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__FORCETOC__ <!-- will force the creation of a Table of Contents --> <!-- __NOTOC__ will force TOC off --> =Dyson (1893)= {| class="HNM82" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black" |- ! style="height: 125px; width: 125px; background-color:#ffeeee;" |[[H_BookTiledMenu#Toroidal_.26_Toroidal-Like|<b>Dyson<br />(1893)</b>]] |} Our focus, here, is on the pioneering work of [http://adsabs.harvard.edu/abs/1893RSPTA.184...43D F. W. Dyson (1893a, Philosophical Transactions of the Royal Society of London. A., 184, 43 - 95)] and [http://adsabs.harvard.edu/abs/1893RSPTA.184.1041D (1893b, Philosophical Transactions of the Royal Society of London. A., 184, 1041 - 1106)]. He used analytic techniques to determine the approximate equilibrium structure of axisymmetric, uniformly rotating, incompressible tori. [http://adsabs.harvard.edu/abs/1974ApJ...190..675W C.-Y. Wong (1974, ApJ, 190, 675 - 694)] extended Dyson's work, using numerical techniques to obtain more accurate equilibrium structures for incompressible tori having solid body rotation. Since then, [http://adsabs.harvard.edu/abs/1981PThPh..65.1870E Y. Eriguchi & D. Sugimoto (1981, Progress of Theoretical Physics, 65, 1870 - 1875)] and [http://adsabs.harvard.edu/abs/1988ApJS...66..315H I. Hachisu, J. E. Tohline & Y. Eriguchi (1987, ApJ, 323, 592 - 613)] have mapped out the full sequence of Dyson-Wong tori, beginning from a bifurcation point on the Maclaurin spheroid sequence. The most challenging aspect of each of these studies has been the development of an analytic and/or computational technique that can be used to accurately determine the gravitational potential of toroidal-shaped configurations. With this in mind, it should be appreciated that, in a paper that preceded his 1974 work, [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W C.-Y. Wong (1973, Annals of Physics, 77, 279)] derived an analytic expression for the ''exact'' potential (inside as well as outside) of axisymmetric, uniform-density tori having an arbitrarily specified ratio of the major to minor (cross-sectional) radii, <math>~R/d</math>. This is an outstanding accomplishment that has received little attention in the astrophysics literature and, therefore, has heretofore been under-appreciated. In a [[Apps/Wong1973Potential#Wong.27s_.281973.29_Analytic_Potential|separate, accompanying discussion]], we detail how Wong accomplished this task. ==External Potential== ===His Derived Expression=== (See an accompanying [[Appendix/Ramblings/Dyson1893Part1|''Ramblings Chapter'']] for additional derivation details.) On p. 62, in §8 of [http://adsabs.harvard.edu/abs/1893RSPTA.184...43D Dyson (1893a)], we find the following approximate expression for the potential at point "P", anywhere exterior to an [http://www.mathematicsdictionary.com/english/vmd/full/t/torusanchorring.htm anchor ring]: <table border="1" cellpadding="5" align="center"> <tr><td align="center" colspan="1"> '''Equation & text copied without modification from <br />p. 62 of [http://adsabs.harvard.edu/abs/1893RSPTA.184...43D Dyson (1893a)]'''<br /> ''The Potential of an Anchor Ring''<br /> Phil. Trans. Royal Soc. London. A., Vol. 184 </td></tr> <tr> <td align="left"> <!-- [[File:DysonExternalPotentialEquation.png|550px|center|To be inserted: the Potential Exterior to an Anchor Ring]] --> Therefore, at any external point, <table border="0" align="center" width="100%" cellpadding="5"> <tr> <td align="right"><math>V</math></td> <td align="center"><math>=</math></td> <td align="left"> <math> \frac{4F}{R + R_1} \biggl\{1 - \frac{1}{8} \frac{a^2}{c^2} \cos^2 \frac{\psi}{2} - \frac{1}{768} \frac{a^4}{c^4} \biggl[ 5 + 8\cos\psi - \cos^2\psi - 4\cos^3\psi - \frac{4c^2}{RR_1} \cos 2\psi \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>+</math> &c.<math>\biggr\}</math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>+~\frac{E(R+R_1)}{RR_1} \biggl\{\frac{a^2}{8c^2} \cos\psi - \frac{1}{192} \frac{a^4}{c^4} \biggl[ 2\cos^2\psi - 4\cos\psi + \frac{2c^2}{RR_1} \cos 2\psi \biggr] + </math> &c.<math>\biggr\} \, ,</math> </td> </tr> </table> where <math>\psi</math> is the angle between <math>R</math> and <math>R_1</math>, and the modulus of the elliptic functions is <table border="0" align="center" width="100%" cellpadding="5"> <tr> <td align="center"><math>\frac{R_1 - R}{R_1 + R} \, .</math></td> </tr> </table> </td> </tr> <tr> <td align="center" colspan="1"> [[File:DysonTorusIllustration03.png|300px|center|Anchor Ring Schematic]]<br /> [http://www.mathematicsdictionary.com/english/vmd/full/t/torusanchorring.htm Anchor ring] schematic, adapted from figure near the top of §2 (on p. 47) of [http://adsabs.harvard.edu/abs/1893RSPTA.184...43D Dyson (1893a)]<br /> [ Here are some [[Apps/DysonWongTori|Related Schematics]] ] </td> </tr> </table> In Dyson's expression, the leading factor of <math>~F</math> is the [https://en.wikipedia.org/wiki/Elliptic_integral#Complete_elliptic_integral_of_the_first_kind complete elliptic integral of the first kind], namely, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~F = F(\mu)</math> </td> <td align="center"> <math>~\equiv</math> </td> <td align="left"> <math>~\int_0^{\pi/2} \frac{d\phi}{\sqrt{1 - \mu^2 \sin^2\phi}} \, ,</math> </td> </tr> </table> where, <math>~\mu \equiv (R_1 - R)/(R_1 + R)</math>. Similarly, <math>~E = E(\mu)</math> is the [https://en.wikipedia.org/wiki/Elliptic_integral#Complete_elliptic_integral_of_the_second_kind complete elliptic integral of the second kind]. ===Comparison With Thin Ring Approximation=== In the limit of <math>~a/c \rightarrow 0</math>, Dyson's expression gives, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~V_\mathrm{Dyson}</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~\frac{4K(\mu)}{R+R_1} \, ,</math> </td> </tr> </table> where we have acknowledged that, in the twenty-first century, the complete elliptic integral of the first kind is more customarily represented by the letter, <math>~K</math>. In a [[Apps/DysonWongTori#RingPotential|separate discussion]], we have shown that the gravitational potential of an infinitesimally thin ring is given precisely by the expression, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\biggl[ \frac{\pi}{GM}\biggr] \Phi_\mathrm{TR}</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~- \frac{2K(k)}{R_1} \, ,</math> </td> </tr> </table> where, <math>~k \equiv [1-(R/R_1)^2]^{1 / 2}</math>. Is Dyson's expression identical to this one when <math>~a/c = 0</math> ? ====Proof==== Taking a queue from our [[2DStructure/ToroidalGreenFunction#Basic_Elements_of_a_Toroidal_Coordinate_System|accompanying discussion of toroidal coordinates]], if we adopt the variable notation, <div align="center"> <math>~\eta \equiv \ln\biggl(\frac{R_1}{R}\biggr) \, ,</math> </div> then we can write, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\cosh\eta = \frac{1}{2}\biggl[e^\eta + e^{-\eta}\biggr]</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~\frac{R^2 + R_1^2}{2RR_1} \, ,</math> </td> </tr> </table> which implies that, <!