When a ball is released from rest in a very long column of viscous liquid, its downward acceleration is 'a' (just after release). Then its acceleration when it has acquired two third of the maximum velocity: 

When the ball is just released, the net force on ball is ${\mathrm{W}}_{\mathrm{eff}}(= \mathrm{mg} - \mathrm{buoyant} \mathrm{force})$
The terminal velocity $\text{'}{\mathrm{v}}_{\mathrm{f}}\text{'}$ of the ball is attained when net force on the ball is zero

$\therefore$ Viscous force $6\mathrm{\pi \eta r} {\mathrm{v}}_{\mathrm{f}} = {\mathrm{W}}_{\mathrm{eff}}$
When the ball acquires $\frac{2}{3}$rd of its maximum velocity 

${\mathrm{v}}_{\mathrm{f}}$
The viscous force is = $\frac{2}{3}{\mathrm{W}}_{\mathrm{eff}}$
Hence net force is ${\mathrm{W}}_{\mathrm{eff}}-\frac{2}{3}{\mathrm{W}}_{\mathrm{eff}} = \frac{1}{3}{\mathrm{W}}_{\mathrm{eff}}$
$\therefore$ Required acceleration = $\frac{\mathrm{a}}{3}$