During WWII DF stood for Defensive Fire or Direction Finding (by radio) (see HF/DF - Royal Navy High Frequency Radio Direction Finding, WW2). Can't help with the abbreviation I but it generally stood for Intelligence. Electronic warfare was in it's early stages so it might refer to the development of gunnery linked to radio direction finding.
Thanks Kevin. We are thinking along the same lines. This was the 1920's so I'm thinking now that it stands for Director Firing and Instruction, but I can't be certain. There is no explanation in the Navy List that I can find, but the abbreviation must have been well understood by people 100 years ago