-- <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\tanh^2\biggl(\frac{\eta}{2}\biggr) = \frac{\cosh\eta - 1}{\cosh\eta+1}</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~\biggl[\frac{R_1 - R}{R_1 + R}\biggr]^2 \, ,</math> </td> </tr> </table> and, --> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\biggl[ \frac{2}{\coth\eta +1} \biggr]^{1 / 2} = [1 - e^{-2\eta}]^{1 / 2}</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~\biggl[ 1 - \biggl(\frac{R}{R_1}\biggr)^2 \biggr]^{1 / 2} \, .</math> </td> </tr> </table> This is the definition of the parameter, <math>~k</math>, in the expression for <math>~\Phi_\mathrm{TR}</math>. Now, if we employ the [https://dlmf.nist.gov/19.8#ii ''Descending Landen Transformation'' for the complete elliptic integral of the first kind], we can make the substitution, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~K(k)</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ (1 + k_1)K(k_1) \, , </math> </td> <td align="center"> where, </td> <td align="right"> <math>~k_1</math> </td> <td align="center"> <math>~\equiv</math> </td> <td align="left"> <math>~ \frac{1-\sqrt{1-k^2}}{1+\sqrt{1-k^2}} \, . </math> </td> </tr> </table> But notice that, <math>~\sqrt{1-k^2} = e^{-\eta}</math>, in which case, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~k_1 </math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \frac{1-e^{-\eta}}{1+e^{-\eta}} </math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \frac{1-R/R_1}{1+R/R_1} </math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \frac{R_1-R}{R_1+R} \, , </math> </td> </tr> </table> which is the definition of the parameter, <math>~\mu</math>, in the expression for <math>~V_\mathrm{Dyson}</math>. Hence, we can write, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\biggl[ \frac{\pi}{GM}\biggr] \Phi_\mathrm{TR}</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~- \frac{2}{R_1} \biggl[(1+k_1)K(k_1) \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~- \frac{2K(\mu)}{R_1} \biggl[1+\frac{R_1-R}{R_1+R} \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~- \frac{4K(\mu)}{R_1+R} \, .</math> </td> </tr> </table> Aside from the adopted sign convention, this is indeed precisely the expression given by <math>~V_\mathrm{Dyson}</math> when <math>~a/c = 0</math> . ===Evaluation=== ====Dyson's Figures==== In his effort to illustrate the behavior of equipotential contours in the space exterior to various ''anchor rings'', Dyson evaluated his expression for the potential up through <math>~\mathcal{O}(\tfrac{a^2}{c^2})</math>; that is, he evaluated the function, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~V_2 \equiv V_\mathrm{Dyson}\biggr|_{\mathcal{O}(a^2/c^2)}</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)\biggr] + \frac{(R + R_1)E(\mu)}{RR_1}\biggl[\frac{1}{8}\biggl(\frac{a^2}{c^2}\biggr) \cos\psi \biggr] \, . </math> </td> </tr> </table> Figures 1, 2, 3, & 6 from [http://adsabs.harvard.edu/abs/1893RSPTA.184...43D Dyson (1893a)] — replicated immediately below — show his resulting set of contours for six cases: Tori (anchor rings) having aspect ratios of <math>~a/c = 0, 1/5, 2/5, 1</math>. Click on an image to view the contour plot at higher resolution. In what follows we present results from our own evaluation of this "V<sub>2</sub>" function for the single case of an anchor ring having <math>~a/c = 2/5</math>. <table border="0" cellpadding="8" align="center"><tr><td align="center"> <table border="1" cellpadding="5" align="center"> <tr><td align="center" colspan="2" bgcolor="lightgreen"> '''Figures 1, 2, 3, & 6 extracted without modification from pp. 63-66 of <br /> {{ Dyson1893figure }} <!-- [http://adsabs.harvard.edu/abs/1893RSPTA.184...43D F. W. Dyson (1893)]'''<br /> ''The Potential of an Anchor Ring''<br /> Phil. Trans. Royal Soc. London. A., Vol. 184 --> <br />[https://doi.org/10.1098/rsta.1893.0002 https://doi.org/10.1098/rsta.1893.0002]<br /> [[File:PermissionsRectYellow.png|75px|link=Appendix/Permissions#Dyson1893]] </td></tr> <tr> <td> [[File:RoverDinfty.png|300px|center|The Potential Exterior to an Anchor Ring; R/d = infinity]] </td> <td> [[File:RoverD5over1.png|300px|center|The Potential Exterior to an Anchor Ring; R/d = 5]] </td> </tr> <tr> <td> [[File:RoverD5over2.png|300px|center|The Potential Exterior to an Anchor Ring; R/d = 1.667]] </td> <td> [[File:RoverDunity.png|300px|center|The Potential Exterior to an Anchor Ring; R/d = 1]] </td> </tr> </table> </td></tr></table> ====Our Attempt to Replicate==== First, let's test the accuracy of [http://adsabs.harvard.edu/abs/1893RSPTA.184...43D Dyson's (1893a)] "series expansion" expression for the elliptic integrals, <math>~K(\mu)</math> and <math>~E(\mu)</math>; in the following table, the high-precision evaluations labeled "''Numerical Recipes''" have been drawn from the tabulated data that is provided in our [[2DStructure/ToroidalCoordinateIntegrationLimits#Evaluation_of_Elliptic_Integrals|accompanying discussion]] of incomplete elliptic integrals. Drawing from our [[Appendix/SpecialFunctions#Complete_Elliptic_Integrals|accompanying set of Key mathematical relations]] — in which <math>~k</math>, rather than <math>~\mu</math>, represents the function modulus — the relevant series-expansion expressions are: <div align="center"> {{ Math/EQ_EllipticIntegral01 }}<br /> {{ Math/EQ_EllipticIntegral02 }} </div> These expressions — up through <math>~\mathcal{O}(\mu^4)</math> — can be found in the middle of p. 58 of [http://adsabs.harvard.edu/abs/1893RSPTA.184...43D Dyson (1893a)]. We strongly suspect that, in constructing the equipotential contours shown in his figures 1-6, Dyson used expressions for <math>~K(\mu)</math> and <math>~E(\mu)</math> that were more accurate than this. For example, we found it necessary to include terms up through <math>~\mathcal{O}(\mu^{10})</math> in order to match to three digits accuracy the potential contour values and coordinate locations reported by Dyson. <table border="1" cellpadding="8" align="center"> <tr> <td align="center" rowspan="2"><math>~\mu</math></td> <td align="center" colspan="2">Numerical Recipes</td> <td align="center" colspan="2">Series expansion up through <math>~\mathcal{O}(\mu^4)</math></td> <td align="center" colspan="2">Series expansion up through <math>~\mathcal{O}(\mu^{10})</math></td> </tr> <tr> <td align="center"><math>~K(\mu)</math></td> <td align="center"><math>~E(\mu)</math></td> <td align="center"><math>~K(\mu)</math></td> <td align="center"><math>~E(\mu)</math></td> <td align="center"><sup>†</sup><math>K(\mu)~</math></td> <td align="center"><math>~E(\mu)</math></td> </tr> <tr> <td align="right">0.34202014</td> <td align="right">1.62002589</td> <td align="right">1.52379921</td> <td align="right">1.6198</td> <td align="right">1.5239</td> <td align="right">1.6200263</td> <td align="right">1.5237989</td> </tr> <tr> <td