ON THE DIRECT NUMERICAL CALCULATION OF ELLIPTIC FUNCTIONS AND INTEGRALS By LOUIS V. KING, M. A., D. So., P. R. 8. Macdonald Professor of Physic, McQill University, Montreal CAMBRIDGE AT THE UNIVERSITY PRESS 1924 DEDICATED TO THE MEMORY OF JAMES HAKKNESS PETBR REDPATH PROFESSOR OP PURE MATHEMATICS, MG ILL UNIVERSITY 1903-1923 PRINTED IN GREAT BRITAIN PREFACE WHILE lecturing in 1913 on the calculation of self and mutual induction, the writer noticed that the integral in the case of co axial circles was of such a form as to make the application of the arithmetico-geometrical means obvious. An extremely elegant formula Proc. Roy. Soc. A, Vol. c. 1921, p. 63. Appendix, this book, No. 35 when put to numerical test justified the method as one well adapted to computation. The calculation of more complex cases of self and mutual induction led to several interesting formulae for the complete third elliptic integral. With a view to making a complete study of the group of recurrence formulae associated with the A. O. M. scales, further work was postponed until a vacation in 1921, when many of the results in cluded in this book were obtained, and were thought by the writer to be new. On later consulting the Collected Works of Legendre, Gauss and Jacobi, the writer found that he had rediscovered many known formulae, but that quite a number, likely to be of service in computation, appeared to be hitherto unknown. In the summer of 1922 the writer decided, as a vacation task, to include in a single monograph the entire theory of elliptic functions associated with the A. G. M. scales, thus com pleting, and to some extent adding to, the work of Gauss left unpublished after his death, and placing before mathematicians in accessible form a mode of approach to Elliptic Function Theory directly related to the art of machine-computation. It is interesting to note in this connection that this subject was to have formed the content of the third volume of Halphens Fonctions Ettiptiqws, unfortunately left incomplete at the time of his death. While the present monograph contains many new formulae and methods of computation, no claim is made as to novelty in fundamental analytical treatment. References to more important books and memoirs consulted by the author are given in foot-notes. A complete bibliography is not given since the reader will find exhaustive references in the Royal Society Index, 1800-1900, Vol. I, Mathematics, and after 1900 in tha mathematical volumes of the International Catalogue qf Sciential Literature. VI PREFACE The author desires to express his thanks to his late mathematical col league, Professor James Harkness of McGill University, for reading over the manuscript and proofs, and also to Mr Arthur Berry of Kings College, Cambridge, for revising proofs, verifying formulae and for calling the writers attention to several slips of calculation in the main text and appendix. Finally, the author wishes to thank his colleague, Dr A. S. Eve, F. R. S., for his aid in obtaining from McGill University that financial assistance which has made possible the publication of this little volume. For the courtesy of the officers of the Cambridge University Press in undertaking to print this book in its usual impeccable style, the author is also deeply grateful. For assistance in the final revision of the proofs, the author is indebted to one of his students, Mr W. L. Robertson. L. V. K. McGiLL UNIVERSITY, April 18, 1924 CONTENTS SECTION PAGK I. INTRODUCTION i II. HISTORICAL NOTE ON LANDENS TRANSFORMATION AND THE VARIOUS SCALES OF MODULI AND AMPLI TUDES 2 III. ON THE SCALE OF ARITHMETICO-GEOMETRICAL MEANS J IV. LANDENS SCALE OF INCREASING AMPLITUDES tun f n l - f n b n a n hin l n . . . 7 i Calculation of F fa k E lc A and K ... 7 ii Calculation of an ,, on ,, dn v, in terms of the argument u . i V. THE HYPERBOLIC SCALE OF INCREASING AMPLITUDES tanh n 1 - n b n a n tanh f n . . . . 10 i Calculation of F j, tt, , etc - .....