1 #LyX 2.1 created this file. For more info see http://www.lyx.org/
6 \use_default_options true
7 \maintain_unincluded_children false
9 \language_package default
14 \font_typewriter default
16 \font_default_family default
17 \use_non_tex_fonts false
23 \default_output_format default
25 \bibtex_command default
26 \index_command default
27 \paperfontsize default
31 \use_package amsmath 1
32 \use_package amssymb 1
35 \use_package mathdots 1
36 \use_package mathtools 1
38 \use_package stackrel 1
39 \use_package stmaryrd 1
40 \use_package undertilde 1
42 \cite_engine_type default
46 \paperorientation portrait
56 \paragraph_separation indent
57 \paragraph_indentation default
58 \quotes_language english
61 \paperpagestyle default
62 \tracking_changes false
71 \begin_layout Standard
74 \boldsymbol{\iota}=\left[\begin{array}{ccc}
75 \iota_{00} & \iota_{01} & \iota_{02}\\
76 \iota_{10} & \iota_{11} & \iota_{12}\\
77 \iota_{20} & \iota_{21} & \iota_{22}
86 \boldsymbol{\iota}=\left[\begin{array}{ccc}
87 {\displaystyle \sum_{i=1}^{n}m_{i}\left(x_{i1}^{2}+x_{i2}^{2}\right)} & -{\displaystyle \sum_{i=1}^{n}m_{i}x_{i0}x_{i1}} & -{\displaystyle \sum_{i=1}^{n}m_{i}x_{i0}x_{i2}}\\
88 -{\displaystyle \sum_{i=1}^{n}m_{i}x_{i0}x_{i1}} & {\displaystyle \sum_{i=1}^{n}m_{i}\left(x_{i0}^{2}+x_{i2}^{2}\right)} & -{\displaystyle \sum_{i=1}^{n}m_{i}x_{i1}x_{i2}}\\
89 -{\displaystyle \sum_{i=1}^{n}m_{i}x_{i0}x_{i2}} & -{\displaystyle \sum_{i=1}^{n}m_{i}x_{i1}x_{i2}} & {\displaystyle \sum_{i=1}^{n}}m_{i}\left(x_{i0}^{2}+x_{i1}^{2}\right)
98 \boldsymbol{\iota}=\sum_{i=1}^{n}m_{i}\left[\begin{array}{ccc}
99 \left(x_{i1}^{2}+x_{i2}^{2}\right) & -x_{i0}x_{i1} & -x_{i0}x_{i2}\\
100 -x_{i0}x_{i1} & \left(x_{i0}^{2}+x_{i2}^{2}\right) & -x_{i1}x_{i2}\\
101 -x_{i0}x_{i2} & -x_{i1}x_{i2} & \left(x_{i0}^{2}+x_{i1}^{2}\right)
110 \boldsymbol{\iota}=\sum_{i=1}^{n}m_{i}\left[\begin{array}{ccc}
111 \left|\mathbf{p}_{i}\right|^{2}-x_{i0}^{2} & -x_{i0}x_{i1} & -x_{i0}x_{i2}\\
112 -x_{i0}x_{i1} & \left|\mathbf{p}_{i}\right|^{2}-x_{i1}^{2} & -x_{i1}x_{i2}\\
113 -x_{i0}x_{i2} & -x_{i1}x_{i2} & \left|\mathbf{p}_{i}\right|^{2}-x_{i2}^{2}
122 \boldsymbol{\iota}=\sum_{i=1}^{n}m_{i}\left(\left[\begin{array}{ccc}
123 \left|\mathbf{p}_{i}\right|^{2} & 0 & 0\\
124 0 & \left|\mathbf{p}_{i}\right|^{2} & 0\\
125 0 & 0 & \left|\mathbf{p}_{i}\right|^{2}
126 \end{array}\right]-\left[\begin{array}{ccc}
127 x_{i0}^{2} & x_{i0}x_{i1} & x_{i0}x_{i2}\\
128 x_{i0}x_{i1} & x_{i1}^{2} & x_{i1}x_{i2}\\
129 x_{i0}x_{i2} & x_{i1}x_{i2} & x_{i2}^{2}
130 \end{array}\right]\right)
138 \boldsymbol{\iota}=\sum_{i=1}^{n}m_{i}\left(\left|\mathbf{p}_{i}\right|^{2}\mathbf{I}-\mathbf{p}_{i}\mathbf{p}_{i}^{\top}\right)
146 \boldsymbol{\iota}=\sum_{i=1}^{n}m_{i}\left|\mathbf{p}_{i}\right|^{2}\mathbf{I}-\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\mathbf{p}_{i}^{\top}
154 \boldsymbol{\iota}=\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}^{\top}\mathbf{p}_{i}\mathbf{I}-\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\mathbf{p}_{i}^{\top}
