Htykuut

In the "Introduction" section of the paper Support-Vector Networks, it mentioned Fisher's solution to a model of two normal distributed populations:

My questions are:

1. How to derive equation (1)? I even doubt that it should be:
   $F_{sq} (x) = sign left [ left ( x - m_1 right ) ^T Sigma_1 ^{-1} left ( x - m_1 right ) - left ( x - m_2 right ) ^T Sigma_2 ^{-1} left ( x - m_2 right ) + ln { dfrac { left | Sigma_1 right |} { left | Sigma_2 right | } } right ]$
   
   because according to Linear discriminant analysis, the solution is:
   




2. Why the number of free parameters in equation (1) is $dfrac {n (n + 3)} {2}$?
   
   In my opinion, $m_1, m_2, Sigma_1, Sigma_2$ are all free parameters, because
   $Sigma_1, Sigma_2$ are symmetric matrices, so the number should be $n + n + dfrac {n left (n + 1 right )}{2} + dfrac {n left (n + 1 right )}{2} = n left (n + 3 right )$




3. Why the number of free parameters in equation (2) is $n$?
   
   We can rewrite Equation (2) as $F_{sq} left ( X right ) = WX + b$, so $W$ and $b$ are both free parameters, then the number should be $n + 1$.