Multiple and Submultiple Prefixes The following prefixes are those recom- mended by the International Committee on Weights and Measures. They have been adopted by the National Bureau of Standards and will find wide use in written technical communication . Prefix Multiple Symbol Pronunciation tera W^ T ter' a giga 109 G ji'ga mega 10« M meg' a kilo 103 k kll'o hecto 102 h hek'to deka 10 Submultiple da dek'a deci io-» d des' 1 centi 10-2 c sen' ti milli 10-3 m mil' 1 micro io-« /A mi' kro nano 10-8 n nan' 6 pico 10-12 P pe'c6 femto 10-15 f fem' to atto io-i« a at' to CORNELL UNIVERSITY LIBRARY 924 067 404 974 Cornell University Library The original of this book is in the Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924067404974 SMITHSONIAN MISCELLANEOUS COLLECTIONS 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 ADVERTISEMENT The 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. These include the Smithsonian Geographical Tables (third edition, reprint, 1918); the Smithsonian Meteorological Tables (fourth revised edition, 1918); the Smithsonian Physical Tables (seventh revised edition, 1921); and the Smithsonian Mathematical Tables: Hyperbolic Functions (second reprint, 192 1). The present volume comprises the most important formulae of many branches of applied mathematics, an illustrated discussion of the methods of mechanical integration, and tables of elliptic functions. The volume has been compiled by Dr. E. P. Adams, of Princeton University. Prof. F. R. Moulton, of the Univer- sity of Chicago, contributed the section on numerical solution of differential equations. The 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. The 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 The original object of this collection 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. There are many such results that are difficult 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 collection 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 readUy 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 elhptic functions, calculated, on Sir George GreenhiU'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 collection, that of the late Professor B. O. Peirce can hardly be improved upon; and the elaborate collection of definite integrals by Bierens de Haan show how inadequate any brief tables of definite integrals would be. A short fist of useful tables of this kind, as well as of other volumes, having an object similar to this one, is appended. Should the plan of this collection meet with favor, it is hoped that suggestions for improving it and making it more generally useful may be received. To Professor Moulton, for contributing the chapter on the Numerical Integration of Differential Equations, and to Sir George Greenhill, for his Intro- duction to the Tables 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 Princeton, New Jersey COLLECTIONS OF MATHEMATICAL FORMULAE, ETC. B. O. Peirce: A Short Table of Integrals, Boston, 1899. G. Petit Bois: Tables d'lntegrales Indefinies, Paris, 1906. T. J. I'A. Bromwich: Elementary Integrals, Cambridge, 1911. D. BiERENS de Haan: NouveUes Tables d'lntegrales Definies, Leiden, 1867. E. Jahnke and F. Emde: Funktionentafeln mit Formeln und Kurven, Leipzig, 1909. G. S. Carr: a Synopsis of Elementary Results in Pure and Applied Mathe- matics, London, 1880. W. Laska: Sammlung von Formeln der reinen und angewandten Mathematik, Braunschweig, 1888-1894. W. LiGOWSKi: Taschenbuch der Mathematik, Berlin, 1893. O. Th. Burklen: Formelsammlung und Repetitorium der Mathematik, Berhn, 1922. F. Auerbach: Taschenbuch fur Mathematiker und Physiker, i. Jahrgang, 1909. Leipzig, 1909. CONTENTS pXdE SysffiOtS . ^ //.. ^ /.-.•. ^ . .■ , viii I. Algebra i II. Geometry 29 III. Trigonometry 61 IV. Vector Analysis 91 V. Curvilinear Coordinates 99 VI. Infinite Series 109 VII. Special Applications of Analysis 145 VIII. Differential Equations 162 EX. Differential Equations (Continued) 191 X. Numerical Solution of Differential Equations 220 XI. Elliptic Fxinctions ... 243 Introduction by Sir George Greenhill, F.R.S. 245 Tables of Elliptic Functions, by Col. R. L. Hippisley ... 259 Index 311 SYMBOLS log logarithm. Whenever used the Naperian logarithm is understood. To find the common logarithm to base lo : logio a = 0.43429 ... log a. log a = 2.30259 . . . logio a. ! Factorial, nl where n is an integer denotes 1.2.3.4 Equivalent notation 12 ^ Does not equal. > Greater than. < Less than. ^ Greater than, or equal to. ^ Less than, or equal to. Binomial coefficient. See 1.51- © — > Approaches. \ aik\ Determinant where aik is the element in the ith row and jfeth column, .,, ^' — ' " ' ' \ Functional determinant. See 1.37. d{Xi, X2 ) I a I Absolute value of a. If a is a real quantity its "numerical value, without regard to sign. If o is a complex quantity, a = a -\- i/3, I a I = modulus of a = -fVa^ + ;8^. i The imaginary = -fV — i. k=n 2 Sign of summation, i.e., ^ak = ai + 02 + as + ■ . . ■ + an. t=n ■T T Product, i.e., 11 (i + T^x) = (i + a;)(i + 2x)(i + 3a;) . . . . (i + nx). I. ALGEBRA 1.00 Algebraic Identities. 1. a" -b" = (a- 6)(o''-i + o"-26 + a^'W + + ab"-^ + b"'^). 2. a" ± 6" = (a + 6)(fl"-^ - a"-^6 + a^'-W - T a6"-2 ± j»-i). w odd: upper sign. n even: lower sign. 3. (x + Oi)(x + 02) (a; + a„) = X" + PiX"-i + PaX"-^ + . . . . + Pn-iX + P„. Pi = ai + a2+ .:.... +an. Pk = sum of all the products of the a's taken ^ at a time. P» = aia2a3 . ... an- 4. (a^ + 62)(a2 + ,82) = {aa T 6/3)2 + (^^ ^ 5^)2. 5. (a2 _ J2)(a2 - /32) = (aa ± b^f - (a/3 ± baf. 6. (o2 + 62 + c2)(a2 + /32 + 72) = (aa + 6/3 + cyf + (67 - /3c)2 + (ca - 7a)2 + (a/3 - a6)2. 7. (a2 + 62 + c2 + cP){a^ + ,32 + 72 + 52) = (^^ + ^,^3 + c7 + (^S)2 + (a;8 - 6a + c5 - £^7)2 + (a7 - 66 - ca + d^f + (a5 + 67 - cjS - c > pi, where c is the root sought. The equation can always be transformed into one in which this condition holds by multipl5dng or dividing the roots by some power of 10 by 1.231. Diminish the roots by pi by 1.233. In the transformed equation Ao(x - px)" + Ai(x - /-i)"-! + .... + Ar^iix - pi) +An = o put p2 ^ An 10 An-1 and diminish the roots by ^2/10, yielding a second transformed equation Bo(x -pi- ^j" + Bi(x -pi- ^V-i + ....+ 5„ = o. 8 MATHEMATICAL FOEMULvE AND ELLIPTIC FUNCTIONS If Bn and Bn-i are of the same sign pi was taken too large and must be dimin- ished. Then take p3 ^ Bn lOO £,_i and continue the operation. The required root will be: c = p, + h^P^ + ^ lO lOO 1.262 Graeffe's Method. This method determines approximate values of all the roots of a numerical equation, complex as well as real. Write the equation of the wth degree /(x) = Oo»" - aiic""! + cW"''' — . . . . ± a„ = o. The product fix)-f{-x) = Aox^'' - ylix2"-2 + ^2a;'"-* - . . . . ± An = o contains only even powers of x. It is an equation of the nth degree in x^. The coefficients are determined by Ao = ai A\ = a-? — 20002 Ai = ai — 201^3 + 2^004 A% = 03" — 202a4 + 2aiai — 2ao(ie Ai = a^ — za^a^ + 2a2a6 — 2aiaj + 2aoa8 The roots of the equation Aoy" - Aiy"-^ + Aiy"-^ - . . . . ± ^„ = o are the squares of the roots of the given equation. Continuing this process we get an equation RoW - i?iM"-i + RiU"-^ - . . . .±Rn = o whose roots are the 2''th powers of the roots of the given equation. Put X = 2'. Let the roots of the given equation be ci, d, . . . . , c„. Suppose first that Ci > C2 > C3 > > c„ Then for large values of X, If the roots are real they may be determined by extracting the Xth roots of these quantities. Whether they are ± is determined by taking the sign which approximately satisfies the equation f{x) = o. Suppose next that complex roots enter so that there are equahties among the absolute values of the roots. Suppose that I Ci I ^ i C2 I ^ I C3 U . . . . ^ I C^ I ; I Cp I > I Cp+1 I ; I <^p+i I ^ I C3H-2 I ^ . . . . 5: I C„ I ALGEBRA 9 Then if X is large enough so that Cp^ is large compared to Cp+.i^, Ci^, c^, .... c^ approximately satisfy the equation: Rau^ - Riu^-^ + i?2M»'~^ - . . . .