The energy due to torsion is

For torsion, Hooke's Law states that the angular displacement is proportional to the torque, resulting in a straight-line plot of torque vs the angle. The area under this graph is the energy:

Essentially, I'll equate these two formulas. Since the energy is supplied at A, , where  is the angle that portion AB (portion BC is free) of shaft ABC turns through.

From the analysis in Chapter 11, the energy stored in AB is

To find the energy stored in the other shaft, I need the torque. The forces between the gears are equal but opposite, so

Using this...

The total energy is equated to the energy from Hooke's Law: