We had vainly tried to trace the fourth letter, which Mrs. Inglethorp had written on the evening preceding her death. 
Mrs. Inglethorp was not a letter writer so she could not have been the author of the fourth letter.
mnli hypothesis: Mrs. Inglethorp was not a letter writer so she could not have been the author of the fourth letter. premise: We had vainly tried to trace the fourth letter, which Mrs. Inglethorp had written on the evening preceding her death. 