±Rp = o and Cjh-i^, Cp+2'^, . . . , Cn'^ approximately satisfy the equation: i2j,M"-P - i2j,+lM"-''-l + /?p+2M"-3'-2 - . . . . ± i2„ = o. Therefore when X is large enough the given equation breaks down into a number of simpler equations. This stage is shown in the process of deriving the suc- cessive equations when certain of the coefficients are obtained from those of the preceding equation simply by squaring. References : Encyklopadie der Math. Wiss. I, i, 3a (Runge). Bairstow : Applied Aerodynamics, pp. 553-560; the solution of a numerical equation of the 8th degree is given by Graeffe's Method, 1.270 Quadratic Equations. The roots are: 1.271 Cubic equations. Substitute where X •? + 2ax ■\-b = o. X\= —a + s/c^ — b X2 = —a — y/c^ — b X1 + Xi = - 20 XiXi = b. a^ > b roots are real, a? < b roots are complex, a^ — b roots are equal. (x) x^ + ax^ + bx -\- c = o. (2) X = y — 3 (3) y' - 2,Py-2q = o 3* = b ^^ 3 2? Roots of (3) : a p>o, q>o, q^> p^ cosh 4> = o& 2 , 3 27 V^3 lO MATHEMATICAL FOEMUL^E AND ELLIPTIC FUNCTIONS yi = 2-\/l> cosh — 3 % = h iy/Jp sinh — ^ 2> yz= — — — i-\/-ip sinh — • ^p>o,q f, cosh d) = — = y/f <\> yi = - 2\/* cosh — 3 % = — — + ^V^ sinh — 2 ""^ 3 ya = — ^^ — iy/zP sinh — • 2 3 If ^ < o sinh 4> = —7= li p>o, q^ < p^ yi = 2-s/ — * sinh — 3 yi . / 1 4> yi= h «v — 3* cosh — 2 3 yi • / , 4> yz = w — '\P cosh — • 2 3 cos (/) = ^ Vi'^ yi = 2vi' cos — y2 = - — + V3^ sin — 2 3 ys = V^ sin — ■ 23 1.272 Biquadratic equations. flo*:* + a\X^ + (hy? + asi*; + ^4 = 0, Substitute X = y ALGEBRA II H = aaCh. — a^ G = a^az — 300^102 + 2ai' F = ao^cii - 40o^aia3 + 6aoai^a2 - 2,ai^ I = 00^4 — 4aia3 + 3^^ F = ati"! - 3£P / = aoOiai + 2a\aiai — a(,ai — a-^a^ — a^ A = P — 2']P = the discriminant G^ + 4^=* = aaKHI - aoJ). Nature of the roots of the biquadratic: A = o Equal roots are present Two roots only equal: T and / are not both zero Three roots are equal : I = J = o Two distinct pairs of equal roots: G = o; ail — i2H^ = o Four roots equal : H = I = J = o. A < o Two real and two complex roots A > o Roots are either all real or aU complex: H < o and a^?! — 12^?^ < o Roots aU real H > o and Oo^I - i2H^ > o Roots all complex. DETERMINANTS 1.300 A determinant of the wth order, with n^ elements, is written: A flu ffll2 ^13 (l21 022 023 asi O32 O33 Oln 02n O371 Onl On2 On3 O-nn Oij I , [i •) n) 1.301 A determinant is not changed in value by writing rows for columns and columns for rows. 1.302 If two columns or two rows of a determinant are interchanged the re- sulting determinant is unchanged in value but is of the opposite sign. 1.303 A determinant vanishes if it has two equal columns or two equal rows. 1.304 If each element of a row or a column is multiplied by the same factor the determinant itself is multiplied by that factor. 12 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS 1.305 A determinant is not changed in value if to each element of a row or column is added the corresponding element of another row or column mul- tiplied by a common factor. 1.306 If each element of the Ith row or colimin consists of the sum of two or more terms the determinant sphts up into the sum of two or more de- terminants having for elements of the Ith row or column the separate terms of the ^th row or column of the given determinant. 1.307 If corresponding elements of two rows or columns of a determinant have a constant ratio the determinant vanishes. 1.308 If the ratio of the differences of corresponding elements in the pth. and qth rows or colimins to the differences of corresponding elements in the fth and ith rows or columns be constant the determinant vanishes. 1.309 If p rows or columns of a determinant whose elements are rational integral functions of x becolne equal or proportional when x = h, the determinant is divisible by (x — h) ^~^. MULTIPLICATION OF DETERMINANTS 1.320 Two determinants of equal order may be multiplied together by the scheme : I «U- \ X \ bill = I <^ij I where Cij = aubji + aabft + + aiJ>in- 1.321 If the two determinants to be multiplied are of unequal order the one of lower order can be raised to one of equal order by bordering it; i.e. : dn ffi2 fliTi Chl 022 (hn O'nl fln2 O-nn IOC CIO O O I GOO GOO .Oil ai2 . . . . Oln .021 022 ... . 02„ Onl 0„2 ^nn 1.322 The product of two determinants may be written: Oil flln X hn hln an\' hnv. DIFFERENTIATION OF DETERMINANTS 1.330 If the elements of a determinant, A, are functions of a variable, t: aA + d 11 d^ a 21 Ow, din 0> n\ fln2 dnn + flu O 12 ^- The sign of any term is determined by the number of inversions from the second suffixes of the leading term, being positive if there is an even number of inversions and negative if there is an odd number of inversions. 1.341 The coefficient of an when the determinant A is fuUy expanded is: i^-A-- 14 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS Aj, is the first minor of the determinant A corresponding to an and is a de- terminant of order « — i. It may be obtained from A by crossing out the row and column which intersect in dj,-, and multiplying by (—1)'+'. 1.342 1.343 flil Ayi + ffi2 A,-2 + . . . . + fljn A,„ = flu Aij + 02iA2,- + + a„iA„,- = 3^A ^ d^u ^ dAij daijdau dan dau o if i=t=7 A if i = j o ii i =|=i Aiii=j is the coefficient of anaki in the complete expansion of the determinant A. It may be obtained from A, except for sign, by crossing out the rows and columns which intersect in aij and aki. 1.344 I A;,- I X I a;,- I = A" I Aii I = A"-i. The determinant | A;,- | is the reciprocal determinant to A. 1.345 _^A_ daijdau Aii Ail _aA _aA _ aA dA dOij daki dan dak,- 1.346 A^- 5'A daijdaudapq 1.347 32 A daijdaki 1.348 If A = 0, Aij Ail Ai3 Akj Akl Ai-g. Apj Apl Apq a^A a_A a_A _ aA aA daij daki dan dakj 1.350 If aij - aji the determinant is symmetrical. In a symmetrical determinant Aij = Aji. 1.351 If aij = —aji the determinant is a skew determinant. In a skew determinant Aij = i-iy-'Aji. ALGEBRA 15 1.352 If Gij = -an, and an = o, the determinant is a skew symmetrical determinant. A skew symmetrical determinant of even order is a perfect square. A skew symmetrical determinant of odd order vanishes. 1.360 A system of linear equations: anXi + anXi + + ai„x„ = ki 021X1 + ai2X2 + + a2nXn = h has a solution: provided that an\Xi + araXi + + a„„a;„ = kn l^-Xi==h^li + h^ii+ + knKi ^ = I Oil I + o- 1.361 If A = o, but all the first minors are not o, 32 A lA ss' OC j .A,,+ V dassdarj u I, 2, n) where ^ may be any one of the integers i, 2, . . . . , n. 1.362 li ki = k2 = . . ■ . . - = kn = o, the linear equations are homogeneous, and if A = o, ^^— = A~ (7 = I, 2, ... m). 1.363 The condition that n linear homogeneous equations in n variables shall be consistent is that the determinant, A, shall vanish. 1.364 If there are n + i linear equations in n variables : aiiXi + ai2X2 + + ainXn = h 021X1 + 022^ + + OinXn — kl OnlXl + an2X2 + + a„nXn = kn ClXl + C2.T2 + +C„.T„ = ^„+l the condition that this system shall be consistent is that the determinant-' On Oj2 (^In ky Oil 023. ^n ^2 On\ On2 ^nn kn Ci C2 Cn kn+l i6 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS 1.370 Functional Determinants. If yi, yh • • • •> y-n are n functions oix\,Xi, , Xn : y* =/ft(a;i, %, , acn) the determinant: / = 3yi dyx dxx dx% dxi dxi dyi dXn dXr, djn dyn dxi doCi dy„ dx„ dyi dXj ^(yi. y2, ■ • ■ .,yn) d{Xi, OC2, . . . ., Xn) is the Jacobian. 1.371 11 yi, y2, , yn are the partial derivatives of a function F{xi, Xi, , «„) : ' dF ,. . the symmetrical determinant: d'^F H = dXi dXj \dxi' dx2 ' ' ' ' dxJ d(Xi, X2, , x„) is the Hessian. 