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==Intermediate Step== ===Objective=== As has been [[#His_Derived_Expression|reprinted above]], on p. 62 of Dyson's [http://adsabs.harvard.edu/abs/1893RSPTA.184...43D ''Part I''] we find his power-series expression for the external potential, namely, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\frac{\pi V_\mathrm{Dyson}}{GM} \biggr|_{\mathcal{O}(a^4/c^4)}</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \frac{4K(\mu)}{R+R_1}\biggl\{ 1 ~-~ \frac{1}{8}\biggl(\frac{a^2}{c^2}\biggr) \cos^2\biggl( \frac{\psi}{2}\biggr) - \frac{1}{768}\biggl(\frac{a}{c}\biggr)^4 \biggl[ 5 ~+~ 8\cos\psi ~-~ \cos^2\psi ~-~ 4\cos^3\psi ~-~ \frac{4c^2}{RR_1} \cos2\psi \biggr] \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ + \frac{(R + R_1)E(\mu)}{RR_1}\biggl\{ \frac{1}{8}\biggl(\frac{a}{c}\biggr)^2 \cos\psi ~-~\frac{1}{192} \biggl(\frac{a}{c}\biggr)^4 \biggl[ 2\cos^2\psi ~-~4\cos\psi ~+~ \frac{2c^2}{RR_1}\cos2\psi \biggr] \biggr\} \, , </math> </td> </tr> </table> where — as in the context of toroidal coordinates — we occasionally will make the substitution, <math>~e^\eta = R_1/R</math>, and therefore, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\mu</math> </td> <td align="center"> <math>~\equiv</math> </td> <td align="left"> <math>~\frac{R_1 - R}{R_1+R} = \frac{e^\eta - 1}{e^\eta + 1} \, . </math> </td> </tr> </table> In order to facilitate matching boundary conditions at the surface of the torus, between the exterior and interior expressions for the gravitational potential, Dyson rewrites this ''Part I'' expression for the external potential and — explicitly evaluating it on the torus surface — sets, <math>~R = a</math>. Specifically, on p. 1049 of Dyson's [http://adsabs.harvard.edu/abs/1893RSPTA.184.1041D ''Part II''] we find equation (6), which reads, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\frac{V}{2\pi a^2}</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \ln\biggl(\frac{8c}{a}\biggr) + \frac{1}{2}\biggl(\frac{a}{c}\biggr) \biggl[\ln\biggl(\frac{8c}{a}\biggr) - \frac{5}{4}\biggr] \cos\chi + \biggl\{ \frac{1}{16} \biggl[ \ln\biggl(\frac{8c}{a}\biggr) - \frac{5}{2} \biggr] + \frac{3}{16} \biggl[\ln\biggl(\frac{8c}{a}\biggr) +\frac{17}{36} - \frac{72}{36}\biggr]\cos2\chi\biggr\}\biggl(\frac{a^2}{c^2}\biggr) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ + \biggl\{ \frac{3}{32}\biggl[ \ln\biggl(\frac{8c}{a}\biggr) - \frac{25}{12}\biggr]\cos\chi + \frac{5}{64}\biggl[ \ln\biggl(\frac{8c}{a}\biggr)+\frac{7}{24} - \frac{48}{24}\biggr]\cos3\chi \biggr\} \biggl(\frac{a^3}{c^3}\biggr) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ + \biggl\{ \frac{9}{256}\biggl[ \ln\biggl(\frac{8c}{a}\biggr) - 2\biggr] + \frac{7}{128}\biggl[ \ln\biggl(\frac{8c}{a}\biggr) - \frac{19}{168} - 2\biggr]\cos2\chi + \frac{35}{1024} \biggl[ \ln\biggl(\frac{8c}{a}\biggr) - 2 + \frac{19}{120}\biggr]\cos4\chi \biggr\} \biggl(\frac{a^4}{c^4}\biggr) ~+~\cdots </math> </td> </tr> </table> In order to obtain this alternate power-series expression, Dyson … * Expresses angular variations in terms of the angle, <math>~\chi</math>, instead of the angle, <math>~\psi</math>; these two angles are identified in the [[#His_Derived_Expression|above schematic]]. * Employs power-series expansions of both elliptic integral functions, <math>~K(\mu)</math> and <math>~E(\mu)</math>. * Uses the binomial theorem to develop a number of other power-series expressions. In what follows we will attempt to demonstrate that this second (''Part II'', equation 6) expression is identical to the first. ===The Ratio R<sub>1</sub>/c=== Note that, via the law of cosines, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~R_1^2</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~(2c)^2 + R^2 - 4Rc\cos\chi</math> </td> </tr> <tr> <td align="right"> <math>~\Rightarrow ~~~\biggl(\frac{R_1}{c}\biggr)^2</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~4 + \biggl( \frac{R}{c}\biggr)^2 - 4\biggl(\frac{R}{c}\biggr)\cos\chi</math> </td> </tr> </table> At the surface of the torus, where <math>~R=a</math>, we therefore have, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\frac{R_1}{c}</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~2\biggl[ 1 - \biggl(\frac{a}{c}\biggr)\cos\chi + \frac{1}{4}\biggl( \frac{a}{c}\biggr)^2 \biggr]^{1 / 2} \, .</math> </td> </tr> </table> ====Low Order==== Employing the [[Appendix/Ramblings/PowerSeriesExpressions#Binomial|binomial theorem]], we can write, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\biggl(\frac{R_1}{c}\biggr)^{-1}</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~\frac{1}{2}\biggl[ 1 - \biggl(\frac{a}{c}\biggr)\cos\chi + \frac{1}{4}\biggl( \frac{a}{c}\biggr)^2 \biggr]^{-1 / 2} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~\approx</math> </td> <td align="left"> <math>~\frac{1}{2}\biggl\{ 1 - \frac{1}{2} \biggl[ - \biggl(\frac{a}{c}\biggr)\cos\chi + \frac{1}{4}\biggl( \frac{a}{c}\biggr)^2 \biggr] + \frac{3}{8}\biggl[ - \biggl(\frac{a}{c}\biggr)\cos\chi + \cancelto{0}{\frac{1}{4}\biggl( \frac{a}{c}\biggr)^2} \biggr]^2\biggr\}</math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~\approx</math> </td> <td align="left"> <math>~\frac{1}{2}\biggl\{ 1 + \frac{1}{2} \biggl(\frac{a}{c}\biggr)\cos\chi - \frac{1}{8}\biggl( \frac{a}{c}\biggr)^2 + \frac{3}{8}\biggl(\frac{a}{c}\biggr)^2\cos^2\chi \biggr\}</math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~\approx</math> </td> <td align="left"> <math>~ \frac{1}{2}\biggl\{ 1 + \frac{1}{2} \biggl(\frac{a}{c}\biggr)\cos\chi +\biggl( \frac{a}{c}\biggr)^2 \biggl[ \frac{3}{8}\cos^2\chi -\frac{1}{8}\biggr] \biggr\} </math> </td> </tr> <tr> <td align="right"> <math>~\Rightarrow~~~ 1 + \frac{a}{c} \biggl(\frac{R_1}{c}\biggr)^{-1}</math> </td> <td align="center"> <math>~\approx</math> </td> <td align="left"> <math>~1 + \frac{1}{2}\biggl( \frac{a}{c}\biggr) + \frac{1}{4} \biggl(\frac{a}{c}\biggr)^2\cos\chi </math> </td> </tr> <tr> <td align="right"> <math>~\Rightarrow~~~ \biggl[ 1 + \frac{a}{c} \biggl(\frac{R_1}{c}\biggr)^{-1} \biggr]^{-1}</math> </td> <td align="center"> <math>~\approx</math> </td> <td align="left"> <math>~1 - \biggl[ \frac{1}{2}\biggl( \frac{a}{c}\biggr) + \frac{1}{4} \biggl(\frac{a}{c}\biggr)^2\cos\chi \biggr] + \biggl[ \frac{1}{2}\biggl( \frac{a}{c}\biggr) + \frac{1}{4} \biggl(\frac{a}{c}\biggr)^2\cos\chi \biggr]^2 </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~\approx</math> </td> <td align="left"> <math>~ 1 - \frac{1}{2}\biggl( \frac{a}{c}\biggr) + \frac{1}{4} \biggl( \frac{a}{c}\biggr)^2 (1 - \cos\chi) </math> </td> </tr> </table> ====Higher Order==== Adopting the shorthand notation, <div align="center"> <math>~\gamma \equiv \frac{1}{2}\biggl(\frac{R_1}{c}\biggr) \, ,</math> and <math>~b \equiv - \biggl(\frac{a}{c}\biggr)\cos\chi + \frac{1}{4}\biggl( \frac{a}{c}\biggr)^2 \, ,</math> </div> and employing the [[Appendix/Ramblings/PowerSeriesExpressions#Binomial|binomial theorem]], we can write, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\gamma = \biggl[ 1 + b \biggr]^{1 / 2}</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ 1 + \frac{1}{2}b - \frac{1}{2^3}b^2 + \frac{1}{2^4}b^3 - \frac{3\cdot 5}{2^7\cdot 3}b^4 + \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ 1 + \frac{1}{2}\biggl[ - \biggl(\frac{a}{c}\biggr)\cos\chi + \frac{1}{4}\biggl( \frac{a}{c}\biggr)^2 \biggr] - \frac{1}{2^3}\biggl[ - \biggl(\frac{a}{c}\biggr)\cos\chi + \frac{1}{4}\biggl( \frac{a}{c}\biggr)^2 \biggr]^2 </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ + \frac{1}{2^4}\biggl[ - \biggl(\frac{a}{c}\biggr)\cos\chi + \frac{1}{4}\biggl( \frac{a}{c}\biggr)^2 \biggr]^3 - \frac{3\cdot 5}{2^7\cdot 3}\biggl[ - \biggl(\frac{a}{c}\biggr)\cos\chi + \frac{1}{4}\biggl( \frac{a}{c}\biggr)^2 \biggr]^4 + \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ 1 ~-~ \frac{1}{2} \biggl(\frac{a}{c}\biggr)\cos\chi + \frac{1}{2^3}\biggl( \frac{a}{c}\biggr)^2 - \frac{1}{2^3}\biggl[\biggl(\frac{a}{c}\biggr)^2\cos^2\chi - \frac{1}{2}\biggl( \frac{a}{c}\biggr)^3 \cos\chi + \frac{1}{2^4}\biggl( \frac{a}{c}\biggr)^4 \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ + \frac{1}{2^4}\biggl[ - \biggl(\frac{a}{c}\biggr)\cos\chi + \frac{1}{4}\biggl( \frac{a}{c}\biggr)^2 \biggr]\biggl[\biggl(\frac{a}{c}\biggr)^2\cos^2\chi - \frac{1}{2}\biggl( \frac{a}{c}\biggr)^3 \cos\chi \biggr] - \frac{3\cdot 5}{2^7\cdot 3}\biggl[ \biggl(\frac{a}{c}\biggr)^4\cos^4\chi \biggr] + \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ 1 ~-~ \frac{1}{2} \biggl(\frac{a}{c}\biggr)\cos\chi + \frac{1}{2^3}\biggl( \frac{a}{c}\biggr)^2 -~\frac{1}{2^3}\biggl(\frac{a}{c}\biggr)^2\cos^2\chi ~+~ \frac{1}{2^4}\biggl( \frac{a}{c}\biggr)^3 \cos\chi ~-~ \frac{1}{2^7}\biggl( \frac{a}{c}\biggr)^4 </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ -~\frac{1}{2^4} \biggl(\frac{a}{c}\biggr)^3\cos^3\chi ~+~ \frac{1}{2^5}\biggl( \frac{a}{c}\biggr)^4 \cos^2\chi + \frac{1}{2^6} \biggl(\frac{a}{c}\biggr)^4\cos^2\chi ~-~ \frac{3\cdot 5}{2^7\cdot 3} \biggl(\frac{a}{c}\biggr)^4\cos^4\chi ~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ 1 ~-~ \frac{1}{2} \biggl(\frac{a}{c}\biggr)\cos\chi + \frac{1}{2^3}\biggl( \frac{a}{c}\biggr)^2 (1-\cos^2\chi) +~ \frac{1}{2^4}\biggl( \frac{a}{c}\biggr)^3 (\cos\chi - \cos^3\chi) +~\frac{1}{2^7}\biggl( \frac{a}{c}\biggr)^4 \biggl[~-~ 1 ~+~ 6 \cos^2\chi ~-~ 5 \cos^4\chi \biggr] ~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) \, . </math> </td> </tr> </table> <span id="gammaInverse">Also, we have,</span> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\frac{1}{\gamma} = 2\biggl(\frac{R_1}{c}\biggr)^{-1} = \biggl[ 1 + b \biggr]^{-1 / 2}</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ 1 -\frac{1}{2}b + \frac{3}{2^3}b^2 - \frac{3\cdot 5}{2^4\cdot 3}b^3 + \frac{3\cdot 5\cdot 7}{2^7\cdot 3}b^4 + \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~1 - \frac{1}{2}\biggl[- \biggl(\frac{a}{c}\biggr)\cos\chi + \frac{1}{4}\biggl( \frac{a}{c}\biggr)^2\biggr] + \frac{3}{2^3}\biggl[- \biggl(\frac{a}{c}\biggr)\cos\chi + \frac{1}{4}\biggl( \frac{a}{c}\biggr)^2\biggr]^2 </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ - \frac{3\cdot 5}{2^4\cdot 3}\biggl[- \biggl(\frac{a}{c}\biggr)\cos\chi + \frac{1}{4}\biggl( \frac{a}{c}\biggr)^2\biggr]^3 + \frac{3\cdot 5\cdot 7}{2^7\cdot 3}\biggl[- \biggl(\frac{a}{c}\biggr)\cos\chi + \frac{1}{4}\biggl( \frac{a}{c}\biggr)^2\biggr]^4 + \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr)</math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~1 + \frac{1}{2}\biggl(\frac{a}{c}\biggr)\cos\chi - \frac{1}{2^3}\biggl( \frac{a}{c}\biggr)^2 + \frac{3}{2^3}\biggl[\biggl(\frac{a}{c}\biggr)^2\cos^2\chi ~-~ \frac{1}{2} \biggl(\frac{a}{c}\biggr)^3\cos\chi ~+~ \frac{1}{2^4}\biggl( \frac{a}{c}\biggr)^4\biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ - \frac{3\cdot 5}{2^4\cdot 3}\biggl[- \biggl(\frac{a}{c}\biggr)\cos\chi + \frac{1}{4}\biggl( \frac{a}{c}\biggr)^2\biggr] \biggl[\biggl(\frac{a}{c}\biggr)^2\cos^2\chi ~-~ \frac{1}{2} \biggl(\frac{a}{c}\biggr)^3\cos\chi \biggr] + \frac{3\cdot 5\cdot 7}{2^7\cdot 3}\biggl[\biggl(\frac{a}{c}\biggr)^4\cos^4\chi \biggr] + \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr)</math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~1 + \frac{1}{2}\biggl(\frac{a}{c}\biggr)\cos\chi - \frac{1}{2^3}\biggl( \frac{a}{c}\biggr)^2 + \frac{3}{2^3}\biggl(\frac{a}{c}\biggr)^2\cos^2\chi ~-~ \frac{3}{2^4}\biggl(\frac{a}{c}\biggr)^3\cos\chi ~+~ \frac{3}{2^7}\biggl( \frac{a}{c}\biggr)^4 </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ + \frac{3\cdot 5}{2^4\cdot 3} \biggl[\biggl(\frac{a}{c}\biggr)^3\cos^3\chi ~-~ \frac{1}{2} \biggl(\frac{a}{c}\biggr)^4\cos^2\chi \biggr] - \frac{3\cdot 5}{2^6\cdot 3} \biggl[\biggl(\frac{a}{c}\biggr)^4\cos^2\chi\biggr] + \frac{3\cdot 5\cdot 7}{2^7\cdot 3}\biggl[\biggl(\frac{a}{c}\biggr)^4\cos^4\chi \biggr] + \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr)</math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~1 + \frac{1}{2}\biggl(\frac{a}{c}\biggr)\cos\chi + \frac{1}{2^3}\biggl(\frac{a}{c}\biggr)^2 \biggl[ 3\cos^2\chi - 1 \biggr] + \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^3\biggl[ 5\cos^3\chi ~-~ 3\cos\chi \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ ~+~ \frac{1}{2^7} \biggl( \frac{a}{c}\biggr)^4 \biggl[ 3 ~-~ 30 \cos^2\chi ~+~ 35 \cos^4\chi \biggr] ~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) \, . </math> </td> </tr> </table> Hence, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~1~+~\biggl( \frac{a}{c}\biggr)\biggl(\frac{R_1}{c}\biggr)^{-1} </math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~1 ~+~\frac{1}{2}\biggl( \frac{a}{c}\biggr) ~+~ \frac{1}{2^2}\biggl(\frac{a}{c}\biggr)^2\cos\chi ~+~ \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^3 \biggl[ 3\cos^2\chi - 1 \biggr] ~+~ \frac{1}{2^5}\biggl(\frac{a}{c}\biggr)^4\biggl[ 5\cos^3\chi ~-~ 3\cos\chi \biggr] ~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) \, . </math> </td> </tr> </table> And, adopting the shorthand notation, <div align="center"> <math>~d \equiv \frac{1}{2}\biggl( \frac{a}{c}\biggr) ~+~ \frac{1}{2^2}\biggl(\frac{a}{c}\biggr)^2\cos\chi ~+~ \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^3 \biggl[ 3\cos^2\chi - 1 \biggr] ~+~ \frac{1}{2^5}\biggl(\frac{a}{c}\biggr)^4\biggl[ 5\cos^3\chi ~-~ 3\cos\chi \biggr] \, ,</math> </div> we have, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\biggl[1~+~\biggl( \frac{a}{c}\biggr)\biggl(\frac{R_1}{c}\biggr)^{-1}\biggr]^{-1} </math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ 1 -d + d^2 - d^3 + d^4 ~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~1 - \frac{1}{2}\biggl( \frac{a}{c}\biggr) ~-~ \frac{1}{2^2}\biggl(\frac{a}{c}\biggr)^2\cos\chi ~-~ \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^3 \biggl[ 3\cos^2\chi - 1 \biggr] ~-~ \frac{1}{2^5}\biggl(\frac{a}{c}\biggr)^4\biggl[ 5\cos^3\chi ~-~ 3\cos\chi \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~+~ \biggl\{ \frac{1}{2}\biggl( \frac{a}{c}\biggr) ~+~ \frac{1}{2^2}\biggl(\frac{a}{c}\biggr)^2\cos\chi ~+~ \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^3 \biggl[ 3\cos^2\chi - 1 \biggr] \biggr\} \biggl\{ \frac{1}{2}\biggl( \frac{a}{c}\biggr) ~+~ \frac{1}{2^2}\biggl(\frac{a}{c}\biggr)^2\cos\chi ~+~ \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^3 \biggl[ 3\cos^2\chi - 1 \biggr] \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~-~ \biggl[ \frac{1}{2}\biggl( \frac{a}{c}\biggr) ~+~ \frac{1}{2^2}\biggl(\frac{a}{c}\biggr)^2\cos\chi \biggr] \biggl[ \frac{1}{2}\biggl( \frac{a}{c}\biggr) ~+~ \frac{1}{2^2}\biggl(\frac{a}{c}\biggr)^2\cos\chi \biggr]^2 ~+~\frac{1}{2^4}\biggl( \frac{a}{c}\biggr)^4 ~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~1 - \frac{1}{2}\biggl( \frac{a}{c}\biggr) ~-~ \frac{1}{2^2}\biggl(\frac{a}{c}\biggr)^2\cos\chi ~-~ \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^3 \biggl[ 3\cos^2\chi - 1 \biggr] ~-~ \frac{1}{2^5}\biggl(\frac{a}{c}\biggr)^4\biggl[ 5\cos^3\chi ~-~ 3\cos\chi \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~+~ \frac{1}{2}\biggl( \frac{a}{c}\biggr) \biggl\{ \frac{1}{2}\biggl( \frac{a}{c}\biggr) ~+~ \frac{1}{2^2}\biggl(\frac{a}{c}\biggr)^2\cos\chi ~+~ \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^3 \biggl[ 3\cos^2\chi - 1 \biggr] \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~+~ \biggl\{\frac{1}{2^2}\biggl(\frac{a}{c}\biggr)^2\cos\chi \biggr\} \biggl\{ \frac{1}{2}\biggl( \frac{a}{c}\biggr) ~+~ \frac{1}{2^2}\biggl(\frac{a}{c}\biggr)^2\cos\chi \biggr\} ~+~ \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^3 \biggl[ 3\cos^2\chi - 1 \biggr] \biggl\{ \frac{1}{2}\biggl( \frac{a}{c}\biggr) \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~-~ \biggl[ \frac{1}{2}\biggl( \frac{a}{c}\biggr) ~+~ \frac{1}{2^2}\biggl(\frac{a}{c}\biggr)^2\cos\chi \biggr] \biggl[ \frac{1}{2}\biggl( \frac{a}{c}\biggr) ~+~ \frac{1}{2^2}\biggl(\frac{a}{c}\biggr)^2\cos\chi \biggr]^2 ~+~\frac{1}{2^4}\biggl( \frac{a}{c}\biggr)^4 ~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~1 - \frac{1}{2}\biggl( \frac{a}{c}\biggr) ~-~ \frac{1}{2^2}\biggl(\frac{a}{c}\biggr)^2\cos\chi ~-~ \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^3 \biggl[ 3\cos^2\chi - 1 \biggr] ~-~ \frac{1}{2^5}\biggl(\frac{a}{c}\biggr)^4\biggl[ 5\cos^3\chi ~-~ 3\cos\chi \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~+~ \frac{1}{2^2}\biggl( \frac{a}{c}\biggr)^2 ~+~ \frac{1}{2^3}\biggl(\frac{a}{c}\biggr)^3\cos\chi ~+~ \frac{1}{2^5}\biggl(\frac{a}{c}\biggr)^4 \biggl[ 3\cos^2\chi - 1 \biggr] ~+~ \frac{1}{2^3}\biggl(\frac{a}{c}\biggr)^3\cos\chi +\frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^4\cos^2\chi ~+~\frac{1}{2^5}\biggl(\frac{a}{c}\biggr)^4 \biggl[ 3\cos^2\chi - 1 \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~-~ \biggl[ \frac{1}{2^2}\biggl( \frac{a}{c}\biggr)^2 ~+~ \frac{1}{2^2}\biggl(\frac{a}{c}\biggr)^3\cos\chi ~+~ \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^4\cos^2\chi \biggr] \biggl[ \frac{1}{2}\biggl( \frac{a}{c}\biggr) ~+~ \frac{1}{2^2}\biggl(\frac{a}{c}\biggr)^2\cos\chi \biggr] ~+~\frac{1}{2^4}\biggl( \frac{a}{c}\biggr)^4 ~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ 1 - \frac{1}{2}\biggl( \frac{a}{c}\biggr) ~+~ \frac{1}{2^2}\biggl(\frac{a}{c}\biggr)^2(1-\cos\chi ) ~+~ \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^3 \biggl[ 2\cos\chi ~+~2 (\cos\chi -1) ~-~ 2( 3\cos^2\chi - 1 ) \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> ~+~ \frac{1}{2^5} \biggl(\frac{a}{c}\biggr)^4 \biggl[ ( 3\cos^2\chi - 1 ) + 2\cos^2\chi ~+~( 3\cos^2\chi - 1 ) ~-~2 \cos\chi ~-~ 4 \cos\chi ~+~2 ~-~ ( 5\cos^3\chi ~-~ 3\cos\chi ) \biggr] ~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ 1 - \frac{1}{2}\biggl( \frac{a}{c}\biggr) ~+~ \frac{1}{2^2}\biggl(\frac{a}{c}\biggr)^2(1-\cos\chi ) ~+~ \frac{1}{2^3}\biggl(\frac{a}{c}\biggr)^3 ( 2\cos\chi ~-~ 3\cos^2\chi ) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> ~+~ \frac{1}{2^5} \biggl(\frac{a}{c}\biggr)^4 ( -~9 \cos\chi ~+~8\cos^2\chi ~-~ 5\cos^3\chi ) ~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) \, . </math> </td> </tr> </table> ===Relationship Between Angles=== Drawing on the Law of Cosines, [[#The_Ratio_R1.2Fc|as above]], we can state that on the torus surface, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~R_1^2</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~(2c)^2 + a^2 - 4ac\cos\chi</math> </td> </tr> </table> Alternatively, applying the Law of Cosines to the angle, <math>~\psi</math>, we have, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~(2c)^2</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~R_1^2 + a^2 - 2aR_1\cos\psi</math> </td> </tr> <tr> <td align="right"> <math>~\Rightarrow ~~~\cos\psi</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~\frac{R_1^2 + a^2 - 4c^2}{2aR_1} \, .</math> </td> </tr> </table> Therefore, anywhere along the surface of the torus, we can switch from one of these angles to the other via the relation, <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{1}{2a}\biggl[ 4c^2 + a^2 - 4ac\cos\chi + a^2 - 4c^2\biggr] \biggl[4c^2 + a^2 - 4ac\cos\chi\biggr]^{-1 / 2} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~- \biggl[\cos\chi ~-~ \frac{1}{2}\biggl( \frac{a}{c}\biggr) \biggr] \biggl[1 - \biggl(\frac{a}{c}\biggr) + \frac{1}{4}\biggl(\frac{a}{c}\biggr)^2 \cos\chi\biggr]^{-1 / 2} \, .</math> </td> </tr> </table> ====Cosine ψ Expansion==== Employing the [[Appendix/Ramblings/PowerSeriesExpressions#Binomial|binomial theorem]], we therefore can write, <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>~- \biggl[\cos\chi ~-~ \frac{1}{2}\biggl( \frac{a}{c}\biggr) \biggr] \biggl\{ 1 - \frac{1}{2} \biggl[- \biggl(\frac{a}{c}\biggr) + \frac{1}{4}\biggl(\frac{a}{c}\biggr)^2 \cos\chi \biggr] + \frac{3}{8}\biggl[- \biggl(\frac{a}{c}\biggr) + \frac{1}{4}\biggl(\frac{a}{c}\biggr)^2 \cos\chi \biggr]^2 </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ - \frac{5}{2^4}\biggl[- \biggl(\frac{a}{c}\biggr)\biggr]^3 + \frac{5\cdot 7}{2^7}\biggl[- \biggl(\frac{a}{c}\biggr)\biggr]^4 + \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~- \biggl[\cos\chi ~-~ \frac{1}{2}\biggl( \frac{a}{c}\biggr) \biggr] \biggl\{ 1 ~+~ \frac{1}{2} \biggl(\frac{a}{c}\biggr) ~-~ \frac{1}{2^3}\biggl(\frac{a}{c}\biggr)^2 \cos\chi ~+~ \frac{3}{8}\biggl[ \biggl(\frac{a}{c}\biggr)^2 ~-~ \frac{1}{4}\biggl(\frac{a}{c}\biggr)^3 \cos\chi ~+~ \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^4 \cos^2\chi \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ +~ \frac{5}{2^4} \biggl(\frac{a}{c}\biggr)^3 ~+~ \frac{5\cdot 7}{2^7} \biggl(\frac{a}{c}\biggr)^4 ~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~- \biggl[\cos\chi ~-~ \frac{1}{2}\biggl( \frac{a}{c}\biggr) \biggr] \biggl\{ 1 ~+~ \frac{1}{2} \biggl(\frac{a}{c}\biggr) ~+~ \frac{1}{2^3} \biggl(\frac{a}{c}\biggr)^2\biggl[ 3 ~-~ \cos\chi \biggr] +~ \frac{1}{2^5} \biggl(\frac{a}{c}\biggr)^3 \biggl[ 10 ~-~ 3 \cos\chi \biggr]~+~ \frac{1}{2^7} \biggl(\frac{a}{c}\biggr)^4 \biggl[3 \cos^2\chi ~+~ 5\cdot 7 \biggr] ~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) \biggr\} </math> </td> </tr> </table> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\Rightarrow ~~~ \cos\psi \biggr|_{\mathcal{O}(a^2/c^2)}</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~- \biggl[\cos\chi ~-~ \frac{1}{2}\biggl( \frac{a}{c}\biggr) \biggr] \biggl\{ 1 ~+~ \frac{1}{2} \biggl(\frac{a}{c}\biggr) ~+~ \frac{1}{2^3} \biggl(\frac{a}{c}\biggr)^2\biggl[ 3 ~-~ \cos\chi \biggr] \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ - \cos\chi \biggl\{ 1 ~+~ \frac{1}{2} \biggl(\frac{a}{c}\biggr) ~+~ \frac{1}{2^3} \biggl(\frac{a}{c}\biggr)^2\biggl[ 3 ~-~ \cos\chi \biggr] \biggr\} + \frac{1}{2}\biggl( \frac{a}{c}\biggr) \biggl\{ 1 ~+~ \frac{1}{2} \biggl(\frac{a}{c}\biggr) \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ -\cos\chi ~+~ \frac{1}{2} \biggl(\frac{a}{c}\biggr)(1-\cos\chi) ~+~ \frac{1}{2^3} \biggl(\frac{a}{c}\biggr)^2\biggl[2 - 3\cos\chi + \cos^2\chi \biggr] \, . </math> </td> </tr> </table> ====Cosine-Squared Expansion==== Letting, <div align="center"> <math>~b \equiv \biggl[- \biggl(\frac{a}{c}\biggr) + \frac{1}{4}\biggl(\frac{a}{c}\biggr)^2 \cos\chi \biggr] \, ,</math> </div> via the [[Appendix/Ramblings/PowerSeriesExpressions#Binomial|binomial theorem]] we have, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\cos^2\psi</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~\biggl[\cos\chi ~-~ \frac{1}{2}\biggl( \frac{a}{c}\biggr) \biggr]^2 \biggl[1 - \biggl(\frac{a}{c}\biggr) + \frac{1}{4}\biggl(\frac{a}{c}\biggr)^2 \cos\chi\biggr]^{-1 } </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~\biggl[\cos\chi ~-~ \frac{1}{2}\biggl( \frac{a}{c}\biggr) \biggr]^2 \biggl\{ 1 - b + b^2 - b^3 + b^4 - \mathcal{O}(b^5) \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~\biggl[\cos\chi ~-~ \frac{1}{2}\biggl( \frac{a}{c}\biggr) \biggr]^2 \biggl\{ 1 - \biggl[- \biggl(\frac{a}{c}\biggr) + \frac{1}{4}\biggl(\frac{a}{c}\biggr)^2 \cos\chi \biggr] + \biggl[- \biggl(\frac{a}{c}\biggr) + \frac{1}{4}\biggl(\frac{a}{c}\biggr)^2 \cos\chi \biggr]^2 </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ - \biggl[- \biggl(\frac{a}{c}\biggr) + \frac{1}{4}\biggl(\frac{a}{c}\biggr)^2 \cos\chi \biggr]^3 + \biggl[- \biggl(\frac{a}{c}\biggr) + \frac{1}{4}\biggl(\frac{a}{c}\biggr)^2 \cos\chi \biggr]^4 + \mathcal{O}\biggl( \frac{a^5}{c^5} \biggr) \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~\biggl[\cos\chi ~-~ \frac{1}{2}\biggl( \frac{a}{c}\biggr) \biggr]^2 \biggl\{ 1 + \biggl[\biggl(\frac{a}{c}\biggr) - \frac{1}{4}\biggl(\frac{a}{c}\biggr)^2 \cos\chi \biggr] + \biggl[\biggl(\frac{a}{c}\biggr)^2 - \frac{1}{2}\biggl(\frac{a}{c}\biggr)^3 \cos\chi + \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^4 \cos^2\chi \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ + \biggl[\biggl(\frac{a}{c}\biggr) ~-~ \frac{1}{4}\biggl(\frac{a}{c}\biggr)^2 \cos\chi \biggr]\biggl[\biggl(\frac{a}{c}\biggr)^2 - \frac{1}{2}\biggl(\frac{a}{c}\biggr)^3 \cos\chi + \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^4 \cos^2\chi \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ + \biggl[\biggl(\frac{a}{c}\biggr)^2 - \frac{1}{2}\biggl(\frac{a}{c}\biggr)^3 \cos\chi + \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^4 \cos^2\chi \biggr]^2 + \mathcal{O}\biggl( \frac{a^5}{c^5} \biggr) \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~\biggl[\cos\chi ~-~ \frac{1}{2}\biggl( \frac{a}{c}\biggr) \biggr]^2 \biggl\{ 1 + \biggl(\frac{a}{c}\biggr) - \frac{1}{4}\biggl(\frac{a}{c}\biggr)^2 \cos\chi + \biggl(\frac{a}{c}\biggr)^2 - \frac{1}{2}\biggl(\frac{a}{c}\biggr)^3 \cos\chi + \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^4 \cos^2\chi </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ ~+~ \biggl(\frac{a}{c}\biggr)^3 ~-~ \frac{1}{2}\biggl(\frac{a}{c}\biggr)^4 \cos\chi ~-~ \frac{1}{4}\biggl(\frac{a}{c}\biggr)^4 \cos\chi + \biggl(\frac{a}{c}\biggr)^4 + \mathcal{O}\biggl( \frac{a^5}{c^5} \biggr) \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \biggl[\cos^2\chi ~-~ \biggl( \frac{a}{c}\biggr)\cos\chi ~+~ \frac{1}{2^2}\biggl( \frac{a}{c}\biggr)^2 \biggr] \biggl\{1 ~+~ \biggl(\frac{a}{c}\biggr) ~+~ \biggl(\frac{a}{c}\biggr)^2\biggl[1 ~-~ \frac{1}{4} \cos\chi \biggr] ~+~ \biggl(\frac{a}{c}\biggr)^3 \biggl[1~-~ \frac{1}{2} \cos\chi \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ + \biggl(\frac{a}{c}\biggr)^4\biggl[1 ~+~ \frac{1}{2^4} \cos^2\chi ~-~ \frac{3}{4} \cos\chi \biggr] + \mathcal{O}\biggl( \frac{a^5}{c^5} \biggr) \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \cos^2\chi \biggl\{1 ~+~ \biggl(\frac{a}{c}\biggr) ~+~ \biggl(\frac{a}{c}\biggr)^2\biggl[1 ~-~ \frac{1}{4} \cos\chi \biggr] ~+~ \biggl(\frac{a}{c}\biggr)^3 \biggl[1~-~ \frac{1}{2} \cos\chi \biggr] + \biggl(\frac{a}{c}\biggr)^4\biggl[1 ~+~ \frac{1}{2^4} \cos^2\chi ~-~ \frac{3}{4} \cos\chi \biggr] \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> ~-~ \biggl( \frac{a}{c}\biggr)\cos\chi \biggl\{1 ~+~ \biggl(\frac{a}{c}\biggr) ~+~ \biggl(\frac{a}{c}\biggr)^2\biggl[1 ~-~ \frac{1}{4} \cos\chi \biggr] ~+~ \biggl(\frac{a}{c}\biggr)^3 \biggl[1~-~ \frac{1}{2} \cos\chi \biggr] \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~+~ \frac{1}{2^2}\biggl( \frac{a}{c}\biggr)^2 \biggl\{1 ~+~ \biggl(\frac{a}{c}\biggr) ~+~ \biggl(\frac{a}{c}\biggr)^2\biggl[1 ~-~ \frac{1}{4} \cos\chi \biggr] \biggr\} ~+~ \mathcal{O}\biggl( \frac{a^5}{c^5} \biggr) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \cos^2\chi ~+~ \biggl(\frac{a}{c}\biggr)\biggl[ \cos^2\chi ~-~\cos\chi \biggr] ~+~ \biggl(\frac{a}{c}\biggr)^2\biggl[ \cos^2\chi ~-~ \frac{1}{4} \cos^3\chi ~-~ \cos\chi ~+~\frac{1}{2^2}\biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> ~+~ \biggl(\frac{a}{c}\biggr)^3\biggl\{ \cos^2\chi \biggl[1~-~ \frac{1}{2} \cos\chi \biggr] ~-~ \cos\chi \biggl[1 ~-~ \frac{1}{4} \cos\chi \biggr]~+~\frac{1}{2^2} \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> ~+~\biggl(\frac{a}{c}\biggr)^4 \biggl\{ \cos^2\chi \biggl[1 ~+~ \frac{1}{2^4} \cos^2\chi ~-~ \frac{3}{4} \cos\chi \biggr] ~-~ \cos\chi \biggl[1~-~ \frac{1}{2} \cos\chi \biggr]~+~\frac{1}{2^2} \biggl[1 ~-~ \frac{1}{4} \cos\chi \biggr] \biggr\} ~+~ \mathcal{O}\biggl( \frac{a^5}{c^5} \biggr) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \cos^2\chi ~+~ \biggl(\frac{a}{c}\biggr)\biggl[ \cos^2\chi ~-~\cos\chi \biggr] ~+~ \biggl(\frac{a}{c}\biggr)^2\biggl[ \frac{1}{2^2} ~-~ \cos\chi~+~ \cos^2\chi ~-~ \frac{1}{4} \cos^3\chi \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math> ~+~ \biggl(\frac{a}{c}\biggr)^3\biggl[ \frac{1}{2^2}~-~ \cos\chi ~+~ \frac{5}{4} \cos^2\chi ~-~ \frac{1}{2} \cos^3\chi \biggr] ~+~\biggl(\frac{a}{c}\biggr)^4 \biggl[ \frac{1}{2^2} ~-~ \frac{17}{2^4} \cos\chi ~+~ \frac{3}{2} \cos^2\chi ~-~ \frac{3}{4} \cos^3\chi ~+~\frac{1}{2^4} \cos^4\chi \biggr] ~+~ \mathcal{O}\biggl( \frac{a^5}{c^5} \biggr) </math> </td> </tr> </table> ====Cosine-Cubed Expansion==== Again, letting, <div align="center"> <math>~b \equiv \biggl[- \biggl(\frac{a}{c}\biggr) + \frac{1}{4}\biggl(\frac{a}{c}\biggr)^2 \cos\chi \biggr] \, ,</math> </div> via the [[Appendix/Ramblings/PowerSeriesExpressions#Binomial|binomial theorem]] we have, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\cos^3\psi</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~\biggl[\frac{1}{2}\biggl( \frac{a}{c}\biggr) ~-~ \cos\chi \biggr]^3 \biggl[1 - \biggl(\frac{a}{c}\biggr) + \frac{1}{4}\biggl(\frac{a}{c}\biggr)^2 \cos\chi\biggr]^{-3 / 2 } </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~\biggl[\frac{1}{2}\biggl( \frac{a}{c}\biggr) ~-~ \cos\chi \biggr]^3 \biggl\{ 1 -\frac{3}{2}\biggl[ b \biggr] + \frac{3\cdot 5}{2^3} \biggl[ b \biggr]^2 - \frac{3\cdot 5\cdot 7}{2^4\cdot 3}\biggl[ b \biggr]^3 + \frac{3\cdot 5\cdot 7\cdot 9}{2^7\cdot 3}\biggl[ b \biggr]^4 + \mathcal{O}(b^5) \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~\biggl[\frac{1}{2}\biggl( \frac{a}{c}\biggr) ~-~ \cos\chi \biggr]^3 \biggl\{ 1 -\frac{3}{2}\biggl[ - \biggl(\frac{a}{c}\biggr) + \frac{1}{4}\biggl(\frac{a}{c}\biggr)^2 \cos\chi \biggr] + \frac{3\cdot 5}{2^3} \biggl[ - \biggl(\frac{a}{c}\biggr) + \frac{1}{4}\biggl(\frac{a}{c}\biggr)^2 \cos\chi \biggr]^2 </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ - \frac{5\cdot 7}{2^4}\biggl[ - \biggl(\frac{a}{c}\biggr) + \frac{1}{4}\biggl(\frac{a}{c}\biggr)^2 \cos\chi \biggr]^3 + \frac{5\cdot 7\cdot 9}{2^7}\biggl[ - \biggl(\frac{a}{c}\biggr) + \frac{1}{4}\biggl(\frac{a}{c}\biggr)^2 \cos\chi \biggr]^4 + \mathcal{O}(b^5) \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~\biggl[\frac{1}{2}\biggl( \frac{a}{c}\biggr) ~-~ \cos\chi \biggr]^3 \biggl\{ 1 + \frac{3}{2}\biggl(\frac{a}{c}\biggr) - \frac{3}{2^3}\biggl(\frac{a}{c}\biggr)^2 \cos\chi + \frac{3\cdot 5}{2^3} \biggl[ \biggl(\frac{a}{c}\biggr)^2 - \frac{1}{2} \biggl(\frac{a}{c}\biggr)^3 \cos\chi + \frac{1}{2^4} \biggl(\frac{a}{c}\biggr)^4 \cos^2\chi \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ - \frac{5\cdot 7}{2^4}\biggl[ - \biggl(\frac{a}{c}\biggr) + \frac{1}{4}\biggl(\frac{a}{c}\biggr)^2 \cos\chi \biggr] \biggl[ \biggl(\frac{a}{c}\biggr)^2 - \frac{1}{2} \biggl(\frac{a}{c}\biggr)^3 \cos\chi + \frac{1}{2^4} \biggl(\frac{a}{c}\biggr)^4 \cos^2\chi \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ + \frac{5\cdot 7\cdot 9}{2^7}\biggl[ \biggl(\frac{a}{c}\biggr)^2 - \frac{1}{2} \biggl(\frac{a}{c}\biggr)^3 \cos\chi + \frac{1}{2^4} \biggl(\frac{a}{c}\biggr)^4 \cos^2\chi \biggr]^2 + \mathcal{O}(b^5) \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~\biggl[\frac{1}{2}\biggl( \frac{a}{c}\biggr) ~-~ \cos\chi \biggr]^3 \biggl\{ 1 + \frac{3}{2}\biggl(\frac{a}{c}\biggr) - \frac{3}{2^3}\biggl(\frac{a}{c}\biggr)^2 \cos\chi + \frac{3\cdot 5}{2^3} \biggl[ \biggl(\frac{a}{c}\biggr)^2 - \frac{1}{2} \biggl(\frac{a}{c}\biggr)^3 \cos\chi + \frac{1}{2^4} \biggl(\frac{a}{c}\biggr)^4 \cos^2\chi \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ + \frac{5\cdot 7}{2^4} \biggl(\frac{a}{c}\biggr) \biggl[ \biggl(\frac{a}{c}\biggr)^2 - \frac{1}{2} \biggl(\frac{a}{c}\biggr)^3 \cos\chi \biggr] - \frac{5\cdot 7}{2^6}\biggl(\frac{a}{c}\biggr)^4 \cos\chi + \frac{3^2\cdot 5\cdot 7}{2^7}\biggl(\frac{a}{c}\biggr)^4 + \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~\biggl[ \cos^2\chi ~-~\biggl( \frac{a}{c}\biggr)\cos\chi + \frac{1}{2^2}\biggl( \frac{a}{c}\biggr)^2 \biggr] \biggl[\frac{1}{2}\biggl( \frac{a}{c}\biggr) ~-~ \cos\chi \biggr] \biggl\{ 1 + \frac{3}{2}\biggl(\frac{a}{c}\biggr) - \frac{3}{2^3}\biggl(\frac{a}{c}\biggr)^2 \cos\chi + \frac{3\cdot 5}{2^3}\biggl(\frac{a}{c}\biggr)^2 </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ - \frac{3\cdot 5}{2^4}\biggl(\frac{a}{c}\biggr)^3 \cos\chi + \frac{3\cdot 5}{2^7} \biggl(\frac{a}{c}\biggr)^4 \cos^2\chi + \frac{5\cdot 7}{2^4} \biggl(\frac{a}{c}\biggr)^3 - \frac{5\cdot 7}{2^5} \biggl(\frac{a}{c}\biggr)^4 \cos\chi - \frac{5\cdot 7}{2^6}\biggl(\frac{a}{c}\biggr)^4 \cos\chi + \frac{3^2\cdot 5\cdot 7}{2^7}\biggl(\frac{a}{c}\biggr)^4 + \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~\biggl\{ -\cos^3\chi + \frac{3}{2}\biggl( \frac{a}{c}\biggr) \cos^2\chi - \frac{3}{2^2}\biggl( \frac{a}{c}\biggr)^2\cos\chi + \frac{1}{2^3}\biggl( \frac{a}{c}\biggr)^3 \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~\times \biggl\{ 1 + \frac{3}{2}\biggl(\frac{a}{c}\biggr) + \frac{3}{2^3} \biggl(\frac{a}{c}\biggr)^2 \biggl[5 - \cos\chi \biggr] + \frac{5}{2^4} \biggl(\frac{a}{c}\biggr)^3 \biggl[ 7 - 3\cos\chi \biggr] + \frac{5}{2^7} \biggl(\frac{a}{c}\biggr)^4 \biggl[ 3\cos^2\chi - 2^2\cdot 7 \cos\chi - 2\cdot 7 \cos\chi + 3^2\cdot 7 \biggr] + \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~-\cos^3\chi \biggl\{ 1 + \frac{3}{2}\biggl(\frac{a}{c}\biggr) + \frac{3}{2^3} \biggl(\frac{a}{c}\biggr)^2 \biggl[5 - \cos\chi \biggr] + \frac{5}{2^4} \biggl(\frac{a}{c}\biggr)^3 \biggl[ 7 - 3\cos\chi \biggr] + \frac{5}{2^7} \biggl(\frac{a}{c}\biggr)^4 \biggl[ 3\cos^2\chi - 2^2\cdot 7 \cos\chi - 2\cdot 7 \cos\chi + 3^2\cdot 7 \biggr] \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~+ \frac{3}{2}\biggl( \frac{a}{c}\biggr) \cos^2\chi \biggl\{ 1 + \frac{3}{2}\biggl(\frac{a}{c}\biggr) + \frac{3}{2^3} \biggl(\frac{a}{c}\biggr)^2 \biggl[5 - \cos\chi \biggr] + \frac{5}{2^4} \biggl(\frac{a}{c}\biggr)^3 \biggl[ 7 - 3\cos\chi \biggr] \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ - \frac{3}{2^2}\biggl( \frac{a}{c}\biggr)^2\cos\chi \biggl\{ 1 + \frac{3}{2}\biggl(\frac{a}{c}\biggr) + \frac{3}{2^3} \biggl(\frac{a}{c}\biggr)^2 \biggl[5 - \cos\chi \biggr] \biggr\} + \frac{1}{2^3}\biggl( \frac{a}{c}\biggr)^3 \biggl\{ 1 + \frac{3}{2}\biggl(\frac{a}{c}\biggr) \biggr\} + \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~-\cos^3\chi \biggl\{ 1 + \frac{3}{2}\biggl(\frac{a}{c}\biggr) + \frac{3}{2^3} \biggl(\frac{a}{c}\biggr)^2 \biggl[5 - \cos\chi \biggr] + \frac{5}{2^4} \biggl(\frac{a}{c}\biggr)^3 \biggl[ 7 - 3\cos\chi \biggr] + \frac{5}{2^7} \biggl(\frac{a}{c}\biggr)^4 \biggl[ 3\cos^2\chi - 2^2\cdot 7 \cos\chi - 2\cdot 7 \cos\chi + 3^2\cdot 7 \biggr] \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~+ \frac{3}{2} \cos^2\chi \biggl\{ \biggl( \frac{a}{c}\biggr) + \frac{3}{2}\biggl(\frac{a}{c}\biggr)^2 + \frac{3}{2^3} \biggl(\frac{a}{c}\biggr)^3 \biggl[5 - \cos\chi \biggr] + \frac{5}{2^4} \biggl(\frac{a}{c}\biggr)^4 \biggl[ 7 - 3\cos\chi \biggr] \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ - \frac{3}{2^2} \cos\chi \biggl\{ \biggl( \frac{a}{c}\biggr)^2 + \frac{3}{2}\biggl(\frac{a}{c}\biggr)^3 + \frac{3}{2^3} \biggl(\frac{a}{c}\biggr)^4 \biggl[5 - \cos\chi \biggr] \biggr\} + \frac{1}{2^3} \biggl( \frac{a}{c}\biggr)^3 + \frac{3}{2^4}\biggl(\frac{a}{c}\biggr)^4 </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ + \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) </math> </td> </tr> </table> ===Coefficients of Elliptic Integrals=== Rewriting the external potential, as provided in the [[#Objective|above-stated objective]], and evaluating it at the torus surface, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\frac{\pi V_\mathrm{Dyson}}{GM} \biggr|_{\mathcal{O}(a^4/c^4)}</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \frac{4K(\mu)}{a+R_1}\biggl\{ t_K \biggr\} + \frac{(a + R_1)E(\mu)}{aR_1}\biggl\{ t_E \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \frac{4K(\mu)}{c} \biggl(\frac{R_1}{c}\biggr)^{-1} \biggl[1 + \biggl(\frac{a}{c}\biggr) \biggl(\frac{R_1}{c}\biggr)^{-1} \biggr]^{-1}\biggl\{ t_K \biggr\} + \frac{E(\mu) }{a}\biggl[1 + \biggl(\frac{a}{c}\biggr) \biggl(\frac{R_1}{c}\biggr)^{-1} \biggr] \biggl\{ t_E \biggr\} \, , </math> </td> </tr> </table> where, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~t_K</math> </td> <td align="center"> <math>~\equiv</math> </td> <td align="left"> <math>~ 1 ~-~ \frac{1}{8}\biggl(\frac{a^2}{c^2}\biggr) \cos^2\biggl( \frac{\psi}{2}\biggr) - \frac{1}{768}\biggl(\frac{a}{c}\biggr)^4 \biggl[ 5 ~+~ 8\cos\psi ~-~ \cos^2\psi ~-~ 4\cos^3\psi ~-~ \frac{4c^2}{RR_1} \cos2\psi \biggr] \, , </math> </td> </tr> </table> and, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~t_E</math> </td> <td align="center"> <math>~\equiv</math> </td> <td align="left"> <math>~ \frac{1}{8}\biggl(\frac{a}{c}\biggr)^2 \cos\psi ~-~\frac{1}{192} \biggl(\frac{a}{c}\biggr)^4 \biggl[ 2\cos^2\psi ~-~4\cos\psi ~+~ \frac{2c^2}{RR_1}\cos2\psi \biggr] \, . </math> </td> </tr> </table> Given our derived power-series expressions for various trigonometric functions, these coefficients can be rewritten as, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~t_K</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ 1 ~-~ \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^2 (1 + \cos\psi) + \frac{1}{2^6\cdot 3}\biggl(\frac{a}{c}\biggr)^3 \biggl(\frac{R_1}{c}\biggr)^{-1}(2\cos^2\psi - 1) - \frac{1}{2^8\cdot 3}\biggl(\frac{a}{c}\biggr)^4 \biggl[ 5 ~+~ 8\cos\psi ~-~ \cos^2\psi ~-~ 4\cos^3\psi \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ 1 ~-~ \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^2 \biggl\{ 1 -\cos\chi ~+~ \frac{1}{2} \biggl(\frac{a}{c}\biggr)(1-\cos\chi) ~+~ \frac{1}{2^3} \biggl(\frac{a}{c}\biggr)^2\biggl[2 - 3\cos\chi + \cos^2\chi \biggr] \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ + \frac{1}{2^6\cdot 3}\biggl(\frac{a}{c}\biggr)^3 \biggl(\frac{R_1}{c}\biggr)^{-1}\biggl\{ 2 \biggl[ \cos^2\chi ~+~ \biggl(\frac{a}{c}\biggr)\biggl( \cos^2\chi ~-~\cos\chi \biggr) \biggr] - 1 \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ - \frac{1}{2^8\cdot 3}\biggl(\frac{a}{c}\biggr)^4 \biggl\{ 5 ~-~ 8\cos\chi ~-~ \cos^2\chi ~+~ 4\cos^3\chi \biggr\} + \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ 1 ~-~ \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^2 \biggl[ 1 -\cos\chi \biggr] ~-~ \frac{1}{2^5}\biggl(\frac{a}{c}\biggr)^3 \biggl[ 1-\cos\chi \biggr] ~-~ \frac{1}{2^7}\biggl(\frac{a}{c}\biggr)^4 \biggl[2 - 3\cos\chi + \cos^2\chi \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ + \frac{1}{2^6\cdot 3} \biggl(\frac{R_1}{c}\biggr)^{-1}\biggl\{ \biggl(\frac{a}{c}\biggr)^3\biggl(2 \cos^2\chi - 1 \biggr) ~+~ 2\biggl(\frac{a}{c}\biggr)^4\biggl( \cos^2\chi ~-~\cos\chi \biggr) \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ - \frac{1}{2^8\cdot 3}\biggl(\frac{a}{c}\biggr)^4 \biggl\{ 5 ~-~ 8\cos\chi ~-~ \cos^2\chi ~+~ 4\cos^3\chi \biggr\} + \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ 1 ~-~ \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^2 \biggl[ 1 -\cos\chi \biggr] + \frac{1}{2^6\cdot 3} \biggl(\frac{a}{c}\biggr)^3\biggl\{ \biggl(\frac{R_1}{c}\biggr)^{-1} \biggl(2 \cos^2\chi - 1 \biggr) ~-~ 2\cdot 3 \biggl( 1-\cos\chi \biggr) \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ + \frac{1}{2^8\cdot 3} \biggl(\frac{a}{c}\biggr)^4 \biggl\{ 2^3 \biggl(\frac{R_1}{c}\biggr)^{-1} \biggl( \cos^2\chi ~-~\cos\chi \biggr) - \biggl[ 5 ~-~ 8\cos\chi ~-~ \cos^2\chi ~+~ 4\cos^3\chi \biggr] ~-~ 2\cdot 3\biggl[2 - 3\cos\chi + \cos^2\chi \biggr] \biggr\} + \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ 1 ~-~ \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^2 ( 1 -\cos\chi ) + \frac{1}{2^6\cdot 3} \biggl(\frac{a}{c}\biggr)^3\biggl[ \biggl(\frac{R_1}{c}\biggr)^{-1} (2 \cos^2\chi - 1 ) -6~+~6\cos\chi \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ + \frac{1}{2^8\cdot 3} \biggl(\frac{a}{c}\biggr)^4 \biggl[ 2^3 \biggl(\frac{R_1}{c}\biggr)^{-1} ( \cos^2\chi ~-~\cos\chi ) ~-17 + 26\cos\chi -5 \cos^2\chi ~-~ 4\cos^3\chi \biggr] + \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) \, ; </math> </td> </tr> </table> and, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~t_E</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \frac{1}{8}\biggl(\frac{a}{c}\biggr)^2 \cos\psi ~-~\frac{1}{2^5\cdot 3} \biggl(\frac{a}{c}\biggr)^3 \biggl(\frac{R_1}{c}\biggr)^{-1} \biggl\{ 2 \biggl[\cos^2\psi\biggr] - 1\biggr\} ~-~\frac{1}{2^5\cdot 3} \biggl(\frac{a}{c}\biggr)^4 \biggl[ \cos^2\psi ~-~2\cos\psi \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \frac{1}{8}\biggl(\frac{a}{c}\biggr)^2 \biggl\{ -\cos\chi ~+~ \frac{1}{2} \biggl(\frac{a}{c}\biggr)(1-\cos\chi) ~+~ \frac{1}{2^3} \biggl(\frac{a}{c}\biggr)^2\biggl[2 - 3\cos\chi + \cos^2\chi \biggr] \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ ~-~\frac{1}{2^5\cdot 3} \biggl(\frac{a}{c}\biggr)^3 \biggl(\frac{R_1}{c}\biggr)^{-1} \biggl\{ 2 \biggl[\cos^2\chi ~+~ \biggl(\frac{a}{c}\biggr)\biggl( \cos^2\chi ~-~\cos\chi \biggr)\biggr] - 1\biggr\} ~-~\frac{1}{2^5\cdot 3} \biggl(\frac{a}{c}\biggr)^4 \biggl[\cos^2\chi ~+~2\cos\chi \biggr] + \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ -\frac{1}{2^3} \biggl(\frac{a}{c}\biggr)^2\cos\chi ~+~ \frac{1}{2^4} \biggl(\frac{a}{c}\biggr)^3(1-\cos\chi) ~+~ \frac{1}{2^6} \biggl(\frac{a}{c}\biggr)^4\biggl[2 - 3\cos\chi + \cos^2\chi \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ ~-~\frac{1}{2^5\cdot 3} \biggl(\frac{a}{c}\biggr)^3 \biggl(\frac{R_1}{c}\biggr)^{-1} \biggl[ 2\cos^2\chi ~+~ 2\biggl(\frac{a}{c}\biggr)\biggl( \cos^2\chi ~-~\cos\chi \biggr) - 1\biggr] ~-~\frac{1}{2^5\cdot 3} \biggl(\frac{a}{c}\biggr)^4 \biggl( \cos^2\chi ~+~2\cos\chi \biggr) + \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ -\frac{1}{2^3} \biggl(\frac{a}{c}\biggr)^2\cos\chi ~+~ \frac{1}{2^4} \biggl(\frac{a}{c}\biggr)^3(1-\cos\chi) ~-~\frac{1}{2^5\cdot 3} \biggl(\frac{a}{c}\biggr)^3 \biggl(\frac{R_1}{c}\biggr)^{-1} \biggl( 2\cos^2\chi - 1\biggr) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ ~-~\frac{1}{2^4\cdot 3} \biggl(\frac{a}{c}\biggr)^4 \biggl(\frac{R_1}{c}\biggr)^{-1} \biggl( \cos^2\chi ~-~\cos\chi \biggr) ~+~ \frac{1}{2^6 \cdot 3} \biggl(\frac{a}{c}\biggr)^4 \biggl[ 6 - 13\cos\chi + \cos^2\chi \biggr] + \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ -\frac{1}{2^3} \biggl(\frac{a}{c}\biggr)^2\cos\chi ~+~ \frac{1}{2^5\cdot 3} \biggl(\frac{a}{c}\biggr)^3 \biggl[ 6(1-\cos\chi) ~-~\biggl(\frac{R_1}{c}\biggr)^{-1} ( 2\cos^2\chi - 1 ) \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ ~+~ \frac{1}{2^6 \cdot 3} \biggl(\frac{a}{c}\biggr)^4 \biggl[ ( 6 - 13\cos\chi + \cos^2\chi ) ~-~4 \biggl(\frac{R_1}{c}\biggr)^{-1} ( \cos^2\chi ~-~\cos\chi ) \biggr] + \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) \, . </math> </td> </tr> </table> Now, inserting to the appropriate order the [[#The_Ratio_R1.2Fc|above expression for the ratio,]] <math>~R_1/c</math> — namely, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\biggl(\frac{R_1}{c}\biggr)^{-1}</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \frac{1}{2} + \frac{1}{2^2} \biggl(\frac{a}{c}\biggr)\cos\chi + \mathcal{O}\biggl(\frac{a^2}{c^2}\biggr) \, , </math> </td> </tr> </table> we have, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~t_K</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ 1 ~-~ \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^2 ( 1 -\cos\chi ) + \frac{1}{2^6\cdot 3} \biggl(\frac{a}{c}\biggr)^3 \biggl\{ \biggl[ \frac{1}{2} + \frac{1}{2^2} \biggl(\frac{a}{c}\biggr)\cos\chi \biggr] (2 \cos^2\chi - 1 ) -6~+~6\cos\chi \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ + \frac{1}{2^8\cdot 3} \biggl(\frac{a}{c}\biggr)^4 \biggl\{ 2^2 ( \cos^2\chi ~-~\cos\chi ) ~-17 + 26\cos\chi -5 \cos^2\chi ~-~ 4\cos^3\chi \biggr\} + \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ 1 ~-~ \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^2 ( 1 -\cos\chi ) + \frac{1}{2^7\cdot 3} \biggl(\frac{a}{c}\biggr)^3 ( -13~+~12\cos\chi +2 \cos^2\chi ) + \frac{1}{2^8\cdot 3} \biggl(\frac{a}{c}\biggr)^4 (2 \cos^3\chi - \cos\chi ) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ + \frac{1}{2^8\cdot 3} \biggl(\frac{a}{c}\biggr)^4 (-17 + 22\cos\chi - \cos^2\chi ~-~ 4\cos^3\chi ) + \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ 1 ~-~ \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^2 ( 1 -\cos\chi ) + \frac{1}{2^7\cdot 3} \biggl(\frac{a}{c}\biggr)^3 ( -13~+~12\cos\chi +2 \cos^2\chi ) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ + \frac{1}{2^8\cdot 3} \biggl(\frac{a}{c}\biggr)^4 (-17 + 21\cos\chi - \cos^2\chi ~-~ 2\cos^3\chi ) + \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) \, ; </math> </td> </tr> </table> and, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~t_E</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ -\frac{1}{2^3} \biggl(\frac{a}{c}\biggr)^2\cos\chi ~+~ \frac{1}{2^5\cdot 3} \biggl(\frac{a}{c}\biggr)^3 \biggl\{ 6(1-\cos\chi) ~+~\biggl[ \frac{1}{2} + \frac{1}{2^2} \biggl(\frac{a}{c}\biggr)\cos\chi \biggr] (1- 2\cos^2\chi ) \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ ~+~ \frac{1}{2^6 \cdot 3} \biggl(\frac{a}{c}\biggr)^4 \biggl\{ ( 6 - 13\cos\chi + \cos^2\chi ) ~+~2 ( \cos\chi - \cos^2\chi) \biggr\} + \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ -\frac{1}{2^3} \biggl(\frac{a}{c}\biggr)^2\cos\chi ~+~ \frac{1}{2^6\cdot 3} \biggl(\frac{a}{c}\biggr)^3 ( 13- 12\cos\chi - 2\cos^2\chi ) ~+~ \frac{1}{2^7\cdot 3} \biggl(\frac{a}{c}\biggr)^4 (\cos\chi - 2\cos^3\chi ) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ ~+~ \frac{1}{2^6 \cdot 3} \biggl(\frac{a}{c}\biggr)^4 ( 6 - 11\cos\chi - \cos^2\chi ) + \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ -\frac{1}{2^3} \biggl(\frac{a}{c}\biggr)^2\cos\chi ~+~ \frac{1}{2^6\cdot 3} \biggl(\frac{a}{c}\biggr)^3 ( 13- 12\cos\chi - 2\cos^2\chi ) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ ~+~ \frac{1}{2^7\cdot 3} \biggl(\frac{a}{c}\biggr)^4 ( 12 -21 \cos\chi - 2\cos^2\chi - 2\cos^3\chi ) + \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) \, . </math> </td> </tr> </table> ===Alternate "Small" Argument of Elliptic Integrals=== <!-- Let's switch from <math>~4K(\mu)/(R_1+R)</math> to <math>~2K(k)/R_1</math>, where, <math>~k \equiv [1 - (R/R_1)^2]^{1 / 2}</math>, and recognize that, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~k' = \sqrt{1-k^2}</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~e^{-\eta} = \frac{R}{R_1} \, .</math> </td> </tr> </table> --> Defining the "small parameter," <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~k'</math> </td> <td align="center"> <math>~\equiv</math> </td> <td align="left"> <math>~ \sqrt{1-\mu^2} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \biggl[1 - \biggl( \frac{e^\eta - 1}{e^\eta + 1}\biggr)^2 \biggr]^{1 / 2} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \biggl[\frac{(e^\eta + 1)^2 - (e^\eta - 1)^2 }{(e^\eta + 1)^2} \biggr]^{1 / 2} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \biggl[\frac{(e^{2\eta} + 2e^\eta+ 1)- (e^{2\eta} - 2e^\eta + 1) }{(e^\eta + 1)^2} \biggr]^{1 / 2} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \biggl[\frac{4e^\eta }{(e^\eta + 1)^2} \biggr]^{1 / 2} = \biggl[\frac{4e^{-\eta} }{(1 + e^{-\eta})^2} \biggr]^{1 / 2} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~2\biggl( \frac{R}{R_1}\biggr)^{1 / 2} \biggl[1 + \frac{R}{R_1} \biggr]^{-1} \, . </math> </td> </tr> </table> At the surface of the torus, where <math>~R=a</math>, we therefore have, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~k'</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ 2\biggl( \frac{a}{c}\biggr)^{1 / 2}\biggl( \frac{R_1}{c}\biggr)^{-1 / 2} \biggl[1 + \frac{a}{c}\cdot \frac{c}{R_1} \biggr]^{-1} </math> </td> </tr> <tr> <td align="right"> <math>~\Rightarrow ~~~ \frac{4}{k'}</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~2\biggl( \frac{a}{c}\biggr)^{- 1 / 2}\biggl( \frac{R_1}{c}\biggr)^{1 / 2} \biggl[1 + \frac{a}{c}\cdot \frac{c}{R_1} \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~2\biggl( \frac{c}{a}\biggr)^{1 / 2} \biggl[ 4 + \biggl( \frac{a}{c}\biggr)^2 - 4\biggl(\frac{a}{c}\biggr)\cos\chi \biggr]^{1 / 4} \biggl\{ 1 + \frac{a}{c}\cdot \biggl[ 4 + \biggl( \frac{a}{c}\biggr)^2 - 4\biggl(\frac{a}{c}\biggr)\cos\chi \biggr]^{-1 / 2} \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~\biggl( \frac{2^3c}{a}\biggr)^{1 / 2} \biggl[ 1 - \biggl(\frac{a}{c}\biggr)\cos\chi + \frac{1}{4}\biggl( \frac{a}{c}\biggr)^2 \biggr]^{1 / 4} \biggl\{ 1 + \frac{a}{2c}\cdot \biggl[ 1 - \biggl(\frac{a}{c}\biggr)\cos\chi + \biggl( \frac{a}{2c}\biggr)^2 \biggr]^{-1 / 2} \biggr\} </math> </td> </tr> <tr> <td align="right"> <math>~\Rightarrow ~~~ \ln \frac{4}{k'}</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~\frac{1}{2} \ln\biggl( \frac{2^3c}{a}\biggr) + \frac{1}{4}\ln\biggl[ 1 - \biggl(\frac{a}{c}\biggr)\cos\chi + \frac{1}{4}\biggl( \frac{a}{c}\biggr)^2 \biggr] + \ln\biggl\{ 1 + \frac{a}{2c}\cdot \biggl[ 1 - \biggl(\frac{a}{c}\biggr)\cos\chi + \biggl( \frac{a}{2c}\biggr)^2 \biggr]^{-1 / 2} \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~\approx</math> </td> <td align="left"> <math>~ \frac{1}{2} \ln\biggl( \frac{2^3c}{a}\biggr) + \frac{1}{4}\ln\biggl[ 1 - \biggl(\frac{a}{c}\biggr)\cos\chi \biggr] + \ln\biggl[ 1 + \frac{a}{2c} \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~\approx</math> </td> <td align="left"> <math>~ \frac{1}{2} \ln\biggl( \frac{2^3c}{a}\biggr) + \frac{a}{2c}\biggl[ 1 - \frac{1}{2} \cos\chi \biggr] </math> </td> </tr> </table> Alternatively, if — [[#Higher_Order|as above]] — we adopt the shorthand notation, <math>~\gamma \equiv R_1/(2c)</math>, we can write, <table border="1" cellpadding="8" width="85%" align="center"> <tr><td align="left"> <div align="center">'''Summary'''</div> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\gamma</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ 1 ~-~ \frac{1}{2} \biggl(\frac{a}{c}\biggr)\cos\chi + \frac{1}{2^3}\biggl( \frac{a}{c}\biggr)^2 (1-\cos^2\chi) +~ \frac{1}{2^4}\biggl( \frac{a}{c}\biggr)^3 (\cos\chi - \cos^3\chi) +~\frac{1}{2^7}\biggl( \frac{a}{c}\biggr)^4 \biggl[~-~ 1 ~+~ 6 \cos^2\chi ~-~ 5 \cos^4\chi \biggr] ~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) \, ; </math> </td> </tr> <tr> <td align="right"> <math>~\frac{1}{\gamma}</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ 1 + \frac{1}{2}\biggl(\frac{a}{c}\biggr)\cos\chi + \frac{1}{2^3}\biggl(\frac{a}{c}\biggr)^2 \biggl[ 3\cos^2\chi - 1 \biggr] + \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^3\biggl[ 5\cos^3\chi ~-~ 3\cos\chi \biggr] ~+~ \frac{1}{2^7} \biggl( \frac{a}{c}\biggr)^4 \biggl[ 3 ~-~ 30 \cos^2\chi ~+~ 35 \cos^4\chi \biggr] ~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) \, . </math> </td> </tr> </table> </td></tr> </table> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~k'</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \biggl( \frac{2a}{c}\biggr)^{1 / 2} \gamma^{-1 / 2} \biggl[1 + \biggl(\frac{a}{c}\biggr) \frac{1}{2\gamma} \biggr]^{-1} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \biggl( \frac{2a}{c}\biggr)^{1 / 2} \gamma^{-1 / 2} \biggl[1 + \biggl(\frac{a}{c}\biggr) \frac{1}{\gamma} + \biggl(\frac{a}{c}\biggr)^2 \frac{1}{4\gamma^2} \biggr]^{-1 / 2} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \biggl( \frac{2a}{c}\biggr)^{1 / 2} \biggl[\gamma + \biggl(\frac{a}{c}\biggr) + \frac{1}{4}\biggl(\frac{a}{c}\biggr)^2 \frac{1}{\gamma} \biggr]^{-1 / 2} </math> </td> </tr> <tr> <td align="right"> <math>~\Rightarrow ~~~(k')^{2m}</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \biggl( \frac{2a}{c}\biggr)^{m } \biggl[\gamma + \biggl(\frac{a}{c}\biggr) + \frac{1}{4}\biggl(\frac{a}{c}\biggr)^2 \frac{1}{\gamma} \biggr]^{-m } = \biggl( \frac{2a}{c}\biggr)^{m } \Gamma^{-m} </math> </td> </tr> <tr> <td align="right"> <math>~\Rightarrow ~~~ \frac{4}{k'}</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \biggl( \frac{2^3c}{a}\biggr)^{1 / 2} \biggl[\gamma + \biggl(\frac{a}{c}\biggr) + \frac{1}{4}\biggl(\frac{a}{c}\biggr)^2 \frac{1}{\gamma} \biggr]^{1 / 2} = \biggl( \frac{2^3c}{a}\biggr)^{1 / 2} \Gamma^{1 / 2} </math> </td> </tr> <tr> <td align="right"> <math>~\Rightarrow ~~~\ln \frac{4}{k'}</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \frac{1}{2} \ln\biggl( \frac{2^3c}{a}\biggr) + \frac{1}{2}\ln \Gamma \, , </math> </td> </tr> </table> where, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\Gamma</math> </td> <td align="center"> <math>~\equiv</math> </td> <td align="left"> <math>~ \gamma + \biggl(\frac{a}{c}\biggr) + \frac{1}{4}\biggl(\frac{a}{c}\biggr)^2 \frac{1}{\gamma} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ 1 ~-~ \frac{1}{2} \biggl(\frac{a}{c}\biggr)\cos\chi + \frac{1}{2^3}\biggl( \frac{a}{c}\biggr)^2 (1-\cos^2\chi) +~ \frac{1}{2^4}\biggl( \frac{a}{c}\biggr)^3 (\cos\chi - \cos^3\chi) +~\frac{1}{2^7}\biggl( \frac{a}{c}\biggr)^4 \biggl[~-~ 1 ~+~ 6 \cos^2\chi ~-~ 5 \cos^4\chi \biggr] ~+~ \biggl(\frac{a}{c}\biggr) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ +~ \frac{1}{2^2}\biggl(\frac{a}{c}\biggr)^2 \biggl\{ 1 + \frac{1}{2}\biggl(\frac{a}{c}\biggr)\cos\chi + \frac{1}{2^3}\biggl(\frac{a}{c}\biggr)^2 \biggl[ 3\cos^2\chi - 1 \biggr] + \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^3\biggl[ 5\cos^3\chi ~-~ 3\cos\chi \biggr] ~+~ \frac{1}{2^7} \biggl( \frac{a}{c}\biggr)^4 \biggl[ 3 ~-~ 30 \cos^2\chi ~+~ 35 \cos^4\chi \biggr] \biggr\} ~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ 1 + \biggl(\frac{a}{c}\biggr)(1~-~ \frac{1}{2} \cos\chi ) + \frac{1}{2^3}\biggl( \frac{a}{c}\biggr)^2 (1-\cos^2\chi) +~ \frac{1}{2^4}\biggl( \frac{a}{c}\biggr)^3 (\cos\chi - \cos^3\chi) +~\frac{1}{2^7}\biggl( \frac{a}{c}\biggr)^4 (-~ 1 ~+~ 6 \cos^2\chi ~-~ 5 \cos^4\chi ) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ +~ \frac{1}{2^2}\biggl(\frac{a}{c}\biggr)^2 + \frac{1}{2^3}\biggl(\frac{a}{c}\biggr)^3\cos\chi + \frac{1}{2^5}\biggl(\frac{a}{c}\biggr)^4 ( 3\cos^2\chi - 1 ) ~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ 1 + \biggl(\frac{a}{c}\biggr) (1~-~ \frac{1}{2} \cos\chi ) + \frac{1}{2^3}\biggl( \frac{a}{c}\biggr)^2 (3-\cos^2\chi) +~ \frac{1}{2^4}\biggl( \frac{a}{c}\biggr)^3 (3\cos\chi - \cos^3\chi) +~\frac{1}{2^7}\biggl( \frac{a}{c}\biggr)^4 (- 5 ~+~ 18 \cos^2\chi ~-~ 5 \cos^4\chi ) ~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) \, . </math> </td> </tr> </table> Now, if we adopt the shorthand notation, <div align="center"> <math>~g \equiv \biggl(\frac{a}{c}\biggr) (1~-~ \frac{1}{2} \cos\chi ) + \frac{1}{2^3}\biggl( \frac{a}{c}\biggr)^2 (3-\cos^2\chi) +~ \frac{1}{2^4}\biggl( \frac{a}{c}\biggr)^3 (3\cos\chi - \cos^3\chi) +~\frac{1}{2^7}\biggl( \frac{a}{c}\biggr)^4 (- 5 ~+~ 18 \cos^2\chi ~-~ 5 \cos^4\chi ) ~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) \, , </math> </div> we also have, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\ln\Gamma = \ln (1 + g)</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ g - \frac{1}{2}g^2 + \frac{1}{3}g^3 - \frac{1}{4}g^4 ~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \biggl(\frac{a}{c}\biggr) (1~-~ \frac{1}{2} \cos\chi ) + \frac{1}{2^3}\biggl( \frac{a}{c}\biggr)^2 (3-\cos^2\chi) +~ \frac{1}{2^4}\biggl( \frac{a}{c}\biggr)^3 (3\cos\chi - \cos^3\chi) +~\frac{1}{2^7}\biggl( \frac{a}{c}\biggr)^4 (- 5 ~+~ 18 \cos^2\chi ~-~ 5 \cos^4\chi ) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ - \frac{1}{2}\biggl[ \biggl(\frac{a}{c}\biggr) (1~-~ \frac{1}{2} \cos\chi ) + \frac{1}{2^3}\biggl( \frac{a}{c}\biggr)^2 (3-\cos^2\chi) +~ \frac{1}{2^4}\biggl( \frac{a}{c}\biggr)^3 (3\cos\chi - \cos^3\chi) \biggr]^2 </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ + \frac{1}{3}\biggl[ \biggl(\frac{a}{c}\biggr) (1~-~ \frac{1}{2} \cos\chi ) + \frac{1}{2^3}\biggl( \frac{a}{c}\biggr)^2 (3-\cos^2\chi) \biggr]^3 - \frac{1}{4}\biggl[ \biggl(\frac{a}{c}\biggr) (1~-~ \frac{1}{2} \cos\chi ) \biggr]^4 ~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \biggl(\frac{a}{c}\biggr) (1~-~ \frac{1}{2} \cos\chi ) + \frac{1}{2^3}\biggl( \frac{a}{c}\biggr)^2 (3-\cos^2\chi) +~ \frac{1}{2^4}\biggl( \frac{a}{c}\biggr)^3 (3\cos\chi - \cos^3\chi) +~\frac{1}{2^7}\biggl( \frac{a}{c}\biggr)^4 (- 5 ~+~ 18 \cos^2\chi ~-~ 5 \cos^4\chi ) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ - \frac{1}{2} \biggl\{ \frac{1}{2^2}\biggl(\frac{a}{c}\biggr)^2 (2~-~ \cos\chi )^2 + \frac{1}{2^4}\biggl( \frac{a}{c}\biggr)^3 (3-\cos^2\chi) (2~-~ \cos\chi ) +~ \frac{1}{2^5}\biggl( \frac{a}{c}\biggr)^4 (3\cos\chi - \cos^3\chi)(2~-~ \cos\chi ) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ ~+~ \frac{1}{2^3}\biggl( \frac{a}{c}\biggr)^3 (3-\cos^2\chi) (1~-~ \frac{1}{2} \cos\chi ) ~+~ \frac{1}{2^6}\biggl( \frac{a}{c}\biggr)^4 (3-\cos^2\chi) (3-\cos^2\chi) ~+~ \frac{1}{2^4}\biggl( \frac{a}{c}\biggr)^4 (3\cos\chi - \cos^3\chi)(1~-~ \frac{1}{2} \cos\chi ) \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ + \frac{1}{3}\biggl[ \biggl(\frac{a}{c}\biggr) (1~-~ \frac{1}{2} \cos\chi ) + \frac{1}{2^3}\biggl( \frac{a}{c}\biggr)^2 (3-\cos^2\chi) \biggr] \biggl[ \biggl(\frac{a}{c}\biggr)^2 (1~-~ \frac{1}{2} \cos\chi )^2 ~+~ \frac{1}{2^2}\biggl( \frac{a}{c}\biggr)^3 (1~-~ \frac{1}{2} \cos\chi ) (3-\cos^2\chi) \biggr] - \frac{1}{2^6} \biggl(\frac{a}{c}\biggr)^4(2~-~ \cos\chi )^4 ~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \frac{1}{2}\biggl(\frac{a}{c}\biggr) (2~-~ \cos\chi ) + \frac{1}{2^3}\biggl( \frac{a}{c}\biggr)^2 (3-\cos^2\chi) ~-~\frac{1}{2^3}\biggl(\frac{a}{c}\biggr)^2 (2~-~ \cos\chi )^2 </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ +~ \frac{1}{2^4}\biggl( \frac{a}{c}\biggr)^3 (3\cos\chi - \cos^3\chi) ~-~ \frac{1}{2^5}\biggl( \frac{a}{c}\biggr)^3 (3-\cos^2\chi) (2~-~ \cos\chi ) ~-~ \frac{1}{2^5}\biggl( \frac{a}{c}\biggr)^3 (3-\cos^2\chi) (2~-~ \cos\chi ) ~+~\frac{1}{2^3\cdot 3}\biggl(\frac{a}{c}\biggr)^3 (2~-~ \cos\chi )^2 (2~-~\cos\chi ) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ ~-~ \frac{1}{2^6}\biggl( \frac{a}{c}\biggr)^4 (3\cos\chi - \cos^3\chi)(2~-~ \cos\chi ) ~-~ \frac{1}{2^7}\biggl( \frac{a}{c}\biggr)^4 (3-\cos^2\chi) (3-\cos^2\chi) ~-~ \frac{1}{2^6}\biggl( \frac{a}{c}\biggr)^4 (3\cos\chi - \cos^3\chi)(2~-~ \cos\chi ) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ ~+~ \frac{1}{2^4\cdot 3}\biggl( \frac{a}{c}\biggr)^4 (2~-~ \cos\chi ) (3-\cos^2\chi)(2~-~ \cos\chi ) +\frac{1}{2^5\cdot 3}\biggl( \frac{a}{c}\biggr)^4 (3-\cos^2\chi) (2~-~ \cos\chi )^2 +~\frac{1}{2^7}\biggl( \frac{a}{c}\biggr)^4 (- 5 ~+~ 18 \cos^2\chi ~-~ 5 \cos^4\chi ) - \frac{1}{2^6} \biggl(\frac{a}{c}\biggr)^4(2~-~ \cos\chi )^4 ~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) </math> </td> </tr> </table> And, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~~(k')^{2}</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \biggl( \frac{2a}{c}\biggr) \Gamma^{-1} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \biggl( \frac{2a}{c}\biggr) (1+g)^{-1} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \biggl( \frac{2a}{c}\biggr) \biggl\{1 ~-~ g ~+~ g^2 ~-~ g^3 \biggr\} ~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \biggl( \frac{2a}{c}\biggr) \biggl\{1 ~-~ \biggl[ \biggl(\frac{a}{c}\biggr) (1~-~ \frac{1}{2} \cos\chi ) + \frac{1}{2^3}\biggl( \frac{a}{c}\biggr)^2 (3-\cos^2\chi) +~ \frac{1}{2^4}\biggl( \frac{a}{c}\biggr)^3 (3\cos\chi - \cos^3\chi) \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ ~+~ \biggl[\biggl(\frac{a}{c}\biggr) (1~-~ \frac{1}{2} \cos\chi ) + \frac{1}{2^3}\biggl( \frac{a}{c}\biggr)^2 (3-\cos^2\chi) \biggr]^2 ~-~ \biggl[ \biggl(\frac{a}{c}\biggr) (1~-~ \frac{1}{2} \cos\chi ) \biggr]^3 \biggr\} ~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \biggl( \frac{2a}{c}\biggr) \biggl\{1 ~-~ \frac{1}{2}\biggl(\frac{a}{c}\biggr) (2~-~ \cos\chi ) + \frac{1}{2^3}\biggl( \frac{a}{c}\biggr)^2\biggl[ (3-\cos^2\chi) ~+~2(2~-~ \cos\chi )^2 \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ +~ \frac{1}{2^4}\biggl( \frac{a}{c}\biggr)^3\biggl[ (3\cos\chi - \cos^3\chi) ~+~ (2~-~ \cos\chi )(3-\cos^2\chi) ~-~ (2~-~ \cos\chi )^3 \biggr] \biggr\} ~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) </math> </td> </tr> </table> And, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~~(k')^{4}</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \biggl( \frac{2a}{c}\biggr)^2 \Gamma^{-2} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \biggl( \frac{2a}{c}\biggr)^2 (1+g)^{-2} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \biggl( \frac{2a}{c}\biggr)^2 \biggl\{1 ~-~ 2g ~+~ 3g^2 \biggr\} ~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \biggl( \frac{2a}{c}\biggr)^2 \biggl\{1 ~-~ 2\biggl[ \biggl(\frac{a}{c}\biggr) (1~-~ \frac{1}{2} \cos\chi ) + \frac{1}{2^3}\biggl( \frac{a}{c}\biggr)^2 (3-\cos^2\chi) \biggr] ~+~ 3\biggl[ \biggl(\frac{a}{c}\biggr) (1~-~ \frac{1}{2} \cos\chi ) \biggr]^2 \biggr\} ~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \biggl( \frac{2a}{c}\biggr)^2 \biggl\{1 -~\biggl(\frac{a}{c}\biggr) (2~-~ \cos\chi ) ~+~ \frac{1}{2^2} \biggl(\frac{a}{c}\biggr)^2\biggl[ 3(2~-~ \cos\chi )^2 ~-~ (3-\cos^2\chi) \biggr] \biggr\} ~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ 4\biggl( \frac{a}{c}\biggr)^2 -~\biggl(\frac{a}{c}\biggr)^3 (8~-~ 4\cos\chi ) ~+~ \biggl(\frac{a}{c}\biggr)^4 ( 9 - 12\cos\chi + 4\cos^2\chi ) ~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) </math> </td> </tr> </table> ===Elliptic Integral Expressions=== Hence, drawing from our set of [[Appendix/EquationTemplates#Complete_Elliptic_Integrals|Key Expressions for the complete elliptic integral of the first kind]], specifically, {{ Math/EQ_EllipticIntegral03 }} we can write, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~K(\mu)</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \ln \frac{4}{k^'} + \frac{1}{2^2}\biggl( \ln\frac{4}{k^'} - 1 \biggr){k'}^2 + \frac{3^2}{2^6} \biggl( \ln\frac{4}{k^'} - \frac{7}{6} \biggr){k'}^4 + \frac{5^2}{2^8} \biggl( \ln\frac{4}{k^'} - \frac{37}{30} \biggr){k'}^6 + \cdots </math> </td> </tr> </table> Now, we recognize that, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\biggl(\frac{a}{2}\biggr) \frac{4K(\mu)}{R_1+a}</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \biggl(\frac{a}{2c}\biggr) 4K(\mu) \biggl[\frac{R_1}{c} + \frac{a}{c} \biggr]^{-1} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \biggl(\frac{a}{c}\biggr) 2K(\mu) \biggl(\frac{R_1}{c}\biggr)^{-1}\biggl[1 + \frac{a}{c} \biggl(\frac{R_1}{c}\biggr)^{-1}\biggr]^{-1} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~\approx</math> </td> <td align="left"> <math>~ \biggl(\frac{a}{c}\biggr) \biggl(\frac{R_1}{c}\biggr)^{-1}\biggl[1 + \frac{a}{c} \biggl(\frac{R_1}{c}\biggr)^{-1}\biggr]^{-1}\biggl\{ 2\ln \frac{4}{k^'} + \frac{1}{2}\biggl( \ln\frac{4}{k^'} - 1 \biggr){k'}^2 \biggr\} \, . </math> </td> </tr> </table> <!-- <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\Rightarrow ~~~\biggl(\frac{a}{2}\biggr) \frac{4K(\mu)}{R_1+R} = 2K(\mu)\biggl[1 + \frac{R_1}{a}\biggr]^{-1}</math> </td> <td align="center"> <math>~\approx</math> </td> <td align="left"> <math>~ \biggl[1 + \frac{R_1}{a}\biggr]^{-1} \biggl\{ 2\ln \frac{4}{k^'} + \frac{1}{2}\biggl( \ln\frac{4}{k^'} - 1 \biggr){k'}^2 \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~\approx</math> </td> <td align="left"> <math>~ \biggl[1 + \frac{R_1}{a}\biggr]^{-1} \biggl\{ \ln\biggl( \frac{2^3c}{a}\biggr) + \frac{1}{4}\biggl[ \ln\biggl( \frac{2^3c}{a}\biggr) - 2 \biggr] 2^2\biggl( \frac{a}{c}\biggr)\biggl( \frac{R_1}{c}\biggr)^{-1} \biggl[1 + \frac{a}{c}\cdot \frac{c}{R_1} \biggr]^{-2} \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \biggl[1 + \frac{R_1}{a}\biggr]^{-1} \biggl\{ \ln\biggl( \frac{2^3c}{a}\biggr) + \frac{1}{2}\biggl[ \ln\biggl( \frac{2^3c}{a}\biggr) - 2 \biggr] \biggl( \frac{a}{c}\biggr)\biggl[ 1 - \biggl(\frac{a}{c}\biggr)\cos\chi + \frac{1}{4}\biggl( \frac{a}{c}\biggr)^2\biggr]^{-1 / 2} \biggl[1 + \frac{a}{c}\cdot \frac{c}{R_1} \biggr]^{-2} \biggr\} </math> </td> </tr> </table> --> Also, drawing from our set of [[Appendix/EquationTemplates#Complete_Elliptic_Integrals|Key Expressions for the complete elliptic integral of the second kind]], specifically, {{ Math/EQ_EllipticIntegral04 }} we have, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~E(\mu)</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ 1 ~+~ \frac{1}{2}\biggl( \ln \frac{4}{k'} - \frac{1}{2}\biggr)(k')^2 ~+~ \frac{3}{2^4}\biggl( \ln \frac{4}{k'} - 1 - \frac{1}{3\cdot 4}\biggr)(k')^4 ~+~ \frac{3^2\cdot 5}{2^7\cdot 3}\biggl( \ln \frac{4}{k'} - 1 - \frac{1}{2\cdot 3} - \frac{1}{2\cdot 3\cdot 5}\biggr)(k')^6 ~+~ \cdots </math> </td> </tr> <tr> <td align="right"> <math>~\Rightarrow ~~~ \biggl(\frac{a}{2}\biggr) \frac{(R_1+R)E(\mu)}{RR_1}</math> </td> <td align="center"> <math>~\approx</math> </td> <td align="left"> <math>~ \frac{1}{2} \biggl[ 1 + \frac{a}{c}\biggl( \frac{R_1}{c}\biggr)^{-1} \biggr] \biggl\{ 1 + \frac{1}{2}\biggl( \ln \frac{4}{k'} - \frac{1}{2}\biggr)(k')^2 \biggr\} \, . </math> </td> </tr> </table> ===External Potential at Torus Surface=== ====Initial Low Resolution==== Hence, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\biggl(\frac{a}{2}\biggr)V_2 </math> </td> <td align="center"> <math>~\approx</math> </td> <td align="left"> <math>~ \biggl[1 - \frac{1}{8}\biggl(\frac{a^2}{c^2}\biggr) \cos^2\biggl( \frac{\psi}{2}\biggr)\biggr] \biggl(\frac{a}{c}\biggr) \biggl(\frac{R_1}{c}\biggr)^{-1}\biggl[1 + \frac{a}{c} \biggl(\frac{R_1}{c}\biggr)^{-1}\biggr]^{-1}\biggl\{ 2\ln \frac{4}{k^'} + \frac{1}{2}\biggl( \ln\frac{4}{k^'} - 1 \biggr){k'}^2 \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ + \biggl[\frac{1}{8}\biggl(\frac{a^2}{c^2}\biggr) \cos\psi \biggr] \frac{1}{2} \biggl[ 1 + \frac{a}{c}\biggl( \frac{R_1}{c}\biggr)^{-1} \biggr] \biggl\{ 1 + \frac{1}{2}\biggl( \ln \frac{4}{k'} - \frac{1}{2}\biggr)(k')^2 \biggr\} </math> </td> </tr> <tr> <td align="right"> <math>~\Rightarrow ~~~ cV_2 </math> </td> <td align="center"> <math>~\approx</math> </td> <td align="left"> <math>~ \biggl[1 - \cancelto{0}{\frac{1}{8}\biggl(\frac{a^2}{c^2}\biggr) \cos^2\biggl( \frac{\psi}{2}\biggr)}\biggr] \biggl(\frac{R_1}{c}\biggr)^{-1}\biggl[1 + \frac{a}{c} \biggl(\frac{R_1}{c}\biggr)^{-1}\biggr]^{-1}\biggl\{ 4\ln \frac{4}{k^'} + \biggl( \ln\frac{4}{k^'} - 1 \biggr){k'}^2 \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ + \biggl[\frac{1}{8}\biggl(\frac{a}{c}\biggr) \cos\psi \biggr] \biggl[ 1 + \frac{a}{c}\biggl( \frac{R_1}{c}\biggr)^{-1} \biggr] \biggl\{ 1 + \frac{1}{2}\biggl( \ln \frac{4}{k'} - \frac{1}{2}\biggr)(k')^2 \biggr\} \, . </math> </td> </tr> </table> Hence, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\Rightarrow ~~~ cV_2 </math> </td> <td align="center"> <math>~\approx</math> </td> <td align="left"> <math>~ \biggl(\frac{R_1}{c}\biggr)^{-1}\biggl[1 + \frac{a}{c} \biggl(\frac{R_1}{c}\biggr)^{-1}\biggr]^{-1}\biggl\{ 4\ln \frac{4}{k^'} + \biggl( \ln\frac{4}{k^'} - 1 \biggr){k'}^2 \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ + \biggl[\frac{1}{8}\biggl(\frac{a}{c}\biggr) \cos\psi \biggr] \biggl[ 1 + \frac{a}{c}\biggl( \frac{R_1}{c}\biggr)^{-1} \biggr] \biggl\{ 1 + \frac{1}{2}\biggl( \ln \frac{4}{k'} - \frac{1}{2}\biggr)(k')^2 \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~\approx</math> </td> <td align="left"> <math>~ \biggl\{ 1 + \frac{1}{2} \biggl(\frac{a}{c}\biggr)\cos\chi +\biggl( \frac{a}{c}\biggr)^2 \biggl[ \frac{3}{8}\cos^2\chi -\frac{1}{8}\biggr] \biggr\} \biggl[1 - \frac{1}{2}\biggl( \frac{a}{c}\biggr) + \frac{1}{4} \biggl( \frac{a}{c}\biggr)^2 (1 - \cos\chi)\biggr] \biggl\{ 2\ln \frac{4}{k^'} + \frac{1}{2}\biggl( \ln\frac{4}{k^'} - 1 \biggr){k'}^2 \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ + \frac{1}{8}\biggl(\frac{a}{c}\biggr) \cos\psi \biggl[ 1 + \frac{1}{2}\biggl( \frac{a}{c}\biggr) + \frac{1}{4} \biggl(\frac{a}{c}\biggr)^2\cos\chi \biggr] \biggl\{ 1 + \frac{1}{2}\biggl( \ln \frac{4}{k'} - \frac{1}{2}\biggr)(k')^2 \biggr\} \, . </math> </td> </tr> </table> To order <math>~(a/c)^1</math>, this gives, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\Rightarrow ~~~ cV_2 </math> </td> <td align="center"> <math>~\approx</math> </td> <td align="left"> <math>~ 2\ln \frac{4}{k^'} + \frac{1}{2}\biggl( \ln\frac{4}{k^'} - 1 \biggr){k'}^2 + \biggl(\frac{a}{c}\biggr)(\cos\chi -1) \ln \frac{4}{k^'} + \frac{1}{8}\biggl(\frac{a}{c}\biggr) \cos\psi </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~\approx</math> </td> <td align="left"> <math>~ 2\ln \frac{4}{k^'} + \biggl(\frac{a}{c}\biggr) \biggl\{ - 1 + \cos\chi \ln \frac{4}{k^'} + \frac{1}{8} \cos\psi \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~\approx</math> </td> <td align="left"> <math>~ 2\ln \frac{4}{k^'} + \biggl(\frac{a}{c}\biggr) \biggl\{ - 1 + \cos\chi \biggl[ \ln \frac{4}{k^'} - \frac{1}{8} \biggr] \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~\approx</math> </td> <td align="left"> <math>~ \ln\biggl( \frac{2^3c}{a}\biggr) + \frac{a}{c}\biggl[ 1- \frac{1}{2} \cos\chi \biggr] + \biggl(\frac{a}{c}\biggr) \biggl\{ \frac{1}{2} \cos\chi \biggl[\ln\biggl( \frac{2^3c}{a}\biggr) - \frac{1}{4} \biggr] -1 \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~\approx</math> </td> <td