This page contains information supplementary to (Garcia, Reznik, and Koiller 2020). We consider Poncelet 3-periodic families interscribed in a pair of ellipses \(\mathcal{E}\) and \(\mathcal{E}_c\) with semiaxes \(a,b\) and \(a_c,b_c\), respectively. The Cayley condition for the pair to admit a Poncelet 3-periodic family is (Georgiev and Nedyalkova 2012):
\[\frac{a_c}a+\frac{b_c}b=1\]
Below we report on triangle centers on (Kimberling 2019) whose loci over Poncelet 3-periodics is an ellipse. These are typically first detected numerically and then proven rigorously, e.g., (Romaskevich 2014),(Tabachnikov and Tsukerman 2014),(Schwartz and Tabachnikov 2016),(Garcia 2019),(Fierobe 2021).
Here we focus on first 200 triangle centers on (Kimberling 2019). We consider 3-periodic families in the confocal (elliptic billiard) and homothetic pair. Appendix A tabulates fit errors for loci in the confocal family, which should be negligible when a given locus is an ellipse.
The family of 3-Periodics in some classic ellipse pairs can be viewed here. All loci herein can be visualized with our interactive app; illustrations and documentation are available on this webpage.
The following explicit expressions were given for the semi-axes \(a_c,b_c\) of the \(N=3\) confocal caustic (Garcia 2019):
\[a_c=\frac{a\left(\delta-b^{2}\right)}{c^2},\;\;\;b_c=\frac{b\left(a^{2}-\delta\right)}{c^2}\] with \(\delta=\sqrt{a^4-a^2b^2+b^4}\) and \(c^2=a^2-b^2\).
Below we provide explicit expressions for the semi-axes \(a_i,b_i\) of loci, i.e., let the locus of center \(X_i\) be described as:
\[\frac{x^2}{a_{i}^2}+\frac{y^2}{b_{i}^2}=1\]
A gallery of loci for all of \(X_1\) thru \(X_{100}\) appears here.
\[a_1=\frac{\delta-b^{2}}a,\;\;\;b_1=\frac{a^{2}-\delta}b\]
The locus of the excenters is an ellipse with axes (Garcia 2019):
\[a_e=\frac{b^{2}+\delta}a,\;\;\; b_e=\frac{a^{2}+\delta}b\]
Notice it is similar to the \(X_1\) locus, i.e., \(a_1/b_1=b_e/a_e\).
Note: The ellipticity of \(X_1\) for confocals was proven in (Romaskevich 2014) and (Garcia 2019).
\[\left(a_2,b_2\right)=k_2\left(a,b\right),\;\textrm{where}\; k_2=\frac{2\delta -a^{2}-b^{2}}{3c^2}\]
Note: The ellipticity of \(X_2\) for any Poncelet family was proven in (Schwartz and Tabachnikov 2016), (Garcia 2019).
\[ a_3=\frac{a^{2}-\delta}{2a},\;\;\; b_3=\frac{\delta-b^{2}}{2b} \]
Additionally, when \(a/b= \frac{\sqrt{2\sqrt{33}+2}}{2} \;{\simeq}\;1.836\), \(b_3=b\), i.e., the top and bottom vertices of the locus of \(X_3\) coincide with the billiard’s.
Note: The ellipticity of \(X_3\) for confocals was proven in (Garcia 2019) and (Fierobe 2021). In (Tabachnikov and Tsukerman 2014), one consideres the ``circumcenter-of-mass’’ which generalizes this result for any \(N{\geq}3\) Poncelet family.