1.372 li yi, yi, , y„ are given as implicit functions of xi, 0C2, Xn by the n equations : Fiiyi, yi, , yn, Xi, od, , a;„) = o then Fn(yi, yt, , yn, xx,ooi., , a;„) = O 3(yi,y2, , yn) ^ , ^y djFuFj,. . ., F„) ^ djFuFi,. ..,Fn) d{xi, Xi, . . ., Xn) d(xi, Xi, . . ., Xn) ' d(yi, yi, . . . , y„) 1.373 If the n functions yi, yi, , y„ are not independent of each other the Jacobian, /, vanishes; and if 7 = o the n functions yi, yi, . . . ., y„ are not independent of each other but are connected by a relation ^(yi. yz, ,yn) = ALGEBRA I 7 1.374 Covariant property. If the variables Xi, X2, . . . ., Xn are transformed by a linear substitution : Xi = an |i + a,-2^2 + + fli„^„ (i = i, 2, . . . ., n) and the functions yi,y2, ,ynoixi,oc2, ,Xn become the functions Vl, Vi, , Vn of ^1, ^2, , ^n : jf _ ^(.Vu Vi, > Vn) _ d{yi, yi, , y„) , . or f = J- 1 an I where | «<,- 1 is the determinant or modulus of the transformation. For the Hessian, E' = H-\ an I \ 1.380 To change the variables in a multiple integral : I = f fF{yi, y2, , yMyidyi dyn to new variables, xi, x^, . . . ., Xn when yi, y^, , yn are given functions of Xl, Xi, yXn'. i=f rj-/" ^'' ' ^i F{x)dxdx. dx. J J dyxi, Xi, , Xn) where F{x) is the result of substituting xi, Xi, . . . ., Xn for y\, y^, . . ., y„ in i?(yi, yz, , )-»). PERMUTATIONS AND COMBINATIONS 1.400 Given « dififerent elements. Represent each by a number, i, 2, 3 , n. The number of permutations of the n different elements is, nPn = n\ e.g., » = 3 : (123), (132), (213), (231), (312), (321) = 6 = 3! 1.401 Given n different elements. The number of permutations in groups of r {r(l,23)(2,3l)(3,I2)(0,I23) = 8 1.407 Given n similar elements. The number of ways they can be combined into m different groups when blank groups are allowed is ALGEBRA I 9 {n + m — i) ! {m — i) !w! e.g., n=-6, OT = 3: Group I 6554443333222221111110000000 Group 2 0102013021403125041326051423 1=28 Group 3 0010210312041320514230615243 1.408 Given n similar elements. The number of ways they can be combined into m different groups when blank groups are not allowed, so that each group shaU contain at least one element, is (n- i)! {m — i)[{n — m) 1.51 BINOMIAL COEFFICIENTS n{n — i) (« — 2) . . . in — k -\- 1) ^- \k) ~ k\{n-k)[~ \n-k)~ " ' '■(:)+t:.)=(:::> 5. (j = oifw<^. II. 1 + 20 MATHEMATICAL FORMULAE AND ELLIPTIC FUNCTIONS 1.62 Table of Binomial Coefficients. C) (:) (:) c) (:) (:) C) (:) (.") (.:) I 2 I 3 3 I 4 6 4 I 5 lO 10 S I 6 IS 20 15 6 I 7 21 35 35 21 7 I 8 28 56 70 56 28 8 I 9 36 84 126 126 84 36 9 I lO 45 120 ' 210 252 210 120 45 10 I II 55 i6s 330 462 462 330 165 55 II I 12 66 220 495 792 924 792, 495 220 66 12 1.521 Glaisher, Mess, of Math. 47, p. 97, 1918, has given a complete table of binomial coefficients, from « = 2 to » = 50, and k = o to k = n. 1.61 Resolution into Partial Fractions. If F{x) and f{x) are two polynomials La x and f{x) is of higher degree than F(x), f{x) Z/ (t>{a) x-a'^ ^{p-i)\ dc^-\{c) x - c\ \_X flj X = aj where ^^'-'-[jBfl The first summation is to be extended for all the simple roots, a, of f{pc) and the second summation for aU the multiple roots, c, of order p, oi f(x). FINITE DIFFERENCES AND SIMS. 1.811 Definitions. 1. A/(x) =f{x + h)-f{x). 2. Aj{x) = Af(x + h)- Af{x). = fix + 2h) - 2f(x + h) +fix). ALGEBEA 21 3. Ay(x) = ^J{x + h)- Ay(x). = f{x + sh) - 3j{x + 2h) + 3fix + h) -j{x). 4. Mix) = fix + nh) - ""-fix + n- ih) + ^^^^^fix + T^h) - 1.812 1. A[cfix)'^ = cAfix) (c a constant). 2. A[/i(^) +Mx) + ....] = A/i(x) + AMx) + 3. A[/i(x) -Mx)-] = Mx) ■ A/2(x) +Mx + h)-AMx) = Mx)-AMx) +f2ix)-AMx) + AMx)-AMx). . A /i W Mx) ■ A/i(x) - Mx) ■ AMx) *• Mx) Mx)-Mx + h) 1.813 The wth difference of a polynomial of the «th degree is constant. If Jix) = a^n + aiX"-^ + + Gn-iX + an A^fix) = n\aji^. 1.82 A'^{ix - h)ix - h - h)ix - h - 2h) ix-h -n-ih)] w(« — i)(» — 2 ) (w — w + i)/?™ = ix — b)ix — b — h)ix — b — iJi) . . . . ix — b — n — m — ih). . I 2. A"*' ■ ix + b)ix + b + h)ix + b + 2h) . . . . ix + b + n-ih) , . niv: + i) (» + 2) (m + w - i)/!" = (—1) ' ix -\- b) ix -\- h ■\- H) ix + b+ 2h) . . . . ix + b+ n + m — ih) 3. A"'a'' = (o* - i)'"a'^ 4. Alog/(x)=log(i+^). . , r. / ■ cM™ . / , ch + Tr\ 5. A" sin icx + a) = ( 2 sm — 1 sm ( ex + d + w 1 • 6. A*" cos icx + a) = I 2 sin —1 cos \cx + d + m 1 • 22 MATHEMATICAL FOEMULyE AND ELLIPTIC FUNCTIONS 1.83 Newton's Interpolation Formula. (a: - g) (x - a - h) (x - a - 2h) . , , + 77-^5 ^JW + ....... (x — a) (x — a — h) (x — a — n — ih) . ,, ^ (x - ffi) {x- a-h) {x- a-nh) ^^^^ , . where | has a value intermediate between the greatest and least of a, {a + wA), and X. 1.831 /(a + nh) = /(a) + ^ A/(.) + ^^ Ay(a) + -J^-zA^ A%a) i[ ■''' ' ' 2! '^^ ' 3!,. + + «A''-y(a) + A"/(a). 1.832 Symbolicallj d 1. A = e ai — I 2. fia + nh) = (1 + A)"/(a) 1.833 If Mo =/(ffl), Ml =/(« + h), U2=f{a+ 2A), . . . ., m^ =/(o + xh), hJ- Mj = (i + A) ^Mo = e '% Mo- 1.840 The operator inverse to the difference, A, is the sum, S. 2 = A-i = _J e'' Si - I 1.841 If AF{x) =/(x), 2/(x) = F{x) + C, where C is an arbitrary constant. 1.842 I. Sc/(») = cXfix). 2. s[/i(x) +Mx) + ...] = s/i(^) + i:mx) + . . . 3. S[/iW • A/2(x)] = Mx) -Mx) - s[/2(x + h)-AMxn ALGEBRA 23 1.843 Indefinite Sums. I. Ii[_(x - b)ix - b - h){x - b - 2h) . . . {x - b - n - lA)] (x-b) (x-b-h) (x-b-nh) +C. {n + i)h {x-\-b){x-\-b + h) . . . . {x + b + n - ih) I I (n- i)h {x + b){x + b + h) . . . . {x + b + n - 2h) / ck A {ex hoi + C. a" - I sm 4. ^ cos {ex + d) = ?7 '- + C. eh 2 sin — 2 5. 2ij sill ('^^ + ^) = ^T ^ + C. I ch \ [ex hoi 2 sm ■ 2 1.844 If f{x) is a polynomial of degree w, 1.845 If f{x) is a polynomial of degree n, /(x) = oox;'' + OiX"~' + .... + o„_ix + an, and S/(x) = F(x) + C, Fix) = Cox"+i + eix" + C2X"-i + . . . . + c„x + c„+i, where (« + i)Aco = Oo (w + i)« ,, , , ; — h%a + nhei = Oi 2! (n + i)n(n - i) „ , n{n - i) ,, , / x, -i^ -^ ¥eo + - — -; «% + {n - ijhei = (h 3! 2! The coefficient c„+i may be taken arbitrarily. 24 MATHEMATICAL FORMULAE AND ELLIPTIC FUNCTIONS 1.850 Definite Sums. From the indefinite sum, S/(x) = F{x) + C, a definite sum is obtained by subtraction, ^/(a;) = F{a + nh) - F{a + w/s). fl+mA 1.861 2/W = M +f(a + h) +f{a +2h) + . . . . +f{a + n - ih) = F{a + nh) - F{d). By means of this formvda many finite siuns may be evaluated. 1.852 a+nh yj(x — b)(x — b — h){x — b — 2h) . . . . {x — b — k — ih) a ^ (a — b + nh) (a — b + n — ih) . . . . (a — b + n — kk) (k + i)h (a — b)(a — b — h) . . . . {a — b — kh) (k+Tjh 1.853 Xjjx ~ a)(x — a — h) , . . . {x — a — k — ih) a n(n — i)(w — 2) . . . . (n — ^),j. = ikTT) ^" 1.854 If f(x) is a polynomial of degree m it can be expressed : f{x) = A(, + Ai(x — a) + AifyX — a){x — a — h) + . . . . + Am{x — a){x — a - h) . . (x - a - m - ih), a+nh 2J-/ \ A .A ^('"' — i) 7 A n(n — i)(n — 2) ,„ f(x) = Ao-n + Ai— -h + A2— — -k' 2 •? . n{n — 1) . . . (n — m) {m + I) 1.855 If fix) is a pol5momial of degree (m — i) or lower, it can be expressed : f{x) = Aa + Ai{x-\- mh) -\- A2{x + mh) {x + m - ih) + .... + Am^iix + mh) ... {x+ 2h) and, 'V1_ fix) ^0 ^ x{x + h)(x + 2h) . . . {x + mh) mh \ a{a + h) . . . (a + m - ih) ALGEBRA 2$ (o + nK) {a + n + m — ih) {m - i)h \ a{a + h) . . . {a + m - 2h) (a + nh) . . . (a + n + m - 2h) A„^i f I I + h [a a + nh 1.866 If f{x) is a polynomial of degree m it can be expressed : /(x) = Ao + Ai(x + mh) + A2{x + mh) {x + m - ih) + . + Amix + mh) ... {x + h) and, a+nh 'V /W A^l I where, ~x{x + h) . . . {x + mh) mh \ a{a + h) . . . . {a + m - ih) 1 (a + nk) {a -{- m -\- n — ih) } a~\-nh + +fi2zi(l--^U^„2- h [a a -^ nh j JSmtx a ^" X a a + h a + 2h a + n - ih 1.86 Euler's Summation Formula. + .... + A,n-ih-^'^ {/("-^Kft) - /(»'-2)(a) i, x=b ' 4>m{z) ^ ^^j^;;^ (^2 x = a 4>m{z) = — r + ^1 -, 77 + Ai -. TT + .... + A^-ih'"-^z. ^ w! [m - 1)1 {m - 2)1 ml(j)m(z), with ^ = I, is the Bernoullian polynomial. Ai = -^, yl2* + 1 = o; the coefficients An are connected with Bernoulli's numbers (6.902), Bk, by the relation, A,, = {- i)*+i ^* (2^)! y4i = . Av = — . Ai = , ^6 = ^^ 2' 12' 720' 30240 26 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS 1.861 b 2^W = I £f^^)^' - \ [ m - f(a) J + ± { /'(6) _ f(a) } -^lW)-/"'w|+^{m-r(a)/-., 1.862 2„ , /" , I ^ 1 dUx I d^Ux , I d^Ux J 2 12 ox 720 dar 30240 ox° SPECIAL FINITE SERIES 1.871 Arithmetical progressions. If s is the sum, a the first term, 5 the common difference, I the last term, and n the nmnber of terms, 5 = a + (a + 5) + (a + 25) + . . . . [a + (« - i)5] I = a + {n — T.) b s = -[2a + (w — 1)8] 1.872 Geometrical progressions. s = a + ap -\- ap^ + + ap'^-^ *" - I s = a- p - I a If p»'-- ■ *= I Bi, B2, B3, ... are Bernoulli's numbers (6.902), f^J are the binomial coefficients (1.51) ; the series ends with the term in w if ^ is even, and with the term in n^ if k is odd. 1.876 I I I I I , I 02 - + -+- + - + . . . . + - = 7 + log« + — 1234 M 2« n{n + 1) 03 n{n + 1) [n + 2) 7 = Euler's constant = 0.5772156649 . . . 