        RowBox[{\(x[0]\), "==", 
          StyleBox["0.6`",
            StyleBoxAutoDelete->True,
            PrintPrecision->1]}], ",", 
        RowBox[{
          RowBox[{
            SuperscriptBox["x", "\[Prime]",
              MultilineFunction->None], "[", "0", "]"}], "==", 
          StyleBox["1.25`",
            StyleBoxAutoDelete->True,
            PrintPrecision->3]}]}], "}"}]], "Output"]
}, Open  ]],

Cell[TextData[{
  "Notice that derivatives are indicated by \"primes\" and we have indicated \
that ",
  StyleBox["x",
    FontSlant->"Italic"],
  " is a function of time."
}], "Text"]
}, Open  ]],

Cell[CellGroupData[{

Cell[TextData[{
  "Now use the command ",
  StyleBox["NDSolve",
    FontFamily->"Courier"],
  " to get the solution."
}], "Subsection"],

Cell[CellGroupData[{

Cell["\<\
I now get the solution. Notice that I must assign values to the \
constants using replacement rules.\
\>", "Subsubsection"],

Cell[TextData[{
  "The nice thing about using replacements rules is that the variables are ",
  StyleBox["not",
    FontSlant->"Italic"],
  " permanently assigned the numerical values."
}], "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(duffsoln1 = 
      NDSolve[duffing /. {\[Gamma] -> 0.15, A -> 0.3}, x, {t, 0, 100}]\)], 
  "Input"],

Cell[BoxData[
    \(NDSolve::"mxst" \( : \ \) 
      "Maximum number of \!\(1000\) steps reached at the point \!\(t\) == \
\!\(58.9464420548711398`\)."\)], "Message"],

Cell[BoxData[
    RowBox[{"{", 
      RowBox[{"{", 
        RowBox[{"x", "\[Rule]", 
          
          TagBox[\(InterpolatingFunction[{{0.`, 58.9464420548711398`}}, "<>"]
              \),
            False,
            Editable->False]}], "}"}], "}"}]], "Output"]
}, Open  ]],

Cell[TextData[{
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " has default limits on almost everything. We can integrate out past 58.9 \
seconds by setting the ",
  StyleBox["MaxSteps",
    FontFamily->"Courier",
    FontWeight->"Bold"],
  " variable to a higher value."
}], "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(duffsoln1 = 
      NDSolve[duffing /. {\[Gamma] -> 0.15, A -> 0.3}, x, {t, 0, 100}, 
        MaxSteps -> 5000]\)], "Input"],

Cell[BoxData[
    RowBox[{"{", 
      RowBox[{"{", 
        RowBox[{"x", "\[Rule]", 
          TagBox[\(InterpolatingFunction[{{0.`, 100.`}}, "<>"]\),
            False,
            Editable->False]}], "}"}], "}"}]], "Output"]
}, Open  ]],

Cell[TextData[{
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " returns an ",
  StyleBox["InterpolatingFunction",
    FontFamily->"Courier",
    FontWeight->"Bold"],
  " as the solution. You can work with ",
  StyleBox["InterpolatingFunction",
    FontFamily->"Courier",
    FontWeight->"Bold"],
  "'s as you would with ",
  StyleBox["Sin",
    FontFamily->"Courier",
    FontWeight->"Bold"],
  " or ",
  StyleBox["Cos",
    FontFamily->"Courier",
    FontWeight->"Bold"],
  " or any other function."
}], "Text"]
}, Open  ]]
}, Open  ]],

Cell[CellGroupData[{

Cell["Plot the solution versus time.", "Subsection"],

Cell["\<\
We use the Plot command to plot the solution. First, let's look at \
a couple of simple examples.\
\>", "Text"],

Cell[CellGroupData[{

Cell["2D Plot Example", "Subsubsection"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Plot[\(x\^2\) Cos[x], {x, 0, 20}]\)], "Input"],

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.5 Mabswid
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.44541 .5762 .45669 .58706 .47603 .60392 .46477 .59279 Metetra
0 .143 .641 r
.46477 .59279 .47603 .60392 .49578 .62032 .48454 .60892 Metetra
0 .231 .722 r
.48454 .60892 .49578 .62032 .51594 .63516 .50473 .6235 Metetra
0 .355 .821 r
.50473 .6235 .51594 .63516 .53649 .64739 .52532 .63549 Metetra
.149 .493 .905 r
.52532 .63549 .53649 .64739 .55738 .65604 .54626 .64394 Metetra
.357 .611 .946 r
.54626 .64394 .55738 .65604 .57853 .66031 .56748 .64803 Metetra
.527 .692 .944 r
.56748 .64803 .57853 .66031 .59986 .65962 .58889 .64721 Metetra
.648 .739 .916 r
.58889 .64721 .59986 .65962 .62128 .65363 .61041 .64113 Metetra
.731 .764 .879 r
.61041 .64113 .62128 .65363 .64268 .64232 .63193 .62978 Metetra
.785 .777 .845 r
.63193 .62978 .64268 .64232 .66398 .62595 .65337 .6134 Metetra
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.65337 .6134 .66398 .62595 .68511 .60504 .67465 .59253 Metetra
.847 .787 .793 r
.67465 .59253 .68511 .60504 .70601 .58038 .69573 .56793 Metetra
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.69573 .56793 .70601 .58038 .72668 .5529 .71658 .54054 Metetra
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.71658 .54054 .72668 .5529 .74713 .52366 .73722 .51141 Metetra
.879 .796 .764 r
.73722 .51141 .74713 .52366 .76739 .49379 .75768 .48163 Metetra
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.75768 .48163 .76739 .49379 .78755 .46437 .77803 .45231 Metetra
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.77803 .45231 .78755 .46437 .8077 .43645 .79837 .42447 Metetra
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.79837 .42447 .8077 .43645 .82795 .41095 .81881 .39903 Metetra
.846 .809 .814 r
.81881 .39903 .82795 .41095 .84844 .38867 .83947 .37678 Metetra
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.83947 .37678 .84844 .38867 .8693 .37025 .86049 .35836 Metetra
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.86049 .35836 .8693 .37025 .89067 .35616 .88201 .34423 Metetra
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.88201 .34423 .89067 .35616 .91267 .34669 .90416 .33468 Metetra
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.30486 .51763 .3169 .52278 .33506 .51919 .32313 .51386 Metetra
.31 .725 .976 r
.32313 .51386 .33506 .51919 .35314 .52002 .34132 .51427 Metetra
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.34132 .51427 .35314 .52002 .37124 .52504 .35951 .51864 Metetra
0 .352 .733 r
.35951 .51864 .37124 .52504 .38943 .53382 .37779 .52657 Metetra
0 .216 .63 r
.37779 .52657 .38943 .53382 .4078 .54578 .39624 .5375 Metetra
0 .141 .583 r
.39624 .5375 .4078 .54578 .42644 .5602 .41495 .55074 Metetra
0 .123 .59 r
.41495 .55074 .42644 .5602 .44541 .5762 .43398 .56545 Metetra
0 .152 .641 r
.43398 .56545 .44541 .5762 .46477 .59279 .45338 .58071 Metetra
0 .223 .721 r
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0 .322 .806 r
.47318 .59551 .48454 .60892 .50473 .6235 .4934 .60884 Metetra
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.4934 .60884 .50473 .6235 .52532 .63549 .51401 .61971 Metetra
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.453 .584 .894 r
.53497 .62722 .54626 .64394 .56748 .64803 .55623 .63063 Metetra
.565 .631 .876 r
.55623 .63063 .56748 .64803 .58889 .64721 .57769 .6294 Metetra
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.59928 .62323 .61041 .64113 .63193 .62978 .6209 .61208 Metetra
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.64246 .59619 .65337 .6134 .67465 .59253 .6639 .57607 Metetra
.816 .739 .779 r
.6639 .57607 .67465 .59253 .69573 .56793 .68517 .55242 Metetra
.838 .754 .771 r
.68517 .55242 .69573 .56793 .71658 .54054 .70624 .52612 Metetra
.855 .77 .767 r
.70624 .52612 .71658 .54054 .73722 .51141 .72712 .49816 Metetra
.867 .786 .769 r
.72712 .49816 .73722 .51141 .75768 .48163 .74784 .46957 Metetra
.876 .804 .776 r
.74784 .46957 .75768 .48163 .77803 .45231 .76847 .4414 Metetra
.878 .823 .789 r
.76847 .4414 .77803 .45231 .79837 .42447 .78909 .41461 Metetra
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.78909 .41461 .79837 .42447 .81881 .39903 .8098 .39007 Metetra
.86 .863 .84 r
.8098 .39007 .81881 .39903 .83947 .37678 .83073 .36855 Metetra
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.83073 .36855 .83947 .37678 .86049 .35836 .852 .35064 Metetra
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.852 .35064 .86049 .35836 .88201 .34423 .87375 .33678 Metetra
.692 .873 .966 r
.87375 .33678 .88201 .34423 .90416 .33468 .89608 .32726 Metetra
.396 .841 .933 r
.29233 .51672 .30486 .51763 .32313 .51386 .31074 .51267 Metetra
.107 .656 .814 r
.31074 .51267 .32313 .51386 .34132 .51427 .32907 .51238 Metetra
.141 0 0 r
.32907 .51238 .34132 .51427 .35951 .51864 .34741 .51563 Metetra
.302 0 0 r
.34741 .51563 .35951 .51864 .37779 .52657 .36583 .52207 Metetra
.383 0 0 r
.36583 .52207 .37779 .52657 .39624 .5375 .38442 .53119 Metetra
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.38442 .53119 .39624 .5375 .41495 .55074 .40325 .54235 Metetra
