Hello GuybrushThreepwood,
Sorry for the delay in getting back to you.
I've had a quick look through a number of the records that I have access to. I can't find a great deal on them in the main staff records held at the National Archives. There is a gap in the main joining records from Apr. 1857 and Feb. 1869 and the registers of leavers doesn't start until 1889. I think the best place to check for information on the Farmers would be the Police Orders, which are held under MEPO 7. Or you could try the Metropolitan Police Historical Collection, to see if they have any information.
With regard to John Farmer. I have found a John Farmer, who joined the police force in 1880 with warrant number 64304 and is shown as dying in 1902. Could this be your chap?
Sorry I can't be of more help.
Yours, BarnabyP