Introduction:
In the given problem, we have an overhanging beam ABC with AB = L and BC = a, where C is the free end. The beam is subjected to a vertical load W at the free end. We are required to determine the location where the maximum moment occurs.

Explanation:
To find the location of the maximum moment, we need to analyze the bending moment diagram of the beam.

1. Bending Moment Distribution:
- As the beam is subjected to a vertical load W at the free end, the bending moment distribution in the beam will be triangular.
- The maximum bending moment occurs at the point where the load is applied, which is at point B.

2. Calculation of Bending Moment at B:
- We can calculate the bending moment at point B by using the equation M = W*a, where M is the bending moment, W is the load, and a is the distance from the point of support to the point where the load is applied.
- In this case, the load is applied at the free end C, so the distance a is equal to BC.
- Therefore, the bending moment at B is M = W*BC.

3. Comparison of Bending Moments:
- Now, let's compare the bending moments at different points on the beam.
- The bending moment at A can be calculated using the equation M = W*L, where L is the total length of the beam.
- The bending moment at C can be calculated using the equation M = W*0, as there is no distance between the load and point C.
- Since BC is shorter than AB, the bending moment at B (M = W*BC) will be greater than the bending moment at A (M = W*L) and C (M = W*0).
- Therefore, the maximum bending moment occurs at point B.

Conclusion:
In conclusion, the maximum moment in the overhanging beam ABC occurs at point B, where the load is applied. This is because the bending moment is directly proportional to the distance from the point of support to the point where the load is applied. As BC is shorter than AB, the bending moment at B is greater than at any other point on the beam.