0 .195 .635 r
.40325 .54235 .41495 .55074 .43398 .56545 .42238 .5548 Metetra
0 .254 .728 r
.42238 .5548 .43398 .56545 .45338 .58071 .44185 .56771 Metetra
0 .335 .812 r
.44185 .56771 .45338 .58071 .47318 .59551 .46171 .58019 Metetra
.122 .414 .861 r
.46171 .58019 .47318 .59551 .4934 .60884 .48196 .59132 Metetra
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.48196 .59132 .4934 .60884 .51401 .61971 .50259 .60023 Metetra
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.50259 .60023 .51401 .61971 .53497 .62722 .52357 .60614 Metetra
.509 .561 .844 r
.52357 .60614 .53497 .62722 .55623 .63063 .54483 .60839 Metetra
.582 .589 .824 r
.54483 .60839 .55623 .63063 .57769 .6294 .56631 .6065 Metetra
.64 .612 .806 r
.56631 .6065 .57769 .6294 .59928 .62323 .58794 .60021 Metetra
.686 .634 .792 r
.58794 .60021 .59928 .62323 .6209 .61208 .60963 .5895 Metetra
.724 .655 .78 r
.60963 .5895 .6209 .61208 .64246 .59619 .6313 .57457 Metetra
.756 .676 .773 r
.6313 .57457 .64246 .59619 .6639 .57607 .65289 .55585 Metetra
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.65289 .55585 .6639 .57607 .68517 .55242 .67435 .53398 Metetra
.809 .721 .768 r
.67435 .53398 .68517 .55242 .70624 .52612 .69567 .50972 Metetra
.831 .747 .771 r
.69567 .50972 .70624 .52612 .72712 .49816 .71683 .48395 Metetra
.85 .776 .779 r
.71683 .48395 .72712 .49816 .74784 .46957 .73787 .4576 Metetra
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.73787 .4576 .74784 .46957 .76847 .4414 .75885 .43157 Metetra
.875 .843 .81 r
.75885 .43157 .76847 .4414 .78909 .41461 .77983 .40676 Metetra
.875 .881 .836 r
.77983 .40676 .78909 .41461 .8098 .39007 .8009 .38394 Metetra
.859 .919 .869 r
.8009 .38394 .8098 .39007 .83073 .36855 .82218 .3638 Metetra
.817 .95 .907 r
.82218 .3638 .83073 .36855 .852 .35064 .84378 .34689 Metetra
.735 .96 .942 r
.84378 .34689 .852 .35064 .87375 .33678 .86581 .33362 Metetra
.605 .931 .961 r
.86581 .33362 .87375 .33678 .89608 .32726 .88838 .32424 Metetra
.173 .703 .699 r
.27927 .51969 .29233 .51672 .31074 .51267 .29785 .51528 Metetra
.061 0 0 r
.29785 .51528 .31074 .51267 .32907 .51238 .31637 .51402 Metetra
.223 0 0 r
.31637 .51402 .32907 .51238 .34741 .51563 .33491 .51572 Metetra
.318 0 0 r
.33491 .51572 .34741 .51563 .36583 .52207 .35354 .52008 Metetra
.352 0 0 r
.35354 .52008 .36583 .52207 .38442 .53119 .37232 .52666 Metetra
.321 0 0 r
.37232 .52666 .38442 .53119 .40325 .54235 .39133 .53492 Metetra
0 .328 .732 r
.39133 .53492 .40325 .54235 .42238 .5548 .4106 .54423 Metetra
0 .399 .84 r
.4106 .54423 .42238 .5548 .44185 .56771 .4302 .55388 Metetra
.142 .456 .885 r
.4302 .55388 .44185 .56771 .46171 .58019 .45015 .56313 Metetra
.293 .492 .881 r
.45015 .56313 .46171 .58019 .48196 .59132 .47045 .57123 Metetra
.404 .514 .857 r
.47045 .57123 .48196 .59132 .50259 .60023 .4911 .57746 Metetra
.485 .531 .83 r
.4911 .57746 .50259 .60023 .52357 .60614 .51208 .58118 Metetra
.545 .546 .806 r
.51208 .58118 .52357 .60614 .54483 .60839 .53333 .58184 Metetra
.593 .561 .788 r
.53333 .58184 .54483 .60839 .56631 .6065 .55481 .57908 Metetra
.633 .577 .775 r
.55481 .57908 .56631 .6065 .58794 .60021 .57645 .57267 Metetra
.667 .595 .766 r
.57645 .57267 .58794 .60021 .60963 .5895 .59817 .56259 Metetra
.697 .615 .76 r
.59817 .56259 .60963 .5895 .6313 .57457 .61993 .54901 Metetra
.725 .637 .759 r
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.752 .662 .761 r
.64166 .53228 .65289 .55585 .67435 .53398 .66332 .51291 Metetra
.778 .691 .768 r
.66332 .51291 .67435 .53398 .69567 .50972 .68489 .49153 Metetra
.803 .725 .778 r
.68489 .49153 .69567 .50972 .71683 .48395 .70637 .46887 Metetra
.826 .764 .794 r
.70637 .46887 .71683 .48395 .73787 .4576 .72777 .44567 Metetra
.847 .811 .814 r
.72777 .44567 .73787 .4576 .75885 .43157 .74915 .42271 Metetra
.861 .864 .84 r
.74915 .42271 .75885 .43157 .77983 .40676 .77055 .40072 Metetra
.859 .92 .87 r
.77055 .40072 .77983 .40676 .8009 .38394 .79206 .38036 Metetra
.826 .968 .894 r
.79206 .38036 .8009 .38394 .82218 .3638 .81376 .36223 Metetra
.741 .986 .899 r
.81376 .36223 .82218 .3638 .84378 .34689 .83575 .34679 Metetra
.592 .949 .868 r
.83575 .34679 .84378 .34689 .86581 .33362 .85812 .33438 Metetra
.403 .856 .807 r
.85812 .33438 .86581 .33362 .88838 .32424 .88097 .32524 Metetra
.26568 .52601 .27927 .51969 .29785 .51528 .28445 .52116 Metetra
.14 0 0 r
.28445 .52116 .29785 .51528 .31637 .51402 .3032 .51869 Metetra
.228 0 0 r
.3032 .51869 .31637 .51402 .33491 .51572 .322 .51845 Metetra
.268 0 0 r
.322 .51845 .33491 .51572 .35354 .52008 .34089 .52021 Metetra
.253 0 0 r
.34089 .52021 .35354 .52008 .37232 .52666 .35993 .52362 Metetra
0 .427 .69 r
.35993 .52362 .37232 .52666 .39133 .53492 .37917 .52828 Metetra
.005 .514 .862 r
.37917 .52828 .39133 .53492 .4106 .54423 .39866 .53369 Metetra
.194 .558 .934 r
.39866 .53369 .4106 .54423 .4302 .55388 .41842 .53931 Metetra
.337 .56 .918 r
.41842 .53931 .4302 .55388 .45015 .56313 .43849 .5446 Metetra
.431 .55 .875 r
.43849 .5446 .45015 .56313 .47045 .57123 .45887 .54898 Metetra
.493 .541 .834 r