162 \boldsymbol{\iota}=\sum_{i=1}^{n}m_{i}\left(\mathbf{p}_{i}-\boldsymbol{\mu}\right)^{\top}\left(\mathbf{p}_{i}-\boldsymbol{\mu}\right)\mathbf{I}-\sum_{i=1}^{n}m_{i}\left(\mathbf{p}_{i}-\boldsymbol{\mu}\right)\left(\mathbf{p}_{i}-\boldsymbol{\mu}\right)^{\top}
170 \boldsymbol{\iota}=\sum_{i=1}^{n}m_{i}\left(\mathbf{p}_{i}^{\top}-\boldsymbol{\mu}^{\top}\right)\left(\mathbf{p}_{i}-\boldsymbol{\mu}\right)\mathbf{I}-\sum_{i=1}^{n}m_{i}\left(\mathbf{p}_{i}-\boldsymbol{\mu}\right)\left(\mathbf{p}_{i}^{\top}-\boldsymbol{\mu}^{\top}\right)
178 \boldsymbol{\iota}=\sum_{i=1}^{n}m_{i}\left(\mathbf{p}_{i}^{\top}\left(\mathbf{p}_{i}-\boldsymbol{\mu}\right)-\boldsymbol{\mu}^{\top}\left(\mathbf{p}_{i}-\boldsymbol{\mu}\right)\right)\mathbf{I}-\sum_{i=1}^{n}m_{i}\left(\mathbf{p}_{i}\left(\mathbf{p}_{i}^{\top}-\boldsymbol{\mu}^{\top}\right)-\boldsymbol{\mu}\left(\mathbf{p}_{i}^{\top}-\boldsymbol{\mu}^{\top}\right)\right)
186 \boldsymbol{\iota}=\sum_{i=1}^{n}m_{i}\left(\mathbf{p}_{i}^{\top}\mathbf{p}_{i}-\mathbf{p}_{i}^{\top}\boldsymbol{\mu}-\boldsymbol{\mu}^{\top}\mathbf{p}_{i}+\boldsymbol{\mu}^{\top}\boldsymbol{\mu}\right)\mathbf{I}-\sum_{i=1}^{n}m_{i}\left(\mathbf{p}_{i}\mathbf{p}_{i}^{\top}-\mathbf{p}_{i}\boldsymbol{\mu}^{\top}-\boldsymbol{\mu}\mathbf{p}_{i}^{\top}+\boldsymbol{\mu}\boldsymbol{\mu}^{\top}\right)
194 \boldsymbol{\iota}=\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}^{\top}\mathbf{p}_{i}-\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}^{\top}\boldsymbol{\mu}-\sum_{i=1}^{n}m_{i}\boldsymbol{\mu}^{\top}\mathbf{p}_{i}+\sum_{i=1}^{n}m_{i}\boldsymbol{\mu}^{\top}\boldsymbol{\mu}\right)\mathbf{I}-\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\mathbf{p}_{i}^{\top}-\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\boldsymbol{\mu}^{\top}-\sum_{i=1}^{n}m_{i}\boldsymbol{\mu}\mathbf{p}_{i}^{\top}+\sum_{i=1}^{n}m_{i}\boldsymbol{\mu}\boldsymbol{\mu}^{\top}\right)
202 \boldsymbol{\iota}=\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}^{\top}\mathbf{p}_{i}-\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}^{\top}\right)\boldsymbol{\mu}-\sum_{i=1}^{n}m_{i}\boldsymbol{\mu}^{\top}\mathbf{p}_{i}+\left(\sum_{i=1}^{n}m_{i}\right)\boldsymbol{\mu}^{\top}\boldsymbol{\mu}\right)\mathbf{I}-\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\mathbf{p}_{i}^{\top}-\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)\boldsymbol{\mu}^{\top}-\sum_{i=1}^{n}m_{i}\boldsymbol{\mu}\mathbf{p}_{i}^{\top}+\left(\sum_{i=1}^{n}m_{i}\right)\boldsymbol{\mu}\boldsymbol{\mu}^{\top}\right)
210 \boldsymbol{\iota}=\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}^{\top}\mathbf{p}_{i}-\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}^{\top}\right)\boldsymbol{\mu}-\sum_{i=1}^{n}m_{i}\boldsymbol{\mu}^{\top}\mathbf{p}_{i}+\left(\sum_{i=1}^{n}m_{i}\right)\boldsymbol{\mu}^{\top}\boldsymbol{\mu}\right)\mathbf{I}+\left(-\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\mathbf{p}_{i}^{\top}+\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)\boldsymbol{\mu}^{\top}+\sum_{i=1}^{n}m_{i}\boldsymbol{\mu}\mathbf{p}_{i}^{\top}-\left(\sum_{i=1}^{n}m_{i}\right)\boldsymbol{\mu}\boldsymbol{\mu}^{\top}\right)
