She became a central figure in College life, particularly during the eight years she served as Vice-Principal. Not just within the College but more widely in the University, she was valued for her calmness, her willingness to take on extra burdens, her staunch common sense, elegant drafting, and ability to keep objectives clearly in mind.

... introduced the Payne-Rayner inequality, an isoperimetric inequality for the first eigenfunction of the Laplacian. More specifically, it gives an exact lower bound of the first eigenvalue by means of some integral norms of the first eigenfunction, and also says that the first eigenfunction satisfies a reverse Hölder inequality. This paper and a second which extended the result to higher dimensions stimulated further advances in isoperimetric problems by others.

An isoperimetric inequality is obtained which relates the $L_{1}$ and $L_{2}$-integrals of the first eigenfunction in the problem of the vibrating clamped membrane.

Previous results of the authors [2] relating the $L_{1}$ and $L_{2}$ integrals of the first eigenfunction in the two dimensional fixed membrane problem are here extended to the analogous problem in higher dimensions.

In many books on elementary mechanics, it is assumed that velocity is a vector quantity on the grounds that it has magnitude and direction. It is, however, very far from obvious that these are sufficient reasons and a rather more thorough investigation of the nature of velocity may provide not only a clearer idea of what velocity is but may also clear up some of the difficulties associated with "relative" velocity.

Alan quite correctly believed our common research interests would ensure joint work would eventuate. That worked exceedingly well of course. Our joint paper and best work on 'Variational Methods for Nonlinear Eigenvalue Problems associated with Thermal Ignition' was, in a strange way, an early forerunner of the now common path-following techniques for such problems. It led me, with others, to develop algorithms to characterise, and estimate, thresholds for self-ignition. Margaret was also a wonderful teacher and mentor.

I went to Oxford for interview. St Hilda's College was my first choice College and Oxford was my first choice University, so I was very worried that I might not be accepted. I very much wanted to make a good impression. St Hilda's College had two mathematics tutors, Margaret Rayner and Irene Moore. They were called Miss Rayner and Miss Moore, although they both had doctorates. Miss Rayner was tutor in Applied Mathematics and was the more senior. ... Lectures at the Mathematical Institute in St Giles' did not require me to wear my gown, but individual tutorials, with either Miss Rayner or Miss Moore at St Hilda's College did. ... Tutorials with Miss Rayner [were] a fearsome intellectual experience, taken in pairs, where the work of the preceding week was taken apart and new challenges were attacked. [After the results of the final examinations were announced] I had a nice letter from Miss Rayner congratulating me. She said that I had done well, except for one paper, and was at the lower end of the Firsts.

Although she modified her earliest ambition to teach in a school, she never lost her interest in school education. One of her first University committees was at the Department of Educational Studies and she had her first training in chairing a large meeting in connection with teacher training in the Area Training Organisation. Her concern with school education was kept alive by work with the Mathematical Association and by membership of a number of governing bodies of schools, both independent and state maintained. For many years she was chief examiner in mathematics for the International Baccalaureate and so became aware of developments in curricula and in assessment in Europe and North America. She found this an exciting task, drawing up syllabuses that would be acceptable to universities all over the world. It was an experience of the greatest value when she later joined the Secondary Examinations Council (and even later the School Examinations and Assessment Council) and took part in approving syllabuses for GCSE and A-level.

... became aware of developments in curricula and in assessment in Europe and North America. It was an experience of the greatest value when she later joined the Secondary Examinations Council and, later still, the Schools Examination and Assessment Council concerned with curriculum development.

1990 Birthday Honours. Commander of the Order of the British Empire (CBE), Miss Margaret Eva Rayner, lately Vice-Chairman, Schools Examination and Assessment Council; Vice-Principal, St Hilda's College.

This chapter opens with a summary of the state of Oxford mathematics at the beginning of the 20th century, with Professors Esson, Elliott, and Love occupying the mathematical Chairs. A wind of change blew across the University with the appointment of the Cambridge mathematician G H Hardy, who held the Savilian Chair of Geometry for eleven years. The chapter concludes with a description of the major rise in mathematical activity following the Second World War.

Dr Rayner was an admired mathematician and a much-loved tutor. In retirement, she turned historian and was the author of the College's centenary history. She will be missed greatly by her former colleagues, students, and friends at St Hilda's.

We are very sad to announce the death of Emeritus Fellow and former Chronicle Editor Dr Margaret Rayner, who continued to be a wonderful source of knowledge and advice after handing over her Editor's red pen to us three years ago. We will ... remember her as an inspiring educator, a dear friend, and a St Hilda's woman to the core.

Margaret Rayner, who has died at the age of 89, was a respected mathematician and much loved tutor, a highly valued member of both St Hilda's College as well as the wider academic community in Oxford, who was known nationally and internationally for her work on school examinations. ... The characteristics that come to mind most immediately when thinking about Margaret are her intellectual curiosity, her energy, her capacity for making friends and her generous hospitality. She cared about people and, in return, they cared about her and were fond of her. She will be much missed.