.45887 .54898 .47045 .57123 .4911 .57746 .47956 .55194 Metetra
.538 .538 .803 r
.47956 .55194 .4911 .57746 .51208 .58118 .50053 .55299 Metetra
.573 .539 .779 r
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.602 .545 .763 r
.52177 .55175 .53333 .58184 .55481 .57908 .54322 .54793 Metetra
.627 .554 .753 r
.54322 .54793 .55481 .57908 .57645 .57267 .56485 .54139 Metetra
.651 .567 .748 r
.56485 .54139 .57645 .57267 .59817 .56259 .5866 .5321 Metetra
.673 .584 .747 r
.5866 .5321 .59817 .56259 .61993 .54901 .60841 .52019 Metetra
.696 .605 .75 r
.60841 .52019 .61993 .54901 .64166 .53228 .63026 .50591 Metetra
.719 .63 .758 r
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.743 .662 .77 r
.65211 .48964 .66332 .51291 .68489 .49153 .67394 .47183 Metetra
.768 .702 .789 r
.67394 .47183 .68489 .49153 .70637 .46887 .69574 .45301 Metetra
.794 .751 .813 r
.69574 .45301 .70637 .46887 .72777 .44567 .71753 .43375 Metetra
.817 .811 .845 r
.71753 .43375 .72777 .44567 .74915 .42271 .73935 .41463 Metetra
.831 .882 .88 r
.73935 .41463 .74915 .42271 .77055 .40072 .76122 .39619 Metetra
.816 .951 .907 r
.76122 .39619 .77055 .40072 .79206 .38036 .78321 .37896 Metetra
.74 .986 .899 r
.78321 .37896 .79206 .38036 .81376 .36223 .80539 .36338 Metetra
.579 .941 .822 r
.80539 .36338 .81376 .36223 .83575 .34679 .82783 .34981 Metetra
.368 .817 .692 r
.82783 .34981 .83575 .34679 .85812 .33438 .8506 .33853 Metetra
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.085 0 0 r
.25155 .53498 .26568 .52601 .28445 .52116 .27054 .52962 Metetra
.155 0 0 r
.27054 .52962 .28445 .52116 .3032 .51869 .28957 .52574 Metetra
.182 0 0 r
.28957 .52574 .3032 .51869 .322 .51845 .30867 .52325 Metetra
.165 0 0 r
.30867 .52325 .322 .51845 .34089 .52021 .32788 .52199 Metetra
.084 0 0 r
.32788 .52199 .34089 .52021 .35993 .52362 .34724 .52173 Metetra
.08 .633 .817 r
.34724 .52173 .35993 .52362 .37917 .52828 .36678 .52221 Metetra
.28 .705 .969 r
.36678 .52221 .37917 .52828 .39866 .53369 .38653 .52313 Metetra
.421 .683 .971 r
.38653 .52313 .39866 .53369 .41842 .53931 .40651 .52415 Metetra
.496 .633 .914 r
.40651 .52415 .41842 .53931 .43849 .5446 .42674 .52491 Metetra
.536 .591 .857 r
.42674 .52491 .43849 .5446 .45887 .54898 .44723 .52509 Metetra
.561 .563 .813 r
.44723 .52509 .45887 .54898 .47956 .55194 .46798 .52434 Metetra
.58 .547 .782 r
.46798 .52434 .47956 .55194 .50053 .55299 .48896 .5224 Metetra
.595 .538 .761 r
.48896 .5224 .50053 .55299 .52177 .55175 .51018 .51901 Metetra
.609 .536 .747 r
.51018 .51901 .52177 .55175 .54322 .54793 .5316 .51402 Metetra
.623 .539 .739 r
.5316 .51402 .54322 .54793 .56485 .54139 .5532 .50734 Metetra
.637 .547 .737 r
.5532 .50734 .56485 .54139 .5866 .5321 .57494 .49896 Metetra
.652 .56 .739 r
.57494 .49896 .5866 .5321 .60841 .52019 .59679 .48894 Metetra
.667 .578 .745 r
.59679 .48894 .60841 .52019 .63026 .50591 .61874 .47744 Metetra
.685 .602 .757 r
.61874 .47744 .63026 .50591 .65211 .48964 .64075 .46468 Metetra
.705 .634 .776 r
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.727 .677 .802 r
.66283 .45093 .67394 .47183 .69574 .45301 .68496 .43652 Metetra
.75 .733 .837 r
.68496 .43652 .69574 .45301 .71753 .43375 .70715 .4218 Metetra
.771 .806 .881 r
.70715 .4218 .71753 .43375 .73935 .41463 .72941 .40711 Metetra
.776 .891 .926 r
.72941 .40711 .73935 .41463 .76122 .39619 .75179 .39282 Metetra
.731 .962 .941 r
.75179 .39282 .76122 .39619 .78321 .37896 .7743 .37923 Metetra
.588 .947 .865 r
.7743 .37923 .78321 .37896 .80539 .36338 .797 .36664 Metetra
.365 .816 .69 r
.797 .36664 .80539 .36338 .82783 .34981 .81993 .35529 Metetra
.81993 .35529 .82783 .34981 .8506 .33853 .84313 .34535 Metetra
.84313 .34535 .8506 .33853 .87377 .32971 .86666 .33694 Metetra
.116 0 0 r
.23692 .54575 .25155 .53498 .27054 .52962 .25615 .53984 Metetra
.126 0 0 r
.25615 .53984 .27054 .52962 .28957 .52574 .27549 .53441 Metetra
.095 0 0 r
.27549 .53441 .28957 .52574 .30867 .52325 .29494 .52943 Metetra
.009 0 0 r
.29494 .52943 .30867 .52325 .32788 .52199 .31453 .52483 Metetra
.154 .69 .695 r
.31453 .52483 .32788 .52199 .34724 .52173 .33426 .52054 Metetra
.369 .827 .924 r
.33426 .52054 .34724 .52173 .36678 .52221 .35416 .51647 Metetra
.528 .831 .999 r
.35416 .51647 .36678 .52221 .38653 .52313 .37422 .51251 Metetra
.592 .754 .955 r
.37422 .51251 .38653 .52313 .40651 .52415 .39447 .50854 Metetra
.61 .678 .889 r
.39447 .50854 .40651 .52415 .42674 .52491 .41491 .50445 Metetra
.614 .622 .835 r
.41491 .50445 .42674 .52491 .44723 .52509 .43554 .50011 Metetra
.614 .583 .795 r
.43554 .50011 .44723 .52509 .46798 .52434 .45636 .49543 Metetra
.614 .558 .768 r
.45636 .49543 .46798 .52434 .48896 .5224 .47738 .4903 Metetra
.615 .542 .749 r
.47738 .4903 .48896 .5224 .51018 .51901 .49858 .48466 Metetra
.617 .534 .737 r
.49858 .48466 .51018 .51901 .5316 .51402 .51996 .47844 Metetra
.621 .531 .732 r
.51996 .47844 .5316 .51402 .5532 .50734 .54152 .47162 Metetra
.625 .534 .731 r
.54152 .47162 .5532 .50734 .57494 .49896 .56323 .46419 Metetra
.631 .542 .735 r
.56323 .46419 .57494 .49896 .59679 .48894 .58511 .45617 Metetra