align="right">0.57357644</td> <td align="right">1.73124518</td> <td align="right">1.43229097</td> <td align="right">1.7239</td> <td align="right">1.4336</td> <td align="right">1.73124518</td> <td align="right">1.43230</td> </tr> <tr> <td align="right">0.76604444</td> <td align="right">1.93558110</td> <td align="right">1.30553909</td> <td align="right">1.8773</td> <td align="right">1.3150</td> <td align="right">1.93558109</td> <td align="right">1.3061</td> </tr> <tr> <td align="right">0.90630779</td> <td align="right">2.30878680</td> <td align="right">1.16382796</td> <td align="right">2.042</td> <td align="right">1.199</td> <td align="right">2.308784</td> <td align="right">1.1700</td> </tr> <tr> <td align="right">0.98480775</td> <td align="right">3.15338525</td> <td align="right">1.04011440</td> <td align="right">2.16</td> <td align="right">1.12</td> <td align="right">3.150</td> <td align="right">1.069</td> </tr> <tr> <td align="left" colspan="7"> <sup>†</sup>We actually used the "descending Landen transformation" to evaluate <math>~K(\mu)</math> through <math>~\mathcal{O}(\mu^{10})</math>. </td> </tr> </table> For <math>~c=1</math> and a specification of the ratio, <math>~a/c</math>, take the following steps to map out an equipotential curve that has <math>~V_2 = V_0</math>: * Choose a value of <math>~R \ge a</math> ** ''Guess'' a value of <math>~(c-R) \le R_1 \le (c+R) ~~~\Rightarrow ~~~ \varpi = (R_1^2 - R^2)/(4c)</math> and, <math>~z = \pm \sqrt{ R_1^2 - (c+\varpi)^2}</math> ** Set <math>~ \cos\psi = (R_1^2 + R^2 - 4c^2)/(2RR_1)</math> ** Evaluate the function, <math>~V_2</math> ** If <math>~V_2 \ne V_0</math> to the desired accuracy, loop back up and guess another value of <math>~R_1</math> * If <math>~V_2 = V_0</math> to the desired accuracy, save the coordinate location, <math>~(\varpi,z)</math>, and loop back up to pick another value of <math>~R</math> <table border="1" align="center" cellpadding="8"> <tr><td align="center" colspan="1" bgcolor="lightgreen"> ''Top Panel''<br />(as above) '''Figure 3 extracted without modification from p. 65 of <br />[http://adsabs.harvard.edu/abs/1893RSPTA.184...43D F. W. Dyson (1893)]'''<br /> ''II. The Potential of an Anchor Ring''<br /> Phil. Trans. Royal Soc. London. A., Vol. 184<br />[https://doi.org/10.1098/rsta.1893.0002 https://doi.org/10.1098/rsta.1893.0002]<br /> [[File:PermissionsRectYellow.png|75px|link=Appendix/Permissions#Dyson1893]] </td></tr> <tr> <td rowspan="1" align="left" valign="bottom"> [[File:RoverD5over2.png|800px|center|The Potential Exterior to an Anchor Ring; R/d = 2.5]] </td> </tr> <tr> <td align="left" valign="bottom"> [[File:DysonCompare01.png|625px|Compare with Dyson]]</td> </tr> </table> ====Tabulated Data==== As the data in the following table documents, we have been able to construct equipotential contours that agree with Dyson, not only qualitatively, but quantitatively. For example: * The dark green contour has been designed to touch the surface of the torus precisely where its outermost edge cuts through the equatorial plane <math>~(\varpi,z) = (1.4,0)</math>. This means that <math>~R = 0.4</math> and <math>~R_1 = 2.4</math>. (These four coordinate values are highlighted in pink in the second major column of the table.) When we plugged these values of <math>~R</math> and <math>~R_1</math> into Dyson's expression for <math>~V_2</math>, we determined that the value of the potential at this point on the torus surface is 0.8551 — see the yellow-highlighted heading of the second major table column. Compare this to the value of 0.855 that Dyson has printed just below the Figure 3 x-axis where a fiducial identifies the coordinate, <math>~\varpi = 1.4</math>. As has been catalogued at the bottom of table column #2, we have found that this dark-green contour touches the vertical axis at the coordinate location, <math>~(\varpi,z) = (0,0.572)</math>, for which, <math>~R_1 = R = 1.1518</math>. * By design — see the coordinate values highlighted in pink in table column #1 — our outermost (pink) contour touches the equatorial plane at <math>~(\varpi,z) = (1.5,0) ~\Rightarrow ~ (R,R_1) = (0.5,2.5)</math>. When we plugged these values of <math>~R</math> and <math>~R_1</math> into Dyson's expression for <math>~V_2</math>, we determined that the value of the potential at this point outside the torus is 0.7737 — see the yellow-highlighted heading of table column #1. Compare this to the value of 0.777 that Dyson has printed just below the Figure 3 x-axis where a fiducial identifies the coordinate, <math>~\varpi = 1.5</math>. As has been catalogued at the bottom of table column #1, we have found that this pink contour touches the vertical axis at the coordinate location, <math>~(\varpi,z) = (0,0.794)</math>, for which, <math>~R_1 = R = 1.2766</math>. * Similarly, we have constructed contours that intersect the equatorial plane at the fiducials marking <math>~\varpi = 0.0</math> (red curve & table column #5), <math>~\varpi = 0.2</math> (light-green curve & table column #6), and <math>~\varpi = 0.4</math> (light-blue curve & table column #7). According to our calculations, they correspond, respectively, to values of the potential, <math>~V_2 = 0.9800</math> (Dyson's corresponding fiducial label is 0.980), <math>~V_2 = 0.9896</math> (Dyson's corresponding fiducial label is 0.990), and <math>~V_2 = 1.0212</math> (Dyson's corresponding fiducial label is 1.021). * Finally, we constructed two contours (blue and orange) by initially specifying the value of the potential, rather than specifying the coordinate values <math>~(R,R_1)</math>. We used the values of the potential that Dyson associated with the fiducials along the vertical axis at <math>~(\varpi,z) = (0.0,0.4)</math> and at <math>~(\varpi,z) = (0.0,0.2)</math>: Respectively, <math>~V_2 = 0.912</math> — blue contour detailed in our table column #3 — and <math>~V_2 = 0.961</math>— orange contour detailed in our table column #4. We determined that these two contour curves intersected the vertical axis at, respectively, <math>~(\varpi,z) = (0.0, 0.402)</math> and <math>~(\varpi,z) = (0.0, 0.204)</math>, that is, at coordinate locations that were nearly identical to the locations labeled by Dyson. <div id="TabulatedData" style="width: 100%; height: 30em; overflow: auto;"> <table border="1" cellpadding="5" cellspacing="5" align="center"> <tr> <th align="center" colspan="7"> Coordinates of Points that Trace Seven Different Equipotential Contours External to the Anchor Ring With <math>~c/a = 5/2</math> </th> </tr> <tr> <th align="center">Column #1</th> <th align="center">Column #2</th> <th align="center">Column #3</th> <th align="center">Column #4</th> <th align="center">Column #5</th> <th align="center">Column #6</th> <th align="center">Column #7</th> </tr> <tr> <td align="center" valign="top"> <!