To find the vernier constant, we need to understand the relationship between the main scale and the vernier scale.$\\$Given:$\\$• $(N + 1)$ divisions of the vernier scale coincide with $N$ divisions of the main scale.$\\$• 1 main scale division (MSD) represents $0.1$ mm.$\\$Let's denote:$\\$• $$Length of $N$ main scale divisions $= N \times 0.1$ mm.$\\$• $$Length of $(N + 1)$ vernier scale divisions $= (N + 1) \times$VSD$, \\$where VSD is the length of one vernier scale division.$\\$Since $N$ main scale divisions coincide with $(N + 1)$ vernier scale divisions:$\\ N \times 0.1 = (N + 1) \times$VSD$\\$Solving for VSD:$\\$VSD$= \frac{N \times 0.1}{N + 1} \\$The vernier constant (VC) is the difference between one main scale division and one vernier scale division:$\\$VC$=$MSD$-$VSD$\\$Substitute the values:$\\$VC$= 0.1 - \frac{N \times 0.1}{N + 1} \\$Simplify:$\\$VC$= \frac{0.1(N + 1) - 0.1N}{N + 1} \\$VC$= \frac{0.1N + 0.1 - 0.1N}{N + 1} \\$VC$= \frac{0.1}{N + 1} \\$Convert VC from mm to cm:$\\$VC$= \frac{0.1}{N + 1} \times \frac{1}{10}$ cm$\\$VC$= \frac{0.01}{N + 1}$ cm$\\$The correct option is:$\\$Option 4: $100(N+1) \\$This corresponds to the vernier constant being $\frac{0.01}{N + 1}$ cm, which matches the given option.