align="left"> <math>~ \ln\biggl( \frac{2^3c}{a}\biggr) + \frac{a}{c} \biggl\{ 1- \frac{1}{2} \cos\chi + \frac{1}{2} \cos\chi \biggl[\ln\biggl( \frac{2^3c}{a}\biggr) - \frac{1}{4} \biggr] -1 \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~\approx</math> </td> <td align="left"> <math>~ \ln\biggl( \frac{2^3c}{a}\biggr) + \frac{1}{2}\biggl(\frac{a}{c}\biggr) \biggl[\ln\biggl( \frac{2^3c}{a}\biggr) - \frac{5}{4} \biggr] \cos\chi \, . </math> </td> </tr> </table> ---- We are trying to match equation (6) in [http://adsabs.harvard.edu/abs/1893RSPTA.184.1041D Dyson's (1893b)] "Part II", that is, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\frac{V}{2\pi a^2}</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \ln\biggl(\frac{8c}{a}\biggr) + \frac{1}{2}\biggl(\frac{a}{c}\biggr) \biggl[\ln\biggl(\frac{8c}{a}\biggr) - \frac{5}{4}\biggr] \cos\chi + \biggl\{ \frac{1}{16} \biggl[ \ln\biggl(\frac{8c}{a}\biggr) - \frac{5}{2} \biggr] + \frac{3}{16} \biggl[\ln\biggl(\frac{8c}{a}\biggr) +\frac{17}{36} - \frac{72}{36}\biggr]\cos2\chi\biggr\}\biggl(\frac{a^2}{c^2}\biggr) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ + \biggl\{ \frac{3}{32}\biggl[ \ln\biggl(\frac{8c}{a}\biggr) - \frac{25}{12}\biggr]\cos\chi + \frac{5}{64}\biggl[ \ln\biggl(\frac{8c}{a}\biggr)+\frac{7}{24} - \frac{48}{24}\biggr]\cos3\chi \biggr\} \biggl(\frac{a^3}{c^3}\biggr) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ + \biggl\{ \frac{9}{256}\biggl[ \ln\biggl(\frac{8c}{a}\biggr) - 2\biggr] + \frac{7}{128}\biggl[ \ln\biggl(\frac{8c}{a}\biggr) - \frac{19}{168} - 2\biggr]\cos2\chi + \frac{35}{1024} \biggl[ \ln\biggl(\frac{8c}{a}\biggr) - 2 + \frac{19}{120}\biggr]\cos4\chi \biggr\} \biggl(\frac{a^4}{c^4}\biggr) ~+~\cdots </math> </td> </tr> </table> ====High Resolution==== <table border="1" width="85%" align="center" cellpadding="8"> <tr><td align="left"> <div align="center">'''Summary'''</div> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~2\biggl(\frac{R_1}{c}\biggr)^{-1}</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ 1 + \frac{1}{2}\biggl(\frac{a}{c}\biggr)\cos\chi + \frac{1}{2^3}\biggl(\frac{a}{c}\biggr)^2 \biggl[ 3\cos^2\chi - 1 \biggr] + \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^3\biggl[ 5\cos^3\chi ~-~ 3\cos\chi \biggr] ~+~ \frac{1}{2^7} \biggl( \frac{a}{c}\biggr)^4 \biggl[ 3 ~-~ 30 \cos^2\chi ~+~ 35 \cos^4\chi \biggr] ~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) \, . </math> </td> </tr> <tr> <td align="right"> <math>~\biggl[1~+~\biggl( \frac{a}{c}\biggr)\biggl(\frac{R_1}{c}\biggr)^{-1}\biggr]^{-1} </math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ 1 - \frac{1}{2}\biggl( \frac{a}{c}\biggr) ~+~ \frac{1}{2^2}\biggl(\frac{a}{c}\biggr)^2(1-\cos\chi ) ~+~ \frac{1}{2^3}\biggl(\frac{a}{c}\biggr)^3 ( 2\cos\chi ~-~ 3\cos^2\chi ) ~+~ \frac{1}{2^5} \biggl(\frac{a}{c}\biggr)^4 ( -~9 \cos\chi ~+~8\cos^2\chi ~-~ 5\cos^3\chi ) ~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) \, . </math> </td> </tr> <tr> <td align="right"> <math>~t_K</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ 1 ~-~ \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^2 ( 1 -\cos\chi ) + \frac{1}{2^7\cdot 3} \biggl(\frac{a}{c}\biggr)^3 ( -13~+~12\cos\chi +2 \cos^2\chi ) + \frac{1}{2^8\cdot 3} \biggl(\frac{a}{c}\biggr)^4 (-17 + 21\cos\chi - \cos^2\chi ~-~ 2\cos^3\chi ) + \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) \, . </math> </td> </tr> <tr> <td align="right"> <math>~1~+~\biggl( \frac{a}{c}\biggr)\biggl(\frac{R_1}{c}\biggr)^{-1} </math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ 1 ~+~\frac{1}{2}\biggl( \frac{a}{c}\biggr) ~+~ \frac{1}{2^2}\biggl(\frac{a}{c}\biggr)^2\cos\chi ~+~ \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^3 ( 3\cos^2\chi - 1 ) ~+~ \frac{1}{2^5}\biggl(\frac{a}{c}\biggr)^4 ( 5\cos^3\chi ~-~ 3\cos\chi ) ~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) \, . </math> </td> </tr> <tr> <td align="right"> <math>~t_E</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ -\frac{1}{2^3} \biggl(\frac{a}{c}\biggr)^2\cos\chi ~+~ \frac{1}{2^6\cdot 3} \biggl(\frac{a}{c}\biggr)^3 ( 13- 12\cos\chi - 2\cos^2\chi ) ~+~ \frac{1}{2^7\cdot 3} \biggl(\frac{a}{c}\biggr)^4 ( 12 -21 \cos\chi - 2\cos^2\chi - 2\cos^3\chi ) + f_{E5}\biggl(\frac{a}{c}\biggr)^5 + \mathcal{O}\biggl(\frac{a^6}{c^6}\biggr) \, . </math> </td> </tr> </table> </td></tr> </table> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\frac{\pi V_\mathrm{Dyson}}{GM/c} </math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ 2K(\mu) \biggl[~2\biggl(\frac{R_1}{c}\biggr)^{-1}\biggr] \biggl[1 + \biggl(\frac{a}{c}\biggr) \biggl(\frac{R_1}{c}\biggr)^{-1} \biggr]^{-1}\biggl\{ t_K \biggr\} + E(\mu) \biggl[1 + \biggl(\frac{a}{c}\biggr) \biggl(\frac{R_1}{c}\biggr)^{-1} \biggr] \biggl\{\biggl(\frac{c}{a}\biggr) t_E \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ 2K(\mu) \biggl\{ ~2\biggl(\frac{R_1}{c}\biggr)^{-1}\biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ \times \biggl\{ \biggl[1 + \biggl(\frac{a}{c}\biggr) \biggl(\frac{R_1}{c}\biggr)^{-1} \biggr]^{-1} \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ \times \biggl\{ t_K \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ + E(\mu) \biggl\{ 1 + \biggl(\frac{a}{c}\biggr) \biggl(\frac{R_1}{c}\biggr)^{-1} \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ \times \biggl\{\biggl(\frac{c}{a}\biggr) t_E \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ 2K(\mu) \biggl\{ 1 + \frac{1}{2}\biggl(\frac{a}{c}\biggr)\cos\chi + \frac{1}{2^3}\biggl(\frac{a}{c}\biggr)^2 \biggl[ 3\cos^2\chi - 1 \biggr] + \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^3\biggl[ 5\cos^3\chi ~-~ 3\cos\chi \biggr] ~+~ \frac{1}{2^7} \biggl( \frac{a}{c}\biggr)^4 \biggl[ 3 ~-~ 30 \cos^2\chi ~+~ 35 \cos^4\chi \biggr] \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ \times \biggl\{ 1 - \frac{1}{2}\biggl( \frac{a}{c}\biggr) ~+~ \frac{1}{2^2}\biggl(\frac{a}{c}\biggr)^2(1-\cos\chi ) ~+~ \frac{1}{2^3}\biggl(\frac{a}{c}\biggr)^3 ( 2\cos\chi ~-~ 3\cos^2\chi ) ~+~ \frac{1}{2^5} \biggl(\frac{a}{c}\biggr)^4 ( -~9 \cos\chi ~+~8\cos^2\chi ~-~ 5\cos^3\chi ) \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ \times \biggl\{ 1 ~-~ \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^2 ( 1 -\cos\chi ) + \frac{1}{2^7\cdot 3} \biggl(\frac{a}{c}\biggr)^3 ( -13~+~12\cos\chi +2 \cos^2\chi ) + \frac{1}{2^8\cdot 3} \biggl(\frac{a}{c}\biggr)^4 (-17 + 21\cos\chi - \cos^2\chi ~-~ 2\cos^3\chi ) \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ + E(\mu) \biggl\{ 1 ~+~\frac{1}{2}\biggl( \frac{a}{c}\biggr) ~+~ \frac{1}{2^2}\biggl(\frac{a}{c}\biggr)^2\cos\chi ~+~ \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^3 \biggl[ 3\cos^2\chi - 1 \biggr] ~+~ \frac{1}{2^5}\biggl(\frac{a}{c}\biggr)^4\biggl[ 5\cos^3\chi ~-~ 3\cos\chi \biggr] \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ \times \biggl\{-\frac{1}{2^3} \biggl(\frac{a}{c}\biggr)\cos\chi ~+~ \frac{1}{2^6\cdot 3} \biggl(\frac{a}{c}\biggr)^2 ( 13- 12\cos\chi - 2\cos^2\chi ) ~+~ \frac{1}{2^7\cdot 3} \biggl(\frac{a}{c}\biggr)^3 ( 12 -21 \cos\chi - 2\cos^2\chi - 2\cos^3\chi ) + f_{E5}\biggl(\frac{a}{c}\biggr)^4 \biggr\} ~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ 2K(\mu) \biggl\{ 1 + \frac{1}{2}\biggl(\frac{a}{c}\biggr)\cos\chi + \frac{1}{2^3}\biggl(\frac{a}{c}\biggr)^2 \biggl[ 3\cos^2\chi - 1 \biggr] + \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^3\biggl[ 5\cos^3\chi ~-~ 3\cos\chi \biggr] ~+~ \frac{1}{2^7} \biggl( \frac{a}{c}\biggr)^4 \biggl[ 3 ~-~ 30 \cos^2\chi ~+~ 35 \cos^4\chi \biggr] \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ \times \biggl\{ 1 ~-~ \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^2 ( 1 -\cos\chi ) + \frac{1}{2^7\cdot 3} \biggl(\frac{a}{c}\biggr)^3 ( -13~+~12\cos\chi +2 \cos^2\chi ) + \frac{1}{2^8\cdot 3} \biggl(\frac{a}{c}\biggr)^4 (-17 + 21\cos\chi - \cos^2\chi ~-~ 2\cos^3\chi ) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ - \frac{1}{2}\biggl( \frac{a}{c}\biggr) ~+~ \frac{1}{2^5}\biggl(\frac{a}{c}\biggr)^3 ( 1 -\cos\chi ) - \frac{1}{2^8\cdot 3} \biggl(\frac{a}{c}\biggr)^4( -13~+~12\cos\chi +2 \cos^2\chi ) +\frac{1}{2^2}\biggl(\frac{a}{c}\biggr)^2(1-\cos\chi ) - \frac{1}{2^6}\biggl(\frac{a}{c}\biggr)^4(1-\cos\chi )^2 </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ ~+~ \frac{1}{2^3}\biggl(\frac{a}{c}\biggr)^3 ( 2\cos\chi ~-~ 3\cos^2\chi ) ~+~ \frac{1}{2^5} \biggl(\frac{a}{c}\biggr)^4 (-~9 \cos\chi ~+~8\cos^2\chi ~-~ 5\cos^3\chi ) \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ + E(\mu) \biggl\{ -\frac{1}{2^3} \biggl(\frac{a}{c}\biggr)\cos\chi \biggl[ 1 ~+~\frac{1}{2}\biggl( \frac{a}{c}\biggr) ~+~ \frac{1}{2^2}\biggl(\frac{a}{c}\biggr)^2\cos\chi ~+~ \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^3 ( 3\cos^2\chi - 1 ) \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ + \frac{1}{2^6\cdot 3} \biggl(\frac{a}{c}\biggr)^2 ( 13- 12\cos\chi - 2\cos^2\chi ) \biggl[ 1 ~+~\frac{1}{2}\biggl( \frac{a}{c}\biggr) ~+~ \frac{1}{2^2}\biggl(\frac{a}{c}\biggr)^2\cos\chi \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ + \frac{1}{2^7\cdot 3} \biggl(\frac{a}{c}\biggr)^3 ( 12 -21 \cos\chi - 2\cos^2\chi - 2\cos^3\chi ) \biggl[ 1 ~+~\frac{1}{2}\biggl( \frac{a}{c}\biggr) \biggr] + f_{E5}\biggl(\frac{a}{c}\biggr)^4 \biggr\} ~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ 2K(\mu) \biggl\{ 1 + \frac{1}{2}\biggl(\frac{a}{c}\biggr)\cos\chi + \frac{1}{2^3}\biggl(\frac{a}{c}\biggr)^2 \biggl[ 3\cos^2\chi - 1 \biggr] + \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^3\biggl[ 5\cos^3\chi ~-~ 3\cos\chi \biggr] ~+~ \frac{1}{2^7} \biggl( \frac{a}{c}\biggr)^4 \biggl[ 3 ~-~ 30 \cos^2\chi ~+~ 35 \cos^4\chi \biggr] \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ \times \biggl\{ 1 - \frac{1}{2}\biggl( \frac{a}{c}\biggr) +\frac{3}{2^4}\biggl(\frac{a}{c}\biggr)^2(1-\cos\chi ) + \frac{1}{2^7\cdot 3} \biggl(\frac{a}{c}\biggr)^3 \biggl[ ( -13~+~12\cos\chi +2 \cos^2\chi ) ~+~ 12 ( 1 -\cos\chi ) ~+~ 48 ( 2\cos\chi ~-~ 3\cos^2\chi ) \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ + \frac{1}{2^8\cdot 3} \biggl(\frac{a}{c}\biggr)^4 \biggl[ (-17 + 21\cos\chi - \cos^2\chi ~-~ 2\cos^3\chi ) - ( -13~+~12\cos\chi +2 \cos^2\chi ) - 12 (1-\cos\chi )^2 ~+~ 24 (-~9 \cos\chi ~+~8\cos^2\chi ~-~ 5\cos^3\chi ) \biggr] \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ + E(\mu) \biggl\{ ~-~ \frac{1}{2^3} \biggl(\frac{a}{c}\biggr)\cos\chi ~ -\frac{1}{2^4} \biggl(\frac{a}{c}\biggr)^2 \cos\chi ~ -\frac{1}{2^5} \biggl(\frac{a}{c}\biggr)^3 \cos^2\chi ~ -\frac{1}{2^7} \biggl(\frac{a}{c}\biggr)^4 ( 3\cos^3\chi - \cos\chi ) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ + \frac{1}{2^6\cdot 3} \biggl(\frac{a}{c}\biggr)^2 ( 13- 12\cos\chi - 2\cos^2\chi ) ~+~\frac{1}{2^7\cdot 3} \biggl(\frac{a}{c}\biggr)^3 ( 13- 12\cos\chi - 2\cos^2\chi ) ~+~ \frac{1}{2^8\cdot 3} \biggl(\frac{a}{c}\biggr)^4 ( 13- 12\cos\chi - 2\cos^2\chi )\cos\chi </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ + \frac{1}{2^7\cdot 3} \biggl(\frac{a}{c}\biggr)^3 ( 12 -21 \cos\chi - 2\cos^2\chi - 2\cos^3\chi ) ~+~\frac{1}{2^8\cdot 3} \biggl(\frac{a}{c}\biggr)^4 ( 12 -21 \cos\chi - 2\cos^2\chi - 2\cos^3\chi ) + f_{E5}\biggl(\frac{a}{c}\biggr)^4 \biggr\} ~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ 2K(\mu) \biggl\{ 1 + \frac{1}{2}\biggl(\frac{a}{c}\biggr)\cos\chi + \frac{1}{2^3}\biggl(\frac{a}{c}\biggr)^2 ( 3\cos^2\chi - 1 ) + \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^3 ( 5\cos^3\chi ~-~ 3\cos\chi ) ~+~ \frac{1}{2^7} \biggl( \frac{a}{c}\biggr)^4 ( 3~-~ 30 \cos^2\chi~+~ 35 \cos^4\chi ) \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ \times \biggl\{ 1 - \frac{1}{2}\biggl( \frac{a}{c}\biggr) +\frac{3}{2^4}\biggl(\frac{a}{c}\biggr)^2(1-\cos\chi ) + \frac{1}{2^7\cdot 3} \biggl(\frac{a}{c}\biggr)^3 \biggl[ ( -13~+~12\cos\chi +2 \cos^2\chi ) ~+~ ( 12 -12\cos\chi ) ~+~ ( 96\cos\chi ~-~ 144\cos^2\chi ) \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ + \frac{1}{2^8\cdot 3} \biggl(\frac{a}{c}\biggr)^4 \biggl[ (-17 + 21\cos\chi - \cos^2\chi ~-~ 2\cos^3\chi ) + ( 13~-~12\cos\chi -2 \cos^2\chi ) + (-12 + 24\cos\chi - 12\cos^2\chi) ~+~ (-~216 \cos\chi ~+~192\cos^2\chi ~-~ 120\cos^3\chi ) \biggr] \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ + E(\mu) \biggl\{ ~-~ \frac{1}{2^3} \biggl(\frac{a}{c}\biggr)\cos\chi + \frac{1}{2^6\cdot 3} \biggl(\frac{a}{c}\biggr)^2\biggl[ ( 13- 12\cos\chi - 2\cos^2\chi ) ~ -12 \cos\chi \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ ~+~\frac{1}{2^7\cdot 3} \biggl(\frac{a}{c}\biggr)^3 \biggl[ ( 13- 12\cos\chi - 2\cos^2\chi ) + ( 12 -21 \cos\chi - 2\cos^2\chi - 2\cos^3\chi ) ~ -~12 \cos^2\chi \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ ~+~ \frac{1}{2^8\cdot 3} \biggl(\frac{a}{c}\biggr)^4 \biggl[ ( 13\cos\chi - 12\cos^2\chi - 2\cos^3\chi ) ~+~( 12 -21 \cos\chi - 2\cos^2\chi - 2\cos^3\chi ) ~ -~6 ( 3\cos^3\chi - \cos\chi ) + 2^8\cdot 3 f_{E5} \biggr] \biggr\} ~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) </math> </td> </tr> </table> That is, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\frac{\pi V_\mathrm{Dyson}}{GM/c} </math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ 2K(\mu) \biggl\{ 1 + \frac{1}{2}\biggl(\frac{a}{c}\biggr)\cos\chi + \frac{1}{2^3}\biggl(\frac{a}{c}\biggr)^2 ( 3\cos^2\chi - 1 ) + \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^3 ( 5\cos^3\chi ~-~ 3\cos\chi ) ~+~ \frac{1}{2^7} \biggl( \frac{a}{c}\biggr)^4 ( 3~-~ 30 \cos^2\chi~+~ 35 \cos^4\chi ) \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ \times \biggl\{ 1 - \frac{1}{2}\biggl( \frac{a}{c}\biggr) +\frac{3}{2^4}\biggl(\frac{a}{c}\biggr)^2(1-\cos\chi ) + \frac{1}{2^7\cdot 3} \biggl(\frac{a}{c}\biggr)^3 ( -1 ~+~ 96\cos\chi~-~ 