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