-- <p class="p1">V_2 = 0.7737</p> --> <table cellspacing="1" cellpadding="5" class="t1"> <tr> <td align="center" colspan="4" bgcolor="yellow">V<sub>2</sub> = 0.7737</td> </tr> <tr> <td valign="middle" align="center"> <p class="p3">R</p> </td> <td valign="middle" align="center"> <p class="p3">R1</p> </td> <td valign="middle" align="center"> <p class="p3"><math>~\varpi</math></p> </td> <td valign="middle" align="center"> <p class="p3">z</p> </td> </tr> <tr> <td valign="middle" bgcolor="pink"> <p class="p4">0.5000</p> </td> <td valign="middle" bgcolor="pink"> <p class="p4">2.5000</p> </td> <td valign="middle" bgcolor="pink"> <p class="p4">1.500</p> </td> <td valign="middle" bgcolor="pink"> <p class="p4">0.000</p> </td> </tr> <tr> <td valign="middle" > <p class="p4">0.5005</p> </td> <td valign="middle" > <p class="p4">2.4990</p> </td> <td valign="middle" > <p class="p4">1.499</p> </td> <td valign="middle" > <p class="p4">0.043</p> </td> </tr> <tr> <td valign="middle" > <p class="p4">0.504</p> </td> <td valign="middle" > <p class="p4">2.4889</p> </td> <td valign="middle" > <p class="p4">1.485</p> </td> <td valign="middle" > <p class="p4">0.137</p> </td> </tr> <tr> <td valign="middle" > <p class="p4">0.510</p> </td> <td valign="middle" > <p class="p4">2.4720</p> </td> <td valign="middle" > <p class="p4">1.463</p> </td> <td valign="middle" > <p class="p4">0.215</p> </td> </tr> <tr> <td valign="middle" > <p class="p4">0.520</p> </td> <td valign="middle" > <p class="p4">2.4445</p> </td> <td valign="middle" > <p class="p4">1.426</p> </td> <td valign="middle" > <p class="p4">0.298</p> </td> </tr> <tr> <td valign="middle" > <p class="p4">0.530</p> </td> <td valign="middle" > <p class="p4">2.4177</p> </td> <td valign="middle" > <p class="p4">1.391</p> </td> <td valign="middle" > <p class="p4">0.358</p> </td> </tr> <tr> <td valign="middle" > <p class="p4">0.550</p> </td> <td valign="middle" > <p class="p4">2.3665</p> </td> <td valign="middle" > <p class="p4">1.324</p> </td> <td valign="middle" > <p class="p4">0.444</p> </td> </tr> <tr> <td valign="middle" > <p class="p4">0.580</p> </td> <td valign="middle" > <p class="p4">2.2940</p> </td> <td valign="middle" > <p class="p4">1.232</p> </td> <td valign="middle" > <p class="p4">0.532</p> </td> </tr> <tr> <td valign="middle" > <p class="p4">0.610</p> </td> <td valign="middle" > <p class="p4">2.2265</p> </td> <td valign="middle" > <p class="p4">1.146</p> </td> <td valign="middle" > <p class="p4">0.592</p> </td> </tr> <tr> <td valign="middle" > <p class="p4">0.640</p> </td> <td valign="middle" > <p class="p4">2.1632</p> </td> <td valign="middle" > <p class="p4">1.067</p> </td> <td valign="middle" > <p class="p4">0.636</p> </td> </tr> <tr> <td valign="middle" > <p class="p4">0.700</p> </td> <td valign="middle" > <p class="p4">2.0465</p> </td> <td valign="middle" > <p class="p4">0.925</p> </td> <td valign="middle" > <p class="p4">0.696</p> </td> </tr> <tr> <td valign="middle" > <p class="p4">0.800</p> </td> <td valign="middle" > <p class="p4">1.8745</p> </td> <td valign="middle" > <p class="p4">0.718</p> </td> <td valign="middle" > <p class="p4">0.749</p> </td> </tr> <tr> <td valign="middle" > <p class="p4">0.9000</p> </td> <td valign="middle" > <p class="p4">1.7240</p> </td> <td valign="middle" > <p class="p4">0.541</p> </td> <td valign="middle" > <p class="p4">0.774</p> </td> </tr> <tr> <td valign="middle" > <p class="p4">1.000</p> </td> <td valign="middle" > <p class="p4">1.5890</p> </td> <td valign="middle" > <p class="p4">0.381</p> </td> <td valign="middle" > <p class="p4">0.786</p> </td> </tr> <tr> <td valign="middle" > <p class="p4">1.100</p> </td> <td valign="middle" > <p class="p4">1.4670</p> </td> <td valign="middle" > <p class="p4">0.236</p> </td> <td valign="middle" > <p class="p4">0.791</p> </td> </tr> <tr> <td valign="middle" > <p class="p4">1.2000</p> </td> <td valign="middle" > <p class="p4">1.3558</p> </td> <td valign="middle" > <p class="p4">0.100</p> </td> <td valign="middle" > <p class="p4">0.793</p> </td> </tr> <tr> <td valign="middle" > <p class="p4">1.277</p> </td> <td valign="middle" > <p class="p4">1.2766</p> </td> <td valign="middle" > <p class="p4">0.000</p> </td> <td valign="middle" > <p class="p4">0.794</p> </td> </tr> </table> <p class="p2"><br></p> <p class="p2"><br></p> </td> <td align="center" valign="top"> <!-- <p class="p1">V_2 = 0.8551</p> --> <table cellspacing="1" cellpadding="5" class="t1"> <tr> <td align="center" colspan="4" bgcolor="yellow">V<sub>2</sub> = 0.8551</td> </tr> <tr> <td valign="middle" align="center"> <p class="p3">R</p> </td> <td valign="middle" align="center"> <p class="p3">R1</p> </td> <td valign="middle" align="center"> <p class="p3"><math>~\varpi</math></p> </td> <td valign="middle" align="center"> <p class="p3">z</p> </td> </tr> <tr> <td valign="middle" bgcolor="pink" > <p class="p4">0.400</p> </td> <td valign="middle" bgcolor="pink" > <p class="p4">2.4000</p> </td> <td valign="middle" bgcolor="pink" > <p class="p4">1.400</p> </td> <td valign="middle" bgcolor="pink" > <p class="p4">0.000</p> </td> </tr> <tr> <td valign="middle" > <p class="p4">0.405</p> </td> <td valign="middle" > <p class="p4">2.3830</p> </td> <td valign="middle" > <p class="p4">1.379</p> </td> <td valign="middle" > <p class="p4">0.144</p> </td> </tr> <tr> <td valign="middle" > <p class="p4">0.410</p> </td> <td valign="middle" > <p class="p4">2.3668</p> </td> <td valign="middle" > <p class="p4">1.358</p> </td> <td valign="middle" > <p class="p4">0.199</p> </td> </tr> <tr> <td valign="middle" > <p class="p4">0.425</p> </td> <td valign="middle" > <p class="p4">2.3190</p> </td> <td valign="middle" > <p class="p4">1.299</p> </td> <td valign="middle" > <p class="p4">0.302</p> </td> </tr> <tr> <td valign="middle" > <p class="p4">0.450</p> </td> <td valign="middle" > <p class="p4">2.2458</p> </td> <td valign="middle" > <p class="p4">1.210</p> </td> <td valign="middle" > <p class="p4">0.398</p> </td> </tr> <tr> <td valign="middle" > <p class="p4">0.480</p> </td> <td valign="middle" > <p class="p4">2.1655</p> </td> <td valign="middle" > <p class="p4">1.115</p> </td> <td valign="middle" > <p class="p4">0.466</p> </td> </tr> <tr> <td valign="middle" > <p class="p4">0.520</p> </td> <td valign="middle" > <p class="p4">2.0690</p> </td> <td valign="middle" > <p class="p4">1.003</p> </td> <td valign="middle" > <p class="p4">0.520</p> </td> </tr> <tr> <td valign="middle" > <p class="p4">0.570</p> </td> <td valign="middle" > <p class="p4">1.9610</p> </td> <td valign="middle" > <p class="p4">0.880</p> </td> <td valign="middle" > <p class="p4">0.557</p> </td> </tr> <tr> <td valign="middle" > <p class="p4">0.620</p> </td> <td valign="middle" > <p class="p4">1.8635</p> </td> <td valign="middle" > <p class="p4">0.772</p> </td> <td valign="middle" > <p class="p4">0.577</p> </td> </tr> <tr> <td valign="middle" > <p class="p4">0.700</p> </td> <td valign="middle" > <p class="p4">1.7240</p> </td> <td valign="middle" > <p class="p4">0.621</p> </td> <td valign="middle" > <p class="p4">0.588</p> </td> </tr> <tr> <td valign="middle" > <p class="p4">0.800</p> </td> <td valign="middle" > <p