\[ \left(a_4,b_4\right)=\left(\frac{k_4}a,\frac{k_4}b\right),\; k_4=\frac{ (a^{2}+b^{2})\delta-2\,a^{2}b^{2} }{c^2} \]
Additionally:
\[ a_5=\frac{- w'_5(a,b)+ w''_5(a,b) \delta}{ w_5(a,b)},\;\;\;b_5=\frac{ w'_5(b,a)-{w''_5(b,a) \delta}}{w_5(b,a)} \]
\[ w'_5(u,v)=u^2(u^2+3v^2),\;\;\;w''_5(u,v)=3u^2+ v^2,\;\;\;w_5(u,v)=4u(u^2-v^2). \]
\[ \left(a_7,b_7\right)=k_7\left(a,b\right),\;\; k_7=\frac{2\delta - a^{2}-b^{2}}{c^2} \]
\[ a_{8}= {\frac{ (b^2-\delta)^2 }{a c^2 }},\;\;\; b_{8}={\frac{ (a^{2} -\delta )^2 }{b c^2 }} \]
\[ \left(a_{10},b_{10}\right)=\left(\frac{k_{10}}a,\frac{k_{10}}b\right),\;\; k_{10}=\frac{ \left( a^{2}+b^{2} \right)\delta-a^{4}-b^{4}}{2 c^2} \]
\[a_{11}=a_c,\;\;\;b_{11}=b_c\]
\[a_{12}=\frac{ - w'_{12} (a,b)+ w''_{12}(a,b) \delta}{ w_{12} (a,b)},\;\;\;b_{12}=\frac{ w'_{12}(b,a)-w''_{12}(b,a)\delta}{ w_{12}(b,a)}\]
\[\begin{align*} w'_{12}(u,v)=& v^2(15\,{u}^{6}+12\,{v}^{2}{u}^{4}+3\,{u}^{2}{v}^{4}+2\,{v}^{6})\\ w''_{12}(u,v)= & 7\,{u}^{6}+12\,{v}^{2}{u}^{4}+11\,{u}^{2}{v}^{4}+2\,{v}^{6} \\ w_{12}(u,v)=& u( 7\,{u}^{6}+11\,{v}^{2}{u}^{4}-11\,{u}^{2}{v}^{4}-7\,{v}^{6}). \end{align*}\]
\[a_{20}= {\frac{a^{2} \left(3b^2- a^{2} \right) -2b^2\,\delta }{ ac^2}}, \;\;\; b_{20}= {\frac{b^{2} \left(b^{2} -3a^2\right) +2a^2\,\delta}{b c^2 }}\]
\[ a_{21}=\frac{-w'_{21}(a,b)+ w''_{21}(a,b) \delta}{ w_{21}(a,b)},\;\;\;b_{21}=\frac{ w'_{21}(b,a)- \, w''_{21}(b,a)\delta}{ w_{21}(b,a)}\\ \]
\[w'_{21}(u,v)={u}^{4}+{u}^{2}{v}^{2}+ \,{v}^{4},\;\;w''_{21}(u,v)= 2( {u}^{2}+{v}^{2}),\;\;w_{21}(u,v)=u\left( 3\,{u}^{2}+5\,{v}^{2} \right)\]
\[\begin{align*} a_{35}=&\frac{-w'_{35}(a,b)+w''_{35}(a,b) \delta}{w_{35}(a,b)},\;\;\;b_{35}=\frac{-w'_{35}(b,a)+w''_{35}(b,a)\delta}{w_{35}(b,a)} \\ w'_{35}(u,v)=&v^2( 11\,{u}^{4}+4\,{u}^{2}{v}^{2}+{v}^{4} ),\;\;\;w''_{35}(u,v)=\left( 7\,{u}^{2}+{v}^{2} \right) \left( {u}^{2}+{v}^{2}\right)\\ w_{35}(u,v)=& u(7\,{u}^{4}+18\,{u}^{2}{v}^{2}+7\,{v}^{4} ) \end{align*} \]
