Table Of ContentThe Collected Papers of William Burnside
Volume 1
Professor Burnside, by Walter Stoneman, 1917
(By courtesy of the National Portrait Gallery, London)
The Collected Papers of William Burnside
Volume 1: Commentary on Burnside's life and work;
Papers 1883-1899
Edited by
Peter M. Neumann
The Queen's College, Oxford
A. J. S. Mann
University of Greenwich
and
Julia C. Tompson
The editors and publisher acknowledge the generous
support of the London Mathematical Society in the
production of these volumes.
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PREFACE
William Burnside (1852-1927) was a mathematician with a very wide range of interests. Although he is
remembered now mainly as a pioneer of group theory and representation theory, this edition of his work is
intended to be comprehensive, to show the complete mathematician contributing to applied mathematics, to ana
lysis, to geometry, to probability and to statistics as well as to algebra. Within the history of modem algebra it
takes its place alongside the collected works of numerous of his contemporaries, such as Camille Jordan, Felix
Klein, F. G. Frobenius, Issai Schur, L. E. Dickson. We are particularly pleased that it is published on the centenary
of the appearance of one of Burnside's most famous papers, the one in which, using character theory, he finally
managed to prove the paq.8-Theorem-the theorem that finite groups whose orders are divisible by at most two
different prime numbers are soluble, a theorem which until 1904 had been a conjecture whose proof using purely
group-theoretic methods had eluded him and his contemporaries for many years.
The papers are arranged in more-or-less chronological order, as listed in 'A bibliography of William Burnside
(1852-1927)' by Ascher Wagner and Verity Mosenthal, Historia Mathematica, Vol 5, 1978, pp. 307-312. Their
labelling is based on the year in which items were thought to have been written. In order that there should not be
two labelling systems in the literature we have used theirs with a few small changes. Those few changes (mainly
additions) are quite natural and should, we believe, cause no confusion.
We have not reprinted Burnside's books. It was tempting to include the 1897 edition of his Theory of groups of
finite order, which is of considerable historical interest and is not widely available now, but it had to be excluded
for reasons of space. The second edition (1911) of that book was reprinted in 1955 by Dover Publications, Inc.,
and is still readily available. The Introduction to the theory ofp robability, which was published posthumously in
1928, has also been reprinted by Dover, in 1959.
Burnside's best known papers speak for themselves. Nevertheless, they need to be read in context. That context
is in two parts-the contemporary mathematical scene and future developments-and our edition contains essays
covering both aspects. It also contains as much personal information about Burnside as is presently available: we
reprint the 1928 obituary by A. R. Forsyth and publish a new article based on recent research by Martin Everett
and colleagues.
Although, as we have said, Burnside's papers speak for themselves, the reader may appreciate a little help in
the use of this edition. The papers have been reproduced photographically and the original pagination has been
maintained (and is recorded in the foot-line of each page). Some papers were photo-reproduced from reprints
in the possession of Pembroke and St John's Colleges, Cambridge, others from bound volumes of the journals.
Because of the need not to break old bindings there are some cases where there is a little distortion of the text on
the page. We hope and believe however that the content is nevertheless readable.
Papers published in some journals (Messenger of Mathematics, Quarterly Journal of Pure and Applied Math
ematics, and one or two others) do not carry the date of receipt of the manuscript. Papers in the Proceedings of
the London Mathematical Society always carried a date when the paper was 'read' and often also carried a date
of receipt. In some of these cases the date when the paper was received or read was replaced in the reprint by a
line saying where the paper was 'extracted from', and we have then given the received and read information in
the footnote to the first page of the paper. This information has been checked with some care and we hope that
readers who need it find it reliable.
Because the papers are the original published text they retain the original misprints and other slips. In a couple
of cases papers have been photo-reproduced from reprints which carry a very few manuscript emendations. In a
few other cases Burnside published errata. We believe we have traced and included all of these.
