Lord Wellesley, the Governor-General of India from 1798 to 1805, played a crucial role in expanding the British Empire in India. He employed several methods to achieve this objective, including:

1. Subsidiary Alliance:
One of the most significant methods used by Lord Wellesley was the Subsidiary Alliance system. Under this system, the British East India Company entered into treaties with various Indian princely states. According to these treaties, the Indian rulers were required to accept British protection and support in exchange for a subsidiary force stationed in their territory. The princely states also had to agree to various other conditions, such as allowing British troops to pass through their lands and accepting British control over their foreign relations. This system allowed the British to effectively control and influence the internal affairs of these states, eventually leading to their incorporation into the British Empire.

2. Outright Wars:
Lord Wellesley also engaged in several military campaigns and outright wars to expand British territory in India. For example, he launched the Fourth Anglo-Mysore War in 1799, which resulted in the defeat of Tipu Sultan and the annexation of his kingdom of Mysore. Similarly, he initiated the Second Anglo-Maratha War in 1803, which led to the British gaining control over large parts of central India.

3. Assumption of territories of previously subordinated rulers:
Lord Wellesley also took advantage of the political instability and succession disputes among Indian rulers to assume control of their territories. For instance, he annexed the territories of the Nizam of Hyderabad and the Nawab of Carnatic after they failed to support the British during the Fourth Anglo-Mysore War. This expansion of British control over Indian territories further strengthened the British Empire in India.

4. Doctrine of Lapse:
Lord Wellesley also implemented the controversial Doctrine of Lapse, which stated that if an Indian ruler died without a natural heir, his territory would be annexed by the British. This policy was used to annex several princely states, including Satara, Sambalpur, and Jhansi. However, it was met with resistance and criticism from Indian rulers and nationalist leaders.

In conclusion, Lord Wellesley employed a combination of methods, including the Subsidiary Alliance system, outright wars, assumption of territories, and the Doctrine of Lapse, to expand the British Empire in India. These methods allowed the British to gain control over numerous princely states and extend their dominion over large parts of the Indian subcontinent.