The Vernier constant (VC), or least count, is defined as the difference between the value of one main scale division (MSD) and one vernier scale division (VSD) .

1. Establish the relationship between VSD and MSD: The problem states that $(N+1)$ VSDs coincide with $N$ MSDs. Therefore, $1 \text{ VSD} = \frac{N}{N+1} \text{ MSD}$.

2. Apply the Vernier Constant formula: $VC = 1 \text{ MSD} - 1 \text{ VSD}$ $VC = 1 \text{ MSD} - \left(\frac{N}{N+1}\right) \text{ MSD}$ $VC = \text{MSD} \left(1 - \frac{N}{N+1}\right) = \frac{\text{MSD}}{N+1}$.

3. Substitute given values and convert units: We are given $1 \text{ MSD} = 0.1 \text{ mm}$. To find the constant in cm, we use the conversion $1 \text{ mm} = 0.1 \text{ cm}$, thus $0.1 \text{ mm} = 0.01 \text{ cm}$ . $VC = \frac{0.01}{N+1} \text{ cm} = \frac{1}{100(N+1)} \text{ cm}$.