142\cos^2\chi ) + \frac{1}{2^8\cdot 3} \biggl(\frac{a}{c}\biggr)^4 (-16 ~-~183\cos\chi ~+~177 \cos^2\chi ~-~ 122\cos^3\chi ) \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ + E(\mu) \biggl\{ ~-~ \frac{1}{2^3} \biggl(\frac{a}{c}\biggr)\cos\chi + \frac{1}{2^6\cdot 3} \biggl(\frac{a}{c}\biggr)^2 ( 13- 24\cos\chi - 2\cos^2\chi ) ~+~\frac{1}{2^7\cdot 3} \biggl(\frac{a}{c}\biggr)^3 ( 25- 33\cos\chi - 16\cos^2\chi - 2\cos^3\chi ) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ ~+~ \frac{1}{2^8\cdot 3} \biggl(\frac{a}{c}\biggr)^4 ( 12 -2 \cos\chi - 14\cos^2\chi - 22\cos^3\chi + 2^8\cdot 3 f_{E5} ) \biggr\} ~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ 2K(\mu) \biggl\{ 1 - \frac{1}{2}\biggl( \frac{a}{c}\biggr) +\frac{3}{2^4}\biggl(\frac{a}{c}\biggr)^2(1-\cos\chi ) + \frac{1}{2^7\cdot 3} \biggl(\frac{a}{c}\biggr)^3 ( -1 ~+~ 96\cos\chi~-~ 142\cos^2\chi ) + \frac{1}{2^8\cdot 3} \biggl(\frac{a}{c}\biggr)^4 (-16 ~-~183\cos\chi ~+~177 \cos^2\chi ~-~ 122\cos^3\chi ) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ + \frac{1}{2}\biggl(\frac{a}{c}\biggr)\cos\chi - \frac{1}{2^2}\biggl( \frac{a}{c}\biggr)^2\cos\chi +\frac{3}{2^5}\biggl(\frac{a}{c}\biggr)^3(\cos\chi-\cos^2\chi ) + \frac{1}{2^8\cdot 3} \biggl(\frac{a}{c}\biggr)^4 ( - \cos\chi ~+~ 96\cos^2\chi~-~ 142\cos^3\chi ) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ + \frac{1}{2^3}\biggl(\frac{a}{c}\biggr)^2 ( 3\cos^2\chi - 1 ) + \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^3 (1- 3\cos^2\chi ) ~+~\frac{3}{2^7}\biggl(\frac{a}{c}\biggr)^4(1-\cos\chi ) ( 3\cos^2\chi - 1 ) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ + \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^3 ( 5\cos^3\chi ~-~ 3\cos\chi ) ~+~ \frac{1}{2^5}\biggl(\frac{a}{c}\biggr)^4 (3\cos\chi ~-~5\cos^3\chi ) + \frac{1}{2^7} \biggl( \frac{a}{c}\biggr)^4 ( 3~-~ 30 \cos^2\chi~+~ 35 \cos^4\chi ) \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ + E(\mu) \biggl\{ ~-~ \frac{1}{2^3} \biggl(\frac{a}{c}\biggr)\cos\chi + \frac{1}{2^6\cdot 3} \biggl(\frac{a}{c}\biggr)^2 ( 13- 24\cos\chi - 2\cos^2\chi ) ~+~\frac{1}{2^7\cdot 3} \biggl(\frac{a}{c}\biggr)^3 ( 25- 33\cos\chi - 16\cos^2\chi - 2\cos^3\chi ) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ ~+~ \frac{1}{2^8\cdot 3} \biggl(\frac{a}{c}\biggr)^4 ( 12 -2 \cos\chi - 14\cos^2\chi - 22\cos^3\chi + 2^8\cdot 3 f_{E5} ) \biggr\} ~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ 2K(\mu) \biggl\{ 1 + \frac{1}{2}\biggl(\frac{a}{c}\biggr)(\cos\chi - 1) +\frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^2 \biggl[3(1-\cos\chi ) - 4 \cos\chi + 2 ( 3\cos^2\chi - 1 )\biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ + \frac{1}{2^7\cdot 3} \biggl(\frac{a}{c}\biggr)^3 \biggl[ 36 (\cos\chi-\cos^2\chi ) + 24 (1 ~-~ 3\cos\chi - 3\cos^2\chi +5\cos^3\chi ) + ( -1 ~+~ 96\cos\chi~-~ 142\cos^2\chi ) \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ + \frac{1}{2^8\cdot 3} \biggl(\frac{a}{c}\biggr)^4 \biggl[ (-16 ~-~183\cos\chi ~+~177 \cos^2\chi ~-~ 122\cos^3\chi ) ~+~18(1-\cos\chi ) ( 3\cos^2\chi - 1 ) + ( - \cos\chi ~+~ 96\cos^2\chi~-~ 142\cos^3\chi ) ~+~ 24 (3\cos\chi ~-~5\cos^3\chi ) + 6 ( 3~-~ 30 \cos^2\chi~+~ 35 \cos^4\chi ) \biggr] \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ + E(\mu) \biggl\{ ~-~ \frac{1}{2^3} \biggl(\frac{a}{c}\biggr)\cos\chi + \frac{1}{2^6\cdot 3} \biggl(\frac{a}{c}\biggr)^2 ( 13- 24\cos\chi - 2\cos^2\chi ) ~+~\frac{1}{2^7\cdot 3} \biggl(\frac{a}{c}\biggr)^3 ( 25- 33\cos\chi - 16\cos^2\chi - 2\cos^3\chi ) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ ~+~ \frac{1}{2^8\cdot 3} \biggl(\frac{a}{c}\biggr)^4 ( 12 -2 \cos\chi - 14\cos^2\chi - 22\cos^3\chi + 2^8\cdot 3 f_{E5} ) \biggr\} ~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ 2K(\mu) \biggl\{ 1 + \frac{1}{2}\biggl(\frac{a}{c}\biggr)(\cos\chi - 1) +\frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^2 (1-7\cos\chi + 6\cos^2\chi ) + \frac{1}{2^7\cdot 3} \biggl(\frac{a}{c}\biggr)^3 (23 ~+~ 60\cos\chi - 250\cos^2\chi +120\cos^3\chi ) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ + \frac{1}{2^8\cdot 3} \biggl(\frac{a}{c}\biggr)^4 ( -16 ~-~94\cos\chi ~+~147 \cos^2\chi ~-~ 194\cos^3\chi + 210 \cos^4\chi ) \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ + E(\mu) \biggl\{ ~-~ \frac{1}{2^3} \biggl(\frac{a}{c}\biggr)\cos\chi + \frac{1}{2^6\cdot 3} \biggl(\frac{a}{c}\biggr)^2 ( 13- 24\cos\chi - 2\cos^2\chi ) ~+~\frac{1}{2^7\cdot 3} \biggl(\frac{a}{c}\biggr)^3 ( 25- 33\cos\chi - 16\cos^2\chi - 2\cos^3\chi ) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ ~+~ \frac{1}{2^8\cdot 3} \biggl(\frac{a}{c}\biggr)^4 ( 12 -2 \cos\chi - 14\cos^2\chi - 22\cos^3\chi + 2^8\cdot 3 f_{E5} ) \biggr\} ~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) </math> </td> </tr> </table> ====Insert Expressions for K and E==== <table border="1" align="center" width="85%" cellpadding="8"> <tr><td align="left"> <div align="center"> '''Summary''' </div> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~k'</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ 2\biggl( \frac{a}{c}\biggr)^{1 / 2}\biggl( \frac{R_1}{c}\biggr)^{-1 / 2} \biggl[1 + \frac{a}{c}\cdot \biggl( \frac{R_1}{c} \biggr)^{-1} \biggr]^{-1} </math> </td> </tr> <tr> <td align="right"> <math>~\Rightarrow ~~~ \frac{4}{k'}</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~2\biggl( \frac{a}{c}\biggr)^{- 1 / 2}\biggl( \frac{R_1}{c}\biggr)^{1 / 2} \biggl[1 + \frac{a}{c}\cdot \biggl( \frac{R_1}{c} \biggr)^{-1} \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~2\biggl( \frac{c}{a}\biggr)^{1 / 2} \biggl[ 4 + \biggl( \frac{a}{c}\biggr)^2 - 4\biggl(\frac{a}{c}\biggr)\cos\chi \biggr]^{1 / 4} \biggl\{ 1 + \frac{a}{c}\cdot \biggl[ 4 + \biggl( \frac{a}{c}\biggr)^2 - 4\biggl(\frac{a}{c}\biggr)\cos\chi \biggr]^{-1 / 2} \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~\biggl( \frac{2^3c}{a}\biggr)^{1 / 2} \biggl[ 1 - \biggl(\frac{a}{c}\biggr)\cos\chi + \frac{1}{4}\biggl( \frac{a}{c}\biggr)^2 \biggr]^{1 / 4} \biggl\{ 1 + \frac{a}{2c}\cdot \biggl[ 1 - \biggl(\frac{a}{c}\biggr)\cos\chi + \biggl( \frac{a}{2c}\biggr)^2 \biggr]^{-1 / 2} \biggr\} </math> </td> </tr> <tr> <td align="right"> <math>~\Rightarrow ~~~ \ln \frac{4}{k'}</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~\frac{1}{2} \ln\biggl( \frac{2^3c}{a}\biggr) + \frac{1}{4}\ln\biggl[ 1 - \biggl(\frac{a}{c}\biggr)\cos\chi + \frac{1}{4}\biggl( \frac{a}{c}\biggr)^2 \biggr] + \ln\biggl\{ 1 + \frac{a}{2c}\cdot \biggl[ 1 - \biggl(\frac{a}{c}\biggr)\cos\chi + \biggl( \frac{a}{2c}\biggr)^2 \biggr]^{-1 / 2} \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~\approx</math> </td> <td align="left"> <math>~ \frac{1}{2} \ln\biggl( \frac{2^3c}{a}\biggr) + \frac{1}{4}\ln\biggl[ 1 - \biggl(\frac{a}{c}\biggr)\cos\chi \biggr] + \ln\biggl[ 1 + \frac{a}{2c} \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~\approx</math> </td> <td align="left"> <math>~ \frac{1}{2} \ln\biggl( \frac{2^3c}{a}\biggr) + \frac{a}{2c}\biggl[ 1 - \frac{1}{2} \cos\chi \biggr] </math> </td> </tr> </table> </td></tr> </table> Remember that ([[#Elliptic_Integral_Expressions|see above]]), <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~K(\mu)</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \ln \frac{4}{k^'} + \frac{1}{2^2}\biggl( \ln\frac{4}{k^'} - 1 \biggr){k'}^2 + \frac{3^2}{2^6} \biggl( \ln\frac{4}{k^'} - \frac{7}{6} \biggr){k'}^4 + \frac{5^2}{2^8} \biggl( \ln\frac{4}{k^'} - \frac{37}{30} \biggr){k'}^6 + \cdots </math> </td> </tr> </table> And ([[#Elliptic_Integral_Expressions|see above]]), <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~E(\mu)</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ 1 ~+~ \frac{1}{2}\biggl( \ln \frac{4}{k'} - \frac{1}{2}\biggr)(k')^2 ~+~ \frac{3}{2^4}\biggl( \ln \frac{4}{k'} - 1 - \frac{1}{3\cdot 4}\biggr)(k')^4 ~+~ \frac{3^2\cdot 5}{2^7\cdot 3}\biggl( \ln \frac{4}{k'} - 1 - \frac{1}{2\cdot 3} - \frac{1}{2\cdot 3\cdot 5}\biggr)(k')^6 ~+~ \cdots </math> </td> </tr> </table> ---- We are trying to match equation (6) in [http://adsabs.harvard.edu/abs/1893RSPTA.184.1041D Dyson's (1893b)] "Part II", that is, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\frac{V}{2\pi a^2}</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \ln\biggl(\frac{8c}{a}\biggr) + \frac{1}{2}\biggl(\frac{a}{c}\biggr) \biggl[\ln\biggl(\frac{8c}{a}\biggr) - \frac{5}{4}\biggr] \cos\chi + \biggl\{ \frac{1}{16} \biggl[ \ln\biggl(\frac{8c}{a}\biggr) - \frac{5}{2} \biggr] + \frac{3}{16} \biggl[\ln\biggl(\frac{8c}{a}\biggr) +\frac{17}{36} - \frac{72}{36}\biggr]\cos2\chi\biggr\}\biggl(\frac{a^2}{c^2}\biggr) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ + \biggl\{ \frac{3}{32}\biggl[ \ln\biggl(\frac{8c}{a}\biggr) - \frac{25}{12}\biggr]\cos\chi + \frac{5}{64}\biggl[ \ln\biggl(\frac{8c}{a}\biggr)+\frac{7}{24} - \frac{48}{24}\biggr]\cos3\chi \biggr\} \biggl(\frac{a^3}{c^3}\biggr) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ + \biggl\{ \frac{9}{256}\biggl[ \ln\biggl(\frac{8c}{a}\biggr) - 2\biggr] + \frac{7}{128}\biggl[ \ln\biggl(\frac{8c}{a}\biggr) - \frac{19}{168} - 2\biggr]\cos2\chi + \frac{35}{1024} \biggl[ \ln\biggl(\frac{8c}{a}\biggr) - 2 + \frac{19}{120}\biggr]\cos4\chi \biggr\} \biggl(\frac{a^4}{c^4}\biggr) ~+~\cdots </math> </td> </tr> </table> ---- =====To First Order===== <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~~(k')^{2}</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \biggl( \frac{2a}{c}\biggr) \biggl\{1 ~-~ \frac{1}{2}\biggl(\frac{a}{c}\biggr) (2~-~ \cos\chi ) + \frac{1}{2^3}\biggl( \frac{a}{c}\biggr)^2\biggl[ (3-\cos^2\chi) ~+~2(2~-~ \cos\chi )^2 \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ +~ \frac{1}{2^4}\biggl( \frac{a}{c}\biggr)^3\biggl[ (3\cos\chi - \cos^3\chi) ~+~ (2~-~ \cos\chi )(3-\cos^2\chi) ~-~ (2~-~ \cos\chi )^3 \biggr] \biggr\} ~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) </math> </td> </tr> </table> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\ln \frac{4}{k'}</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \frac{1}{2} \ln\biggl( \frac{2^3c}{a}\biggr) + \frac{1}{2}\ln \Gamma </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \frac{1}{2} \ln\biggl( \frac{2^3c}{a}\biggr) + \frac{1}{2}\biggl\{ \frac{1}{2}\biggl(\frac{a}{c}\biggr) (2~-~ \cos\chi ) + \frac{1}{2^3}\biggl( \frac{a}{c}\biggr)^2 (3-\cos^2\chi) ~-~\frac{1}{2^3}\biggl(\frac{a}{c}\biggr)^2 (2~-~ \cos\chi )^2 ~+~ \mathcal{O}\biggl(\frac{a^3}{c^3}\biggr) \biggr\} </math> </td> </tr> </table> Hence, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\frac{\pi V_\mathrm{Dyson}}{GM/c} </math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ 2K(\mu) \biggl\{ 1 + \frac{1}{2}\biggl(\frac{a}{c}\biggr)(\cos\chi - 1) \biggr\} + E(\mu) \biggl\{ ~-~ \frac{1}{2^3} \biggl(\frac{a}{c}\biggr)\cos\chi \biggr\} ~+~ \mathcal{O}\biggl(\frac{a^2}{c^2}\biggr) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~\approx</math> </td> <td align="left"> <math>~ \biggl[ 1 + \frac{1}{2}\biggl(\frac{a}{c}\biggr)(\cos\chi - 1) \biggr] \biggl\{ 2K(\mu) \biggr\} ~-~ \frac{1}{2^3} \biggl(\frac{a}{c}\biggr)\cos\chi \biggl\{~E(\mu) \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~\approx</math> </td> <td align="left"> <math>~ \biggl[ 1 + \frac{1}{2}\biggl(\frac{a}{c}\biggr)(\cos\chi - 1) \biggr] \biggl\{ 2\ln \frac{4}{k'} + \frac{1}{2}\biggl[ \ln\frac{4}{k'} - 1 \biggr] k'^2 \biggr\} ~-~ \frac{1}{2^3} \biggl(\frac{a}{c}\biggr)\cos\chi \biggl\{~1 + \frac{1}{2}\cancelto{0}{\biggl[ \ln\frac{4}{k'} - \frac{1}{2} \biggr] k'^2}\biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~\approx</math> </td> <td align="left"> <math>~ \biggl\{ 2\ln \frac{4}{k'} + \frac{1}{2}\biggl[ \ln\frac{4}{k'} - 1 \biggr] \biggl(\frac{2a}{c}\biggr) \biggr\} + \frac{1}{2}\biggl(\frac{a}{c}\biggr)(\cos\chi - 1) \biggl\{ 2\ln \frac{4}{k'} +\frac{1}{2} \cancelto{0}{\biggl[ \ln\frac{4}{k'} - 1 \biggr] k'^2} \biggr\} ~-~ \frac{1}{2^3} \biggl(\frac{a}{c}\biggr)\cos\chi \biggl\{~1 + \frac{1}{2}\cancelto{0}{\biggl[ \ln\frac{4}{k'} - \frac{1}{2} \biggr] k'^2}\biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~\approx</math> </td> <td align="left"> <math>~ 2\ln \frac{4}{k'} + \biggl(\frac{a}{c}\biggr)\biggl[ \ln\frac{4}{k'} - 1 \biggr] + \biggl(\frac{a}{c}\biggr)(\cos\chi - 1) \biggl\{ \ln \frac{4}{k'} \biggr\} ~-~ \frac{1}{2^3} \biggl(\frac{a}{c}\biggr)\cos\chi </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~\approx</math> </td> <td align="left"> <math>~ 2\ln \frac{4}{k'} + \biggl(\frac{a}{c}\biggr)\biggl[ \ln\frac{4}{k'} - 1 + (\cos\chi - 1) \biggl( \ln \frac{4}{k'} \biggr) ~-~ \frac{1}{2^3} \cos\chi \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~\approx</math> </td> <td align="left"> <math>~ 2\ln \frac{4}{k'} + \biggl(\frac{a}{c}\biggr)\biggl[ - 1 + \cos\chi \biggl( \ln \frac{4}{k'} \biggr) ~-~ \frac{1}{2^3} \cos\chi \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~\approx</math> </td> <td align="left"> <math>~ \ln\biggl( \frac{2^3c}{a}\biggr) + \frac{1}{2}\biggl(\frac{a}{c}\biggr) \biggl\{ (2~-~ \cos\chi ) - 2 + \cos\chi \biggl[ \ln\biggl(\frac{2^3c}{a}\biggr)\biggr] ~-~ \frac{1}{4} \cos\chi \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~\approx</math> </td> <td align="left"> <math>~ \ln\biggl( \frac{2^3c}{a}\biggr) + \frac{1}{2}\biggl(\frac{a}{c}\biggr) \biggl[ \ln\biggl(\frac{2^3c}{a}\biggr) ~-~ \frac{5}{4} \biggr]\cos\chi </math> </td> </tr> </table> =====To Second Order===== <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\frac{\pi V_\mathrm{Dyson}}{GM/c} </math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \biggl\{ 1 + \frac{1}{2}\biggl(\frac{a}{c}\biggr)(\cos\chi - 1) +\frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^2 (1-7\cos\chi + 6\cos^2\chi ) \biggr\} \biggl\{ 