1.877 '^ = ^ I 03 = — 12 19 a* = 1 / x(i-x) (2 -x) . . . «5 = ^ 20 ? + ? I I tt'' Ji + - + ... - + ^2- g W+I 63 (« + i) (« + 2) (w + 3) . (^-i)! . (^ — I — x)dx 1.878 ^,Lj.L. 4--=C - i3 ■'' 2^ 3' n^ (« + i) (w + 2) Cs ~ (« + i) (ra + 2) (w + 3) CO C = 2|-3 = 1.2020569032 i = i (^_i)!/i I I , ^V 28 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS 1.879 Stirling's Formula. log (w!) = log \/27r + (n +-jlogn — n A, (2^-4)! i r • • • . + -^211-2 5tZ5 o< 0< I. The coefficients Ak are given in 1.86. 1.88 1. i + i! + 2-2! + 3-3! + . . . . + n-nl = (n + i)l 2. I-2-3 + 2-3-4+3-4-S + . . . .+n{n+i) {n + 2) =-n{n+i) (« + 2) C» + 3). 4 3. I-2-3 . . . .r+ 2-3-4 . . . (r + i) + + n{n + i) {n + 2) ....(« + r - i) _ n{n + 1) (n + 2) . . . . (n + r) r+i 4. i-^ + 2(^ + i) + 3(^ + 2) + •. +n{p + n- i) = T«(W + l)(3^ + 2W - 2). S- /»•? + (^ - i) (? - i) + (? - 2) (g - 2) + ip-n) (q-n) = ^w[6/>g - (« - i) (3;- + 3g - 2M + i)]. o- I H 1 — 7 c + ••••-{ — 7 ^ ; ^ • a a{a + i) a{a + i) . . . . (a + n — 1) b{h + 1) . . . . (b + n) a - I (i + I — a)a{a + i) . . . . {a + n — i) b + 1 — a II. GEOMETRY 2.00 Transformation of coordinates in a plane. 2.001 Change of origin. Let x, y he. a. system of rectangular or oblique coor- dinates with origin at O. Referred to x, y the coordinates of the new origin 0' are a, b. Then referred to a parallel system of coordinates with origin at O' the coordinates are x', y' . X = x' + a y = y + b. 2.002 Origin unchanged. Directions of axes changed. ObUque coordinates. Let CO be the angle between the x - y axes measured counter-clockwise from the X- to the y-axis. Let the x'-axis make an angle a with the a;-axis and the y'-axis an angle /3 with the x-axis. All angles are measured counter-clockwise from the :*;-axis. Then a; sin oj = x' sin (oj - a) + y' sin (w - /3) y sin oj = x' sin a + y' sin j3 w' = |8 - a. 2.003 Rectangular axes. Let both new and old axes be rectangular, the new axes being turned through an angle d with respect to the old axes. Then a, = J,a =0,^ = ^ + 0. X = x' cos 6 - y' sin y = a;' sin -f- y' cos 0. 2.010 Polar coordinates. Let the y-axis make an angle co with the a;-axis and let the X-ZX& be the initial line for a system of polar coordinates r, 6. All angles are measured in a counter-clockwise direction from the a;-axis. r sin (w — d) X ^ ^~~~" sin oi sin 6 y = r -. ■^ sm w 2.011 If the X, y axes are rectangular, w = -, X = r cos d y = r sin 0. 29 30 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS 2.020 Transformation of coordinates in three dimensions. 2.021 Change of origin. Let x, 31, s be a system of rectangular or oblique coor- dinates with origin at 0. Referred to x, y, z the coordinates of the new origin 0' are a, b, c. Then referred to a parallel system of coordinates with origin at O' the coordinates are x', y', z' . X = x' -\- a y = y' + b z = z' + c 2.022 Transformation from one to another rectangular system. Origin un- changed. The two systems are x, y, z and x' y' z'. Referred to x, y, z the direction cosines of x' are h, nix, «i Referred to x, y, z the direction cosines of y' are h, nh, th Referred to x, y, z the direction cosines of s' are I3, nh, M3 The two systems are connected by the scheme : x' / z' X h • h y mi nh ms z Wi th ns X = hx' -\- hy' + Isz' y = mix' + nhy' + m^' z = n\x' + thy' -\- Uiz' ly + m^ + ni^ = I H + mi' + fh^ = 1 li -\- mi + ni = 1 hnii + hnh + hni3 = o OTiWi + nh'th + mifii = o mh + thh + nsh = o x' = hx + m\y -{■ tiiZ y' = hx + nhy + n^ z' = I3X + msy + fisz h' + V + li=i m-^ + mi -\- mi = t ni + ni + ni = I hh + W1OT2 -1- W1W2 = o hh + W2W3 + fhns = o hh + W3OT1 -|- «3«i = o 2.023 If the transformation from one to another rectangular system is a rotation through an angle Q about an axis which makes angles a, jS, 7 with x, y, z re- spectively, 2 cos d-h + nti + nz— I GEOMETRY 31 cos^ a cos^ COS'' 7 m2 + m — h — I 713 + h — nii — 1 li + m^ — «3 — i 2.024 Transformation from a rectangular to an oblique system, x, y, z rec- tangular system : x', y', z' oblique system. cos xx' = /i cos yx' = m\ cos xy' = h. cos xz' = h cos yy' = W2 cos yz' = mz cos zx' = Wi cos zy' = W2 cos zz' = n% X = hx' + ^23'' + te' y = OTix' + Wsy' + Wsz' z = Ki.r' + ruiy' + wjz' cos y'z' = hk + W2W3 + W2W3 cos z'.v' = Wi + W3W1 + W3«i cos x'y' = hh + mitth + «iW2 7i^ + Wi^ + Wi^ = I ?3^ + mi + ni = 1 2.025 Transformation from one to another oblique system. cos xx' = li cos yx' = mi cos zx' = «i cos X)) = fc COS yy' = W2 cos zy' = «2 A = cos xz' = k COS yz' = wJs cos 2Z' = Ms /l ?2 ^3 Wl«2W3 ic = /ix' + hy' + Isz' y = mix' + m^y' + m^z' z = «ix' + fhy' + «3z' A-x' = (OT2W3 - •»W3«2)x + (W2/3 - n3h)y + (km^ - hm^)z, A-y' = (ot3«i - WiW3)x + (nih - nih)y + ihmi - limi)z, A-z' = (;;;i«-2 - 7n«ih)x + (ihh - thh)y + {hm^ - hmi)z. l^ + m^ + n^ + 2W1M1 cos yz + 2«iZi cos zx + 2/1^1 cos xy = i, 4^ + m«' + Wa^ + 2W2«2 cos rs + ^n^i cos s.v + 2/2W2 cos xy = i, U + W32 + «3^ + 2W3M3 cos yz + 2niz cos zx + 2/3W3 cos xy = I. X + y cos xy + z cos xz = /ix' + h.y' + ^sz', y + X cos xy + z cos zy = m\x' + wsy' + m^' , z + X cos xz + y cos zy = mx' + my' + Wsz'. 32 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS 2.026 Transformation from one to another oblique system. If rix, Tiy, Uz are the normals to the planes yz, zx, xy and Ux', fiy', nj the normals to the planes y'z' , z'x', x'y', X cos xnx = x' cos x'ltx + y' cos y'ux + z' cos z'ux- y cos yuy = x' cos x'ny + y' cos y'fiy + z' cos z'riy. z cos zriz = x' cos x'n^ + y' cos y'uz + %' cos z'lUz. x' cos x'nx' = X cos xnj -\- y cos yux' + z cos zux'- y' cos y'ny = x cos xny ■\- y cos yWj,' + z cos 2!m„'. 2' cos z'wz' = a; cos xnz + y cos ywj' + z cos zw/. 2.030 Transformation from rectangular to spherical polar coordinates. r, the radius vector to a point makes an angle 6 with the z-axis, the projection of r on the x-y plane makes an angle (j) -with the x-axis. X = r sin 6 cos (j) r^ = x^ + y^ + z^ z y = r sin d sin (j) B = cos-i , „, „ = V X + 3' + z z = r cos ± ^ ^y ffl = tan~' - X 2.031 Transformation from rectangular to cyUndrical coordinates. p, the perpendicular from the z-axis to a point makes an angle 6 with the x-z plane. a; = p cos p = Vx^ + y^ y = p sin = tan~i - X z = z 2.032 Curvilinear coordinates in general. See 4.0 2.040 Eulerian Angles. Oxyz and Ox'y'z' are two systems of rectangular axes with the same origin 0. OK is perpendicular to the plane zOz' drawn so that if Oz is vertical, and the projection of Oz' perpendicular to Oz is directed to the south, then OK is directed to the east. Angles z'Oz = B, ^K = ^, yVK = ^. GEOMETRY 33 The direction cosines of the two systems of axes are given by the following scheme : X y z x' i COS COS d COS r^ — sin sin \j/ - cos 4> COS sin i// - sin <^ cos ^ COS 4> sin sin <^ cos & cos ^ + cos <^ sin 4' - sin (^ cos 9 sin ^ + cos ^ cos ^ sin oo^^ ^k k= I the equation of the asymptote is y = 2^ o-kx^ k = o GEOMETRY 41 If of the first degree in x, this represents a rectilinear asymptote; if of a higher degree, a curvilinear as3anptote. 2.250 Singular Points. If the equation of the curve is F {x, y) = o, singular points are those for which dx dy ' Put, ~ dx^ dy^ \dx dy) If Ao the singular point is an isolated point with no real branch of the curve through it. A = o the singular point is an osculating point, or a cusp. The curve has two branches, with a common tangent, which meet at the singular point. -, dF dF d^F d^F d^F . ^ 'dx' 'dy' 'd?' df' dxTy simultaneously vanish at a point the singular point is one of higher order. PLANE CURVES, POLAR COORDINATES 2.270 The equation of the curve is given in the form, r=m. In figure 2, OP = r, angle XOP = 6, angle XTP = r, angle pPt = 4>. 2.271 d is measured in the counter-clockwise direction from the initial line, OX, and s, the arc, is so chosen as to increase with d. The angle is measured in the counter-clockwise direction from the positive radius vector to the positive tangent. Then, T=d + (t>. 