.639 .556 .744 r
.58511 .45617 .59679 .48894 .61874 .47744 .60713 .44762 Metetra
.649 .577 .76 r
.60713 .44762 .61874 .47744 .64075 .46468 .62929 .43861 Metetra
.661 .607 .783 r
.62929 .43861 .64075 .46468 .66283 .45093 .65159 .42922 Metetra
.675 .649 .816 r
.65159 .42922 .66283 .45093 .68496 .43652 .67402 .41956 Metetra
.691 .709 .862 r
.67402 .41956 .68496 .43652 .70715 .4218 .6966 .40974 Metetra
.701 .789 .919 r
.6966 .40974 .70715 .4218 .72941 .40711 .71932 .3999 Metetra
.685 .881 .971 r
.71932 .3999 .72941 .40711 .75179 .39282 .74221 .39014 Metetra
.591 .93 .956 r
.74221 .39014 .75179 .39282 .7743 .37923 .76526 .38057 Metetra
.39 .847 .796 r
.76526 .38057 .7743 .37923 .797 .36664 .7885 .37131 Metetra
.7885 .37131 .797 .36664 .81993 .35529 .81194 .36242 Metetra
.81194 .36242 .81993 .35529 .84313 .34535 .83561 .35398 Metetra
.072 0 0 r
.83561 .35398 .84313 .34535 .86666 .33694 .85952 .34602 Metetra
.106 0 0 r
.22185 .55736 .23692 .54575 .25615 .53984 .24133 .55087 Metetra
.058 0 0 r
.24133 .55087 .25615 .53984 .27549 .53441 .261 .5438 Metetra
.261 .5438 .27549 .53441 .29494 .52943 .28084 .53619 Metetra
.28084 .53619 .29494 .52943 .31453 .52483 .30084 .52809 Metetra
.423 .867 .811 r
.30084 .52809 .31453 .52483 .33426 .52054 .321 .51959 Metetra
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.321 .51959 .33426 .52054 .35416 .51647 .3413 .51078 Metetra
.693 .878 .967 r
.3413 .51078 .35416 .51647 .37422 .51251 .36173 .50176 Metetra
.703 .785 .915 r
.36173 .50176 .37422 .51251 .39447 .50854 .38229 .49266 Metetra
.69 .705 .859 r
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.673 .645 .814 r
.40297 .48359 .41491 .50445 .43554 .50011 .42378 .47467 Metetra
.658 .603 .781 r
.42378 .47467 .43554 .50011 .45636 .49543 .44471 .466 Metetra
.645 .572 .758 r
.44471 .466 .45636 .49543 .47738 .4903 .46577 .45767 Metetra
.634 .551 .743 r
.46577 .45767 .47738 .4903 .49858 .48466 .48697 .44975 Metetra
.626 .537 .733 r
.48697 .44975 .49858 .48466 .51996 .47844 .50832 .44229 Metetra
.62 .528 .729 r
.50832 .44229 .51996 .47844 .54152 .47162 .52983 .43533 Metetra
.615 .526 .73 r
.52983 .43533 .54152 .47162 .56323 .46419 .55152 .42886 Metetra
.612 .528 .736 r
.55152 .42886 .56323 .46419 .58511 .45617 .57339 .42286 Metetra
.611 .537 .747 r
.57339 .42286 .58511 .45617 .60713 .44762 .59546 .41728 Metetra
.61 .553 .766 r
.59546 .41728 .60713 .44762 .62929 .43861 .61773 .41205 Metetra
.611 .579 .793 r
.61773 .41205 .62929 .43861 .65159 .42922 .64023 .40709 Metetra
.613 .618 .832 r
.64023 .40709 .65159 .42922 .67402 .41956 .66295 .40229 Metetra
.612 .675 .886 r
.66295 .40229 .67402 .41956 .6966 .40974 .68589 .39753 Metetra
.599 .752 .951 r
.68589 .39753 .6966 .40974 .71932 .3999 .70905 .3927 Metetra
.546 .833 .998 r
.70905 .3927 .71932 .3999 .74221 .39014 .73243 .38767 Metetra
.404 .844 .937 r
.73243 .38767 .74221 .39014 .76526 .38057 .75602 .38232 Metetra
.194 .72 .72 r
.75602 .38232 .76526 .38057 .7885 .37131 .77981 .37655 Metetra
.77981 .37655 .7885 .37131 .81194 .36242 .80378 .37028 Metetra
.071 0 0 r
.80378 .37028 .81194 .36242 .83561 .35398 .82792 .36343 Metetra
.111 0 0 r
.82792 .36343 .83561 .35398 .85952 .34602 .85222 .35596 Metetra
.057 0 0 r
.20641 .56874 .22185 .55736 .24133 .55087 .22615 .56167 Metetra
.22615 .56167 .24133 .55087 .261 .5438 .24616 .55294 Metetra
.24616 .55294 .261 .5438 .28084 .53619 .26642 .54269 Metetra
.446 .863 .719 r
.26642 .54269 .28084 .53619 .30084 .52809 .28687 .53108 Metetra
.651 .969 .879 r
.28687 .53108 .30084 .52809 .321 .51959 .30748 .51836 Metetra
.762 .957 .935 r
.30748 .51836 .321 .51959 .3413 .51078 .32822 .50483 Metetra
.786 .879 .915 r
.32822 .50483 .3413 .51078 .36173 .50176 .34905 .49082 Metetra
.771 .793 .871 r
.34905 .49082 .36173 .50176 .38229 .49266 .36996 .47668 Metetra
.745 .721 .83 r
.36996 .47668 .38229 .49266 .40297 .48359 .39092 .46275 Metetra
.718 .665 .797 r
.39092 .46275 .40297 .48359 .42378 .47467 .41194 .44938 Metetra
.694 .622 .771 r
.41194 .44938 .42378 .47467 .44471 .466 .433 .43685 Metetra
.672 .589 .753 r
.433 .43685 .44471 .466 .46577 .45767 .45413 .42543 Metetra
.653 .563 .741 r
.45413 .42543 .46577 .45767 .48697 .44975 .47535 .41533 Metetra
.636 .544 .734 r
.47535 .41533 .48697 .44975 .50832 .44229 .49667 .40668 Metetra
.621 .531 .732 r
.49667 .40668 .50832 .44229 .52983 .43533 .51814 .39958 Metetra
.607 .523 .734 r
.51814 .39958 .52983 .43533 .55152 .42886 .53979 .39402 Metetra
.593 .52 .741 r
.53979 .39402 .55152 .42886 .57339 .42286 .56165 .38994 Metetra
.581 .522 .754 r
.56165 .38994 .57339 .42286 .59546 .41728 .58375 .38721 Metetra
.568 .532 .775 r
.58375 .38721 .59546 .41728 .61773 .41205 .60611 .38563 Metetra
.554 .55 .805 r
.60611 .38563 .61773 .41205 .64023 .40709 .62877 .38496 Metetra
.536 .581 .847 r
.62877 .38496 .64023 .40709 .66295 .40229 .65174 .38489 Metetra
.511 .627 .902 r
.65174 .38489 .66295 .40229 .68589 .39753 .67501 .38509 Metetra
.463 .688 .964 r
.67501 .38509 .68589 .39753 .70905 .3927 .69857 .38521 Metetra
.368 .741 .989 r
