'On 31 January 1770, Cook claimed possession of Queen Charlotte Sound, South Island New Zealand, “and the adjacent lands” in the name of King George III.'
He had been instructed about another Continent :
'You are also with the Consent of the Natives to take Possession of Convenient Situations in the Country in the Name of the King of Great Britain'.
https://www.foundingdocs.gov.au/resources/transcripts/nsw1_doc_1768.pdfPrior to his departure for New South Wales, Phillip received his Instructions (composed by Lord Sydney) from King George III, dated 25 April 1787.
'Instructions for Our Trusty George R and well beloved Arthur Phillip Esq. Our Captain General and Governor in Chief, in and over
Our Territory of New South Wales. With these Our Instructions you will receive Our Commission under Our Great seal constituting and appointing you to be Our Captain General and Governor in Chief of
Our Territory called New South Wales extending from the Northern Cape or Extremity of the Coast called Cape York'.
However, there was no property of the king.'Previously Cook had landed on the Australian mainland in five places yet there is
no mention of a claim of annexation in any of these places in his journals or those written by his officers. This indicates that the Possession Island entry was made after the event by others without sufficient investigation.
Why did Cook take possession on what is now known as Possession Island, which was clearly inhabited, without any acknowledgement of the native people?
In Banks’s journal there is a suggestion that Cook had first called this famous landmark Passage Isle and then conveniently renamed it Possession Island.
Nowhere in Cook’s original journal is there mention of the name New South Wales, so again this was clearly added at a later date.
Sydney Parkinson who was watching his Captain and party ashore writes: 'The captain, and some others, went to the top of a hill, and, seeing a clear passage, hoisted a jack, and fired a volley, which was answered by the marines below, and the marines by three vollies from the ship, and three cheers from the main shrouds'. This indicates the significance of the event was after days of being locked by reef strewn passages they had at last found a way clear and were on their way home, and had nothing to do with annexation.
https://navyhistory.org.au/possession-island/So Governor Phillip had no legal property in Sydney for the convict colony. The only place available was NZ.
https://www.alamy.com/hills-and-farmland-tory-channel-queen-charlotte-sound-marl...