2.272 tan (j) r da dr sin (j) rdd ds cos CJ) dr ~ ds 42 MATHEMATICAL FORMULA AND ELLIPTIC PUNCTIONS dr dd 2.273 dr „ sm p 3^ + r cos v tan T = — cos -rs — r sin Q 2.274 Pi? = r i/ 1 + y—- I = polar tangent PV = v/r^ + (-^J = polar normal OR = r'^-T- = polar subtangent dr OV = -ja = polar subnormal. 2.276 OQ = — , = = p = distance of tangent from origin. dr OS = — , = = distance of normal from origin. 2.276 If M = -, the curve r = f(6) is concave or convex to the origin according as d^u is positive or negative. At a point of inflexion this quantity vanishes and changes sign. 2.280 The radius of curvature is, "Am ,^ .- . dh H + 2 /^Y _ dh \dd) *" dd^ 2.281 If M = - the radius of curvature is r fduY 1 3 P = GEOMETRY 43 2.282 If the equation of the curve is given in the form, r=f{s) where s is the arc measured from a fixed point of the curve, \2 v-(i)' dh /drV 'd? + [ds) -' 2.283 If p is the perpendicular from the origin upon the tangent to the curve, dr dH P-'dp '■ P = ^ + d? 2.284 If M = i r 2-285 dT^ + " = ^y 2.286 Polar coordinates of the center of curvature, n, ^i: ^^ ~ ( , /dry dh (dry dr [ddj + dd 'Jdry_ dh- Ue) ' dd' 2.287 If 2C is the chord of curvature (2.225): dr p tanx = rl J ^d'uV 2.290 Rectilinear Asymptotes. If r approaches » as approaches an angle a, and if /-(a - d) approaches a limit, b, then the straight line r sin (a — 6) = b is an asymptote to the curve r =f{d). 44 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS 2.295' Intrinsic Equation of a plane curve. An intrinsic equation of a plane curve is one giving the radius of curvature, p, as a function of the arc, s, p=m If T is the angle between the a;-axis and the positive tangent (2.271) : ds dT = lis) nds f X = xo+ I cos T-ds y = yi>+ I sin T-ds. 2.300 The general equation of the second degree: anx^ + 2aj2xy + a^'f + 2a\zx + 202331 + 033 = A = On O12 Oi3 021 (h2 O23 O31 0-32 O33 Ahk = Minor of om- r Criterion giving the nature of the curve: Ohk = Okh ^33 + ^33 = A^O ^33<0 ^33>0 Parabola Hyperbola aiiA <0 or 022^ >0 Ellipse Imaginary Curve A =0 ^33<0 ^33>0 ^11 <0 or ^22 >0 An = A22 = ■ Pair of Real Straight Lines Intersectic Pair of Imaginary Lines m Finite Real Pair of Imaginary Parallel Lines Double Line (Pascal: Repertorium der hoheren Mathematik, II, i, p. 228) GEOMETRY 45 2.400 Parabola (Fig. 3). 2.401 0, Vertex; F, Focus; ordinate through D, Direc- trix. Equation of parabola, origin at 0, 'f= 4ax X = OM, y = MP, OF = OD = a FL = 2a = semi latus rectum. FP = D'P. 2.402 FP = FT = MD = x + a. D' u n' \ \ 1^"^"^ r' q} ^ f y\ T D \ n F M N Fig. 3 iVP = 2Va(a+!t;), TM = 2X, ilfiV = 2ff, OA'' = x + 2a. OiV' = \/^ (* + 2a), Oe = a;\/^, 05 = (X + 2a)y/- F£ perpendicular to tangent TP. + x F5 = Va(a + x), TP = 2rB = 2Vx(a + x). FB^ = FT X FO = FP -x FO. The tangents TP and f/P' at the extremities of a focal chord PFP' meet on the directrix at U at right angles. T = angle ZTP- tan T » X The tangent at P bisects the angles FPD' and PC/D'. 2.403 Radius of curvature: ^ 2(x + a)^ ^ I iVP'_ Coordinates of center of curvature: ^ = 3X+ 20, 77 = - 2x\/'^- Equation of Evolute: 27a/ = 4(x - 2aY- 46 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS 2.404 Length of arc of parabola measured from vertex, ^x{x + a) + a log \\Ji +~+ y ^j • Area OPMO = - xy. 3 2.405 Polar equation of parabola: r = FP, e = angle XFP, 2a r = — 7- I — cos 2.406 Equation of Parabola in terms of p, the perpendicular from F upon the tangent, and r, the radius vector FP: I = semi latus rectum. 2.410 ElUpse (Fig. 4). 1 p' /\^^ — ^ f /n p 1 V-^ 1 F' \v^^^ N' /v--^ A FA T Fig. 4 2.411 O, Centre; F, F', Foci. Equation of Ellipse origin at 0: T- y'- — I- — = I o2 ^ 62 X = OM, y = MP, a = OA, b = OB. GEOMETRY 47 2.412 Parametric Equations of Ellipse, X = a cos 4>, y = b sin . 2.417 Polar Equation of EUipse, r = F'P, e = angle XF'P, a(i - e^) Y ^ ^ — I — e cos 9 2.418 »■ = OP, B = angle XOP, Vi — ^ cos^ Q 2.419 Equation of Ellipse in t?rms of p, the perpendicular from F upon the tangent at P, and ;•, the radius vector FP: I 2 1 p^ r a I = semi latus rectum. 2.420 Hyperbola (Fig. s). 2.421 0, Center; F, F', Foci. Equation of hyperbola, origin at O, X = OM, y = MP, a = OA = OA'. 2.422 Parametric Equations of hyperbola, X = a cosh u, y = b sinh u. or X = a sec e b' 2.428 Polar Equation of hyperbola: r^F'P, e = XF'P, r = a ^ ^ r = OP, e = XOP, r^ = e cos u — I e^ cos^ 6 — 1 2.429 Equation of right-hand branch of hyperbola in terms of p, the perpen- dicular from F upon the tangent at P and r, the radius vector FP, I 2 I p' r a I = semi latus rectum. GEOMETRY SI 2.450 Cycloids and Trochoids. If a circle of radius a rolls on a straight line as base the extremity of any radius, a, describes a cycloid. The rectangular equation of a cycloid is: X = a{4> — sin ), where the a;-axis is the base with the origin at the initial point of contact. 4> is the angle turned through by the moving circle. (Fig. 6.) FiG.6 A = vertex of cycloid. C = center of generating circle, drawn tangent at A. The tangent to the cycloid at P is parallel to the chord AQ. Arc AP = 2 X chord AQ. The radius of curvature at P is parallel to the chord QD and equal to 2 x chord QD. PQ = circular arc AQ. Length of cycloid: s = 8a; a = CA. Area of cycloid: 5 = sira'. 2.451 A point on the radius, b>a, describes a prolate trochoid. A point, bo Prolate trochoid, d, y = (& ± a) sin 4> — a sin . ±2a 20 2.453 In the epicycloid put b = a. The curve becomes a Cardioid: (x2 + y2)2 _ 6(j2(x2 -I- y^) + 8a'x = 3a*. 2.454 Catenary. The equation may be written: T ^ -'^ 1. y = - a(e» 4- e «). ■'2 2. y = a cosh -• y ± -\/3^ — a^ 3. x = alog The radius of curvature, which is equal to the length of the normal, is: p = a cosh^ "— ^ a 2.455 Spiral of Archimedes. A point moving uniformly along a line which rotates uniformly about a fixed point describes a spiral of Archimedes. The equation is: r= ad, or ■\/x? 4- y2 = a tan~i -• X The polar subtangent = polar subnormal = a. Radius of curvature: ^ 6(2 + 6') ~ r''+2a^' 2.456 Hyperbolic spiral: GEOMETRY 53 2.457 Parabolic spiral: r^ = a^d. 2.458 Logarithmic or equiangular spiral: ;■ = ae"^, n = cot a = const., a = angle tangent to curve makes with the radius vector. 2.459 Lituus: rVd = a. 2.460 Neoid: 2.461 Cissoid: 2.462 Cassinoid: r = a-\-hd. r = 2a tan 6 sin d, (x2 + / + a2)2 = ^a^x" + V^, r* - 2aV2 cos 20 = b'^ - a*. 2.463 Lemniscate (6 = a in Cassinoid): (x? + y^y = 2aKx^ - y^), r^ = 20? cos 20. 2.464 Conchoid: 3?f = (b + yy{a? - f). 2.465 Witch of Agnesi: x'y = 4.a^{2a - y). 2.466 Tractrix: ,. i,,n « + ^^«^ -f dy y -f dx Va? - y^' aVa? - / SOLID GEOMETRY 2.600 The Plane. The general equation of the plane is: ^x + 5y + Cz + D = o. 2.601 I, m, n are the direction cosines of the normal to the plane and p is the perpendicular distance from the origin upon the plane. A, B, C I, m, n p = Ix + my + nz, ■ E 54 MATHEMATICAL FORMULAE AND ELLIPTIC FUNCTIONS 2.602 The perpendicular^ from the point xi, yi, zy upon the plane Ax -\- By ■\- Cz + D = o is: Axi + Bvi + Czi + D 2.603 6 is the angle between the two planes: Aix + Biy + Ciz + A = o, A2X + Biy + C2Z + A = o, :4i^2 + B1B2 + C1C2 cos = VAi^ + Bx^ + Ci2 VAi + W+C^ 2.604 Equation of the plane passing through the three points (xi, yi, Z\) {x2, yi, Zi) (»3, ys, 23): yi si I + y 2i Xi I + z xi yi I = xi yi Zi 3-2 Z2 I Zi X2 I ^2 y2 I X2 3'2 Z2 ys Z3 I Z3 X3 I X3 ys I xs ys Z3 THE RIGHT LINE 2.620 The equations of a right line passing through the point Xi, yi, Zi, and whose direction cosines are I, m, n are: X — xi _ y — y\ _ z — zi I m n 2.621 Q is the angle between the two lines whose direction cosines are l\, m\, Wj and I2, fiH, fh'. cos d = hh + OT1W2 + «lM2, sin^ 6 = {hnh - hmiY + {mifh - ffhniY + {nik - thhY- 2.622 The direction cosines of the normal to the plane defined by the two lines whose direction cosines are h, mi, wi and h, fthrh are: OT1W2 — W2W1 fiih — W2/1 hnh — Ijm-i sin ' sin ' sin d 2.623 The shortest distance between the two lines: X - xi _ y - yi _ z - zi x - Xj _ y - yi _ z - Zj k Wi Wi h W2 is: d {xi - ^2) (wiW2 - W2W]) + (yi - 312) (wi/2 - ndi) + (zi - Z2) {hnh - hmi) { (mifh - nnniY + {nik - n^hY + {hnh - hmiY\^ ' 2.624 The direction cosines of the shortest distance between the two lines are: {mini - fiimi), (nih - n4\), (hni2 - knti) [{mith - nhniy + {nih - nil^)^ + {hnh - hmiy]^ GEOMETRY 55 2.625 The perpendicular distance from the point X2, yi, zi to the line: a: - Xi _ y - yi _ z - Zi is: tZ = { (X2 - Xi)2 + (y2 - yi)2 + (z2 - zi)^}* - {?i(a;2 - Xi) + OTi(y2 - yi) + Wi(2j - Zi)} . 