218 \boldsymbol{\mu}=\frac{{\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}}{{\displaystyle \sum_{k=1}^{n}}m_{k}}=\frac{{\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}}{M}
226 \begin_layout Standard
229 \boldsymbol{\iota}=\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}^{\top}\mathbf{p}_{i}-\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}^{\top}\right)\boldsymbol{\mu}-\sum_{i=1}^{n}m_{i}\boldsymbol{\mu}^{\top}\mathbf{p}_{i}+M\boldsymbol{\mu}^{\top}\boldsymbol{\mu}\right)\mathbf{I}+\left(-\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\mathbf{p}_{i}^{\top}+\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)\boldsymbol{\mu}^{\top}+\sum_{i=1}^{n}m_{i}\boldsymbol{\mu}\mathbf{p}_{i}^{\top}-M\boldsymbol{\mu}\boldsymbol{\mu}^{\top}\right)
237 \boldsymbol{\iota}=\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}^{\top}\mathbf{p}_{i}-\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}^{\top}\right)\frac{{\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}}{M}-\sum_{i=1}^{n}m_{i}\left(\frac{{\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}}{M}\right)^{\top}\mathbf{p}_{i}+M\left(\frac{{\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}}{M}\right)^{\top}\left(\frac{{\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}}{M}\right)\right)\mathbf{I}+\left(-\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\mathbf{p}_{i}^{\top}+\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)\boldsymbol{\mu}^{\top}+\sum_{i=1}^{n}m_{i}\boldsymbol{\mu}\mathbf{p}_{i}^{\top}-M\boldsymbol{\mu}\boldsymbol{\mu}^{\top}\right)
245 \boldsymbol{\iota}=\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}^{\top}\mathbf{p}_{i}-\frac{\left({\displaystyle \sum_{i=1}^{n}m_{i}\mathbf{p}_{i}^{\top}}\right)\left({\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}\right)}{M}-\sum_{i=1}^{n}m_{i}\left(\frac{{\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}}{M}\right)^{\top}\mathbf{p}_{i}+\frac{\left({\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}^{\top}\right)\left({\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}\right)}{M}\right)\mathbf{I}+\left(-\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\mathbf{p}_{i}^{\top}+\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)\boldsymbol{\mu}^{\top}+\sum_{i=1}^{n}m_{i}\boldsymbol{\mu}\mathbf{p}_{i}^{\top}-M\boldsymbol{\mu}\boldsymbol{\mu}^{\top}\right)
253 \boldsymbol{\iota}=\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}^{\top}\mathbf{p}_{i}-\frac{1}{M}\sum_{i=1}^{n}m_{i}\left({\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}^{\top}\right)\mathbf{p}_{i}\right)\mathbf{I}+\left(-\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\mathbf{p}_{i}^{\top}+\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)\boldsymbol{\mu}^{\top}+\sum_{i=1}^{n}m_{i}\boldsymbol{\mu}\mathbf{p}_{i}^{\top}-M\boldsymbol{\mu}\boldsymbol{\mu}^{\top}\right)
261 \boldsymbol{\iota}=\mathbf{I}^{\top}\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}^{\top}\mathbf{p}_{i}-\frac{1}{M}\sum_{i=1}^{n}m_{i}\left({\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}^{\top}\right)\mathbf{p}_{i}\right)^{\top}+\left(-\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\mathbf{p}_{i}^{\top}+\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)\boldsymbol{\mu}^{\top}+\sum_{i=1}^{n}m_{i}\boldsymbol{\mu}\mathbf{p}_{i}^{\top}-M\boldsymbol{\mu}\boldsymbol{\mu}^{\top}\right)
