There were two merchant ships named ARACHNE around in 1840. one was registered in Portsmouth. The ship you are looking for was registered in Liverpool. Her Port Rotation number was 62/7662. Port number 62 was Liverpool. No key exists to connect the Port Rotation Number to the name of a ship.
Port 39 was Exeter and port 64 was London.
Suggest you scour Lloyd's register 1839 to 1841 to see if you can find a ship with that name or similar.
https://hec.lrfoundation.org.uk/archive-library/lloyds-register-of-ships-onlineBe aware that Lloyd's reg doesn't contain details of all ships.
I would also email Liverpool Maritime Museum to see if they can help.
Newspapers can also give some information.. Several Liverpool newspapers reported that a vessel named as ARACHNE arrived in Liverpool from Mauritius 08/04/1840. ( Lloyd's List says 07/04/1840) No mention as to where she was registered but the dates fit.
Nettie. The ship we are looking for is a merchant ship. the ship you refer to is a Royal Navy vessel.
Crew Agreements 1840 of Liverpool ships can be obtained from the National Archives
https://discovery.nationalarchives.gov.uk/details/r/C4064706 For total confimation that this is the correct vessel. You would need to visit Kew to view. They reopen 21st July.