.69857 .38521 .70905 .3927 .73243 .38767 .72242 .38487 Metetra
.206 .718 .889 r
.72242 .38487 .73243 .38767 .75602 .38232 .74652 .38374 Metetra
.74652 .38374 .75602 .38232 .77981 .37655 .77084 .38148 Metetra
.069 0 0 r
.77084 .38148 .77981 .37655 .80378 .37028 .79532 .37784 Metetra
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.79532 .37784 .80378 .37028 .82792 .36343 .81994 .3726 Metetra
.112 0 0 r
.81994 .3726 .82792 .36343 .85222 .35596 .84463 .36563 Metetra
.1907 .57878 .20641 .56874 .22615 .56167 .2107 .57114 Metetra
.2107 .57114 .22615 .56167 .24616 .55294 .23106 .56082 Metetra
.453 .859 .688 r
.23106 .56082 .24616 .55294 .26642 .54269 .25173 .54803 Metetra
.662 .972 .826 r
.25173 .54803 .26642 .54269 .28687 .53108 .27265 .53306 Metetra
.791 .988 .891 r
.27265 .53306 .28687 .53108 .30748 .51836 .29373 .51633 Metetra
.837 .936 .892 r
.29373 .51633 .30748 .51836 .32822 .50483 .31493 .49834 Metetra
.834 .863 .866 r
.31493 .49834 .32822 .50483 .34905 .49082 .33619 .47963 Metetra
.812 .793 .834 r
.33619 .47963 .34905 .49082 .36996 .47668 .35747 .46077 Metetra
.783 .732 .806 r
.35747 .46077 .36996 .47668 .39092 .46275 .37874 .44232 Metetra
.754 .683 .783 r
.37874 .44232 .39092 .46275 .41194 .44938 .39998 .42481 Metetra
.725 .642 .765 r
.39998 .42481 .41194 .44938 .433 .43685 .42121 .40873 Metetra
.698 .608 .752 r
.42121 .40873 .433 .43685 .45413 .42543 .44243 .39448 Metetra
.672 .58 .744 r
.44243 .39448 .45413 .42543 .47535 .41533 .46368 .38237 Metetra
.647 .558 .739 r
.46368 .38237 .47535 .41533 .49667 .40668 .485 .37263 Metetra
.623 .54 .739 r
.485 .37263 .49667 .40668 .51814 .39958 .50644 .36539 Metetra
.599 .526 .743 r
.50644 .36539 .51814 .39958 .53979 .39402 .52805 .36065 Metetra
.574 .516 .752 r
.52805 .36065 .53979 .39402 .56165 .38994 .54989 .35831 Metetra
.548 .511 .767 r
.54989 .35831 .56165 .38994 .58375 .38721 .572 .35818 Metetra
.52 .512 .788 r
.572 .35818 .58375 .38721 .60611 .38563 .59444 .35995 Metetra
.486 .52 .818 r
.59444 .35995 .60611 .38563 .62877 .38496 .61723 .36323 Metetra
.445 .537 .858 r
.61723 .36323 .62877 .38496 .65174 .38489 .64039 .36755 Metetra
.388 .565 .906 r
.64039 .36755 .65174 .38489 .67501 .38509 .66395 .37237 Metetra
.305 .598 .948 r
.66395 .37237 .67501 .38509 .69857 .38521 .68787 .37712 Metetra
.184 .617 .943 r
.68787 .37712 .69857 .38521 .72242 .38487 .71214 .38122 Metetra
.042 .589 .837 r
.71214 .38122 .72242 .38487 .74652 .38374 .73671 .38408 Metetra
.069 0 0 r
.73671 .38408 .74652 .38374 .77084 .38148 .76152 .38517 Metetra
.118 0 0 r
.76152 .38517 .77084 .38148 .79532 .37784 .78649 .38405 Metetra
.115 0 0 r
.78649 .38405 .79532 .37784 .81994 .3726 .81156 .38035 Metetra
.071 0 0 r
.81156 .38035 .81994 .3726 .84463 .36563 .83663 .37384 Metetra
.17484 .58638 .1907 .57878 .2107 .57114 .19509 .57821 Metetra
.447 .864 .72 r
.19509 .57821 .2107 .57114 .23106 .56082 .21579 .56644 Metetra
.663 .972 .826 r
.21579 .56644 .23106 .56082 .25173 .54803 .23687 .55134 Metetra
.799 .994 .872 r
.23687 .55134 .25173 .54803 .27265 .53306 .25824 .53334 Metetra
.858 .959 .872 r
.25824 .53334 .27265 .53306 .29373 .51633 .27979 .51301 Metetra
.87 .902 .852 r
.27979 .51301 .29373 .51633 .31493 .49834 .30145 .49102 Metetra
.858 .843 .828 r
.30145 .49102 .31493 .49834 .33619 .47963 .32314 .46812 Metetra
.837 .789 .806 r
.32314 .46812 .33619 .47963 .35747 .46077 .3448 .44508 Metetra
.81 .741 .787 r
.3448 .44508 .35747 .46077 .37874 .44232 .36638 .42265 Metetra
.782 .699 .773 r
.36638 .42265 .37874 .44232 .39998 .42481 .38788 .40152 Metetra
.753 .663 .762 r
.38788 .40152 .39998 .42481 .42121 .40873 .40928 .38233 Metetra
.723 .63 .755 r
.40928 .38233 .42121 .40873 .44243 .39448 .43062 .36561 Metetra
.692 .602 .751 r
.43062 .36561 .44243 .39448 .46368 .38237 .45193 .35175 Metetra
.661 .577 .75 r
.45193 .35175 .46368 .38237 .485 .37263 .47327 .34106 Metetra
.628 .554 .753 r
.47327 .34106 .485 .37263 .50644 .36539 .4947 .33368 Metetra
.593 .535 .759 r
.4947 .33368 .50644 .36539 .52805 .36065 .51628 .32963 Metetra
.555 .518 .769 r
.51628 .32963 .52805 .36065 .54989 .35831 .5381 .32878 Metetra
.512 .504 .784 r
.5381 .32878 .54989 .35831 .572 .35818 .56022 .33088 Metetra
.464 .494 .805 r
.56022 .33088 .572 .35818 .59444 .35995 .5827 .33554 Metetra
.407 .488 .831 r
.5827 .33554 .59444 .35995 .61723 .36323 .60559 .34226 Metetra
.338 .488 .861 r
.60559 .34226 .61723 .36323 .64039 .36755 .62892 .35041 Metetra
.252 .491 .891 r
.62892 .35041 .64039 .36755 .66395 .37237 .6527 .35929 Metetra
.148 .496 .906 r
.6527 .35929 .66395 .37237 .68787 .37712 .67693 .36815 Metetra
.032 .496 .882 r
.67693 .36815 .68787 .37712 .71214 .38122 .70156 .37618 Metetra
0 .484 .801 r
.70156 .37618 .71214 .38122 .73671 .38408 .72654 .38261 Metetra
.122 0 0 r
.72654 .38261 .73671 .38408 .76152 .38517 .75178 .38671 Metetra
.12 0 0 r
.75178 .38671 .76152 .38517 .78649 .38405 .7772 .38785 Metetra
.07 0 0 r
.7772 .38785 .78649 .38405 .81156 .38035 .80268 .38553 Metetra
.80268 .38553 .81156 .38035 .83663 .37384 .82812 .37944 Metetra
.413 .86 .802 r
.15896 .5905 .17484 .58638 .19509 .57821 .17943 .58188 Metetra