\[ a_{36}=\frac{w'_{36}(a,b)+ w''_{36}(a,b)\delta }{w_{36}(a,b)},\;\;\;b_{36}=\frac{ -w'_{36}(b,a) -w''_{36}(b,a)\delta}{w(b,a)}\]
\[w'_{36}(u,v)= {v}^{2} \left( {u}^{2}+{v}^{2} \right),\;\;\;w''_{36}(u,v)= 3\,{u}^{2}-{v}^{2},\;\;\; w_{36}(u,v)=3u \left(u^2-v^2\right)\]
\[a_{40}=\frac{c^2}a,\;\;\;b_{40}=\frac{c^2}b\]
Additionally:
\[a_{46}=\frac{w'_{46}(a,b)+w''_{46}(a,b) \delta}{ w_{46}(a,b) },\;\;\;b_{46}=\frac{-w'_{46}(b,a) - w''_{46}(b,a)\delta}{ w_{46}(b,a) }\]
\[\begin{align*} w'_{46}(u,v)=& v^2(3\,{u}^{2}-{v}^{2} )(u^2-v^2),\;\;\;w''_{46}(u,v)= (5\,{u}^{2}+{v}^{2})(u^2-v^2) \\ w_{46}(u,v)=& v(5\,{u}^{4}-6\,{u}^{2}{v}^{2}+5\,{v}^{4} ) \end{align*}\]
\[a_{55}= {\frac{a\left(\delta-b^{2} \right)}{a^{2}+b^{2}}},\;\; b_{55} = { \frac{b \left( a^{2}-\delta \right)}{a^{2}+b^{2}}}\]
\[a_{56}= \frac{-w'_{56}(a,b) + w''_{56}(a,b) \delta}{ w_{56}(a,b) },\;\;\; b_{56}= \frac{ w'_{56}(b,a) - w''_{56}(b,a) \delta}{ w_{56}(b,a) } \]
\[\begin{align*} w'_{56}(u,v)=&v^2( {u}^{4}-{u}^{2}{v}^{2}+2\,{v}^{4}),\;\;\;w''_{56}(u,v)= 5\,{u}^{4}-5\,{u}^{2}{v}^{2}+2\,{v}^{4} \\ w_{56}(u,v)=&u(5\,{u}^{4}-6\,{u}^{2}{v}^{2}+5\,{v}^{4} ) \end{align*} \]
\[ \left(a_{57},b_{57}\right)=k_{57}\left(a,b\right),\;\; k_{57}=\frac{c^2}{\delta} %a_{57}=\frac{a (a^2- b^2)}{ \delta}, \;\;\; %b_{57}=\frac{b (a^2- b^2)}{ \delta}\]
\[\left(a_{63},b_{63}\right)=k_{63}\left(a,b\right),\;\; k_{63}=\frac{c^2}{a^2+b^2} %a_{63}={\frac{a \left( a^{2}-a^{2} \right) }{a^{2}+a^{2}}},\;\;\; %b_{63}= {\frac{b \left( a^{2}-a^{2} \right) }{a^{2}+a^{2}}}\]
\[\begin{align*} a_{65}=& \frac{w'_{65}(a,b) + w''_{65}(a,b) \delta}{ w_{65}(a,b)},\;\;\; b_{65}=\frac{-w'_{65}(b,a)- w''_{65}(b,a) \delta}{ w_{65}(b,a)} \end{align*}\]
\[\begin{align*} w'_{65}(u,v)=& {u}^{4}{v}^{2}+{u}^{2}{v}^{4}+2\,{v}^{6},\;\;\; w''_{65}(u,v)={u}^{4}-3\,{u}^{2}{v}^{2}-2\,{v}^{4}\\ w_{65}(u,v)=&u\left( {u}^{2}-{v}^{2} \right) ^{2} \end{align*}\]