2K(\mu) \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ + \biggl\{ ~-~ \frac{1}{2^3} \biggl(\frac{a}{c}\biggr)\cos\chi + \frac{1}{2^6\cdot 3} \biggl(\frac{a}{c}\biggr)^2 ( 13- 24\cos\chi - 2\cos^2\chi ) \biggr\} \biggl\{ E(\mu) \biggr\} ~+~ \mathcal{O}\biggl(\frac{a^3}{c^3}\biggr) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \biggl\{ 1 + \frac{1}{2}\biggl(\frac{a}{c}\biggr)(\cos\chi - 1) +\frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^2 (1-7\cos\chi + 6\cos^2\chi ) \biggr\} \biggl\{ 2\ln \frac{4}{k^'} + \frac{1}{2}\biggl( \ln\frac{4}{k^'} - 1 \biggr){k'}^2 + \frac{3^2}{2^5} \biggl( \ln\frac{4}{k^'} - \frac{7}{6} \biggr){k'}^4 \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ + \biggl\{ ~-~ \frac{1}{2^3} \biggl(\frac{a}{c}\biggr)\cos\chi + \frac{1}{2^6\cdot 3} \biggl(\frac{a}{c}\biggr)^2 ( 13- 24\cos\chi - 2\cos^2\chi ) \biggr\} \biggl\{ 1 ~+~ \frac{1}{2}\biggl( \ln \frac{4}{k'} - \frac{1}{2}\biggr)(k')^2 \biggr\} ~+~ \mathcal{O}\biggl(\frac{a^3}{c^3}\biggr) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \biggl\{ 1 + \frac{1}{2}\biggl(\frac{a}{c}\biggr)(\cos\chi - 1) +\frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^2 (1-7\cos\chi + 6\cos^2\chi ) \biggr\} \biggl\{ 2\biggl[\frac{1}{2} \ln\biggl( \frac{2^3c}{a}\biggr) + \frac{1}{2}\ln \Gamma\biggr] + \frac{1}{2}\biggl[ \frac{1}{2} \ln\biggl( \frac{2^3c}{a}\biggr) + \frac{1}{2}\ln \Gamma - 1 \biggr]{k'}^2 + \frac{3^2}{2^5} \biggl[ \frac{1}{2} \ln\biggl( \frac{2^3c}{a}\biggr) + \frac{1}{2}\ln \Gamma - \frac{7}{6} \biggr] {k'}^4 \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ + \biggl\{ ~-~ \frac{1}{2^3} \biggl(\frac{a}{c}\biggr)\cos\chi + \frac{1}{2^6\cdot 3} \biggl(\frac{a}{c}\biggr)^2 ( 13- 24\cos\chi - 2\cos^2\chi ) \biggr\} \biggl\{ 1 ~+~ \frac{1}{2}\biggl[ \frac{1}{2} \ln\biggl( \frac{2^3c}{a}\biggr) + \frac{1}{2}\ln \Gamma - \frac{1}{2}\biggr](k')^2 \biggr\} ~+~ \mathcal{O}\biggl(\frac{a^3}{c^3}\biggr) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \biggl\{ 1 + \frac{1}{2}\biggl(\frac{a}{c}\biggr)(\cos\chi - 1) +\frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^2 (1-7\cos\chi + 6\cos^2\chi ) \biggr\} \biggl\{ \biggl[ \ln\biggl( \frac{2^3c}{a}\biggr) + \ln \Gamma\biggr] + \frac{1}{2^2}\biggl[ \ln\biggl( \frac{2^3c}{a}\biggr) + \ln \Gamma - 2 \biggr] \biggl( \frac{2a}{c}\biggr) \biggl[ 1 ~-~ \frac{1}{2}\biggl(\frac{a}{c}\biggr) (2~-~ \cos\chi )\biggr] + \frac{3}{2^6} \biggl[ 3 \ln\biggl( \frac{2^3c}{a}\biggr) + \cancelto{0}{3\ln \Gamma} - 7 \biggr] 4\biggl( \frac{a}{c}\biggr)^2 \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ + \biggl\{ ~-~ \frac{1}{2^3} \biggl(\frac{a}{c}\biggr)\cos\chi + \frac{1}{2^6\cdot 3} \biggl(\frac{a}{c}\biggr)^2 ( 13- 24\cos\chi - 2\cos^2\chi ) \biggr\} \biggl\{ 1 ~+~ \frac{1}{2^2}\biggl[ \ln\biggl( \frac{2^3c}{a}\biggr) + \cancelto{0}{\ln \Gamma} - 1 \biggr] \biggl( \frac{2a}{c}\biggr) \biggl[ 1 ~-~\cancelto{0}{ \frac{1}{2}\biggl(\frac{a}{c}\biggr)} (2~-~ \cos\chi )\biggr] \biggr\} ~+~ \mathcal{O}\biggl(\frac{a^3}{c^3}\biggr) \, , </math> </td> </tr> </table> where, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\ln\Gamma</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \frac{1}{2}\biggl(\frac{a}{c}\biggr) (2~-~ \cos\chi ) + \frac{1}{2^3}\biggl( \frac{a}{c}\biggr)^2( -1 ~+~ 4\cos\chi -2\cos^2\chi ) ~+~ \mathcal{O}\biggl(\frac{a^3}{c^3}\biggr) \, . </math> </td> </tr> </table> Hence, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\frac{\pi V_\mathrm{Dyson}}{GM/c} </math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \biggl\{ 1 + \frac{1}{2}\biggl(\frac{a}{c}\biggr)(\cos\chi - 1) +\frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^2 (1-7\cos\chi + 6\cos^2\chi ) \biggr\} \biggl\{ \biggl[ \ln\biggl( \frac{2^3c}{a}\biggr) + \frac{1}{2}\biggl(\frac{a}{c}\biggr) (2~-~ \cos\chi ) + \frac{1}{2^3}\biggl( \frac{a}{c}\biggr)^2( -1 ~+~ 4\cos\chi -2\cos^2\chi ) \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ + \frac{1}{2}\biggl( \frac{a}{c}\biggr) \biggl[ \ln\biggl( \frac{2^3c}{a}\biggr) - 2 + \frac{1}{2}\biggl(\frac{a}{c}\biggr) (2~-~ \cos\chi ) \biggr] \biggl[ 1 ~-~ \frac{1}{2}\biggl(\frac{a}{c}\biggr) (2~-~ \cos\chi )\biggr] + \frac{3}{2^4}\biggl( \frac{a}{c}\biggr)^2 \biggl[ 3 \ln\biggl( \frac{2^3c}{a}\biggr) - 7 \biggr] \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ + \biggl\{ ~-~ \frac{1}{2^3} \biggl(\frac{a}{c}\biggr)\cos\chi + \frac{1}{2^6\cdot 3} \biggl(\frac{a}{c}\biggr)^2 ( 13- 24\cos\chi - 2\cos^2\chi ) \biggr\} \biggl\{ 1 ~+~ \frac{1}{2}\biggl( \frac{a}{c}\biggr)\biggl[ \ln\biggl( \frac{2^3c}{a}\biggr) - 1 \biggr] \biggr\} ~+~ \mathcal{O}\biggl(\frac{a^3}{c^3}\biggr) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \biggl\{ 1 + \frac{1}{2}\biggl(\frac{a}{c}\biggr)(\cos\chi - 1) +\frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^2 (1-7\cos\chi + 6\cos^2\chi ) \biggr\} \biggl\{ \biggl[ \ln\biggl( \frac{2^3c}{a}\biggr) + \frac{1}{2}\biggl(\frac{a}{c}\biggr) (2~-~ \cos\chi ) + \frac{1}{2^3}\biggl( \frac{a}{c}\biggr)^2( -1 ~+~ 4\cos\chi -2\cos^2\chi ) \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ + \frac{1}{2}\biggl( \frac{a}{c}\biggr) \biggl[ \ln\biggl( \frac{2^3c}{a}\biggr) - 2 \biggr] + \frac{1}{2^2}\biggl( \frac{a}{c}\biggr)^2 (2~-~ \cos\chi ) - \frac{1}{2^2}\biggl( \frac{a}{c}\biggr)^2 \biggl[ \ln\biggl( \frac{2^3c}{a}\biggr) - 2 \biggr] (2~-~ \cos\chi ) + \frac{3}{2^4}\biggl( \frac{a}{c}\biggr)^2 \biggl[ 3 \ln\biggl( \frac{2^3c}{a}\biggr) - 7 \biggr] \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ ~-~ \frac{1}{2^3} \biggl(\frac{a}{c}\biggr)\cos\chi ~+~ \frac{1}{2^6\cdot 3} \biggl(\frac{a}{c}\biggr)^2 ( 13- 24\cos\chi - 2\cos^2\chi ) ~-~ \frac{1}{2^4} \biggl(\frac{a}{c}\biggr)^2\cos\chi \biggl[ \ln\biggl( \frac{2^3c}{a}\biggr) - 1 \biggr] ~+~ \mathcal{O}\biggl(\frac{a^3}{c^3}\biggr) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \ln\biggl( \frac{2^3c}{a}\biggr) + \frac{1}{2}\biggl(\frac{a}{c}\biggr) (2~-~ \cos\chi ) ~-~ \frac{1}{2^3} \biggl(\frac{a}{c}\biggr)\cos\chi + \frac{1}{2}\biggl( \frac{a}{c}\biggr) \biggl[ \ln\biggl( \frac{2^3c}{a}\biggr) - 2 \biggr] + \frac{1}{2}\biggl(\frac{a}{c}\biggr)(\cos\chi - 1) \biggl[ \ln\biggl( \frac{2^3c}{a}\biggr) \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ ~+~ \frac{1}{2^3}\biggl( \frac{a}{c}\biggr)^2( -1 ~+~ 4\cos\chi -2\cos^2\chi ) + \frac{1}{2^2}\biggl( \frac{a}{c}\biggr)^2 (2~-~ \cos\chi ) - \frac{1}{2^2}\biggl( \frac{a}{c}\biggr)^2 \biggl[ \ln\biggl( \frac{2^3c}{a}\biggr) - 2 \biggr] (2~-~ \cos\chi ) + \frac{3}{2^4}\biggl( \frac{a}{c}\biggr)^2 \biggl[ 3 \ln\biggl( \frac{2^3c}{a}\biggr) - 7 \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ + \frac{1}{2^2}\biggl(\frac{a}{c}\biggr)^2(\cos\chi - 1) (2~-~ \cos\chi ) +\frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^2 (1-7\cos\chi + 6\cos^2\chi ) \ln\biggl( \frac{2^3c}{a}\biggr) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ ~+~ \frac{1}{2^6\cdot 3} \biggl(\frac{a}{c}\biggr)^2 ( 13- 24\cos\chi - 2\cos^2\chi ) ~-~ \frac{1}{2^4} \biggl(\frac{a}{c}\biggr)^2\cos\chi \biggl[ \ln\biggl( \frac{2^3c}{a}\biggr) - 1 \biggr] ~+~ \mathcal{O}\biggl(\frac{a^3}{c^3}\biggr) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \ln\biggl( \frac{2^3c}{a}\biggr) + \frac{1}{2^3}\biggl(\frac{a}{c}\biggr)\biggl\{ 4(2~-~ \cos\chi ) ~-~ \cos\chi + \biggl[ 4 \ln\biggl( \frac{2^3c}{a}\biggr) - 8 \biggr] + \biggl[ 4\ln\biggl( \frac{2^3c}{a}\biggr) \biggr] (\cos\chi - 1) \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~~+~ \frac{1}{2^6\cdot 3} \biggl( \frac{a}{c}\biggr)^2\biggl\{ 24( -1 ~+~ 4\cos\chi -2\cos^2\chi ) + 48 (2~-~ \cos\chi ) - 48\biggl[ \ln\biggl( \frac{2^3c}{a}\biggr) - 2 \biggr] (2~-~ \cos\chi ) + 36\biggl[ 3 \ln\biggl( \frac{2^3c}{a}\biggr) - 7 \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ + 48(\cos\chi - 1) (2~-~ \cos\chi ) ~+~12 (1-7\cos\chi + 6\cos^2\chi ) \ln\biggl( \frac{2^3c}{a}\biggr) ~+~ ( 13- 24\cos\chi - 2\cos^2\chi ) ~-~12\cos\chi \biggl[ \ln\biggl( \frac{2^3c}{a}\biggr) - 1 \biggr] \biggr\} ~+~ \mathcal{O}\biggl(\frac{a^3}{c^3}\biggr) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \ln\biggl( \frac{2^3c}{a}\biggr) + \frac{1}{2^3}\biggl(\frac{a}{c}\biggr)\biggl\{ 4(2~-~ \cos\chi ) ~-~ \cos\chi + \biggl[ 4 \ln\biggl( \frac{2^3c}{a}\biggr) - 8 \biggr] + \biggl[ 4\ln\biggl( \frac{2^3c}{a}\biggr) \biggr] (\cos\chi - 1) \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~~+~ \frac{1}{2^6\cdot 3} \biggl( \frac{a}{c}\biggr)^2\biggl\{ (-24 + 96\cos\chi-48\cos^2\chi) + (96~-~ 48\cos\chi ) ~+~ \biggl[2- \ln\biggl( \frac{2^3c}{a}\biggr) \biggr] (96~-~ 48 \cos\chi ) + \biggl[ 108 \ln\biggl( \frac{2^3c}{a}\biggr) - 252 \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ + 48(3\cos\chi - \cos^2\chi - 2 ) ~+~ (12 - 84\cos\chi + 72\cos^2\chi ) \ln\biggl( \frac{2^3c}{a}\biggr) ~+~ ( 13- 24\cos\chi - 2\cos^2\chi ) ~+~12\cos\chi ~-~12\cos\chi \biggl[ \ln\biggl( \frac{2^3c}{a}\biggr) - \biggr] \biggr\} ~+~ \mathcal{O}\biggl(\frac{a^3}{c^3}\biggr) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \ln\biggl( \frac{2^3c}{a}\biggr) + \frac{1}{2}\biggl(\frac{a}{c}\biggr)\biggl[ \ln\biggl( \frac{2^3c}{a}\biggr) ~-~ \frac{5}{4}\biggr]\cos\chi </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~~+~ \frac{1}{2^6\cdot 3} \biggl( \frac{a}{c}\biggr)^2\biggl\{ -24 ~+~ 96\cos\chi -48\cos^2\chi + 96~-~ 48\cos\chi + 192 - 252 -96\cos\chi ~+~ ( 13- 12\cos\chi - 2\cos^2\chi ) + (144\cos\chi - 48\cos^2\chi -96 ) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ +36 \biggl[ \ln\biggl( \frac{2^3c}{a}\biggr)\biggr]\cos\chi ~+~(12 - 84\cos\chi + 72\cos^2\chi -96 + 108) \biggl[ \ln\biggl( \frac{2^3c}{a}\biggr)\biggr] \biggr\} ~+~ \mathcal{O}\biggl(\frac{a^3}{c^3}\biggr) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \ln\biggl( \frac{2^3c}{a}\biggr) + \frac{1}{2}\biggl(\frac{a}{c}\biggr)\biggl[ \ln\biggl( \frac{2^3c}{a}\biggr) ~-~ \frac{5}{4}\biggr]\cos\chi </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~~+~ \frac{1}{2^6\cdot 3} \biggl( \frac{a}{c}\biggr)^2\biggl\{ -71 ~+~84\cos\chi -98\cos^2\chi ~+~(24 - 48\cos\chi + 72\cos^2\chi ) \biggl[ \ln\biggl( \frac{2^3c}{a}\biggr)\biggr] \biggr\} ~+~ \mathcal{O}\biggl(\frac{a^3}{c^3}\biggr) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \ln\biggl( \frac{2^3c}{a}\biggr) + \frac{1}{2}\biggl(\frac{a}{c}\biggr)\biggl[ \ln\biggl( \frac{2^3c}{a}\biggr) ~-~ \frac{5}{4}\biggr]\cos\chi ~+~ \frac{1}{2^6\cdot 3} \biggl( \frac{a}{c}\biggr)^2\biggl\{ -71 ~+~84\cos\chi -98\cos^2\chi ~+~24\ln\biggl( \frac{2^3c}{a}\biggr)(1 - 2\cos\chi + 3\cos^2\chi ) \biggr\} ~+~ \mathcal{O}\biggl(\frac{a^3}{c^3}\biggr) </math> </td> </tr> </table> In an effort to compare this expression with equation (6) from Dyson's (1893b) "Part II", we should make the substitutions, <div align="center"> <math>~\ln\biggl(\frac{2^3c}{a}\biggr) \rightarrow (\lambda +2)</math> and <math>~2\cos^2\chi \rightarrow 1 + \cos2\chi \, .</math> </div> This means, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\frac{\pi V_\mathrm{Dyson}}{GM/c}\biggr|_{\mathcal{O}(a^2/c^2)} </math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ ~+~ \frac{1}{2^6\cdot 3} \biggl( \frac{a}{c}\biggr)^2\biggl\{ -71 ~+~84\cos\chi -49(1+\cos2\chi ) ~+~24(\lambda + 2)(1 - 2\cos\chi ) ~+~36(\lambda+2)(1 + \cos2\chi ) \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ ~+~ \frac{1}{2^6\cdot 3} \biggl( \frac{a}{c}\biggr)^2\biggl\{ -71 ~+~84\cos\chi -49 -49 \cos2\chi ~+~24(\lambda + 2 -2\lambda \cos\chi - 4\cos\chi) ~+~36(\lambda+2 +\lambda\cos 2\chi + 2\cos 2\chi) \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ ~+~ \frac{1}{2^6\cdot 3} \biggl( \frac{a}{c}\biggr)^2\biggl\{ -71 ~+~84\cos\chi -49 -49 \cos2\chi ~+~24\lambda + 48 -48\lambda \cos\chi - 96\cos\chi ~+~36\lambda+72 +36\lambda\cos 2\chi + 72\cos 2\chi \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ ~+~ \frac{1}{2^6\cdot 3} \biggl( \frac{a}{c}\biggr)^2\biggl\{ 60\lambda -48\lambda \cos\chi - 12\cos\chi +36\lambda\cos 2\chi + 23\cos 2\chi \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ ~+~ \biggl( \frac{a}{c}\biggr)^2\biggl\{ \frac{5\lambda}{16} - \frac{(4\lambda + 1)}{16}~\cos\chi +\frac{3(\lambda+\tfrac{23}{36})}{16}\cos 2\chi \biggr\} \, . </math> </td> </tr> </table> This expression differs from the 2<sup>nd</sup>-order term in Dyson's equation (6) by the amount, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\Delta \biggr|_{\mathcal{O}(a^2/c^2)} </math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \biggl( \frac{a}{c}\biggr)^2\biggl\{ \frac{5\lambda}{16} - \frac{(4\lambda + 1)}{16}~\cos\chi +\frac{3(\lambda+\tfrac{23}{36})}{16}\cos 2\chi \biggr\} - \biggl(\frac{a}{c}\biggr)^2 \biggl\{ \frac{\lambda - \frac{1}{2}}{16} + \frac{3(\lambda + \frac{17}{36})}{16}\cos2\chi \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \frac{1}{16\cdot 12}\biggl( \frac{a}{c}\biggr)^2\biggl\{ 60\lambda - (48\lambda + 12)~\cos\chi +(36\lambda+23)\cos 2\chi \biggr\} - \frac{1}{16\cdot 12}\biggl(\frac{a}{c}\biggr)^2 \biggl\{ 12\lambda - 6 + (36\lambda + 17) \cos2\chi \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \frac{1}{16\cdot 12}\biggl( \frac{a}{c}\biggr)^2\biggl\{ 48\lambda + 6 - (48\lambda + 12)~\cos\chi +(6)\cos 2\chi \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \frac{1}{2^5}\biggl( \frac{a}{c}\biggr)^2\biggl\{ (\cos 2\chi -1)- (8\lambda + 2)~(1+\cos\chi) \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \frac{1}{2^4}\biggl( \frac{a}{c}\biggr)^2\biggl\{ (\cos\chi - 1)- (4\lambda + 1)~ \biggr\} (1+\cos\chi) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \frac{1}{2^4}\biggl( \frac{a}{c}\biggr)^2 (\cos\chi - 2- 4\lambda ) (1+\cos\chi) </math> </td> </tr> </table>
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