Let circle C be the image of ${x^2} + {y^2} - 2x + 4y - 4 - 0$ in the line $2x - 3y + 5 = 0$ and A be the point on C such that OA is parallel to x-axis and A lies on the right hand side of the centre O of C. If $B(\alpha ,\beta )$ , with $\beta < 4$ , lies on C such that the length of the arc AB is ${(1/6)^{{\rm{th }}}}$ of the perimeter of C, then $\beta - \sqrt 3 \alpha $ is equal to