2.626 The direction cosines of the line passing through the two points %\, yi, Zi and Xi, y-i, Z2 are: fe - -Ti), (y2-yi), (z2 - zi) {(x2-.r0^+(y2-yi)^+(z2-zi)=ip' 2.627 The two luies: X = m\Z + ^1, x = nhz + ^'2, and y = Miz + gi, y = W2Z + 92, intersect at a point if, (wi - nh) (qi - ?2) - (wi - «2) (^1 - P2) = o. The coordinates of the point of intersection are: mip2—m2pi «iff2— «2?i p2 — p\ qi— gi x = — —, y = , z = = nil -W2 ni-W2 mi — m2 Wi — W2 The equation of the plane containing the two lines is then (wi - fii) (x - OTi2 - p^ = (oti - TO2) (y- «iz - gi). SURFACES 2.640 A single equation in x, y, z represents a surface: F(x, y, z) = o. 2.641 The direction cosines of the normal to the surface are: dF_ dF BF dx dy dz I, m, n = , iQpK2 /dpy /dFV ] * " m-m-m 2.642 The perpendicular from the origin upon the tangent plane at x, y, z is,: p = lx + my + nz. 2.643 The two principal radii of curvature of the surface F (x, y, z) = o are given by the two roots of: S6 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS where: k d^ p dx^ dxdy k d^F dxdy P d d^F d^F dxdz dydz dF dx dF by d^F dxdz d^F dydz k fF p^ dz^ dF dz dF dx dF dy dF dz ^'-m-m-m 2.644 The coordinates of each center of curvature are: ^~* + jfe dx' p dF "^y+iTy' f=z + p dF k dz 2.645 The envelope of a family of surfaces: I. File, y, z, a) = o is found by eliminating a between (i) and dF 2. da = o. 2.646 The characteristic of a surface is a curve defined by the" two equations (i) and (2) in 2.645. 2.647 The envelope of a family of surfaces with two variable parameters, a, /3, is obtained by eliminating a and /3 between: I. Fix, y, z, a, /3) = o. dF da dF 3- d^ = °- 2.648 The equations of a surface may be given in the parametric forni: X =Mu, v), y =f2(u, v), z =/3(m, v). The equation of a tangent plane at Xi, yi, zi is: where {X — xi) -Tf — ^ -r \y - y\) -r? — r + (2 - 21) -r-, — r , etc. See 1.370. dih /3) 3(m, v) a/2 du a/2 dv du dfz dv GEOMETRY 57 2.649 The direction cosines to the normal to the surface in the form 2.648 axe: a(/2, /3) a(/3, /o djfu f,) , d{u, v) ' d(u, v) ' d(u, v) I, m, n = ^^ — ^-^ — ( djhm , / a (/a, /O V ^ / a(/i, m \ i 2.650 If the equation of the surface is: z =/(», y), the equation of the tangent plane at xi, yi, 2i is: 2.651 The direction cosines of the normal to the surface in the form 2.660 are: -m P Pi P2 2.654 If p and p' are the radii of curvature in any two mutually perpendicular planes, and pi and p2 the two principal radii of curvature: I _i I I _I p p' ~ Pi P2 2.655 Gauss's measure of the curvature of a surface is: I I P P1P2 SPACE CURVES 2.670 The equations of a space curve may be given in the forms: (a) Fi(x, y, z) = o, Fiix, y, z) = o. (b) x=Mt), y=m, z=Mt). (c) y = 4>{x), z = \l/(x). 58 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS 2.671 The direction cosines of the tangent to a space curve in the form (a) are: dFi dFi dFi dFi , dy dz dz dy 1= 2^ . m = n = „ where T is the positive root of: _faFiaF2 dFi dFi dFi dFi ] ^ \ dx dx dy dy dz dz j 2.672 The direction cosines of the tangent to a space curve in the form (b) are: x', y', z' I, m, n = ~ dFr dz dFi dx dFi dx dFi T dFi dFi dFi dFi dx dy dy dx {^'2 + y'2 + 2'2jJ' where the accents denote differentials with respect to t. 2.673 If s, the length of arc measured from a fixed point on the curve is the parameter, t: . dx dy dz ' ' ds ds ds 2.674 The principal radius of curvature of a space curve in the form (b) is: (x'^ + y'^ + z") g P = {{y'z" - z'y"Y + {z'x" - x'z"y + ix'y" - y'x"Y\i <-'2 - (a;"2 + 3,"2 + s"2 _ j"2)i where the double accents denote second differentials with respect to t, and s, the length of arc, is a function of t. 2.675 When t = s: \2 \ i r [{ThiW-m 2.676 The direction cosines of the principal normal to the space curve in the form (b) are: „ z' {z'x" -x'z")-y' ix'y" -y'x") I _ - , , x'(x'y" - y'x") - z'jy'z" - z'y") m ^ , GEOMETRY 59 n , y'iy'z" - z'y") - x'jz'x" - x'z") where L = [x'^ + y'^ + z''\i\{y'z" - z'y"f + (z'x" - x'z' J + (x'y" - y'x"fY. 2.&n The direction cosines of the binormal to the curve in the form (b) are: /" = y z — z y m ,, X y — y X where S = {{y'z" - z'y"f + {z'x" - x'z"Y + ix'y" - y'x"y]K 2.678 If s, the distance measured along the curve from a fixed point on it is the parameter, t: „ cPx , (Py , dh ^ = pd?"^ ^ Pi?' "" = "di^' where p is the principal radius of curvature; and /" _ (^ — — "^y] " P\ds d? ~ ds dsY . ,, /dz d^x dx dh\ ^ ^PyislF^'Tsd?)' ,, /dx d?y dy esc x = -. , cot x = » cos X COS X sin x tan x sec^ X = 1 + tan^a;, csc^a; = i + cot^x, sin^x + cos^a; = i, versin a; = i - cos a;, coversin a; = i - sin a;, haversin x = sin^ - • 2 o A-l • ■ ^ \ , /I — cos 2X „. / .X ,X 3.01 sm X = - sin (- x) = V , = 2V/ cos^ cos* -» » 2 V 2 2 a 2 tan - . X X tan x 2 = 2 sin - cos 2 2 VT+T^ i + tan^^ 2 Vi + cot^x ^^^ ^ _ cot X tan - + cot x 2 2 . = cot -• (i — cos x) = tan -• (i + cos x), = sin y cos (x — y) + cos y sin (x — y), = cos y sin (x + y) — sin y cos (x + y), 3.02 cos X = cos (- x) == y = 1-2 sin^^j 2 = cos^ sin^ - = 2 cos^ I = I — tan^- •v/i + tan^ X 2 I + tan^ - I + tan x tan - tan x cot i 2 2 2 , X ^ X cot tan - 2 2 cot X sin 2X X , X ^T j_ rot'x 2 sin x cot - + tan - V I -I- coi x 2 2 = cos y cos (x + y) + sin y sin (x + y), - cos y cos (x — 3))— sin ji sin (x — y), = |-(e" + fi-^^). 6i 62 MATHEMATICAL FORMULAE AND ELLIPTIC FUNCTIONS sin 2.V I — cos 2X 3.03 tan x = - tan (- x) = I + cos 2X sin 2X . /i — cos 2x _ sin (x + y) + sin (x — y) V I + cos 2x cos {x + y) + cos (x — 31)' cos {x — y) — cos (x + y) sin (x + y) — sin (x — y) = cot X — 2 cot 2X, , •v tv JL- tan - tan - 2 tan - 22 2 /y /Y" 'V I — tan - I + tan - i — tan^ - 2 2 2 I — tan - I + tan - 2 , 2 i + e^ 3.04 The values of five trigonometric functions in terms of the sixth are given in the following table. (Fpr signs, see 3.05.) sin X = cos X = a tan X = a cot X = a sec X = o CSC X = fl a a I Va^- I a I a Vi -a^ a sm X = Vi + a2 I Vi + a2 a a Va2_ I cos X = Vi -a^ •• a Vi + o2 a I a Vi + a^ I a a a Vi-a^ a a Va^-i I Va^- I a a tan X = Vi -a^ Va2- I Vi -a^ a I Va^- I cot X = Vi -a^ I a I Vi + a2 a a Vi + a2 sec X = Vi -a" I a Va^- I Vi +a2 a Vi + a2 CSC X = Vi -a^ Va^- I a 3.05 The trigonometric functions are periodic, the periods of the sin, cos, sec, CSC being 27r, and those of the tan and cot, t. Their signs may be determined from the following table. In using formulas giving any of the trigonometric TRIGONOMETRY 63 functions by the root of some quantity, the proper sign may be taken from this table. 0° TT 2 0-90° TT 2 TT TT 2 90° - 180° T 180° 2 180° - 270° 2 270° -7r - 27r 2 270° - 360° 27r 360° sin + I + - — I - cos I + - — I - + I tan + ±00 - + ±00 - cot Too + - Too + - =Foo sec I + ±00 - — I - ±00 + I CSC =Fco + I + ±00 - — I =Fa) 3.10 Functions of Half an Angle. (See 3.05 for signs.) . /i — cos X X = ±v ) T 2 3.101 3.102 3.103 sin cos :Vi + sin X =F Vi — sin x \ = ±V/^-(r -^=^^ ^ 2\ ±Vi + tan^ x' + cos .T ±v I + sin a; ± Vi - sin x { , '^ 2\ ±Vi + tan^ x) tan I — COS :V ■ — J I + COS X 64 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS _ sin :»; _ i — cos x I + cos X sin X ' ±V I + tan^ a; - I tan X 3.11 Functions of the Sum and Difference of Two Angles, 3.111 sin (a; ± y) = sin x cos y ± cos x sin y, = cos x COS y (tan x ± tan y), tan X ± tan y . , . tan a; T tan y ^ ' = - < COS (x + ji) + COS (x - y) \ (tan a; ± tan y). 3.112 COS (a; ± y) = cos x cos y =F sin x sin y, = cos X cos y (i T tan a; tan y), cot a; T tan y / _ x = — : : COS \X =F y) cot a; ± tan y cot y T tan a; . . ^ < = sin (« T y ) , cot y tan a; t i = cos X sin y (cot y T tan a;). 