class="p4">1.5712</p> </td> <td valign="middle" > <p class="p4">0.457</p> </td> <td valign="middle" > <p class="p4">0.588</p> </td> </tr> <tr> <td valign="middle" > <p class="p4">0.900</p> </td> <td valign="middle" > <p class="p4">1.4360</p> </td> <td valign="middle" > <p class="p4">0.313</p> </td> <td valign="middle" > <p class="p4">0.581</p> </td> </tr> <tr> <td valign="middle" > <p class="p4">1.000</p> </td> <td valign="middle" > <p class="p4">1.3147</p> </td> <td valign="middle" > <p class="p4">0.182</p> </td> <td valign="middle" > <p class="p4">0.575</p> </td> </tr> <tr> <td valign="middle" > <p class="p4">1.100</p> </td> <td valign="middle" > <p class="p4">1.2050</p> </td> <td valign="middle" > <p class="p4">0.061</p> </td> <td valign="middle" > <p class="p4">0.572</p> </td> </tr> <tr> <td valign="middle" > <p class="p4">1.1518</p> </td> <td valign="middle" > <p class="p4">1.1518</p> </td> <td valign="middle" > <p class="p4">0.000</p> </td> <td valign="middle" > <p class="p4">0.572</p> </td> </tr> </table> <p class="p2"><br></p> </td> <td align="center" valign="top"> <!-- <p class="p1">V_2 = 0.9120</p> --> <table cellspacing="1" cellpadding="5" class="t1"> <tr> <td align="center" colspan="4" bgcolor="pink">V<sub>2</sub> = 0.9120</td> </tr> <tr> <td valign="middle" align="center"> <p class="p3">R</p> </td> <td valign="middle" align="center"> <p class="p3">R1</p> </td> <td valign="middle" align="center"> <p class="p3"><math>~\varpi</math></p> </td> <td valign="middle" align="center"> <p class="p3">z</p> </td> </tr> <tr> <td valign="middle" > <p class="p4">1.0776</p> </td> <td valign="middle" > <p class="p4">1.0776</p> </td> <td valign="middle" bgcolor="pink" > <p class="p4">0.000</p> </td> <td valign="middle" bgcolor="yellow" > <p class="p4">0.402</p> </td> </tr> <tr> <td valign="middle" > <p class="p4">1.000</p> </td> <td valign="middle" > <p class="p4">1.1582</p> </td> <td valign="middle" > <p class="p4">0.085</p> </td> <td valign="middle" > <p class="p4">0.404</p> </td> </tr> <tr> <td valign="middle" > <p class="p4">0.950</p> </td> <td valign="middle" > <p class="p4">1.2135</p> </td> <td valign="middle" > <p class="p4">0.143</p> </td> <td valign="middle" > <p class="p4">0.409</p> </td> </tr> <tr> <td valign="middle" > <p class="p4">0.900</p> </td> <td valign="middle" > <p class="p4">1.2715</p> </td> <td valign="middle" > <p class="p4">0.202</p> </td> <td valign="middle" > <p class="p4">0.416</p> </td> </tr> <tr> <td valign="middle" > <p class="p4">0.800</p> </td> <td valign="middle" > <p class="p4">1.3979</p> </td> <td valign="middle" > <p class="p4">0.328</p> </td> <td valign="middle" > <p class="p4">0.435</p> </td> </tr> <tr> <td valign="middle" > <p class="p4">0.700</p> </td> <td valign="middle" > <p class="p4">1.5401</p> </td> <td valign="middle" > <p class="p4">0.470</p> </td> <td valign="middle" > <p class="p4">0.458</p> </td> </tr> <tr> <td valign="middle" > <p class="p4">0.600</p> </td> <td valign="middle" > <p class="p4">1.7040</p> </td> <td valign="middle" > <p class="p4">0.636</p> </td> <td valign="middle" > <p class="p4">0.477</p> </td> </tr> <tr> <td valign="middle" > <p class="p4">0.550</p> </td> <td valign="middle" > <p class="p4">1.7970</p> </td> <td valign="middle" > <p class="p4">0.732</p> </td> <td valign="middle" > <p class="p4">0.480</p> </td> </tr> <tr> <td valign="middle" > <p class="p4">0.500</p> </td> <td valign="middle" > <p class="p4">1.8998</p> </td> <td valign="middle" > <p class="p4">0.840</p> </td> <td valign="middle" > <p class="p4">0.474</p> </td> </tr> <tr> <td valign="middle" > <p class="p4">0.475</p> </td> <td valign="middle" > <p class="p4">1.9560</p> </td> <td valign="middle" > <p class="p4">0.900</p> </td> <td valign="middle" > <p class="p4">0.464</p> </td> </tr> <tr> <td valign="middle" > <p class="p4">0.440</p> </td> <td valign="middle" > <p class="p4">2.0410</p> </td> <td valign="middle" > <p class="p4">0.993</p> </td> <td valign="middle" > <p class="p4">0.440</p> </td> </tr> <tr> <td valign="middle" > <p class="p4">0.400</p> </td> <td valign="middle" > <p class="p4">2.1510</p> </td> <td valign="middle" > <p class="p4">1.117</p> </td> <td valign="middle" > <p class="p4">0.383</p> </td> </tr> </table> <p class="p2"><br></p> </td> <td align="center" valign="top"> <!-- <p class="p1">V_2 = 0.9610</p> --> <table cellspacing="1" cellpadding="5" class="t1"> <tr> <td align="center" colspan="4" bgcolor="pink">V<sub>2</sub> = 0.9610</td> </tr> <tr> <td valign="middle" align="center"> <p class="p3">R</p> </td> <td valign="middle" align="center"> <p class="p3">R1</p> </td> <td valign="middle" align="center"> <p class="p3"><math>~\varpi</math></p> </td> <td valign="middle" align="center"> <p class="p3">z</p> </td> </tr> <tr> <td valign="middle" > <p class="p4">1.0206</p> </td> <td valign="middle" > <p class="p4">1.0206</p> </td> <td valign="middle" bgcolor="pink" > <p class="p4">0.000</p> </td> <td valign="middle" bgcolor="yellow" > <p class="p4">0.204</p> </td> </tr> <tr> <td valign="middle" > <p class="p4">0.9500</p> </td> <td valign="middle" > <p class="p4">1.0937</p> </td> <td valign="middle" > <p class="p4">0.073</p> </td> <td valign="middle" > <p class="p4">0.210</p> </td> </tr> <tr> <td valign="middle" > <p class="p4">0.900</p> </td> <td valign="middle" > <p class="p4">1.1488</p> </td> <td valign="middle" > <p class="p4">0.127</p> </td> <td valign="middle" > <p class="p4">0.221</p> </td> </tr> <tr> <td valign="middle" > <p class="p4">0.800</p> </td> <td valign="middle" > <p class="p4">1.2685</p> </td> <td valign="middle" > <p class="p4">0.242</p> </td> <td valign="middle" > <p class="p4">0.257</p> </td> </tr> <tr> <td valign="middle" > <p class="p4">0.700</p> </td> <td valign="middle" > <p class="p4">1.4030</p> </td> <td valign="middle" > <p class="p4">0.370</p> </td> <td valign="middle" > <p class="p4">0.304</p> </td> </tr> <tr> <td valign="middle" > <p class="p4">0.600</p> </td> <td valign="middle" > <p class="p4">1.5572</p> </td> <td valign="middle" > <p class="p4">0.516</p> </td> <td valign="middle" > <p class="p4">0.355</p> </td> </tr> <tr> <td valign="middle" > <p class="p4">0.550</p> </td> <td valign="middle" > <p class="p4">1.6440</p> </td> <td valign="middle" > <p class="p4">0.600</p> </td> <td valign="middle" > <p class="p4">0.378</p> </td> </tr> <tr> <td valign="middle" > <p class="p4">0.500</p> </td> <td valign="middle" > <p class="p4">1.7395</p> </td> <td valign="middle" > <p class="p4">0.694</p> </td> <td valign="middle" > <p class="p4">0.395</p> </td> </tr> <tr> <td valign="middle" > <p class="p4">0.450</p> </td> <td valign="middle" > <p class="p4">1.8462</p> </td> <td valign="middle" > <p class="p4">0.801</p> </td> <td valign="middle" > <p class="p4">0.404</p> </td> </tr> <tr> <td valign="middle" > <p class="p4">0.410</p> </td> <td valign="middle" > <p class="p4">1.9690</p> </td> <td valign="middle" > <p class="p4">0.929</p> </td> <td valign="middle" > <p class="p4">0.394</p> </td> </tr> </table> <p class="p2"><br></p> <p class="p2"><br></p> </td> <td align="center" valign="top"> <!