Burnside's writing is clear and taut. Modem readers should have little difficulty with it once they have mastered
the technical terminology of his time-which is not very different from that of our own. There are, perhaps, one
vi PREFACE
or two points where a word of warning might be helpful. Not all French mathematicians have first initial M. In
his rendering of French names Burnside used M. quite properly to mean Monsieur (the English equivalent is Mr).
And when he wrote "Borchardt's Journal" he was using the convention that a journal was known by its editor.
Nowadays it is always referred to as Mathematische Annalen.
We owe debts of gratitude to a considerable number of people and institutions. First, Walter Feit (who for several
decades has pressed for an edition of Burnside's work), June Barrow-Green, Charlie Curtis, Mike Newman, Ron
Solomon, Martin Everett and Kathy Young whom we thank for their contributions and their patience. We thank
the London Mathematical Society for financial and moral support. We thank Susan Oakes, Administrator of the
London Mathematical Society, for help with finding photographs. We are very grateful to the Master and Fellows
of Pembroke College (Cambridge) and of St John's College (Cambridge) and to the librarians of those colleges
and of the Radcliffe Science Library (Oxford) and the Mathematical Institute (Oxford) for permission to use
their material and for practical assistance. We are very grateful to staff and former staff at OUP-particularly
to Elizabeth Johnston, Alison Jones, Richard Lawrence, Anita Petrie, and the many others working behind the
scenes-who have coped so well with what turned out to be a much more complicated and lengthy project than
they or we had anticipated.
This project was considerably more than half completed when our colleague Julia Tompson found herself
compelled to withdraw for personal reasons. By this time she had done a huge amount of editorial and organisational
work, both as a member of OUP and as a scholar and historian of mathematics. We are most grateful to her and,
in recognition of the very large amount of time and effort she invested, we have insisted that she remain listed as
co-editor.
AJSM, PMN: Greenwich and Oxford, March 2004
ACKNOWLEDGEMENTS
The papers have been reproduced from journals and reprints in the collections of Pembroke College, Cambridge,
St John's College, Cambridge, the Radcliffe Science Library, Oxford, and the Mathematical Institute, Oxford,
by permission of the Master and Fellows of Pembroke College, Cambridge, the Master and Fellows of St John's
College, Cambridge, and the librarians of these libraries. The obituary by Forsyth is reprinted from the Journal of
the London Mathematical Society, 3 (1928) by permission of the Society.
Picture credits
Frontispiece (Vol. 1) photograph by Walter Stoneman, by courtesy of the National Portrait Gallery, London.
Portrait p. 789 by permission of the Master and Fellows of Pembroke College, Cambridge.
Portraits Vol. 2, p. ii and p. 111 by permission of the London Mathematical Society.
Photograph of Burnside's tankard (p. 90) by courtesy of Jean Marshall.
Photographs of Pembroke College Cambridge (p. 92) by permission of the Master and Fellows of Pembroke
College, Cambridge.
Photograph of Lady Margaret Eight, 1872 (p. 93) by permission of the Master and Fellows of St John's College,
Cambridge.
Photograph of the Royal Naval College (p. 96) by courtesy of the University of Greenwich.
Reproduction of the illuminated address (pp. 98 and 99) by permission of the Master and Fellows of Pembroke
College, Cambridge.
Reproduction of Burnside's letter to Larmor (pp. 102 and 103) by permission of the Master and Fellows of
St John's College, Cambridge.
Photograph of Nine Croft, Poolewe (p. 104) by courtesy of Jean Marshall.
Photographs of Burnside's grave (p. 107) by Tony Mann.
