Problem Statement:
A wedge of mass m and angle of inclination θ is attached to a massless pulley. The surface between the wedge and the mass M is frictionless. The system is in equilibrium. We are required to find the compression in the spring under equilibrium conditions.

Solution:
To solve this problem, we will consider the forces acting on each component of the system and use the conditions of equilibrium.

Forces Acting on the Wedge:
1. Weight (mg): The weight of the wedge acts vertically downwards and can be resolved into two components:
- Perpendicular component: mg * cos(θ)
- Parallel component: mg * sin(θ)

2. Normal Force (N): The normal force exerted by the surface on the wedge acts perpendicular to the surface. Its magnitude is equal to the perpendicular component of the weight: N = mg * cos(θ)

3. Tension in the String (T): The tension in the string acts horizontally and is equal to the force required to balance the parallel component of the weight of the wedge: T = mg * sin(θ)

Forces Acting on the Mass (M):
1. Weight (Mg): The weight of the mass M acts vertically downwards.

2. Tension in the String (T): The tension in the string acts vertically upwards and is equal to the force required to balance the weight of the mass: T = Mg

Conditions of Equilibrium:
For the system to be in equilibrium, the sum of the forces acting on each component of the system should be zero.

Equilibrium Conditions for the Wedge:
1. In the vertical direction:
- N - mg * cos(θ) = 0

2. In the horizontal direction:
- T - mg * sin(θ) = 0

Equilibrium Conditions for the Mass (M):
1. In the vertical direction:
- T - Mg = 0

Compression in the Spring:
From the equilibrium condition for the mass (M), we have:
T = Mg

Substituting this value of T into the equilibrium condition for the wedge, we get:
Mg - mg * sin(θ) = 0

Simplifying the equation, we find:
M = m * sin(θ)

The compression in the spring, denoted by x, is related to the force exerted by the spring (F) and the spring constant (k) through Hooke's Law:
F = kx

Since the force exerted by the spring is equal to the weight difference between the wedge and the mass, we have:
F = (m - M)g

Equating the two expressions for F, we get:
kx = (m - M)g

Substituting the values of M and m from the equilibrium condition, we have:
kx = (m - m * sin(θ))g

Simplifying the equation and solving for x, we find:
x = (1 - sin(θ)) * g / k

Therefore, the compression in the spring under equilibrium conditions is given by:
x = (1 - sin(θ)) * g / k