The Beatles had recorded two sessions for the BBC radio show Side By Side just three days earlier, but such was the demand for more music from them that they recorded a third on this day.
The recording of I'll Be On My Way is the only one by The Beatles known to exist. It was written by Lennon-McCartney for Billy J Kramer and the Dakotas, another act managed by Brian Epstein. The group recorded it at Abbey Road on 14 March as the b-side to their cover of The Beatles' Do You Want To Know A Secret.
This episode of Side By Side was broadcast from 5pm on Monday 24 June 1963 on the BBC Light Programme. I'll Be On My Way, meanwhile, was released in 1994 on the Live At The BBC collection.