3.113 ^ , . , tan a: =h tan y tan (a; ± y) = -— — —^ I =F tan X tan y cot y ± cot X cot X cot y T I sin ix ± sin 2y cos 2x + cos ay 3.114 , , ^ cot X cot y T I cot (a; ± y ) = — : : — > cot y ± cot X sin 2a; T sin 2y cos 2X — cos 2y 3.115 The cosine and sine of the sum of any number of angles in terms of the sine and cosine of the angles are given by the real and imaginary parts of cos (a;i + ^2 +. • . • + Xr) + « sin (a;i + *^2 + • • • . + ^n) = (cos a;i + i sin x\) (cos a;2 + i sin ^2) (cos a;„ + i sin «„) TRIGONOMETRY 65 3.12 Sums and Differences of Trigonometric Functions. 3.121 sin a; ± sin y = 2 sin |(x ± y) cos |(x T y), = (cos X + COS y) tan |(x ± y), = (cos y - COS x) cot K^ ^ y)) tan \{x ± y) , . . , = 7 ,, _^ . (sm X =F sm 7). tan ^{x=F y) ^ ^ 3.122 cos a; + cos y = 2 cos ^(x + y) cos §(x — y), _ sin a; ± sin y tan I (x ± 31) cot §(x + y) , s = -; TT 7 (cos y — cos %). tan i (x - y) 3.123 cos X - cos 31 = 2 sin i (y + :*;) sin \{y — %) = —(sin X ± sin y) tan |(x =F y). o fo^ ^ .. sin (x± y) 3.124 tan x ± tan y = -' ^^ - 3.130 I. cos X ■ cos y sin (x ± y) ,, ^ , \ = -: — ; _ : (tan x T tan y), sm(xT)() = tan y tan (x ± y)(cot y T tan x), I =F tan X tan y cot (x ± y) ' = (i T tan X tan y) tan (x ± y). « ^«.- i ^ sm (x ± y) 3.125 cot X ± cot y = ± -. — ^ . ^' • smx sm y sin X ± sin y , , , ; = tan |(x ± y). cos X + cos y ^ V ^/ sin X ± sin y , , , ^ \ = - cot |(x T y). cos X — cos y sin X + sin y tan \{x-\- y) sin X — sin y tan \{x — y^ 66 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS 3.140 1. sin^ X + sin^ y = i - cos (x + y) cos (x - y). 2. sin^ x — sin^ y = cos^ y — cos^ x = sin (x + y) sin (x — y). 3. cos^ X — sin^ y = cos (x + y) cos {x — y). 4. sin^ (x + y) + sin^ {x - y) = i - cos 2X cos 2y. 5. sin^ (^ + y) — sin^ {x — y) = sin 2x sin 2y. 6. cos^ (x + y) + cos^ (a; - y) = i + cos 2x cos 2y. 7. cos^ (^ + y) - cos^ (^ - y) = - sin 2X sin 2y. 3.150 1. cos nx cos mx = \ cos (w — w)x + | cos (w + »w)x. 2. sin nx sin wx = J cos (« — m)x — \ cos {n + m)x. 3. cos nx sin mx = J sin (m + m)x — | sin (n — m)x. 3.160 I. 2 gi+ta = gi ^cos y + i sin y). fli+iH = flrx {cos (y log a) + i sin (y log a)}. (cos X ± f sin x) " = cos nx ± i sin nx [De Moivre's Theorem], sin (x ± iy) = sin x cosh y ± i cos x sinh y. cos (x ± iy) = cos x cosh y T i sin x sinh y. cos X = Ke'"" + e"''')- sin X = (e'^ - e-'"). 2 g'^ = cos X + i sin x. g-ii _ (-OS X — J sin X. 3.170 Sines and Cosines of Multiple Angles. 3.171 n an even integer: f . («2 - 22) . , , (»2 _ 22) (m2 - 42) . ^ sin nx = n cos x < sm x ; — sm'' x H ; sin° x — . »2 M^r^a _ 2^) ^2(^2 _ 22) (^2 — ^\ cos wx = I ; sm'' X -\ ; sm* X — sm" x + 2! 4! 6! TRIGONOMETRY 67 3.172 n an odd integer: {n? -1') . , , in' - i2) (w2 - 32) 3! S! cos wx = COS X < 1 ; sin^ X + '^ sin* x I 2! 4/ 3.173 n an even integer: — —X I (ft — 2) sin nx = {-lY cos x { 2"~i sin""' x - - — ; — ^2""^ sin""' x \ i! 2! 3! + cos nx = (—1)2 < 2"~' sin" x ^, 2""^ sin""^ x H ^^ — r-^ 2"~° sin ""* x ' t! 5 n(n — i) (n — %) , . 3 2""' sin""" X + . . . . 3.174 n an odd integer : sin wx = (-1) 2 { 2""^ sin" x : 2"~' sin"~^ x H — ^^ — — ^ 2" 6 cin"— * ■ 2 7.-0 sin"-» X i! 2! 2"~' sin""^ X + . . . n(n - 3) (« - , • c 3! cos MX = (— 1)^~ COS X ^ 2"~i sin"~' x j — 2"~' sin"~^ x (« - 3) (« - 4) „ , . „ , (re - 4) (« - s) (w - 6) . . . 2! 3! + ^ ^^T ^ 2 "-5 sin"~^ X - ^ ^^^-^ — , 2""' sin"~' x + !■ 3.175 n any integer : I X < 2"~1 ( (n - 3) (w - 4) ,„_5 „„„„_6 „ (w - 4) (w - 5) (« - 6) sin nx = sin X < 2"~^ cos"~^ x j— 2""" cos""'' x 3! M , „ n(n — i) cos MX = 2"~1 cos" X : 2"~^ C0S"~^ X H ^ r^^ 2""^ COS"~* .T l! 2! + «(m - 4) (« - 5) „ - ^ , V Hv^-^ 0/ 2n-7 cos"-^ X + , 68 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS 3.176 sin 2* = 2 sin x cos .v. sin 3x = sin X (3 — 4 sin^ x)' = sin x{4 cos^ x — i). sin 4X = sin x{8 cos^ x — 4. cos x). sin 5a; = sin x(s — 20 sin^ x + 16 sin* x) = sin x-(i6 cos* x — 12 cos^ x + i). sin 6x = sin x(32 cos^ x — 32 cos^ x + 6 cos x). 3.177 cos 2x = cos^ X — sirf x = 1 — 2 sw? X = 2 COS^ X — I. cos 3X = COS x(4 cos^ x — 3)-^v'cs ?c-Jcrcsx = cos x(i - 4 sin^x). cos 4X = 8 cos* X — 8 cos^ x + i. cos sx = cos x(i6 cos* x — 20 cos^ ^ + s) = cos x(i6 sin* x — 12 sin^ x + i). cos 6x = 32 cos* X — 48 cos* X + 18 cos^ X — I. 2 tan X 3.178 tan 2x = cot 2X = I — tan^ X cot^ X — I 2 cot X 3.180 Integral Powers of Sine and Cosine. 3.181 n an even integer : (—1)2 < / \ 1 wfw — l) sin" X = ^^^^i \ cos MX - w cos (n — 2)x H j — cos (w - 4)x 2! w(w - 1) (w - 2) / <-N , , / n: j_ ^' — ^^ ^-p ^ COS (w - 6)x + + (-1)' 2 ■ ■ ■ ©■©' if . r \ . n{n - I.) , V " X = ■^^: { COS nX + n cos (W - 2)X -\ ^^ ; COS (« - 4)x COS" y> = ■■: — ~: \ Lus /^^ t « ^^s V'^ — :^;;^ -1 r 2" ^ [ 2: , n{n - i) (n - 2) , , i n\ + -^ -T COS (w - 6)x + ., . . + i y-r— ; 3! TRIGONOMETRY 69 3.182 n an odd integer: n—j ■ « (~l) ^ / • • / \ , **(" - l) • / \ Sin" :»; = ^^zr~ ] sm wx — n sin (w — 2)x -\ ■ sin (w — 4)21; 2 [ 2. ^ -. sin (w - 6)x + . . . . + (-1) 2 -; r — - — ■ — — sin X I . if . f \ , nin — i) , > cos" X = — zi { cos nx + n cos [n — 2)x -\ ; — cos (n — Ajx 2" ■■ [ 2! -\ ^ 1 cos (w - 6)X + + 7 r -. 7- cos X (=^)^(^)' 3.183 3.184 sin^ X = |(i — cos 2x). sin' X = 3(3 sin X — sin 3X). sin* X = |(cos 4X — 4 cos 2x + 3). sin^ X = re (sin 5x — 5 sin 3X + 10 sin x). sin* X = — -^(cos 6x — 6 cos 4X+ 15 cos 2x — 10). cos^ X = 5(1 + cos 2X). cos' X = i(3 cos X + cos 3x). cos* X = 5(3 + 4 cos 2x + cos 4x). COS* X = iV(io COS X + 5 cos 3x + cos sx). cos* X = -^(lo + 15 cos 2x + 6 COS 4X + cos 6x). INVERSE CIRCULAR FUNCTIONS 3.20 The inverse circular and logarithmic functions are multiple valued; i.e., if o = - tan"' 2XV I — X = cot"' = j log (x + V x^ — i) • 3.22 X COS"' X = IT — COS"' (—x) = sin"' x = —cos"' (2X? — i) 2 2 _ Vi -x2 = 2 cos" 1^/1+^ == sin-' Vi -x2 = tan"' • = 2 tan-' \/- — - = - tan"' { — -, \ = cot"': y 1 + X 2 = - tan - \ 5 > = cuL - / ; + X 2 1 2X^ - I J V I - x'' 3.23 = i log (x + Vx^ — l)= TT— i log (Vx^ — I — x). tan"' X = - tan"' (-x) = sin"' — -==^= = cos ' ■ Vi + x^ Vi + x^ I . , 2X TT I + x^ = - sin"' ; = cot"' X = sec ' Vi + x^ TT ^ _i I I _i I - » tan ' - = - cos ' 5 2 X 2 I + x-^ , f I + Vi + xM ? . _j f Vi + X- - I = 2 cos"' { ==^— } = 2 sin ' < =r:=^ — f = ^ sill \ 2V1 + X^ J I 2V1 + 1 ^ , 2X ,. _i f Vl + X^ - I - tan"' J = 2 tan ' 2 I — X'' [ X J x + c = — tan"' c + tan I — ex I., I— ix I., i + X I.. I + ix = - Z log r- = - J log ■: = » log :- • 2 I + tX 2 ° t - X 2 I-tX TRIGONOMETRY 7 1 3.25 I. sin~^ X ± sin~' y = sm~'{x-\/i — y^ ± yVi — x^}. 2. cos~i X ± cos~^ y = cos~'{x3' =F \/(i - «^) (i - y^)}. 3. sin~^ a; ± cos~i y = sin^^jx^' ± VCi - x^) (i - y'^l = cos^'jj'-v/i — x^ =F xVi — y'^\- 4. tan~^ X ± tan~' y = tan~^ _ • I =F xy xy zi= I •^ ■^ y^ X y=Fx = cot 1 — xy ± 1 HYPERBOLIC TUNCTIONS 3.30 Formulas for the hyperbolic functions may be obtained from the corre- sponding formulas for the circular functions by replacing x by ix and using the following relations: 1. sin ix = \i{e'' — e~^) = i sinh x, 2. cos ix = Ke"" + e~^) = cosh x. i(e'^-i) . tan tx = — 5 = t tanh x. cot tx = —I -5 = — t coth X. sec ix = = sech x. 2% . CSC tx = = — t cscn X. sin""' ix = i sinh~' x = i log (a; + Vi + 3C^). ' ia; = — »■ cosh~' a; = -j log {x + \/i + o^). COS /I -4- X - ' I — .r 10. /x + I cot~' ix = — i coth~' X = — i log y ^ 72 MATHEMATICAL FORMULvE AND ELLIPTIC FUNCTIONS 3.310 The values of five hyperbolic functions in terms of the sixth are given in the following table : sinh X = a cosh X = a tanh X = a coth X ^ a sech X = a csch x = a sinh X = cosh X = tanh X = coth X = sech X csch X = Vi a Vi + a2 Va ^ + 1 a I Vi + a2 Va'- Va^- Va" Va' Vi — o^ Va^ Vi- Va' Vi -a' Va'- Vi -a^ Va Vi- I a Vi — o^ Vi — a^ Vi + a' a I Vi + a^ Vi + a a" Vi+a^ Vi — a^ 3.311 Periodicity of the Hyperbolic Functions. The functions sinh x, cosh x, sech x, csch x have an imaginary period 2Ti, e.g. : cosh X = cosh {x + 27rin), where n is any integer. The functions tanh x, coth x have an imaginary period iri. The values of the h}^erboUc functions for the argument o, —i, iri, - — > are given in the following table : o TV . — I 2 iri ^7' sinh o i o — i cosh I o — I tanh o co-i o 00 -i coth 00 00 o sech I a> — I 00 csch 00 — i 00 i TRIGONOMETRY 73 3.320 I. sinh ? = \/' cosh X — I , 1 4 /cosh X + I 2. cosh -X = y - 2 ^ , I cosh X — I sinh x . /cosh x — i tanh -X = — ^-j = — = V — 2 sinh X cosh x + i V cosh x + i 3.33 1. sinh (x ± y) = sinh x cosh y ± cosh x sinh y. 2. cosh (x ± y) = cosh X cosh y ± sinh x sinh y. ^ , / ^ tanh X ± tanh y 7.. tanh (x ± y) = -. ~- • ^ I ± tanh X tanh y ,, - > coth X coth y ± I 4. coth (x ± y) = — -r ^ coth y ± coth x 3.34 1. sinh X + sinh y = 2 sinh ^{x + y) cosh i(^ — y)- 2. . sinh X — sinh y = 2 cosh ^{x + y) sinh ^(x — y). 