Editing Appendix/Ramblings/Dyson1893Part1 (section)

===Step 5===
Differentiating a second time gives,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{d}{dc}\biggl[\frac{1}{c}\cdot \frac{d\mathfrak{I}(r,\theta,c)}{dc}\biggr] </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\frac{d}{dc}\biggl\{
\biggl[ \frac{(R_1+R)E\left(\mu\right)}{c^2RR_1}\biggr] \cos\psi~-~\frac{2K(\mu)}{c^2 (R_1+R)}(1+\cos\psi) 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
~-~\frac{2}{c^3}\biggl\{
\biggl[ \frac{(R_1+R)E\left(\mu\right)}{RR_1}\biggr] \cos\psi~-~\frac{2K(\mu)}{ (R_1+R)}(1+\cos\psi) 
\biggr\}
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
~-~\frac{2}{c^3}\biggl\{
\biggl[ \frac{(R_1+R)E\left(\mu\right)}{RR_1}\biggr] \cos\psi~-~\frac{2K(\mu)}{ (R_1+R)}(1+\cos\psi) 
\biggr\}
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
~-~\frac{2}{c^3}\biggl\{
\biggl[ \frac{(R_1+R)E\left(\mu\right)}{RR_1}\biggr] \cos\psi~-~\frac{2K(\mu)}{ (R_1+R)}(1+\cos\psi) 
\biggr\}
~+~\frac{1}{c^2}\biggl[ \frac{(R_1+R)E\left(\mu\right)}{RR_1}~-~\frac{2K(\mu)}{ (R_1+R)}\biggr] \frac{d(\cos\psi)}{dc}
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\frac{1}{c^3}\biggl[ \frac{4K(\mu)}{ (R_1+R)}\biggr]\biggl[(1+\cos\psi) -  \biggl(\frac{c}{2}\biggr) \frac{d(\cos\psi)}{dc}\biggr]
~-~\frac{2}{c^3}\biggl[ \frac{(R_1+R)E\left(\mu\right)}{RR_1}\biggr] \biggl[ \cos\psi -  \biggl(\frac{c}{2}\biggr) \frac{d(\cos\psi)}{dc}\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ 
~+~\frac{\cos\psi}{c^2}\cdot \frac{d}{dc}\biggl[ \frac{(R_1+R)E\left(\mu\right)}{RR_1}\biggr] 
~-~\frac{(1+\cos\psi)}{2c^3} \biggl\{\biggl[ \frac{E(\mu)(R+R_1)}{RR_1} \biggr] \cos\psi ~-~\biggl[ \frac{4K(\mu)}{R+R_1} \biggr] \cos^2\frac{\psi}{2} \biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\frac{1}{c^3}\biggl[ \frac{4K(\mu)}{ (R_1+R)}\biggr]\biggl[\frac{(1+\cos\psi)^2}{2^2}  + (1+\cos\psi) -  \biggl(\frac{c}{2}\biggr) \frac{d(\cos\psi)}{dc}\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ 
~-~\frac{2}{c^3}\biggl[ \frac{(R_1+R)E\left(\mu\right)}{RR_1}\biggr] \biggl[\frac{(\cos\psi+\cos^2\psi)}{4} ~+~ \cos\psi ~-~  \biggl(\frac{c}{2}\biggr) \frac{d(\cos\psi)}{dc}\biggr]
~+~\frac{\cos\psi}{c^2}\cdot \frac{d}{dc}\biggl[ \frac{(R_1+R)E\left(\mu\right)}{RR_1}\biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\frac{1}{4c^3}\biggl[ \frac{4K(\mu)}{ (R_1+R)}\biggr]\biggl[5+6\cos\psi + \cos^2\psi  -  2c \cdot \frac{d(\cos\psi)}{dc}\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ 
~-~\frac{1}{2c^3}\biggl[ \frac{(R_1+R)E\left(\mu\right)}{RR_1}\biggr] \biggl[5\cos\psi+\cos^2\psi ~-~  2c \cdot \frac{d(\cos\psi)}{dc}\biggr]
~+~\frac{\cos\psi}{c^2}\cdot \frac{d}{dc}\biggl[ \frac{(R_1+R)E\left(\mu\right)}{RR_1}\biggr] \, .
</math>
  </td>
</tr>
</table>

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

<tr>
  <td align="right">
<math>~
\frac{d}{dc}\biggl[ \frac{(R_1+R)E\left(\mu\right)}{RR_1}\biggr]
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{(R_1+R)}{RR_1}\biggr]\frac{dE\left(\mu\right)}{d\mu} \cdot \frac{d\mu}{dc}
~+~
E\left(\mu\right)\frac{d}{dc}\biggl[ \frac{(R_1+R)}{RR_1}\biggr]
</math>
  </td>
</tr>