.65 .969 .878 r
.17943 .58188 .19509 .57821 .21579 .56644 .20045 .56888 Metetra
.792 .988 .891 r
.20045 .56888 .21579 .56644 .23687 .55134 .22191 .55183 Metetra
.858 .958 .872 r
.22191 .55183 .23687 .55134 .25824 .53334 .24371 .53128 Metetra
.88 .913 .845 r
.24371 .53128 .25824 .53334 .27979 .51301 .26571 .50793 Metetra
.881 .866 .82 r
.26571 .50793 .27979 .51301 .30145 .49102 .28781 .48262 Metetra
.869 .823 .8 r
.28781 .48262 .30145 .49102 .32314 .46812 .30991 .45625 Metetra
.851 .784 .785 r
.30991 .45625 .32314 .46812 .3448 .44508 .33194 .42976 Metetra
.83 .748 .774 r
.33194 .42976 .3448 .44508 .36638 .42265 .35384 .40405 Metetra
.805 .715 .766 r
.35384 .40405 .36638 .42265 .38788 .40152 .37559 .37997 Metetra
.777 .685 .762 r
.37559 .37997 .38788 .40152 .40928 .38233 .39719 .35824 Metetra
.747 .657 .761 r
.39719 .35824 .40928 .38233 .43062 .36561 .41867 .3395 Metetra
.713 .629 .763 r
.41867 .3395 .43062 .36561 .45193 .35175 .44006 .32422 Metetra
.676 .603 .767 r
.44006 .32422 .45193 .35175 .47327 .34106 .46144 .31274 Metetra
.634 .578 .773 r
.46144 .31274 .47327 .34106 .4947 .33368 .48287 .30523 Metetra
.587 .552 .783 r
.48287 .30523 .4947 .33368 .51628 .32963 .50445 .3017 Metetra
.533 .528 .795 r
.50445 .3017 .51628 .32963 .5381 .32878 .52626 .30203 Metetra
.471 .503 .809 r
.52626 .30203 .5381 .32878 .56022 .33088 .54838 .30591 Metetra
.4 .479 .825 r
.54838 .30591 .56022 .33088 .5827 .33554 .57089 .31289 Metetra
.317 .455 .842 r
.57089 .31289 .5827 .33554 .60559 .34226 .59385 .32237 Metetra
.222 .433 .856 r
.59385 .32237 .60559 .34226 .62892 .35041 .61731 .33362 Metetra
.12 .415 .862 r
.61731 .33362 .62892 .35041 .6527 .35929 .64127 .34581 Metetra
.018 .402 .854 r
.64127 .34581 .6527 .35929 .67693 .36815 .66573 .35802 Metetra
0 .401 .829 r
.66573 .35802 .67693 .36815 .70156 .37618 .69066 .36929 Metetra
0 .416 .79 r
.69066 .36929 .70156 .37618 .72654 .38261 .71597 .37867 Metetra
0 .453 .748 r
.71597 .37867 .72654 .38261 .75178 .38671 .74159 .38525 Metetra
0 .516 .718 r
.74159 .38525 .75178 .38671 .7772 .38785 .76738 .38826 Metetra
.034 .605 .709 r
.76738 .38826 .7772 .38785 .80268 .38553 .79323 .38708 Metetra
.182 .713 .723 r
.79323 .38708 .80268 .38553 .82812 .37944 .81899 .3813 Metetra
.6 .934 .954 r
.1432 .59026 .15896 .5905 .17943 .58188 .16387 .58126 Metetra
.76 .959 .935 r
.16387 .58126 .17943 .58188 .20045 .56888 .18516 .56732 Metetra
.837 .936 .893 r
.18516 .56732 .20045 .56888 .22191 .55183 .20696 .5488 Metetra
.87 .901 .852 r
.20696 .5488 .22191 .55183 .24371 .53128 .22912 .52633 Metetra
.881 .866 .82 r
.22912 .52633 .24371 .53128 .26571 .50793 .25152 .50072 Metetra
.88 .834 .797 r
.25152 .50072 .26571 .50793 .28781 .48262 .27401 .47292 Metetra
.872 .805 .781 r
.27401 .47292 .28781 .48262 .30991 .45625 .29649 .44397 Metetra
.86 .779 .771 r
.29649 .44397 .30991 .45625 .33194 .42976 .31886 .41493 Metetra
.843 .755 .765 r
.31886 .41493 .33194 .42976 .35384 .40405 .34107 .3868 Metetra
.823 .732 .764 r
.34107 .3868 .35384 .40405 .37559 .37997 .36307 .36053 Metetra
.799 .709 .766 r
.36307 .36053 .37559 .37997 .39719 .35824 .38487 .33693 Metetra
.77 .687 .771 r
.38487 .33693 .39719 .35824 .41867 .3395 .4065 .3167 Metetra
.735 .663 .78 r
.4065 .3167 .41867 .3395 .44006 .32422 .428 .30036 Metetra
.693 .638 .79 r
.428 .30036 .44006 .32422 .46144 .31274 .44944 .28828 Metetra
.643 .611 .803 r
.44944 .28828 .46144 .31274 .48287 .30523 .47091 .28064 Metetra
.582 .581 .817 r
.47091 .28064 .48287 .30523 .50445 .3017 .49251 .27747 Metetra
.509 .546 .83 r
.49251 .27747 .50445 .3017 .52626 .30203 .51433 .27861 Metetra
.422 .508 .842 r
.51433 .27861 .52626 .30203 .54838 .30591 .53646 .28374 Metetra
.323 .466 .848 r
.53646 .28374 .54838 .30591 .57089 .31289 .559 .29236 Metetra
.215 .422 .848 r
.559 .29236 .57089 .31289 .59385 .32237 .58201 .30381 Metetra
.106 .38 .84 r
.58201 .30381 .59385 .32237 .61731 .33362 .60555 .31729 Metetra
.006 .348 .825 r
.60555 .31729 .61731 .33362 .64127 .34581 .62965 .33187 Metetra
0 .333 .809 r
.62965 .33187 .64127 .34581 .66573 .35802 .65428 .34651 Metetra
0 .342 .799 r
.65428 .34651 .66573 .35802 .69066 .36929 .67943 .36014 Metetra
0 .382 .801 r
.67943 .36014 .69066 .36929 .71597 .37867 .705 .37166 Metetra
0 .458 .82 r
.705 .37166 .71597 .37867 .74159 .38525 .7309 .38005 Metetra
.035 .568 .856 r
.7309 .38005 .74159 .38525 .76738 .38826 .757 .38442 Metetra
.192 .701 .899 r
.757 .38442 .76738 .38826 .79323 .38708 .78315 .38405 Metetra
.381 .832 .931 r
.78315 .38405 .79323 .38708 .81899 .3813 .8092 .37848 Metetra
.685 .885 .972 r
.12768 .58495 .1432 .59026 .16387 .58126 .14851 .5757 Metetra
.785 .884 .917 r
.14851 .5757 .16387 .58126 .18516 .56732 .17003 .56115 Metetra
.835 .865 .867 r
.17003 .56115 .18516 .56732 .20696 .5488 .19209 .54171 Metetra
.859 .843 .828 r
.19209 .54171 .20696 .5488 .22912 .52633 .21456 .51806 Metetra
.869 .823 .8 r
.21456 .51806 .22912 .52633 .25152 .50072 .23727 .49106 Metetra
.872 .805 .781 r
.23727 .49106 .25152 .50072 .27401 .47292 .26009 .46176 Metetra
.87 .789 .769 r