\[\begin{align*} a_{72}=& \frac{w'_{72}(a,b) - w''_{72}(a,b) \delta }{w_{72}(a,b)},\;\;\; b_{72}= \frac{-w'_{72}(b,a) + w''_{72}(b,a) \delta}{w_{72}(b,a) } \end{align*}\]
\[w'_{72}(u,v)= {u}^{6}+2\,{u}^{2}{v}^{4}+{v}^{6},\;\; w''_{72}(u,v)= (3\,{u}^{2}+{v}^{2}){v}^{2},\;\; w_{72}(u,v)= u\left(u^2-v^2 \right)^{2}\]
\[a_{78}= \frac{w'_{78}(a,b) -w''_{78}(a,b) \delta }{w_{78}(a,b) },\;\;\; b_{78}= \frac{-w'_{78}(b,a)+w''_{78}(b,a) \delta}{w_{78}(b,a)}\]
\[\begin{align*} w'_{78}(u,v)=&5\,{u}^{6}-4\,{u}^{4}{v}^{2}+{u}^{2}{v}^{4}+2\,{v}^{6},\;\;\;w''_{78}(u,v)= 2\,{v}^{2} \left( {u}^{2}+{v}^{2}\right)\\ w_{78}(u,v)=&u \left( 5\,{u}^{4}-6\,{v}^{2}{u}^{2}+5\,{v}^{4} \right)\ \end{align*}\]
\[a_{79}= \frac{-w'_{79}(a,b) + w''_{79}(a,b) \delta }{w_{79}(a,b) }, \;\;\; b_{79}= \frac{ w'_{79}(b,a)- w''_{79}(b,a) \delta }{ w_{79}(b,a) }\]
\[\begin{align*} w'_{79}(u,v)=&{v}^{2} 11\,{u}^{4}+4\,{v}^{2}{u}^{2}+{v}^{4},\;\;\;w_{79}''(u,v)= \left( 3\,{u}^{4}+12\,{u}^{2}{v}^{2}+{v}^{4} \right) \\ w_{79}(u,v)=&u \left( u^2-v^2 \right) \left( 3\,{u}^{2}+5\,{v}^{2} \right) \end{align*}\]
\[a_{80}={\frac{ \left( \delta-b^{2} \right) \left( a^{2}+b^{2} \right)}{a c^2}},\;\;\; b_{80}={\frac{ \left( a^{2}-\delta \right) \left( a^{2}+b^{2} \right)}{b c^2}}\]
\[a_{84}=\frac{\left(b^{2}+\delta \right) c^2}{a^{3}},\;\;\; b_{84}=\frac{\left( a^{2}+\delta \right) c^2}{b^{3}}\]
\[a_{88}=a,\;\;\;b_{88}=b\]
\[a_{90}= \frac{ w'_{90}(a,b) + w''_{90}(a,b)\delta }{w_{90}(a,b)},\;\;\; b_{90}= \frac{w'_{90}(b,a)+w''_{90}(b,a) \delta }{ w_{90}(b,a)}\]
\[\begin{align*} w'_{90}(u,v)=&v^2(3u^2-v^2) (u^2-v^2),\;\;\; w''_{90}(u,v)=u^4-v^4 \\ w_{90}(u,v)=& u \left( {u}^{4}+2\,{u}^{2}{v}^{2}-7\,{v}^{4} \right) \end{align*}\]
\[a_{100}=a,\;\;\;b_{100}=b\]
Note: From \(X_{101}\) thru \(X_{200}\), also elliptic are the loci of \(X_k\) \(k\)in 104, 119, 140, 142, 144, 145, 149, 153, 162, 165, 190, 191, 200.
