A solid sphere of volume V and density is falling through a liquid...

When the sphere attains terminal velocity, it moves with a constant velocity, meaning the net force acting on it is zero.

The forces acting on the sphere are:
1. Downward weight, $W = V\rho g$
2. Upward buoyant force, $F_B = V\sigma g$
3. Upward viscous drag force, $F_v$

For equilibrium at terminal velocity:
$W = F_B + F_v$

Rearranging for the viscous drag force:
$F_v = W - F_B$
$F_v = V\rho g - V\sigma g = V(\rho - \sigma)g$

Answer: $V(\rho - \sigma)g$