CONTENTS
VOLUMEl
Preface v
Acknowledgements vii
List of contributors xv
Commentary on Burnside's Life and Work 1
Foreword 3
Water Feit
Bibliography of William Burnside 7
The context of Burnside's contributions to group theory 15
Peter M. Neumann
A still unsettled question 39
M. F. Newman
Burnside and finite simple groups 45
Ronald M. Solomon
Burnside's research on representation theory 55
Charles W. Curtis
Burnside's applied mathematics 63
June Barrow-Green
'William Burnside', reprinted from the Journal of the London Mathematical Society, Vol. 3 (1928) 71
A. R. Forsyth
Notes on Burnside's life 89
Martin G. Everett, A. J. S. Mann and Kathy Young
The Papers 1883-1899 111
1
[1883a] The elliptic functions of K, &c 113
[1883b] Note on centre of pressure of a plane polygon 117
[1884a] On certain spherical harmonics 119
[1887a] On the trisection of the periods for Weierstrass's elliptic functions 125
[1887b] On the partition of energy between the translatory and rotational motions of a set of
non-homogeneous elastic spheres 129
[1888a] On a simplified proof of Maxwell's theorem 137
X CONTENTSOFVOLUMEl
[1888b] Note on the potential of an elliptic cylinder 141
[1888c] On deep-water waves resulting from a limited original disturbance 147
[1889a] On the small wave-motions of a heterogeneous fluid under gravity 163
[1889b] Mathematical notes 1: Geometrical interpretation of a condition of integrability 169
[1889c] Mathematical notes 2: Propagation of energy in the electro-magnetic field 171
[1889d] The lines of zero length on a surface as curvilinear coordinates 173
[1889e] On the resultant of two finite displacements of a rigid body 179
[1890a] On the differential equation of confocal sphero-conics 185
[1890b] On the surfaces whose lines of curvature are all plane 189
[1890c] On a property of plane isothermal curves 197
[1890d] Note on a paper relating to the theory of functions 203
[1891a] On a case of streaming motion 207
[1891b] Note on the addition theorem for hyperbolic functions 209
" [1891c] On a property of linear substitutions 213
[1891d] On functions determined from their discontinuities, and a certain form of boundary condition 217
[1891e] On a certain Riemann's surface 231
[1891t] On the form of closed curves of the third class 239
[1891g] Algebraical notes. I: On the Jacobian of two quadratics. II: On a system of simultaneous
equations 241
[1891h] Two notes on Weierstrass's P(u). 1: Forms of the addition equation. 2: On P(u) considered
as a covariant of a quartic 245
[1891k] On a class of automorphic functions 249
[1892a] On the form of hyperelliptic integrals of the first order, which are expressible as the sum
of two elliptic integrals 289
[1892b] On the application of Abel's theorem to elliptic integrals of the first kind 303
[1892c] On the linear transformation of the elliptic differential 311
[1892d] Prof. Burnside's paper on the partition of energy, R.S.E., July 1887 319
[1892e] Further note on automorphic functions 321
[1892t] Note on pseudo-elliptic integrals 337
[1892g] On the division of the periods of elliptic functions by 9 345
[1892h] Note on the equation y2 = x(x4 - 1) 353
[1893a] On a problem of conformal representation 357
[1893b] Note on linear substitutions 377
[1893c] On the finite displacement of a rigid body 381
[1893d] Note on functions of a real variable 385
[1893e] Note on the theory of groups 389
[1893t] On the curve of intersection of two quadrics 397
[1893g] Notes on the theory of groups of finite order. I: On the proof of Sylow's Theorem. II: On the
possibility of simple groups whose orders are the products of four primes 401
[1893h] On a property of certain determinants 411
[1894a] On Green's function for a system of non-intersecting spheres 415
[1894b] On an application of the theory of groups to Kirkman's problem 423
[1894c] On a class of groups defined by congruences 431
[1894d] On a system of linear congruences 459
[1894e] On certain composite groups 467
[1894t] On the kinematics of non-Euclidean space 483
[1894g] Note on ternary substitutions of determinant unity with integral coefficients 507
[1894h] On a class of groups defined by congruences ( second paper) 511
[1895a] Notes on the theory of groups of finite order (Notes III-VII) 561
