Many thanks to all those who helped me with this, particularly DebbieG and Spendlove.
I had previously thought that Martin Farnell Esq. from Coton-in-the-elms (abt 1750-1815), who is the object of this search, and his wife Frances Matthews had one daughter, Phoebe.
However, it appears from his will that Phoebe was Frances' daughter from a previous marriage. This is stated in one place and, in another place, he refers to Phoebe as his adopted daughter. I supposed this would mean that Matthews was probably the surname of Frances' first husband and that Phoebe was born as Phoebe Matthews. Based on this, (bingo) I found her on the IGI christened on 9 May 1796 at Lullington, Derbyshire, mother Frances Matthews.
This is probably not new to most people on Rootschat, but it shows that one has to look at all sorts of documents to find the information one is searching for.
What I'm wondering now is is whether such an adoption would have been an official thing in those days or not?
Alan