Starting from the sum (3.31) and replacing it by the

Step 1 of 2

If we place the square in the plane \(z = 0\), centered on the origin with its sides parallel to the \(x\) and \(y\) axes,

the sum (3.31) takes the form \(I=\sum m_\alpha \rho _{\alpha }^{2}=\sum m_\alpha \left (x_{\alpha }^{2}+y_{\alpha }^{2}  \right )=2\sum m_\alpha x_{\alpha }^{2}\),

where the last expression holds because, by symmetry, the two terms of the previous expression are equal.