.26009 .46176 .27401 .47292 .29649 .44397 .28289 .43125 Metetra
.863 .775 .763 r
.28289 .43125 .29649 .44397 .31886 .41493 .30556 .40066 Metetra
.853 .762 .762 r
.30556 .40066 .31886 .41493 .34107 .3868 .32803 .37107 Metetra
.838 .749 .766 r
.32803 .37107 .34107 .3868 .36307 .36053 .35027 .34347 Metetra
.818 .736 .774 r
.35027 .34347 .36307 .36053 .38487 .33693 .37228 .31875 Metetra
.792 .722 .786 r
.37228 .31875 .38487 .33693 .4065 .3167 .39406 .2976 Metetra
.758 .706 .802 r
.39406 .2976 .4065 .3167 .428 .30036 .41569 .2806 Metetra
.713 .685 .822 r
.41569 .2806 .428 .30036 .44944 .28828 .43723 .26812 Metetra
.653 .659 .843 r
.43723 .26812 .44944 .28828 .47091 .28064 .45877 .26036 Metetra
.575 .624 .863 r
.45877 .26036 .47091 .28064 .49251 .27747 .48041 .25735 Metetra
.478 .578 .877 r
.48041 .25735 .49251 .27747 .51433 .27861 .50226 .25891 Metetra
.362 .521 .88 r
.50226 .25891 .51433 .27861 .53646 .28374 .52442 .26471 Metetra
.235 .456 .87 r
.52442 .26471 .53646 .28374 .559 .29236 .54699 .27424 Metetra
.11 .391 .847 r
.54699 .27424 .559 .29236 .58201 .30381 .57005 .28679 Metetra
.003 .337 .819 r
.57005 .28679 .58201 .30381 .60555 .31729 .59365 .30152 Metetra
0 .302 .794 r
.59365 .30152 .60555 .31729 .62965 .33187 .61783 .31744 Metetra
0 .294 .784 r
.61783 .31744 .62965 .33187 .65428 .34651 .64258 .33346 Metetra
0 .319 .793 r
.64258 .33346 .65428 .34651 .67943 .36014 .66787 .34842 Metetra
0 .378 .825 r
.66787 .34842 .67943 .36014 .705 .37166 .69362 .36115 Metetra
.025 .472 .877 r
.69362 .36115 .705 .37166 .7309 .38005 .71972 .37057 Metetra
.168 .595 .937 r
.71972 .37057 .7309 .38005 .757 .38442 .74604 .37568 Metetra
.345 .724 .985 r
.74604 .37568 .757 .38442 .78315 .38405 .77242 .37574 Metetra
.524 .831 1 r
.77242 .37574 .78315 .38405 .8092 .37848 .79871 .37023 Metetra
.699 .795 .924 r
.11253 .57415 .12768 .58495 .14851 .5757 .13348 .56475 Metetra
.772 .8 .876 r
.13348 .56475 .14851 .5757 .17003 .56115 .15514 .54998 Metetra
.813 .796 .836 r
.15514 .54998 .17003 .56115 .19209 .54171 .17739 .53023 Metetra
.837 .79 .806 r
.17739 .53023 .19209 .54171 .21456 .51806 .20007 .50621 Metetra
.852 .784 .785 r
.20007 .50621 .21456 .51806 .23727 .49106 .22301 .47879 Metetra
.86 .779 .771 r
.22301 .47879 .23727 .49106 .26009 .46176 .24607 .44903 Metetra
.863 .775 .763 r
.24607 .44903 .26009 .46176 .28289 .43125 .2691 .41805 Metetra
.863 .772 .761 r
.2691 .41805 .28289 .43125 .30556 .40066 .29201 .387 Metetra
.858 .77 .764 r
.29201 .387 .30556 .40066 .32803 .37107 .3147 .35697 Metetra
.849 .768 .772 r
.3147 .35697 .32803 .37107 .35027 .34347 .33715 .32897 Metetra
.835 .766 .786 r
.33715 .32897 .35027 .34347 .37228 .31875 .35934 .30389 Metetra
.813 .763 .806 r
.35934 .30389 .37228 .31875 .39406 .2976 .3813 .28244 Metetra
.781 .757 .832 r
.3813 .28244 .39406 .2976 .41569 .2806 .40307 .2652 Metetra
.732 .745 .862 r
.40307 .2652 .41569 .2806 .43723 .26812 .42472 .25253 Metetra
.662 .723 .894 r
.42472 .25253 .43723 .26812 .45877 .26036 .44636 .24466 Metetra
.564 .684 .92 r
.44636 .24466 .45877 .26036 .48041 .25735 .46809 .24159 Metetra
.435 .623 .931 r
.46809 .24159 .48041 .25735 .50226 .25891 .49 .24316 Metetra
.286 .542 .918 r
.49 .24316 .50226 .25891 .52442 .26471 .51222 .24904 Metetra
.137 .452 .883 r
.51222 .24904 .52442 .26471 .54699 .27424 .53484 .25869 Metetra
.013 .371 .838 r
.53484 .25869 .54699 .27424 .57005 .28679 .55794 .27141 Metetra
0 .312 .8 r
.55794 .27141 .57005 .28679 .59365 .30152 .58158 .28635 Metetra
0 .285 .779 r
.58158 .28635 .59365 .30152 .61783 .31744 .60581 .3025 Metetra
0 .29 .781 r
.60581 .3025 .61783 .31744 .64258 .33346 .63063 .31875 Metetra
0 .327 .807 r
.63063 .31875 .64258 .33346 .66787 .34842 .656 .33394 Metetra
.013 .395 .85 r
.656 .33394 .66787 .34842 .69362 .36115 .68184 .34687 Metetra
.136 .487 .902 r
.68184 .34687 .69362 .36115 .71972 .37057 .70806 .35643 Metetra
.287 .59 .946 r
.70806 .35643 .71972 .37057 .74604 .37568 .73452 .36163 Metetra
.444 .685 .967 r
.73452 .36163 .74604 .37568 .77242 .37574 .76106 .36168 Metetra
.584 .758 .961 r
.76106 .36168 .77242 .37574 .79871 .37023 .78751 .35608 Metetra
.687 .713 .869 r
.09783 .5577 .11253 .57415 .13348 .56475 .11885 .54829 Metetra
.746 .728 .835 r
.11885 .54829 .13348 .56475 .15514 .54998 .14059 .53367 Metetra
.785 .736 .808 r
.14059 .53367 .15514 .54998 .17739 .53023 .16293 .51425 Metetra
.812 .743 .788 r
.16293 .51425 .17739 .53023 .20007 .50621 .18571 .49068 Metetra
.83 .749 .774 r
.18571 .49068 .20007 .50621 .22301 .47879 .20877 .46383 Metetra
.844 .755 .765 r
.20877 .46383 .22301 .47879 .24607 .44903 .23196 .4347 Metetra
.853 .762 .762 r
.23196 .4347 .24607 .44903 .2691 .41805 .25514 .40438 Metetra
.858 .77 .764 r
.25514 .40438 .2691 .41805 .29201 .387 .27819 .37397 Metetra
.86 .779 .771 r
.27819 .37397 .29201 .387 .3147 .35697 .30105 .34455 Metetra
.857 .789 .784 r
.30105 .34455 .3147 .35697 .33715 .32897 .32366 .31708 Metetra
.848 .8 .804 r
.32366 .31708 .33715 .32897 .35934 .30389 .34602 .29242 Metetra
.83 .81 .831 r
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