Consider 3-periodics in an ellipse pair such that \(a_c=a/2\) and \(b_c=b/2\). From \(X_{1}\) to \(X_{200}\), the loci of 27 (resp. 4) triangle centers are ellipses (resp. circles). Their axes are given by:
\[ a_3= \frac{c^2}{4a},\;\;\;b_3=\frac{c^2}{4b} \]
\[ a_4= \frac{c^2}{2a},\;\;\;b_4=\frac{c^2}{2b} \]
\[ a_5= \frac{c^2}{8a},\;\;\;b_5=\frac{c^2}{8b} \]
\[ a_6= \frac{ac^2}{2(a^2+b^2)},\;\;\;b_6=\frac{bc^2}{2(a^2+b^2)} \]
\[ r_{13}= \frac{a-b}{2},\;\;\;r_{14}= \frac{a+b}{2},\;\;\;r_{15}= \frac{(a-b)^2}{2(a+b)},\;\;\;r_{16}= \frac{(a+b)^2}{2(a-b)} \]
\[ a_{17}=\frac{c^2}{2(a+3b)},\;\;\;b_{17}= \frac{c^2}{2(3a+b)} \]
\[ a_{18}=\frac{c^2}{2(a-3b)},\;\;\;b_{18}= \frac{c^2}{2(3a-b)} \]
\[ a_{20}= \frac{c^2}a,\;\;\;b_{20}=\frac{c^2}b \]
\[ a_{32}= \frac {a \left( c^2 \right) \left( 3\,a^{2}+5\,b^{2} \right)}{2(3\,a^{4}+2\,a^{2}b^{2}+3\,b^{4})},\;\;\; b_{32}= \frac {b \left( c^2 \right) \left( 3\,a^{2}+5\,b^{2} \right)}{2(3\,a^{4}+2\,a^{2}b^{2}+3\,b^{4})} \]
\[ a_{39}=\frac { \left( a^{2}-b^{2} \right) a}{2( a^{2}+3\,b^{2})},\;\;\; b_{39}=\frac{ \left( a^{2}-b^{2} \right) b}{2(3 \,a^{2}+\,b^{2})}\]
\[a_{61}= \frac {\left( c^2\right)\left( 3\,a-b \right)}{2( 3a^{2}+2\,ab+3\,b^{2})},\;\;\; b_{61}= \frac { \left( c^2 \right)\left( a-3\,b \right)}{2(3\,a^{2}+2\,ab+3\,b^{2})}\]
\[a_{62}=\,{\frac{\left( c^2\right)\left(3\,a+b\right)}{2(3\,a^{2}-2\,ab+3\,b^{2})}},\;\;\; b_{62}={\frac{ \left( c^2 \right) \left( a+3b\right) }{2(3\,a^{2}-2\,ab+3\,b^{2})}}\]
\[a_{69}={\frac { \left( c^2 \right)a}{a^{2}+b^{2}}},\;\;\; b_{69}=\frac{\left( c^2 \right)b}{a^{2}+b^{2}} \]
\[ a_{76}=\frac{\left( c^2 \right)a}{a^{2}+3b^{2}},\;\;\; b_{76}= \frac{ \left( c^2 \right)b}{3a^{2}+b^{2}} \]
\[a_{83} = \frac { \left( c^2 \right) a}{5a^{2}+3b^{2}},\;\;\; b_{83} = \frac{ \left( c^2 \right) b}{3a^{2}+5b^{2}} \]
\[ a_{98}= \frac{a^2+b^2}{2a},\;\;\;b_{98}= \frac{a^2+b^2}{2b} \]
\[ a_{99}=a,\;\;\;b_{99}=b \]
\[ a_{114}= \frac{a^2+b^2}{4a},\;\;\;b_{114}= \frac{a^2+b^2}{4b} \]
\[ a_{115}= \frac{a }{2},\;\;\;b_{115}= \frac{b}{2} \]
\[ a_{140}= \frac{c^2}{16a},\;\;\;b_{140}= \frac{c^2}{16b} \]
\[ a_{141}=\frac{c^2 a}{4(a^2+b^2)}\;\;\;b_{141}= \frac{c^2 b}{4(a^2+b^2)} \]
\[ a_{147}=\frac{a^2+b^2}a,\;\;\;b_{147}= \frac{a^2+b^2}b \]
\[ a_{148}=2a,\;\;\;b_{148}=2b \]
\[ a_{182}=\frac{c^4}{8a(a^2+b^2)},\;\;\; b_{182}= \frac{c^4}{8b(a^2+b^2)} \]
\[ a_{187}= \frac{a(a^2+3b^2)}{2c^2},\;\;\; b_{187}= \frac{b(3a^2+b^2)}{2c^2} \]
\[ a_{190}= a,\;\;\;b_{190}=b \]
\[ a_{193}=\frac{2c^2 a}{a^2+b^2},\;\;\; b_{193}=\frac{2c^2 b}{a^2+b^2} \]
\[ a_{194}=\frac{2c^2 a}{a^2+3b^2},\;\;\; b_{194}=\frac{2c^2 b}{3a^2+b^2} \]
Let the least-squares fit error be defined by:
\[\text{err}^2(a_i,b_i)= \sum_{k=1}^M{\left[\left(\frac{x_k}{a_i}\right)^2+\left(\frac{y_k}{b_i}\right)^2-1\right]^2}\]
The 29 centers with elliptic loci amongst the first 100 in Kimberling’s Encyclopedia (Kimberling 2019). These are listed in ascending order of (negligible) fit error, \(a/b=1.5\), \(M=1500\). Columns \(\hat{a},\hat{b}\) show the estimated semi-axes’ lengths. Notice for \(X_{100}\) and \(X_{88}\) these are identical to \(a,b\), since these points lie on the EB.