269 \boldsymbol{\iota}=\mathbf{I}^{\top}\left(\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}^{\top}\mathbf{p}_{i}\right)^{\top}-\left(\frac{1}{M}\sum_{i=1}^{n}m_{i}\left({\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}^{\top}\right)\mathbf{p}_{i}\right)^{\top}\right)+\left(-\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\mathbf{p}_{i}^{\top}+\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)\boldsymbol{\mu}^{\top}+\sum_{i=1}^{n}m_{i}\boldsymbol{\mu}\mathbf{p}_{i}^{\top}-M\boldsymbol{\mu}\boldsymbol{\mu}^{\top}\right)
277 \boldsymbol{\iota}=\mathbf{I}\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}^{\top}\mathbf{p}_{i}-\frac{1}{M}\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)^{\top}\left({\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}\right)\right)+\left(\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)\boldsymbol{\mu}^{\top}+\sum_{i=1}^{n}m_{i}\boldsymbol{\mu}\mathbf{p}_{i}^{\top}-\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\mathbf{p}_{i}^{\top}-M\boldsymbol{\mu}\boldsymbol{\mu}^{\top}\right)
285 \boldsymbol{\iota}=\mathbf{I}\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}^{\top}\mathbf{p}_{i}-\frac{1}{M}\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)^{\top}\left({\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}\right)\right)+\left(\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)\left(\frac{{\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}}{M}\right)^{\top}+\sum_{i=1}^{n}m_{i}\left(\frac{{\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}}{M}\right)\mathbf{p}_{i}^{\top}-\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\mathbf{p}_{i}^{\top}-M\left(\frac{{\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}}{M}\right)\left(\frac{{\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}}{M}\right)^{\top}\right)
293 \boldsymbol{\iota}=\mathbf{I}\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}^{\top}\mathbf{p}_{i}-\frac{1}{M}\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)^{\top}\left({\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}\right)\right)+\left(\frac{1}{M}\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)\left({\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}\right)^{\top}+\sum_{i=1}^{n}m_{i}\left(\frac{{\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}}{M}\right)\mathbf{p}_{i}^{\top}-\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\mathbf{p}_{i}^{\top}-\frac{1}{M}\left({\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}\right)\left({\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}\right)^{\top}\right)
301 \boldsymbol{\iota}=\mathbf{I}\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}^{\top}\mathbf{p}_{i}-\frac{1}{M}\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)^{\top}\left({\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}\right)\right)+\left(\frac{1}{M}\sum_{i=1}^{n}m_{i}\left({\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}\right)\mathbf{p}_{i}^{\top}-\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\mathbf{p}_{i}^{\top}\right)
