Let the coefficients of the middle terms in the expansion of $\left( \frac { 1 } { \sqrt { 6 } } + \beta x \right) ^ { 4 } , ( 1 - 3 \beta x ) ^ { 2 }$ and $\left( 1 - \frac { \beta } { 2 } x \right) ^ { 6 } , \beta > 0$, respectively form the first three terms of an A.P. If $d$ is the common difference of this A.P., then $50 - \frac { 2 d } { \beta ^ { 2 } }$ is equal to $\_\_\_\_$.