Editing Appendix/Ramblings/Dyson1893Part1 (section)

===Step 3===

Subsequently, Dyson was able to obtain analytic expressions for successive derivatives of the function, <math>~\mathfrak{I}(r,\theta,c) </math>, by first demonstrating that
[[File:CommentButton02.png|right|100px|Comment by J. E. Tohline on 17 September 2018:  In the middle of p. 61 of Dyson(1893a), there appears to be a typographical error in the expression for the derivative of R<sub>1</sub> with respect to ''c''; as we have indicated here, the numerator should be 4cR<sub>1</sub> instead of 4cR.]]
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{dR}{dc}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{4c^2 + R^2 - R_1^2}{4cR} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{dR_1}{dc}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{4c^2 + R_1^2 - R^2}{4cR_1} \, ,</math> &nbsp; &nbsp; &nbsp; and,
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{d\mu}{dc}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\mu}{c} \cos\psi \, ,</math>
  </td>
</tr>
</table>
where &#8212; as shown above in the [http://www.mathematicsdictionary.com/english/vmd/full/t/torusanchorring.htm Anchor ring] schematic &#8212; <math>~\psi</math> is the angle between <math>~R</math> and <math>~R_1</math> for which (according to the law of cosines),
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\cos\psi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{R^2 + R_1^2 - 4c^2}{2RR_1} \, .</math>
  </td>
</tr>
</table>
It will be useful for us to note that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{d(\cos\psi)}{dc}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{d}{dc}\biggl[
\frac{R^2 + R_1^2 - 4c^2}{2RR_1}
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{2RR_1}\frac{d}{dc}\biggl[
R^2 + R_1^2 - 4c^2
\biggr]
~+~(R^2 + R_1^2 - 4c^2)
\frac{d}{dc}\biggl[
\frac{1}{2RR_1}
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{2RR_1}\biggl[2R\frac{dR}{dc} + 2R_1\frac{dR_1}{dc} - 8c
\biggr]
~+~(R^2 + R_1^2 - 4c^2)\biggl[
- \frac{1}{2R^2R_1} \frac{dR}{dc} - \frac{1}{2RR_1^2}\frac{dR_1}{dc}
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{4cRR_1}\biggl[ 4c^2 + R^2 - R_1^2 + 4c^2 + R_1^2 - R^2 - 2^4c^2
\biggr]
~-~(R^2 + R_1^2 - 4c^2)\biggl\{
\biggl[ \frac{4c^2 + R^2 - R_1^2}{8cR^3R_1}\biggr] + \biggl[ \frac{4c^2 + R_1^2 - R^2}{8cR_1^3 R} \biggr]
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-~\frac{2c}{RR_1}
~-~\frac{\cos\psi}{4cR^2 R_1^2} \biggl[4c^2(R_1^2 + R^2) + 2R_1^2 R^2 - R_1^4 - R^4 
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\cos\psi}{4cR^2 R_1^2} \biggl[
R_1^4 + R^4- 4c^2(R_1^2 + R^2) - 2R_1^2 R^2 
\biggr]
-~\frac{2c}{RR_1}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{4cR^2 R_1^2}\biggl\{
\cos\psi 
[R_1^4 + R^4 - 2R_1^2 R^2 - 4c^2(R_1^2 + R^2) ]
-~8c^2R R_1 \biggr\}
</math>
  </td>
</tr>
</table>

But,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~[R_1^4 + R^4 - 2R_1^2 R^2 - 4c^2(R_1^2 + R^2)]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(R_1^2 - R^2)^2 
- 4c^2(R_1^2 + R^2)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
[(R_1+R)(R_1-R)]^2 
- 4c^2(R_1^2 + R^2)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(R_1^2 + 2R_1 R +R^2)(R_1-R)^2 
- 4c^2(R_1^2 + R^2)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
2R_1 R(R_1-R)^2 
+(R_1^2 + R^2)(R_1-R)^2 
- 4c^2(R_1^2 + R^2)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
2R_1 R(R_1-R)^2 
+(R_1^2 + R^2)[R_1^2 -2R_1 R + R^2 - 4c^2]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
2R_1 R[ (R_1-R)^2 
+(R_1^2 + R^2)(\cos\psi - 1)]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
2R_1 R[  (R_1^2 + R^2)\cos\psi - 2R_1 R]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(2R_1 R)^2 \biggl[ \cos^2\psi  + \biggl(\frac{2c^2 }{R_1 R}\biggr)\cos\psi - 1 \biggr]
</math>
  </td>
</tr>
</table>
Hence,

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
\frac{d(\cos\psi)}{dc}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{(2R_1 R)^2 }{4cR^2 R_1^2}\biggl\{
\cos\psi 
\biggl[ \cos^2\psi  + \biggl(\frac{2c^2 }{R_1 R}\biggr)\cos\psi - 1 \biggr]
-~\frac{8c^2R R_1}{(2R_1 R)^2 } \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{c}\biggl\{
\cos\psi 
\biggl[ \cos^2\psi  + \biggl(\frac{2c^2 }{R_1 R}\biggr)\cos\psi - 1 \biggr]
-~\frac{2c^2}{R_1 R } \biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~
\Rightarrow ~~~ 2c\cdot \frac{d(\cos\psi)}{dc}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
2\cos\psi \biggl[ \cos^2\psi  + \biggl(\frac{2c^2 }{R_1 R}\biggr)\cos\psi - 1 \biggr]
-~\frac{4c^2}{R_1 R } 
</math>
  </td>
</tr>
</table>