\[ \begin{array}{|c|c|c|c|c|} \hline \text{rank} & X_i & \hat{a} & \hat{b} & \sum{err{}^2} \\ \hline 1 & 100 & 1.50 & 1.00 & 5.9\times 10^{-14} \\ 2 & 80 & 1.65 & 0.77 & 8.1\times 10^{-14} \\ 3 & 46 & 1.47 & 1.18 & 8.3\times 10^{-14} \\ 4 & 36 & 2.57 & 2.34 & 1.\times 10^{-13} \\ 5 & 88 & 1.50 & 1.00 & 1.\times 10^{-13} \\ 6 & 56 & 1.05 & 0.74 & 1.1\times 10^{-13} \\ 7 & 20 & 1.18 & 2.43 & 1.1\times 10^{-13} \\ 8 & 72 & 0.75 & 0.35 & 1.1\times 10^{-13} \\ 9 & 63 & 0.58 & 0.38 & 1.2\times 10^{-13} \\ 10 & 40 & 0.83 & 1.25 & 1.3\times 10^{-13} \\ 11 & 78 & 1.12 & 0.28 & 1.3\times 10^{-13} \\ 12 & 57 & 0.96 & 0.64 & 1.4\times 10^{-13} \\ 13 & 79 & 0.85 & 0.68 & 1.4\times 10^{-13} \\ 14 & 4 & 0.98 & 1.48 & 1.4\times 10^{-13} \\ 15 & 65 & 0.90 & 0.58 & 1.4\times 10^{-13} \\ 16 & 7 & 0.79 & 0.52 & 1.5\times 10^{-13} \\ 17 & 11 & 1.14 & 0.24 & 1.7\times 10^{-13} \\ 18 & 5 & 0.44 & 0.50 & 1.7\times 10^{-13} \\ 19 & 84 & 1.09 & 5.25 & 1.8\times 10^{-13} \\ 20 & 12 & 0.55 & 0.38 & 1.8\times 10^{-13} \\ 21 & 1 & 0.64 & 0.30 & 1.8\times 10^{-13} \\ 22 & 2 & 0.26 & 0.17 & 2.3\times 10^{-13} \\ 23 & 3 & 0.10 & 0.48 & 2.5\times 10^{-13} \\ 24 & 55 & 0.44 & 0.09 & 2.9\times 10^{-13} \\ 25 & 8 & 0.48 & 0.07 & 3.3\times 10^{-13} \\ 26 & 10 & 0.08 & 0.11 & 3.6\times 10^{-13} \\ 27 & 90 & 3.93 & 0.34 & 4.\times 10^{-13} \\ 28 & 21 & 0.19 & 0.05 & 4.6\times 10^{-13} \\ 29 & 35 & 0.33 & 0.03 & 4.8\times 10^{-13} \\ \hline \end{array} \] The remaining 71 centers amongst the first 100 in (Kimberling 2019) whose loci are not elliptic, still ordered by increasing fit errors, which now are several orders of magnitude higher. Notice \(X_9\) whose locus is a point has the highest of all errors.
