The question asks us to identify the form of emergency that is not mentioned in the Constitution of India. Let`s look at each option:

Option 1: National Emergency. This is mentioned in the Constitution of India under Article 352. It can be declared when there is a threat to the security of the country, whether it is due to war, external aggression, or armed rebellion.

Option 2: State Emergency in terms of Proclamation of President`s Rule in a State. This is also mentioned in the Constitution of India under Article 356. It allows the President to impose direct Central government control in a state in certain situations where there is a breakdown of the constitutional machinery.

Option 3: Financial Emergency. This is mentioned in the Constitution of India under Article 360. It can be declared when there is a threat to the financial stability or credit of India.

Option 4: Health Emergency. This option is not mentioned in the Constitution of India. While the Constitution does provide for certain emergency provisions, such as National Emergency, State Emergency, and Financial Emergency, it does not specifically mention a Health Emergency.

Therefore, the correct answer is option 4.