The annexation of the Cook Islands, specifically Rarotonga, by New Zealand in 1901 was influenced by several key conditions and circumstances surrounding the Rarotonga ariki (chiefs). Here are the main points regarding the conditions under which the ariki agreed to annexation:

1. Political Instability: The Cook Islands were experiencing significant political instability and internal conflict among the various tribes and chiefs. This instability made the idea of annexation appealing to some leaders who believed that New Zealand's governance could bring order and stability.

2. Protection from External Threats: The ariki were concerned about potential threats from foreign powers, particularly Germany and France, who were expanding their influence in the Pacific. By agreeing to annexation, the Rarotonga ariki hoped to secure protection under New Zealand's administration.

3. Economic Considerations: The ariki recognized that annexation could lead to economic benefits, including improved trade opportunities and access to resources. They believed that New Zealand's governance could facilitate development and improve living conditions for their people.

4. Formal Agreement: The formal agreement for annexation was reached after discussions between the ariki and representatives of the New Zealand government. The ariki were assured that their customs and rights would be respected under the new administration.

5. Desire for Self-Governance: Some of the ariki saw annexation as a pathway to greater self-governance. They hoped that by aligning with New Zealand, they could eventually gain more autonomy and control over their affairs.

6. Cultural Considerations: The ariki were also motivated by a desire to preserve their cultural identity and traditions. They sought assurances that their customs would be respected under New Zealand rule.

The annexation was formalized on September 11, 1901, and while it brought some benefits, it also led to significant changes in governance and the imposition of foreign laws, which had lasting impacts on the Cook Islands' society and culture.

For more detailed historical context, you may refer to resources on the history of the Cook Islands and New Zealand's colonial policies.