-- <p class="p1">V_2 = 0.9800</p> --> <table cellspacing="1" cellpadding="5" class="t1"> <tr> <td align="center" colspan="4" bgcolor="yellow">V<sub>2</sub> = 0.9800</td> </tr> <tr> <td valign="middle" align="center"> <p class="p3">R</p> </td> <td valign="middle" align="center"> <p class="p3">R1</p> </td> <td valign="middle" align="center"> <p class="p3"><math>~\varpi</math></p> </td> <td valign="middle" align="center"> <p class="p3">z</p> </td> </tr> <tr> <td valign="middle" bgcolor="pink" > <p class="p4">1.0000</p> </td> <td valign="middle" bgcolor="pink" > <p class="p4">1.0000</p> </td> <td valign="middle" bgcolor="pink" > <p class="p4">0.000</p> </td> <td valign="middle" bgcolor="pink" > <p class="p4">0.000</p> </td> </tr> <tr> <td valign="middle" > <p class="p4">0.900</p> </td> <td valign="middle" > <p class="p4">1.1053</p> </td> <td valign="middle" > <p class="p4">0.103</p> </td> <td valign="middle" > <p class="p4">0.072</p> </td> </tr> <tr> <td valign="middle" > <p class="p4">0.800</p> </td> <td valign="middle" > <p class="p4">1.2225</p> </td> <td valign="middle" > <p class="p4">0.214</p> </td> <td valign="middle" > <p class="p4">0.147</p> </td> </tr> <tr> <td valign="middle" > <p class="p4">0.700</p> </td> <td valign="middle" > <p class="p4">1.3543</p> </td> <td valign="middle" > <p class="p4">0.336</p> </td> <td valign="middle" > <p class="p4">0.222</p> </td> </tr> <tr> <td valign="middle" > <p class="p4">0.600</p> </td> <td valign="middle" > <p class="p4">1.5050</p> </td> <td valign="middle" > <p class="p4">0.476</p> </td> <td valign="middle" > <p class="p4">0.293</p> </td> </tr> <tr> <td valign="middle" > <p class="p4">0.550</p> </td> <td valign="middle" > <p class="p4">1.5897</p> </td> <td valign="middle" > <p class="p4">0.556</p> </td> <td valign="middle" > <p class="p4">0.325</p> </td> </tr> <tr> <td valign="middle" > <p class="p4">0.500</p> </td> <td valign="middle" > <p class="p4">1.6827</p> </td> <td valign="middle" > <p class="p4">0.645</p> </td> <td valign="middle" > <p class="p4">0.352</p> </td> </tr> <tr> <td valign="middle" > <p class="p4">0.450</p> </td> <td valign="middle" > <p class="p4">1.7865</p> </td> <td valign="middle" > <p class="p4">0.747</p> </td> <td valign="middle" > <p class="p4">0.372</p> </td> </tr> <tr> <td valign="middle" > <p class="p4">0.400</p> </td> <td valign="middle" > <p class="p4">1.9050</p> </td> <td valign="middle" > <p class="p4">0.867</p> </td> <td valign="middle" > <p class="p4">0.377</p> </td> </tr> </table> <p class="p2"><br></p> </td> <td align="center" valign="top"> <!-- <p class="p1">V_2 = 0.9896</p> --> <table cellspacing="1" cellpadding="5" class="t1"> <tr> <td align="center" colspan="4" bgcolor="yellow">V<sub>2</sub> = 0.9896</td> </tr> <tr> <td valign="middle" align="center"> <p class="p3">R</p> </td> <td valign="middle" align="center"> <p class="p3">R1</p> </td> <td valign="middle" align="center"> <p class="p3"><math>~\varpi</math></p> </td> <td valign="middle" align="center"> <p class="p3">z</p> </td> </tr> <tr> <td valign="middle" bgcolor="pink" > <p class="p4">0.8000</p> </td> <td valign="middle" bgcolor="pink" > <p class="p4">1.2000</p> </td> <td valign="middle" bgcolor="pink" > <p class="p4">0.200</p> </td> <td valign="middle" bgcolor="pink" > <p class="p4">0.000</p> </td> </tr> <tr> <td valign="middle" > <p class="p4">0.7950</p> </td> <td valign="middle" > <p class="p4">1.2062</p> </td> <td valign="middle" > <p class="p4">0.206</p> </td> <td valign="middle" > <p class="p4">0.034</p> </td> </tr> <tr> <td valign="middle" > <p class="p4">0.780</p> </td> <td valign="middle" > <p class="p4">1.2248</p> </td> <td valign="middle" > <p class="p4">0.223</p> </td> <td valign="middle" > <p class="p4">0.068</p> </td> </tr> <tr> <td valign="middle" > <p class="p4">0.760</p> </td> <td valign="middle" > <p class="p4">1.2503</p> </td> <td valign="middle" > <p class="p4">0.246</p> </td> <td valign="middle" > <p class="p4">0.099</p> </td> </tr> <tr> <td valign="middle" > <p class="p4">0.730</p> </td> <td valign="middle" > <p class="p4">1.2895</p> </td> <td valign="middle" > <p class="p4">0.282</p> </td> <td valign="middle" > <p class="p4">0.134</p> </td> </tr> <tr> <td valign="middle" > <p class="p4">0.700</p> </td> <td valign="middle" > <p class="p4">1.3305</p> </td> <td valign="middle" > <p class="p4">0.320</p> </td> <td valign="middle" > <p class="p4">0.166</p> </td> </tr> <tr> <td valign="middle" > <p class="p4">0.650</p> </td> <td valign="middle" > <p class="p4">1.4022</p> </td> <td valign="middle" > <p class="p4">0.386</p> </td> <td valign="middle" > <p class="p4">0.213</p> </td> </tr> <tr> <td valign="middle" > <p class="p4">0.600</p> </td> <td valign="middle" > <p class="p4">1.4796</p> </td> <td valign="middle" > <p class="p4">0.457</p> </td> <td valign="middle" > <p class="p4">0.256</p> </td> </tr> <tr> <td valign="middle" > <p class="p4">0.550</p> </td> <td valign="middle" > <p class="p4">1.5633</p> </td> <td valign="middle" > <p class="p4">0.535</p> </td> <td valign="middle" > <p class="p4">0.294</p> </td> </tr> <tr> <td valign="middle" > <p class="p4">0.500</p> </td> <td valign="middle" > <p class="p4">1.6552</p> </td> <td valign="middle" > <p class="p4">0.622</p> </td> <td valign="middle" > <p class="p4">0.328</p> </td> </tr> <tr> <td valign="middle" > <p class="p4">0.450</p> </td> <td valign="middle" > <p class="p4">1.7573</p> </td> <td valign="middle" > <p class="p4">0.721</p> </td> <td valign="middle" > <p class="p4">0.353</p> </td> </tr> <tr> <td valign="middle" > <p class="p4">0.400</p> </td> <td valign="middle" > <p class="p4">1.8737</p> </td> <td valign="middle" > <p class="p4">0.838</p> </td> <td valign="middle" > <p class="p4">0.366</p> </td> </tr> </table> <p class="p2"><br></p> <p class="p2"><br></p> </td> <td align="center" valign="top"> <!-- <p class="p1">V_2 = 1.0212</p> --> <table cellspacing="1" cellpadding="5" class="t1"> <tr> <td align="center" colspan="4" bgcolor="yellow">V<sub>2</sub> = 1.0212</td> </tr> <tr> <td valign="middle" align="center"> <p class="p3">R</p> </td> <td valign="middle" align="center"> <p class="p3">R1</p> </td> <td valign="middle" align="center"> <p class="p3"><math>~\varpi</math></p> </td> <td valign="middle" align="center"> <p class="p3">z</p> </td> </tr> <tr> <td valign="middle" bgcolor="pink" > <p class="p4">0.6000</p> </td> <td valign="middle" bgcolor="pink" > <p class="p4">1.4000</p> </td> <td valign="middle" bgcolor="pink" > <p class="p4">0.400</p> </td> <td valign="middle" bgcolor="pink" > <p class="p4">0.000</p> </td> </tr> <tr> <td valign="middle" > <p class="p4">0.5950</p> </td> <td valign="middle" > <p class="p4">1.4078</p> </td> <td valign="middle" > <p class="p4">0.407</p> </td> <td valign="middle" > <p class="p4">0.048</p> </td> </tr> <tr> <td valign="middle" > <p class="p4">0.580</p> </td> <td valign="middle" > <p class="p4">1.4315</p> </td> <td