\[ \begin{array}{|c|c|c|} \hline \text{rank} & X_i & \sum{err{}^2} \\ \hline 30 & 37 & 1.9\times 10^{-3} \\ 31 & 6 & 6.2\times 10^{-3} \\ 32 & 45 & 8.\times 10^{-3} \\ 33 & 60 & 8.3\times 10^{-3} \\ 34 & 86 & 1.3\times 10^{-2} \\ 35 & 71 & 1.6\times 10^{-2} \\ 36 & 41 & 1.8\times 10^{-2} \\ 37 & 81 & 1.8\times 10^{-2} \\ 38 & 58 & 3.3\times 10^{-2} \\ 39 & 62 & 4.\times 10^{-2} \\ 40 & 31 & 4.3\times 10^{-2} \\ 41 & 89 & 4.4\times 10^{-2} \\ 42 & 42 & 4.5\times 10^{-2} \\ 43 & 17 & 5.6\times 10^{-2} \\ 44 & 13 & 8.3\times 10^{-2} \\ 45 & 83 & 9.\times 10^{-2} \\ 46 & 48 & 1.1\times 10^{-1} \\ 47 & 32 & 1.3\times 10^{-1} \\ 48 & 61 & 1.3\times 10^{-1} \\ 49 & 82 & 1.3\times 10^{-1} \\ 50 & 43 & 1.5\times 10^{-1} \\ 51 & 51 & 1.6\times 10^{-1} \\ 52 & 85 & 1.6\times 10^{-1} \\ 53 & 19 & 1.7\times 10^{-1} \\ 54 & 27 & 1.9\times 10^{-1} \\ 55 & 69 & 1.9\times 10^{-1} \\ 56 & 47 & 2.5\times 10^{-1} \\ 57 & 39 & 2.7\times 10^{-1} \\ 58 & 38 & 2.8\times 10^{-1} \\ 59 & 16 & 3.4\times 10^{-1} \\ 60 & 52 & 5.3\times 10^{-1} \\ 61 & 28 & 5.6\times 10^{-1} \\ 62 & 25 & 7.5\times 10^{-1} \\ 63 & 99 & 8.2\times 10^{-1} \\ 64 & 14 & 8.3\times 10^{-1} \\ 65 & 98 & 8.5\times 10^{-1} \\ \hline \end{array} \]
\[ \begin{array}{|c|c|c|} \hline \text{rank} & X_i & \sum{err{}^2} \\ \hline 66 & 44 & 1.7 \\ 67 & 22 & 1.7 \\ 68 & 34 & 1.7 \\ 69 & 75 & 1.9 \\ 70 & 54 & 2.1 \\ 71 & 53 & 2.2 \\ 72 & 29 & 2.4 \\ 73 & 15 & 2.5 \\ 74 & 33 & 2.6 \\ 75 & 24 & 2.6 \\ 76 & 74 & 3.2 \\ 77 & 97 & 3.3 \\ 78 & 67 & 4. \\ 79 & 23 & 4.6 \\ 80 & 91 & 5.2 \\ 81 & 96 & 5.2 \\ 82 & 76 & 5.8 \\ 83 & 18 & 7.6 \\ 84 & 73 & 8.5 \\ 85 & 94 & 9.3 \\ 86 & 70 & 9.3 \\ 87 & 92 & 1.2\times 10^1 \\ 88 & 77 & 1.4\times 10^1 \\ 89 & 95 & 1.5\times 10^1 \\ 90 & 49 & 1.8\times 10^1 \\ 91 & 87 & 1.8\times 10^1 \\ 92 & 59 & 1.9\times 10^1 \\ 93 & 64 & 3.9\times 10^1 \\ 94 & 68 & 3.9\times 10^1 \\ 95 & 26 & 3.9\times 10^1 \\ 96 & 66 & 3.9\times 10^1 \\ 97 & 50 & 3.9\times 10^1 \\ 98 & 93 & 3.9\times 10^1 \\ 99 & 30 & 3.9\times 10^1 \\ 100 & 9 & 3.9\times 10^1 \\ \hline \end{array} \]
Any comments, ideas, corrections, suggestions, and proofs contributed are very welcome. Email me at: dreznik _theat_ gmail _thedot_ com.