Bernoulli's equation relates the pressure, velocity, and height in a flowing fluid and is derived from the principle of conservation of energy. The correct form of Bernoulli's equation is:

$$P + \rho g h + \frac{1}{2} \rho v^2 = \text{constant}$$

where:

• P is the pressure within the fluid.
• $\rho$ is the density of the fluid.
• g is the acceleration due to gravity.
• h is the height above a reference point.
• v is the velocity of the fluid.

Looking at the options given:

1. Option A: $$P + \frac{1}{2} \rho g h + \frac{1}{2} \rho v^2 = \text{constant}$$
2. Option B: $$P + m g h + \frac{1}{2} m v^2 = \text{constant}$$
3. Option C: $$P + \rho g h + \rho v^2 = \text{constant}$$
4. Option D: $$P + \rho g h + \frac{1}{2} \rho v^2 = \text{constant}$$

Option D correctly represents Bernoulli's equation in its proper form. Therefore, the correct answer is:

Option D