mhih<iiUhhh\>''J.\.<l'<ir U ). If f^ SAUTHSONIAN MISCELLANEOUS COLLECT IONS Volume 74^ Number 1 SMITHSONIAN MATHEMATICAL FORMULAE AND TABLES OF ELLIPTIC FUNCTIONS Mathematical Formulae Prepared by EDWIN P. ADAMS, Ph.D. PROFESSOR OF PHYSICS, PRINCETON UNIVERSITY Tables OF Elliptic Functions Prepared under the Direction of Sir George Greenhill, Bart. COL. R. L. HIPPISLEY, C.B. PUBLICATION 2672 CITY OF WASHINGTON PUBLISHED BY THE SMITHSONIAN INSTITUTION 1922 ^f 2\;,'\%^ Qh4' ^12 .o^^>|f c4^'^t'h.^ ADVERTISEMENT Xhe Smithsonian Institution has maintained for many years a group of publications in the nature of handy books of Information on geographical, meteorological, physical, and mathematical subjects. Xhese include the Smithsonian Geographical Xables (third edition, reprint, 1918); the Smithsonian Meteorological Xables (fourth revised edition, 1918); the Smithsonian Physical Xables i,se\-enth revised edition, 1921); and the Smithsonian Mathematical Xables: Hyperbolic Functions (second reprint, 1921). Xhe present volume comprises the most important formulae of many branches of applied mathematics, an iUustrated discussion of the methods of mechanical Integration, and tables of elliptic functions. Xhe volume has been compiled by Dr. E. P. -\dams, of Princeton University. Prof. F. R. Äloulton, of the Univer- sity of Chicago, contributed the section on numerical Solution of differential equations. Xhe tables of elliptic functions were prepared by Col. R. L. Hippisley, C. B., under the direction of Sir George Greenhill, Bart., who has contributed the introduction to these tables. Xhe Compiler, Dr. Adams, and the Smithsonian Institution are indebted to many physicists and mathematicians, especially to Dr. H. L. Curtis and col- leagues of the Bureau of Standards, for advice, criticism, and Cooperation in the preparation of this volume. Charles D. Walcott, Secretary of the Smithsonian Institution. May, ig22. PREFACE Xhe original object of this coUection of mathematical formulae was to bring together, compactly, some of the more useful results of mathematical analysis for the benefit of those who regard mathematics as a tool, and not as an end in itseK. Xhere are many such results that are diSicult to remember, for one who is not constantly using them, and to find them one is obliged to look through a number of books which may not immediately be accessible. A cöUection of formulae, to meet the object of the present one, must be largely a matter of individual selection; for this reason this volume is issued in an interleaved edition, so that additions, meeting individual needs, may be made, and be readily available for reference. It was not originally intended to include any tables of functions in this volume, but merely to give references to such tables. An exception was made, however, in favor of the tables of elliptic functions, calculated, on Sir George Greenhin's new plan, by Colonel Hippisley, which were fortunately secured for this volume, inasmuch as these tables are not otherwise available. In Order to keep the volume within reasonable bounds, no tables of indefinite and definite Integrals have been included. For a brief coUection, that of the late Professor B. O. Peirce can hardly be improved upon; and the elaborate coUection of definite Integrals by Bierens de Haan show how inadequate any brief tables of definite Integrals would be. A short hst of useful tables of this kind, as weU as of other volumes, having an object similar to this one, is appended. Should the plan of this coUection meet with favor, it is hoped that suggestions for improving it and making it more generaUy useful may be received. Xo Professor Moulton, for contributing the chapter on the Numerical Integration of Differential Equations, and to Sir George Greenhill, for his Intro­ duction to the Xables of Elliptic Functions, I wish to express my gratitude. And I wish also to record my obligations to the Secretary of the Smithsonian In­ stitution, and to Dr. C. G. Abbot, Assistant Secretary of the Institution, for the way in which they have met all my suggestions with regard to this volume. E. P. Adams Peinceton, New Jersey COLLECXIOXS OF AL-VXHEMAXICAL FORMULAE, EXC. B. O. Peirce: .\ Short Xable of Integrals, Boston, 1899. G. Petit Bois: Xables d'Integrales Indefinies, Paris, 1906. X. J. I'A. Bromwich: Elementary Integrals, Cambridge, 1911. D. BiEREXS DE Haax: XouveUes Xables d'Integrales Definies, Leiden, 1867. E. J-AHNKE and F. Emde: Funktionentafeln mit Formeln und Kurven, Leip2ig, 1909. G. S. Carr: -A. Synopsis of Elementary Results in Pure and Applied Mathe­ matics, London, 1880. W. Lask-a: Sammlung von Formeln der reinen und angewandten Mathematik, Brannschweig, 1888-1894. W. Ligowski: Xaschenbuch der Mathematik, Berlin, 1893. O. Xh. BtJRKiEX: Formelsammlung und Repetitorium der Mathematik, Berlin, 1922. F. .\tJERB-ACH: Xaschenbuch für Mathematiker und PhysDier, i. Jahrgang, 1909. Leip2ig, 1909. CONTENTS PAGE SYiEBOLS viii I. Algebr.\ I IL Geometry 29 III. Trigoxometry . 61 I\'. VeCTOR .\XALYSIS gi V. CraviLiXE-AR Coördinates 99 VI. Ineinite Sektes 109 VII. Speclal .Applications of Analysis 145 \XII. DlFEERENTLAL EqU-ATIONS 162 IX. DiTFEREXTiAL Equations (Continued) .... 191 X. X'uMERiC-A.L Solution of Differential Equations ... .220 XI. Elliptic Fl-nctioxs . .243 Introduction by Sir George Greenhill, F.R.S. 245 Xables of EUiptic Functions, by Col. R. L. Hippisley . 259 Index 311 SYMBOLS log logarithm. "Whenever used the Naperian logarithm is understood. Xo find the common logarithm to base lo: logio a = 0.43429 . log a. log a = 2.30259 . logio a. 1 Factorial. n\ where n is an integer denotes 1.2.3.4 n. Equivalent notation lü: =1^ Does not equal. > Greater than. < Less than. ^ Greater than, or equal to. ^ Less than, or equal to. Binomial coeflScient. See 1.51- © —> Approaches. I öi/i I Determinant where a« is the element in the jth row and ^th column, ^, ^'—'''"'{ Functional determinant. See 1.37. 3(«i, Xi ) I a \ Absolute value of a. If 0 is a real quantity its numerical value, without regard to sign. If 0 is a complex quantity, a = a + iß, \ a \ = modulus of a = +-\/a^ + ß^. i Xhe imaginary = -t-V — i. ^ Sign of summation, i.e., ^aj = ßi 4- Oa + «s + • . • -h ßn- II Product, i.e., J | (i -1- Tix) = (i -f x)(i + 2a;) (i -h 3x) . (i + nx). I. ALGEBRA 1.00 Algebraic Identities. I. a" - b" = {a - b)[,a'--^ + a"--b + a"~%' -f- . + ab"-- + ö"-i)- 2. a" ± 6» = (a -1- 6)((j"-i - a"--b + a"-^b- - T ab"-- ± b"-^). n odd: upper sign. n even: lower sign. 3. {x + ai)(.v -h a-j) .... (.^ + fln) = a;" + PiX""! -1- P^"-^ -f- . + Pn-iX -I- P„. Pi = öl -I- 02 + . + a„. Pi = sum of aU the products of the a's taken ^ at a time. P„ = 010203 . ffn. 4. (a^ + i2)(a2 -F j82) = (aa T &j3)2 -1- {aß ± &a)2. 5- (a- - 62)(a2 - jS^) = (aa ± bßf - (a/3 ± ba)\ 6. (a^ -h 6^ -F c2)(a2 -h jS^ -i- 7') = (aa + 6/3-1- ct)^ + {by - ßcf + (ca - 70)^ -1- (a/3 - aby. 7. (a^ + 62 + c^ -1- (f2)(a2 -f /32 -H 72 + S'O = (aa -f- 6(3 + c7 -1- dbf + {aß -ba + c8- dyy + (07 -bd-ca + dßf + {ab + by - cß - da)\ 8. (ac - bdY + {ad + bcf = (ac + 6(f)2 + {ad - bcf. 9. {a -f- 6)(6 -h c)(c + a) = (a + 6 -h c)(ö6 + bc + ca)-abc. IG. (a + h){b + c){c + fl) = 0^(6 -[- c) -|- b\c + a) + c^C« -1- 6) -|- 2tt6c. II. (a -f J)(6 -f c){c -h a) = 6c(6 -f- c) -1- ca(c + a) + ab{a + b) + 2abc. 12. 3(a + b){b + c)(c + a) = {a + b + cy- {a^ + W + (?). 13. {b -a){c- a){c -b) = a\c - b) + b\a - c) + c^b - a). 14. {b - a){c - a){c - 6) = a(62 - c^) -|- &(c2 - a^) + c(a2 - 6^). 15. (6 - a)(c - a)(c - b) = bc{c - b) + ca{a - c) + ab{b - a). 16. (a - by +{b- cY + (c - a)2 = 2C(a - 6)(a - c) + {b - a){b - c) + {c-a){c- 6)]. 17. aä(62 - c^) + ¥{c^ - a^) + (?{a^ - 6«) = (a - &)(& - c)(a - c){ab + bc + ca). 18. (a -t- & + c){a^ + b^ + c2) = bc{b + c) + ca{c + a) + ab{a + b)+a^ + P + c\ 19. (a + ö -I- c){bc + ca + ab) = a^b -\-c) + b\c + a) + c^a •\-h)+ 2,abc. 20. {b + c- a){c + a- b){a + b - c) = a^{b + c) + ¥{c + a) + c\a + b) -(a3 + ¥ + (? + 2abc). 2 MATHEMATICAL PORMUL^ AND ELLIPTIC EUNCTIONS 21. (a + 6 + c)( - a + 6 -I- c){a -b + c){a + b - c) = 2(6V + cV + aW) -{a' + ¥ + (^). 22. {a + b + c + d)''+{a + b-c-df+{a + c-b-d)^+{a + d-b-c)^ = 4(a2 + 62 + c2 + ^2). If A = aa + by + cß B = aß + ba + cy C = ay + bß + ca 23. (a + 6 + c){a + ß + y)=A+B + C. 24. [«2 -1- &2 _^ c2 - (a6 + bc + ca)] [a^ + ß^ + y^ - {aß + ßy-\- 7a)] = ^2 + .B2 + C2 - {AB + BC + CA). 25. (a= -h 63 + c3 - 3a6c)(a' + jS« + 7« - 30)87) = ^^ + .B' -|- C - a^-BC. algebraic equations 1.200 Xhe expression f{x) = aox" + aix"~i + a^x"-'^ + + a,j_ix + a„ is an integral rational function, or a polynomial, of the «th degree in x. 1.201 Xhe equation }{x) = o has n roots which may be real or complex, dis- tinct or repeated. 1.202 If the roots of the equation f{x) = o are Ci, C2, . ., Cn, j{x) = ao(x — Ci){x — o^ {x — Cn) 1.203 Symmetrie functions of the roots are expressions giving certain com- binations of the roots in terms of the coefficients.