John P. Frank, a lawyer and law professor who helped shape the argument in the landmark Brown v. Board of Education case and represented Ernesto Miranda in the United States Supreme Court, died in September 2002 in Scottsdale, Ariz. He was 84. Mr. Frank - then a law professor at Yale - advised Thurgood Marshall of the NAACP Legal Defense and Educational Fund Inc., who argued the Brown case and later joined the court. Mr, Frank earned undergraduate and law degrees from the University of Wisconsin and an advanced degree from the Yale Law School, where he taught from 1949 to 1954. Frank then moved to Arizona, joining the Phoenix firm of Lewis & Roca. At the firm, he represented all sorts of clients, including the Democratic Party of Arizona, unions and corporations. But his most famous client was probably Mr. Miranda, who was convicted on kidnapping and rape charges based largely on his signed confession. In 1966, the Supreme Court overturned the conviction, holding that the confession had not been voluntary because it was not preceded by the now-familiar warnings against self-incrimination, beginning ''You have the right to remain silent.'' Frank was survived by his wife, Lorraine; five children, John Peter Frank of Jefferson, Wis., Gretchen Frank of Los Angeles, Karen Frank of Oakland, Calif., Andrew Frank of San Diego, and Nancy Frank of Oakland; and six grandchildren.