maximum and minimum moduli, and functions of finite order in an angle. In 1936, Cartwright became director of studies in mathematics at Girton College. In 1938, she began work on a new project which had a major impact on the direction of her research. The Radio Research Board of the Department of Scientific and Industrial Research produced a memorandum regarding certain differential equations which came out of modelling radio and radar work. They asked the London Mathematical Society if they could help find a mathematician who could work on these problems and she became interested. The dynamics lying behind the problems were unfamiliar to Cartwright, so she approached Littlewood for help with this aspect. They began to collaborate studying the equations, which greatly surprised the two: The fine structure described here is today seen to be a typical instance of the butterfly effect. The collaboration led to important results which have greatly influenced the direction that the modern theory of dynamical systems has taken. Although the duo did not supply the answer in time, they succeeded in directing the engineers' attention away from faulty equipment towards practical ways of compensating for the electrical "noise" - or erratic fluctuations - being produced. In 1945, Cartwright simplified Hermite's elementary proof of the irrationality of . She set her version of the proof as a Tripos question, later published in an appendix to Sir Harold Jeffreys' book Scientific Inference. In 1947, she was elected to be a Fellow of the Royal Society; although she was not the first woman to be elected to that Society, she was the first female mathematician. Cartwright was appointed Mistress of Girton in 1948 and a Reader in the Theory of Functions in Cambridge in 1959 until 1968. From 1957 to 1960, she was president of the Cambridge Association of University Women. After retiring from Girton, she was a visiting professor at Brown University from 1968 to 1969 and at Claremont Graduate School from 1969 to 1970. Cartwright died in Cambridge, on 3 April 1998 at the age of 97. Recognition Cartwright was the first woman: to serve on the Council of the Royal Society to be President of the Mathematical Association (in 1951) to be President of the London Mathematical Society (in 1961–62) to receive the Sylvester Medal (in 1964) to receive the De Morgan Medal, the highest award of the London Mathematical Society (in 1968) In