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

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

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

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
~-~K(\mu)\biggl[ \frac{(R_1+R)}{cRR_1}\biggr] \cos\psi
+E(\mu)\biggl[ \frac{(R_1+R)}{cRR_1}\biggr] \cos\psi
~+~
\frac{E\left(\mu\right)}{2cR_1^2R^2} \biggl\{ 
R_1^3 \cos\psi ~-~R R_1^2 ~+~ R^3\cos\psi ~-~ R_1 R^2
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
~-~K(\mu)\biggl[ \frac{(R_1+R)}{cRR_1}\biggr] \cos\psi
+E(\mu)\biggl[ \frac{(R_1+R)}{cRR_1}\biggr] \cos\psi
~+~
\frac{E\left(\mu\right)}{2cR_1^2R^2} \biggl\{ 
(R_1^3 + R^3) \cos\psi ~-~R R_1(R_1 + R)
\biggr\} 
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
~-~K(\mu)\biggl[ \frac{(R_1+R)}{cRR_1}\biggr] \cos\psi
+E(\mu)\biggl[ \frac{(R_1+R)}{cRR_1}\biggr] \cos\psi
~+~
\frac{E\left(\mu\right)(R_1+R)}{cR_1 R} \biggl\{ 
\cos^2\psi
~+~ \frac{2c^2\cos\psi }{R_1R}
~-~\frac{1}{2}(1+\cos\psi)
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
~-~K(\mu)\biggl[ \frac{(R_1+R)}{cRR_1}\biggr] \cos\psi
~+~
\frac{E\left(\mu\right)(R_1+R)}{cR_1 R} \biggl[ 
\cos^2\psi
~+~\frac{1}{2}(\cos\psi - 1)
~+~ \frac{2c^2\cos\psi }{R_1R}
\biggr] \, .
</math>
  </td>
</tr>
</table>

We therefore have,

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

<tr>
  <td align="right">
<math>~\frac{1}{c} \cdot \frac{d}{dc}\biggl[\frac{1}{c}\cdot \frac{d\mathfrak{I}(r,\theta,c)}{dc}\biggr] </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\frac{1}{4c^4}\biggl[ \frac{4K(\mu)}{ (R_1+R)}\biggr]\biggl\{ 
5+6\cos\psi + \cos^2\psi  
~-~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 }
\biggr\}
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ 
~+~\frac{\cos\psi}{c^4} \biggl\{ 
\frac{E\left(\mu\right)(R_1+R)}{R_1 R} \biggl[ 
\cos^2\psi
~+~\frac{1}{2}(\cos\psi - 1)
~+~ \frac{2c^2\cos\psi }{R_1R}
\biggr] 
~-~K(\mu)\biggl[ \frac{(R_1+R)}{RR_1}\biggr] \cos\psi
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\frac{1}{c^4}\biggl[ \frac{K(\mu)}{ (R_1+R)}\biggr]\biggl\{ 
5+6\cos\psi + \cos^2\psi  
~-~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 }
\biggr\}
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ 
~-~\frac{1}{2c^4} \biggl[ 
\frac{E\left(\mu\right)(R_1+R)}{R_1 R}\biggr] \biggl[ 
~-~2\cos^3\psi
~-~\cos^2\psi ~+~ \cos\psi
~-~ \frac{4c^2\cos^2\psi }{R_1R}
\biggr] 
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\frac{1}{c^4}\biggl[ \frac{K(\mu)}{ (R_1+R)}\biggr]\biggl\{ 
5 ~+~ 8\cos\psi ~+~ \cos^2\psi  
~-~2\cos^3\psi  
~+~\frac{4c^2}{R_1 R }
~-~ \biggl[ \frac{R_1^2 + R^2 - 4c^2 }{R_1 R} \biggr]\cos^2\psi
~-~ \biggl[ \frac{2R_1R }{R_1 R} \biggr]\cos^2\psi
~-~ \biggl[ \frac{8c^2 }{R_1 R} \biggr]\cos^2\psi
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ 
~-~\frac{1}{2c^4}\biggl[ \frac{(R_1+R)E\left(\mu\right)}{RR_1}\biggr] \biggl\{
8\cos\psi
~-~4\cos^3\psi  ~+~ \biggl(\frac{4c^2 }{R_1 R}\biggr)(1-2\cos^2\psi) 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\frac{1}{c^4}\biggl[ \frac{K(\mu)}{ (R_1+R)}\biggr]\biggl\{ 
5 ~+~ 8\cos\psi ~-~ \cos^2\psi  
~-~4\cos^3\psi  
~+~\frac{4c^2}{R_1 R }(1-2\cos^2\psi)
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ 
~-~\frac{1}{c^4}\biggl[ \frac{(R_1+R)E\left(\mu\right)}{RR_1}\biggr] \biggl\{
4\cos\psi
~-~2\cos^3\psi  ~+~ \biggl(\frac{2c^2 }{R_1 R}\biggr)(1-2\cos^2\psi) 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\frac{1}{c^4}\biggl[ \frac{K(\mu)}{ (R_1+R)}\biggr]\biggl\{ 
5 ~+~ 8\cos\psi ~-~ \cos^2\psi  
~-~4\cos^3\psi  
~-~\biggl( \frac{4c^2}{R_1 R } \biggr) \cos 2\psi
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ 
~+~\frac{1}{c^4}\biggl[ \frac{(R_1+R)E\left(\mu\right)}{RR_1}\biggr] \biggl\{
~-~4\cos\psi
~+~2\cos^3\psi  ~+~ \biggl(\frac{2c^2 }{R_1 R}\biggr)\cos 2\psi 
\biggr\} \, ,
</math>
  </td>
</tr>
</table>
which exactly matches the second equation from the top of p. 62 in [http://adsabs.harvard.edu/abs/1893RSPTA.184...43D Dyson (1893a)].