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途中計算は、こちら。
\[
\left( {n_x}^2-n^2+{s_x}^2 n^2 \right) \left( {n_y}^2-n^2+{s_y}^2 n^2 \right) \left( {n_z}^2-n^2+{s_z}^2 n^2 \right) + {s_x}^2 {s_y}^2 {s_z}^2 n^6 + {s_x}^2 {s_y}^2 {s_z}^2 n^6 \\
\quad - \left( {n_x}^2-n^2+{s_x}^2 n^2 \right) {s_y}^2 {s_z}^2 n^4 - \left( {n_y}^2-n^2+{s_y}^2 n^2 \right) {s_x}^2 {s_z}^2 n^4 - \left( {n_z}^2-n^2+{s_z}^2 n^2 \right) {s_x}^2 {s_y}^2 n^4 \\
= \left( {n_x}^2-n^2 \right) \left( {n_y}^2-n^2 \right) \left( {n_z}^2-n^2 \right) + \left( {n_x}^2-n^2 \right) \left( {n_y}^2-n^2 \right) {s_z}^2 n^2 + \left( {n_y}^2-n^2 \right) \left( {n_z}^2-n^2 \right) {s_x}^2 n^2 + \left( {n_z}^2-n^2 \right) \left( {n_x}^2-n^2 \right) {s_y}^2 n^2 \\
\quad + \left( {n_x}^2-n^2 \right) {s_y}^2 {s_z}^2 n^4
+ \left( {n_y}^2-n^2 \right) {s_z}^2 {s_x}^2 n^4
+ \left( {n_z}^2-n^2 \right) {s_x}^2 {s_y}^2 n^4
+ {s_x}^2 {s_y}^2 {s_z}^2 n^6 + {s_x}^2 {s_y}^2 {s_z}^2 n^6 + {s_x}^2 {s_y}^2 {s_z}^2 n^6 \\
\quad - \left( {n_x}^2-n^2 \right) {s_y}^2 {s_z}^2 n^4
- \left( {n_y}^2-n^2 \right) {s_z}^2 {s_x}^2 n^4
- \left( {n_z}^2-n^2 \right) {s_x}^2 {s_y}^2 n^4
- {s_x}^2 {s_y}^2 {s_z}^2 n^6 - {s_x}^2 {s_y}^2 {s_z}^2 n^6 - {s_x}^2 {s_y}^2 {s_z}^2 n^6 \\
= \left( {n_x}^2-n^2 \right) \left( {n_y}^2-n^2 \right) \left( {n_z}^2-n^2 \right)
+ \left( {n_x}^2-n^2 \right) \left( {n_y}^2-n^2 \right) {s_z}^2 n^2
+ \left( {n_y}^2-n^2 \right) \left( {n_z}^2-n^2 \right) {s_x}^2 n^2
+ \left( {n_z}^2-n^2 \right) \left( {n_x}^2-n^2 \right) {s_y}^2 n^2 \\
= {n_x}^2{n_y}^2{n_z}^2 -n^2{n_y}^2{n_z}^2-n^2{n_z}^2{n_x}^2-n^2{n_x}^2{n_y}^2+n^4{n_x}^2+n^4{n_y}^2+n^4{n_z}^2-n^6 \\
\quad + {s_z}^2 n^2{n_x}^2{n_y}^2-{s_z}^2 n^4{n_y}^2 - {s_z}^2 n^4{n_x}^2+{s_z}^2 n^6
+ {s_x}^2 n^2{n_y}^2{n_z}^2-{s_x}^2 n^4{n_z}^2 - {s_x}^2 n^4{n_y}^2+{s_x}^2 n^6
+ {s_y}^2 n^2{n_z}^2{n_x}^2-{s_y}^2 n^4{n_x}^2 - {s_y}^2 n^4{n_z}^2+{s_y}^2 n^6 \\
= {n_x}^2{n_y}^2{n_z}^2 \left( {s_x}^2 +{s_y}^2 +{s_z}^2 \right)
+ \left( {s_z}^2 - 1 \right) n^2{n_x}^2{n_y}^2
+ \left( {s_x}^2 - 1 \right) n^2{n_y}^2{n_z}^2
+ \left( {s_y}^2 - 1 \right) n^2{n_z}^2{n_x}^2-n^6 \\
\quad +\left( 1-{s_y}^2 - {s_z}^2 \right) n^4{n_x}^2
+\left( 1-{s_z}^2 - {s_x}^2 \right) n^4{n_y}^2
+\left( 1-{s_x}^2 - {s_y}^2 \right) n^4{n_z}^2
+\left( {s_z}^2 +{s_x}^2 +{s_y}^2 \right) n^6 \\
= {s_x}^2{n_x}^2{n_y}^2{n_z}^2 +{s_y}^2{n_x}^2{n_y}^2{n_z}^2 +{s_z}^2{n_x}^2{n_y}^2{n_z}^2
- \left( {s_x}^2 + {s_y}^2 \right) n^2{n_x}^2{n_y}^2
- \left( {s_y}^2 + {s_z}^2 \right) n^2{n_y}^2{n_z}^2
- \left( {s_z}^2 + {s_x}^2 \right) n^2{n_z}^2{n_x}^2-n^6 \\
\quad +{s_x}^2 n^4{n_x}^2
+{s_y}^2 n^4{n_y}^2
+{s_z}^2 n^4{n_z}^2
+ n^6 \\
= \left( {n_y}^2{n_z}^2 - n^2{n_z}^2 - n^2{n_y}^2 + n^4 \right) {s_x}^2{n_x}^2
+\left( {n_z}^2{n_x}^2 - n^2{n_x}^2 - n^2{n_z}^2 + n^4 \right) {s_y}^2{n_y}^2
+\left( {n_x}^2{n_y}^2 - n^2{n_y}^2 - n^2{n_x}^2 + n^4 \right) {s_z}^2{n_z}^2 \\
= \left( {n_y}^2 - n^2 \right) \left( {n_z}^2 - n^2 \right) {s_x}^2{n_x}^2
+\left( {n_z}^2 - n^2 \right) \left( {n_x}^2 - n^2 \right) {s_y}^2{n_y}^2
+\left( {n_x}^2 - n^2 \right) \left( {n_y}^2 - n^2 \right) {s_z}^2{n_z}^2
\]
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