309 \boldsymbol{\iota}^{\top}=\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}^{\top}\mathbf{p}_{i}-\frac{1}{M}\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)^{\top}\left({\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}\right)\right)\mathbf{I}+\left(\frac{1}{M}\sum_{i=1}^{n}m_{i}\left({\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}\right)\mathbf{p}_{i}^{\top}-\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\mathbf{p}_{i}^{\top}\right)^{\top}
317 \boldsymbol{\iota}=\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}^{\top}\mathbf{p}_{i}-\frac{1}{M}\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)^{\top}\left({\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}\right)\right)\mathbf{I}+\left(\frac{1}{M}\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)\left({\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}\right)^{\top}-\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\mathbf{p}_{i}^{\top}\right)
325 \begin_layout Standard
326 --------------------------------------------------------------------------------
327 ----------------------------------------
330 \begin_layout Standard
333 \boldsymbol{\iota}_{n}=\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}^{\top}\mathbf{p}_{i}-\frac{1}{M_{n}}\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)^{\top}\left({\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}\right)\right)\mathbf{I}+\left(\frac{1}{M_{n}}\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)\left({\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}\right)^{\top}-\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\mathbf{p}_{i}^{\top}\right)
341 \boldsymbol{\iota}_{n}=\left(\sum_{i=1}^{n}m_{i}\left|\mathbf{p}_{i}\right|^{2}-\frac{1}{M_{n}}\left|{\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}\right|^{2}\right)\mathbf{I}+\left(\frac{1}{M_{n}}\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)\left({\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}\right)^{\top}-\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\mathbf{p}_{i}^{\top}\right)
349 \boldsymbol{\iota}_{n}=\left(\alpha_{n}-\frac{1}{M_{n}}\boldsymbol{\beta}_{n}^{\top}\boldsymbol{\beta}_{n}\right)\mathbf{I}+\left(\frac{1}{M_{n}}\boldsymbol{\beta}_{n}\boldsymbol{\beta}_{n}^{\top}-\boldsymbol{\gamma}_{n}\right)
357 \boldsymbol{\iota}_{n+1}=\left(\alpha_{n+1}-\frac{1}{M_{n}+m_{n+1}}\boldsymbol{\beta}_{n+1}^{\top}\boldsymbol{\beta}_{n+1}\right)\mathbf{I}+\left(\frac{1}{M_{n}+m_{n+1}}\boldsymbol{\beta}_{n+1}\boldsymbol{\beta}_{n+1}^{\top}-\boldsymbol{\gamma}_{n+1}\right)
365 \boldsymbol{\iota}_{n+1}-\boldsymbol{\iota}_{n}=\left(\alpha_{n+1}-\alpha_{n}\right)\mathbf{I}-\frac{1}{M_{n+1}}\left(\boldsymbol{\beta}_{n+1}^{\top}\boldsymbol{\beta}_{n+1}\mathbf{I}-\boldsymbol{\beta}_{n+1}\boldsymbol{\beta}_{n+1}^{\top}\right)+\frac{1}{M_{n}}\left(\boldsymbol{\beta}_{n}^{\top}\boldsymbol{\beta}_{n}\mathbf{I}-\boldsymbol{\beta}_{n}\boldsymbol{\beta}_{n}^{\top}\right)-\left(\boldsymbol{\gamma}_{n+1}-\boldsymbol{\gamma}_{n}\right)