valign="middle" > <p class="p4">0.428</p> </td> <td valign="middle" > <p class="p4">0.097</p> </td> </tr> <tr> <td valign="middle" > <p class="p4">0.570</p> </td> <td valign="middle" > <p class="p4">1.4477</p> </td> <td valign="middle" > <p class="p4">0.443</p> </td> <td valign="middle" > <p class="p4">0.120</p> </td> </tr> <tr> <td valign="middle" > <p class="p4">0.540</p> </td> <td valign="middle" > <p class="p4">1.4978</p> </td> <td valign="middle" > <p class="p4">0.488</p> </td> <td valign="middle" > <p class="p4">0.171</p> </td> </tr> <tr> <td valign="middle" > <p class="p4">0.500</p> </td> <td valign="middle" > <p class="p4">1.5688</p> </td> <td valign="middle" > <p class="p4">0.553</p> </td> <td valign="middle" > <p class="p4">0.224</p> </td> </tr> <tr> <td valign="middle" > <p class="p4">0.450</p> </td> <td valign="middle" > <p class="p4">1.6663</p> </td> <td valign="middle" > <p class="p4">0.644</p> </td> <td valign="middle" > <p class="p4">0.275</p> </td> </tr> <tr> <td valign="middle" > <p class="p4">0.400</p> </td> <td valign="middle" > <p class="p4">1.7767</p> </td> <td valign="middle" > <p class="p4">0.749</p> </td> <td valign="middle" > <p class="p4">0.312</p> </td> </tr> </table> </td></tr></table> </div> ==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> ==Interior Potential== In equation (9) on p. 1050 of his "Part II", [http://adsabs.harvard.edu/abs/1893RSPTA.184.1041D Dyson (1893b)] presents the following power-series expression for the gravitational potential at points ''inside'' the torus: <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~V</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ 2\pi a^2 \biggl\{ L + \frac{1}{2}\biggl( 1 - \frac{R^2}{a^2}\biggr) + \frac{a}{c}\biggl[ \frac{(L-1)}{2}\biggl(\frac{R}{a}\biggr) - \frac{R^3}{8a^3} \biggr]\cos\chi </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~+~ \frac{a^2}{c^2}\biggl[ - ~\frac{(L - \tfrac{1}{4})}{16} ~+~ \frac{(L-1)}{8} \biggl(\frac{R^2}{a^2} \biggr) ~-~ \frac{3}{64} \biggl( \frac{R^4}{a^4}\biggr) ~+~ \frac{3(L-\tfrac{5}{4})}{16} \biggl(\frac{R^2}{a^2}\biggr) \cos 2\chi ~-~\frac{5}{96} \biggl( \frac{R^4}{a^4}\biggr) \cos 2\chi \biggr] ~+~ \cdots \biggr\} </math> </td> </tr> </table> where, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~L</math> </td> <td align="center"> <math>~\equiv</math> </td> <td align="left"> <math>~\ln\biggl(\frac{8c}{a}\biggr) \, .</math> </td> </tr> </table> Note that, for the example illustrated above, <math>~a/c = 2/5</math> and, hence, <math>~L = \ln(20) = 2.99573</math>. Therefore, at any point on the surface of this example torus, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\frac{V}{2\pi a^2}</math> </td> <td align="center"> <math>~\approx</math> </td> <td align="left"> <math>~ \ln(20) + \frac{2}{5}\biggl[ \frac{\ln(20)-1}{2} - \frac{1}{8} \biggr]\cos\chi </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~+~ \frac{2^2}{5^2}\biggl[ - ~\frac{4\ln(20) - 1}{64} ~+~ \frac{\ln(20)-1}{8} ~-~ \frac{3}{64} ~+~ \frac{12\ln(20)-15}{64} \cos 2\chi ~-~\frac{10}{3\cdot 64} \cos 2\chi \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \ln(20) + \frac{2}{40}\biggl[ 4\ln(20)- 5 \biggr]\cos\chi ~+~ \frac{2^2}{2^6\cdot 5^2}\biggl\{ - ~[4\ln(20) - 1] ~+~ [8\ln(20)-8] ~-~ 3 ~+~ [12\ln(20)-15] \cos 2\chi ~-~\biggl(\frac{10}{3} \biggr) \cos 2\chi \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \ln(20) + \frac{1}{20}\biggl[ 4\ln(20)- 5 \biggr]\cos\chi ~+~ \frac{1}{2^4\cdot 5^2}\biggl\{ 4\ln(20)-10 ~+~ \frac{1}{3}\biggl[ 36\ln(20)-55 \biggr] \cos 2\chi \biggr\} </math> </td> </tr> </table> ===Red Contour=== Let's construct an equipotential contour that extends the red contour into the ''interior'' region. Let's begin by evaluating Dyson's ''interior'' potential expression at the coordinate location where the red contour touches the surface of the torus. According to Column #5, this point on the surface has coordinates, <math>~(\varpi,z) = (0.867, 0.377)</math>; or, equivalently, <math>~R = a = 0.4</math> and, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\cos\chi</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \frac{1-\varpi}{R} = 0.3325 </math> </td> </tr> <tr> <td align="right"> <math>~\Rightarrow ~~~ \chi</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \cos^{-1}(0.3325) = 1.2318 \, . </math> </td> </tr> </table> Hence, for this ''specific'' point on the torus surface, we find, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\frac{V}{2\pi a^2}</math> </td> <td align="center"> <math>~\approx</math> </td> <td align="left"> <math>~ 3 + \biggl[\frac{7}{20}\biggr]\cos\chi ~+~ \frac{1}{2^4\cdot 5^2}\biggl\{ 2 ~+~ \biggl[ \frac{53}{3} \biggr] \cos 2\chi \biggr\} = 3.0870 \, . </math> </td> </tr> </table> <table border="1" align="center" cellpadding="8"> <tr> <th align="center">Contour<br />(column #)</th> <th align="center"><math>~V_2</math> (external)</th> <th align="center">Surface <math>~\varpi</math></th> <th align="center"><math>~\chi</math> (radians)</th> </tr> <tr> <td align="center">blue (3)</td> <td align="right">0.9120</td> <td align="right">1.117</td> <td align="right">1.8676</td> </tr> <tr> <td align="center">orange (4)</td> <td align="right">0.961</td> <td align="right">0.929</td> <td align="right">1.392</td> </tr> <tr> <td align="center">red (5)</td> <td align="right">0.9800</td> <td align="right">0.867</td> <td align="right">1.2318</td> </tr> <tr> <td align="center">light green (6)</td> <td align="right">0.9800</td> <td align="right">0.838</td> <td align="right">1.1538</td> </tr> <tr> <td align="center">light blue (7)</td> <td align="right">1.0212</td> <td align="right">0.749</td> <td align="right">0.8925</td> </tr> </table> =See Also= * [http://adsabs.harvard.edu/abs/2016AJ....152...35F T. Fukushima (2016, AJ, 152, article id. 35, 31 pp.)] — ''Zonal Toroidal Harmonic Expansions of External Gravitational Fields for Ring-like Objects'' * [http://adsabs.harvard.edu/abs/2016ApJ...829...45K W.-T. Kim & S. Moon (2016, ApJ, 829, article id. 45, 22 pp.)] — ''Equilibrium Sequences and Gravitational Instability of Rotating Isothermal Rings'' * [http://adsabs.harvard.edu/abs/2011MNRAS.411..557B E. Y. Bannikova, V. G. Vakulik & V. M. Shulga (2011, MNRAS, 411, 557 - 564)] — ''Gravitational Potential of a Homogeneous Circular Torus: a New Approach'' * [http://adsabs.harvard.edu/abs/2008MNRAS.389..156P D. Petroff & S. Horatschek (2008, MNRAS, 389,156 - 172)] — ''Uniformly Rotating Homogeneous and Polytropic Rings in Newtonian Gravity'' <table border="0" cellpadding="8" align="center" width="90%"> <tr><td align="left"> The following quotes have been taken from [http://adsabs.harvard.edu/abs/2008MNRAS.389..156P Petroff & Horatschek (2008)]:<p></p> <b>§1:</b> "The problem of the self-gravitating ring captured the interest of such renowned scientists as Kowalewsky (1885), Poincaré (1885a,b,c) and Dyson (1892, 1893). Each of them tackled the problem of an axially symmetric, homogeneous ring in equilibrium by expanding it about the thin ring limit. In particular, Dyson provided a solution to fourth order in the parameter <math>~\sigma = a/b</math>, where <math>~a = r_t</math> provides a measure for the radius of the cross-section of the ring and <math>~b = \varpi_t</math> the distance of the cross-section's centre of mass from the axis of rotation."