373 \boldsymbol{\iota}_{n+1}-\boldsymbol{\iota}_{n}=\left(\alpha_{n+1}-\alpha_{n}\right)\mathbf{I}-\frac{M_{n}}{M_{n+1}}\frac{1}{M_{n}}\left(\boldsymbol{\beta}_{n+1}^{\top}\boldsymbol{\beta}_{n+1}\mathbf{I}-\boldsymbol{\beta}_{n+1}\boldsymbol{\beta}_{n+1}^{\top}\right)+\frac{1}{M_{n}}\left(\boldsymbol{\beta}_{n}^{\top}\boldsymbol{\beta}_{n}\mathbf{I}-\boldsymbol{\beta}_{n}\boldsymbol{\beta}_{n}^{\top}\right)-\left(\boldsymbol{\gamma}_{n+1}-\boldsymbol{\gamma}_{n}\right)
381 \boldsymbol{\iota}_{n+1}-\boldsymbol{\iota}_{n}=\left(\alpha_{n+1}-\alpha_{n}\right)\mathbf{I}-\frac{1}{M_{n}}\left[\left(\frac{M_{n}}{M_{n}+m_{n+1}}\boldsymbol{\beta}_{n+1}^{\top}\boldsymbol{\beta}_{n+1}+\boldsymbol{\beta}_{n}^{\top}\boldsymbol{\beta}_{n}\right)\mathbf{I}-\left(\frac{M_{n}}{M_{n}+m_{n+1}}\boldsymbol{\beta}_{n+1}\boldsymbol{\beta}_{n+1}^{\top}+\boldsymbol{\beta}_{n}\boldsymbol{\beta}_{n}^{\top}\right)\right]-\left(\boldsymbol{\gamma}_{n+1}-\boldsymbol{\gamma}_{n}\right)
389 \alpha_{n+1}-\alpha_{n}=\sum_{i=1}^{n+1}m_{i}\left|\mathbf{p}_{i}\right|^{2}-\sum_{i=1}^{n}m_{i}\left|\mathbf{p}_{i}\right|^{2}
397 \alpha_{n+1}-\alpha_{n}=m_{n+1}\left|\mathbf{p}_{n+1}\right|^{2}
405 \frac{M_{n}}{M_{n}+m_{n+1}}\boldsymbol{\beta}_{n+1}^{\top}\boldsymbol{\beta}_{n+1}+\boldsymbol{\beta}_{n}^{\top}\boldsymbol{\beta}_{n}=\frac{M_{n}}{M_{n}+m_{n+1}}\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}+m_{n+1}\mathbf{p}_{n+1}\right)^{\top}\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}+m_{n+1}\mathbf{p}_{n+1}\right)+\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)^{\top}\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)
413 =\frac{M_{n}}{M_{n}+m_{n+1}}\left(\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)^{\top}+m_{n+1}\mathbf{p}_{n+1}^{\top}\right)\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}+m_{n+1}\mathbf{p}_{n+1}\right)+\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)^{\top}\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)
421 =\frac{M_{n}}{M_{n}+m_{n+1}}\left(\boldsymbol{\beta}_{n}^{\top}+m_{n+1}\mathbf{p}_{n+1}^{\top}\right)\left(\boldsymbol{\beta}_{n}+m_{n+1}\mathbf{p}_{n+1}\right)+\boldsymbol{\beta}_{n}^{\top}\boldsymbol{\beta}_{n}
429 =\frac{M_{n}}{M_{n}+m_{n+1}}\boldsymbol{\beta}_{n}^{\top}\boldsymbol{\beta}_{n}+2m_{n+1}\frac{M_{n}}{M_{n}+m_{n+1}}\boldsymbol{\beta}_{n}^{\top}\mathbf{p}_{n+1}+m_{n+1}^{2}\frac{M_{n}}{M_{n}+m_{n+1}}\mathbf{p}_{n+1}^{\top}\mathbf{p}_{n+1}+\boldsymbol{\beta}_{n}^{\top}\boldsymbol{\beta}_{n}
437 =\left(\frac{2M_{n}+m_{n+1}}{M_{n}+m_{n+1}}\right)\boldsymbol{\beta}_{n}^{\top}\boldsymbol{\beta}_{n}+\frac{2m_{n+1}M_{n}}{M_{n}+m_{n+1}}\boldsymbol{\beta}_{n}^{\top}\mathbf{p}_{n+1}+\frac{m_{n+1}^{2}M_{n}}{M_{n}+m_{n+1}}\mathbf{p}_{n+1}^{\top}\mathbf{p}_{n+1}
445 =\frac{1}{M_{n}+m_{n+1}}\left[\boldsymbol{\beta}_{n}^{\top}\left(\left(2M_{n}+m_{n+1}\right)\boldsymbol{\beta}_{n}+2m_{n+1}M_{n}\mathbf{p}_{n+1}\right)+m_{n+1}^{2}M_{n}\mathbf{p}_{n+1}^{\top}\mathbf{p}_{n+1}\right]