<p></p> <b>§7:</b> "In their work on homogeneous rings, Poincaré and Kowalewsky, whose results disagreed to first order, both had made mistakes as Dyson has shown. His result to fourth order is also erroneous as we point out in Appendix B." </td></tr> </table> * [http://adsabs.harvard.edu/abs/2006IJMPB..20.3113C P. H. Chavanis (2006, International Journal of Modern Physics B, 20, 3113 - 3198)] — ''Phase Transitions in Self-Gravitating Systems'' * [http://adsabs.harvard.edu/abs/2003MNRAS.339..515A M. Ansorg, A. Kleinwächter & R. Meinel (2003, MNRAS, 339, 515)] — ''Uniformly Rotating Axisymmetric Fluid Configurations Bifurcating from Highly Flattened Maclaurin Spheroids'' * [http://adsabs.harvard.edu/abs/2001A%26A...375.1091L M. Lombardi & G. Bertin (2001, Astronomy & Astrophysics, 375, 1091 - 1099)] — ''Boyle's Law and Gravitational Instability'' * [http://adsabs.harvard.edu/abs/1996MNRAS.282..234K W. Kley (1996, MNRAS, 282, 234)] — ''Maclurin Discs and Bifurcations to Rings'' * [http://adsabs.harvard.edu/abs/1994ApJ...420..247W J. W. Woodward, J. E. Tohline, & I. Hachisu (1994, ApJ, 420, 247 - 267)] — ''The Stability of Thick, Self-Gravitating Disks in Protostellar Systems'' * [http://adsabs.harvard.edu/abs/1991ApJ...374..610B I. Bonnell & P. Bastien (1991, ApJ, 374, 610 - 622)] — ''The Collapse of Cylindrical Isothermal and Polytropic Clouds with Rotation'' * [http://adsabs.harvard.edu/abs/1990ApJ...361..394T J. E. Tohline & I. Hachisu (1990, ApJ, 361, 394 - 407)] — ''The Breakup of Self-Gravitating Rings, Tori, and Thick Accretion Disks'' * [http://adsabs.harvard.edu/abs/1988A%26A...200..127S F. Schmitz (1988, Astronomy & Astrophysics, 200, 127 - 134)] — ''Equilibrium Structures of Differentially Rotating Self-Gravitating Gases'' * [http://adsabs.harvard.edu/abs/1985Ap%26SS.109...45V P. Veugelen (1985, Astrophysics & Space Science, 109, 45 - 55)] — ''Equilibrium Models of Differentially Rotating Polytropic Cylinders'' * [http://adsabs.harvard.edu/abs/1984MNRAS.208..279A M. A. Abramowicz, A. Curir, A. Schwarzenberg-Czerny, & R. E. Wilson (1984, MNRAS, 208, 279 - 291)] — ''Self-Gravity and the Global Structure of Accretion Discs'' * [http://adsabs.harvard.edu/abs/1983A%26A...119..109B P. Bastien (1983, Astronomy & Astrophysics, 119, 109 - 116)] — ''Gravitational Collapse and Fragmentation of Isothermal, Non-Rotating, Cylindrical Clouds'' * [http://adsabs.harvard.edu/abs/1981PThPh..65.1870E Y. Eriguchi & D. Sugimoto (1981, Progress of Theoretical Physics, 65, 1870 - 1875)] — ''Another Equilibrium Sequence of Self-Gravitating and Rotating Incompressible Fluid'' * [http://adsabs.harvard.edu/abs/1980ApJ...236..160T J. E. Tohline (1980, ApJ, 236, 160 - 171)] — ''Ring Formation in Rotating Protostellar Clouds'' * [http://adsabs.harvard.edu/abs/1980PThPh..63.1957F T. Fukushima, Y. Eriguchi, D. Sugimoto, & G. S. Bisnovatyi-Kogan (1980, Progress of Theoretical Physics, 63, 1957 - 1970)] — ''Concave Hamburger Equilibrium of Rotating Bodies'' * [http://adsabs.harvard.edu/abs/1978MNRAS.184..709K J. Katz & D. Lynden-Bell (1978, MNRAS, 184, 709 - 712)] — ''The Gravothermal Instability in Two Dimensions'' * [http://adsabs.harvard.edu/abs/1977ApJ...214..584M P. S. Marcus, W. H. Press, & S. A. Teukolsky (1977, ApJ, 214, 584- 597)] — ''Stablest Shapes for an Axisymmetric Body of Gravitating, Incompressible Fluid'' (includes torus with non-uniform rotation) <ol type="a"><li>Shortly after their equation (3.2), Marcus, Press & Teukolsky make the following statement: "… we know that an ''equilibrium'' incompressible configuration must rotate uniformly on cylinders (the famous "Poincaré-Wavre" theorem, cf. Tassoul 1977, &Sect;4.3) …"</li></ol> * [http://adsabs.harvard.edu/abs/1976ApJ...207..736H C. J. Hansen, M. L. Aizenman, & R. L. Ross (1976, ApJ, 207, 736 - 744)] — ''The Equilibrium and Stability of Uniformly Rotating, Isothermal Gas Cylinders'' * [http://adsabs.harvard.edu/abs/1974ApJ...190..675W C.-Y. Wong (1974, ApJ, 190, 675 - 694)] — ''Toroidal Figures of Equilibrium'' * [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W C.-Y. Wong (1973, Annals of Physics, 77, 279 - 353)] — ''Toroidal and Spherical Bubble Nuclei'' * [http://adsabs.harvard.edu/abs/1964ApJ...140.1056O J. Ostriker (1964, ApJ, 140, 1056)] — ''The Equilibrium of Polytropic and Isothermal Cylinders'' * [http://adsabs.harvard.edu/abs/1964ApJ...140.1067O J. Ostriker (1964, ApJ, 140, 1067)] — ''The Equilibrium of Self-Gravitating Rings'' * [http://adsabs.harvard.edu/abs/1964ApJ...140.1529O J. Ostriker (1964, ApJ, 140, 1529)] — ''On the Oscillations and the Stability of a Homogeneous Compressible Cylinder'' * [http://adsabs.harvard.edu/abs/1965ApJS...11..167O J. Ostriker (1965, ApJ Supplements, 11, 167)] — ''Cylindrical Emden and Associated Functions'' * [http://adsabs.harvard.edu/abs/1942ApJ....95...88R Gunnar Randers (1942, ApJ, 95, 88)] — ''The Equilibrium and Stability of Ring-Shaped 'barred SPIRALS'.'' * [https://www.amazon.com/Theory-Potential-W-D-Macmillan/dp/0486604861/ref=sr_1_2?s=books&ie=UTF8&qid=1503444466&sr=1-2&keywords=the+theory+of+the+potential William Duncan MacMillan (1958)], ''The Theory of the Potential'', New York: Dover Publications * [https://archive.org/details/foundationsofpot033485mbp Oliver Dimon Kellogg (1929)], ''Foundations of Potential Theory'', Berlin: Verlag Von Julius Springer * [http://adsabs.harvard.edu/abs/1917RSPSA..93..148R Lord Rayleigh (1917, Proc. Royal Society of London. Series A, 93, 148-154)] — ''On the Dynamics of Revolving Fluids'' * [http://adsabs.harvard.edu/abs/1893RSPTA.184.1041D F. W. Dyson (1893, Philosophical Transaction of the Royal Society London. A., 184, 1041 - 1106)] — ''The Potential of an Anchor Ring. Part II.'' <ol type="a"><li>In this paper, Dyson derives the gravitational potential ''inside'' the ring mass distribution</li></ol> * [http://adsabs.harvard.edu/abs/1893RSPTA.184...43D F. W. Dyson (1893, Philosophical Transaction of the Royal Society London. A., 184, 43 - 95)] — ''The Potential of an Anchor Ring. Part I.''<ol type="a"><li>In this paper, Dyson derives the gravitational potential ''exterior to'' the ring mass distribution</li></ol> * [http://adsabs.harvard.edu/abs/1885AN....111...37K S. Kowalewsky (1885, Astronomische Nachrichten, 111, 37)] — ''Zusätze und Bemerkungen zu Laplace's Untersuchung über die Gestalt der Saturnsringe'' * Poincaré (1885a, C. R. Acad. Sci., 100, 346), (1885b, Bull. Astr., 2, 109), (1885c, Bull. Astr. 2, 405). — references copied from paper by [http://adsabs.harvard.edu/abs/1974ApJ...190..675W Wong (1974)] {{ SGFfooter }}
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