3. cosh X + cosh y = 2 cosh |(x + y) cosh ^(x — y). 4. cosh X — cosh y = 2 sinh ^{x + y) sinh §(x — y). c. tanh X + tanh y = — ; '■ — r— ■ •' cosh .T cosh y i 1- i u sinh (.V - y) 6. tanh x — tanh y = — ; r^ — cosh .T cosh y .1, , .T, sinh (x + y) 7. coth X + coth y = -^-j :— r^ — ' sinh X smh y 8. coth X - coth y = - ^!" ~ -^ • sinh X sinh y 74 ^ MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS 3.35 1. sinh {x + y) + sinh (x — y) = 2 sinh x cosh y. 2. sinh (x + y) — sinh {x — y) =2 cosh x sinh y. 3. cosh (» + >>) + cosh (x — y) = 2 cosh x cosh y. 4. cosh (x + y) — cosh {x — y) = 2 sinh x sinh y. , 1 , , sinh X ± sinh y 5- tanhi(x±3') = — r ; r— • •^ '■^ ■" cosh X + cosh y ,, , , , sinh x T sinh y 6. coth §(x ± y) = — r r — cosh X — cosh y tanh X + tanh y sinh (x + y). '■ tanh X — tanh y sinh (x — y). coth X + coth y _ sinh (x + y) coth X — coth y sinh (x — y) 3.36 I 2 sinh (x + y) + cosh (x + y) = (cosh x + sinh x) (cosh y + sinh y). sinh (x + y) sinh (x - y) = sinh^ x - sinh^ y = cosh^ X - cosh^ y. 3. cosh (x + y) cosh (x - y) = cosh^ x + sinh^ y = sinh^ X + cosh^ y. , I + tanh \x 4.- smh X + cosh x = : — {-3- • ^ I — tanh fx 5. (sinh X + cosh x)" = cosh wx + sinh nx. 3.37 1. e" = cosh X + sinh x. 2. g-^ = cosh X - sinh x. 3. sinh X = ^e^ — e~^). 4. cosh X = |(e^ + e~^). TRIGONOMETRY 75 3.38 I. 2. sinli 2:»; = 2 sinh x cosh x, 2 tanh X I — tanh^ X cosh 2a; = cosh^ X + sinh^ x = 2 cosh^ ^ — ij = 1 + 2 sinh^ X, _ I + tanh^ X I — tanh^ X , 2 tanh X ^. tanh 2x = r5 — •^ I + tanh^ X 4. sinh 3a; = 3 sinh a- + 4 sinh' x. 5. cosh 3X = 4 cosh' ^ — 3 cosh x. , , ^ tanh X + tanh' a; 6. tanh 3X =^^ ; — r? • 1 + 3 tanh'' X 3.40 Inverse HyperboUc Functions. The hyperbohc functions being periodic, the inverse functions are multiple valued (3.311). In the following formulas the periodic constants are omitted, the principal values only being given. I. sinh~^ X = log (x + V.v^ + i) = cosh~^ 'v^+i. 2. cosh~i X = log (x + Vx^ — i) = sinh~^ Vx^ — i. 3. tanh-i x = logy^--^• 4. coth~' X = log y ^^ = tanh""' X S sech"! X = log( - + y/-^ ~ ^) ^ cosh~i -• 6. csch~' X = log (~ + y -J + I J = sinh"' — 3.41 ___ 1. sinh"' X ± sinh"' y = sinh"'(xV i + y^ ± 3'V i + x^). 2. cosh"' X ± cosh"' y = cosh~' (.vv ± V (x^ — i){y^ — i)). ->. tanh"' X ± tanh"' y = tanh"' '— "^ 1 ±xy 76 MATHEMATICAL FORMULAE AND ELLIPTIC FUNCTIONS 3.42 I. ^'''^" "k + t) = ^^^^" \{^ - tj' x^ — I ^ , , X — I = tanh~' —„ = 2 tanh~i x^ + 1 X + i' = log X. 2. cosh~i CSC 2x = — sinh~^ cot '2x; = — tanh~i cos 2X, = log tan X. 3. tanh-i tan^ f- ^ ?j = 1 jog CSC X. 4- tanh^i tan^ - = - log sec x. 22° 3.43 The Gudermannian. If, 1. cosh X = sec 6. 2. smh X = tan 0. 3. e"' = sec + tan 6 = tan ( ~ + - ) • 4. x = log tan f^ + -j- S- = gd x. ■ 3.44 1. sinh X = tan gd x, 2. cosh X = sec gd x. 3. tanh X = sin gd x. tanh - = tan - gd x- 2 2 ° ^ _ I + sin gd X cos gd X cos ( — hgdx) sm (j + gdx) TRIGONOMETRY 77 6. tanh~' tan a; = | gd 2x. 7. tan~i tanh x = | gd~^ 2X. 3.50 SOLUTION OF OBLIQUE PLANE TRIANGLES a, b, c = Sides of triangle, a, j3, y = angles opposite to a, b, c, respectively, A = area of triangle, s = ^{a + b + c). Given Sought Formula a, b, c a. A a, b, a j3 a, a, /? sm - a = V ^ ~ • 2 y be I . /s{s - a) ^°^i«=V be ■ tania=v/'^^-/^^'r)- 2 V s{s - a) c^ + i2 - a^ cos a = 1 20c A = Vs{s - a){s - b){s - -c). a 6 sin a When a>b, B<— and but one value results. When b>a 2 /3 has two values. 7 7 = 180° -(a + ;8). c a sin 7 c = — -. sin a A A = \ ab sin y. b J sin /3 = — : sm a y y = 180° - (a + jS). c a sin 7 a sin (a + j8) sm a sm q; 78 MATHEMATICAL FOEMULyE AND ELLIPTIC FUNCTIONS Given Sought Formula A ^ I L • 1 „ sin jS sin 7 A A = - ab sm y = - a' ^ -• 2 2 sin a , , a sin 7 a, 0, y a tan a b — a cos y a,^ f(a + /3) = 90° - It. tan lia - /3) = ^ cot §7 c = (a^ + b^ — 2ab cos 7)^. = {{a + by~4ab cos^ ^y}^ = l(a-by + 4ab sin^ iy]K ^ - ^ X. i JL ^T sin §7 = 7 where tan rf> = 2 V 00 r cos

tan i(a + 7). cos ^{a - c) 2V -r I J cos c = cos a cos J + sin o sin b cos 7. cos a cos (6 — 0) cos C = — Pi cos y cos b cos (a — (/)) cos (j) hav c = hav {a — b) + sin a sin b hav 7 sin tan 7 tan a = -- sin jS = tan 18 = tan -(a + /S) = 2 tan i(Q; - (8) = sin {b - e) sin 7 sin b sin c sin a sin b sin a sin tan 7 sin (a — (/)) cos |(o — b) cot §7 cos 5(0 + 6) sin |(a — 6) cot I7 sin J(a + 6) c, a, /3 tan 6 = cos c tan a tan (^ = cos c tan |3 cos 7 = — cos a cos jS + sin a sin ;8 cos c. cos a cos (|8 + 6) cos 7 - tan a cos e cos P cos (a + «/>) COS ct> tan c sin e sin ((3 + 6) Given a, b, y a, b, c €, 7 TRIGONOMETRY Sought b a, b < e e Formula tan c sin 4> , ia.n c BUI (^ tan = -: — 7 — --7T* sin (a + . ^ V 2 ^/ V 2 -'Z cos (x + sy) = • *=o sin I sin (2M + 2)X 10. '^(-i)'^-^sin {2k - i)x = (-!)"■ cos 2 cos X 2W + I xl II. T^(-i)*cos/fex= -^+ (-1)" ^^ — ^ — -■ A=i 2C0S- r sm x(i — r" cos nx) — (i — r cos x)r" sm nx 1 — 2r cos X + r'' 12. ^j r* sin ^x = S. , (i — '' cos x) (i — r" cos nx) + r"'^^ sin x sin nx r" cos ^x = ; — 5 ' • I — 2r cos X + f* 14- 2 (^* '^"^ 5) = '''^' "" ~ (^ ''''' 5) ■ S. 2(2*sin^5) =(2"sin Jj -sin^x. i^ TEIGONOMETRY 83 16. Xv~l '•^^ ~i- = ^; cot ~ — 2 cot 2.V. 2« 2" 2" n— I Sk^2Tr y/nl mr . mr\ cos = I + cos h sin — ^ • « 2 \ 2 2 / W-I _ ■^^ . k^ 27r v" / nir . inv\ 18. 7, sin = I + cos — -sin — • <^ n 2 \ 2 2 / k=i »— I sin — = cot ■ 2« 4=1 Sj A- 2^"+^ - I IX -Titan^ -T = 7, — T + 4 cot-' 2.v r- cot —- • 2^k 2* ^'2 2 2 h-o 3.62 «— I 5 ^TT esc — Watson (Phil. Mag. 31, p. iii, 1916) has obtained an asymptotic expansion for this sum, and has given the following approximation: 5'„ = 2w{o.732935S992 logio(2w) - 0.1806453871} 0.087266 0.01035 0.004 , 0.005 n 11^ rr' n' Values of Sn are tabulated by integers from w = 2 to w = 30, and from « = 30 to « = 100 at intervals of 5. The expansion of (kr _ §\ n 2) where s. (Rayleigh, Theory of Sound, I, p. 278.) cos X cosh X = — I. xi = 1. 875104, X2 = 4.694098, X3 = 7-854757, Xi = 10.995541, Xi = 14.137168, xe = 17.278759, Xn = 2(2^ — i)7r n>6. I — (i + x^) cos X = o. Xi = 1. 102506, X2 = 4-754761, ^3 = 7-837964, Xi = 11.003766, X6 = I4.I32185, Xe = 17.282097. (Schlomilch: Ubungsbuch, I, p. 354.) IS e 6 - cot 6 = o, = 49° 17' 36".S. (1. c. p. 355.) TRIGONOMETRY 87 3.810 The smallest root of 6 — cos 6 = o, e = 42° 20' 47 ".3- 3.811 The smallest root of xe'' — 2 = 0, is X = 0.8526. 3.812 The smallest root of log (i + x) - fx = o, IS 3.813 The first roots are: (1. c. p. 353.) (1. c. p. 353.) (1. c. p. 353.) 3.814 The first roots are: ^ = 0.73360. tan X — X + - = o. X Xi = 4.480, X2 = 7.723, X3 = 10.90, Xi = 14.07. (Collo, Annalen der Physik, 65, p. 45, 1921.) r cot X + X — = 0. X Xi = o, Xi = 2.744, X3 = 6. 117, Xi = 9.317, Xi = 12.48, xe = 15.64, X7 = 18.80. (CoUo, 1. c.) 3.90 Special Tables. sin d, cos d: The British Association Report for 1916 contains the following tables: Table I, p. 60. sin 6, cos 6, 6 expressed in radians from 6 = o to 6 = 1.600, interval o.ooi, 10 decimal places. Table II, p. 88. — sin 6, 1 — cos d, 6 = o.ooooi to 6 = o.ooioo, interval o.ooooi, 10 decimal places. 88 MATHEMATICAL FORMULAE AND ELLIPTIC FUNCTIONS Table III, p. 90. sin 6, cos 6; 6 = o.i to 6 = lo.o, interval o.i, 15 decimal places. J. Peters (Abh. d. K. P. Akad. der Wissen., Berlin, 1911) has given sines and cosines for every sexagesimal second to 21 places. hav 6, logio hav 6: Bowditch, American Practical Navigator, five- place tables, 0° - 180°, for 15" intervals. Tables for Solution of Spherical Triangles. Aquino's Altitude and Azimuth Tables, London,. 1918. Reprinted in Hydro- graphic Ofl&ce Pubhcation, No. 200, Washington, 1918. Hyperbolic Functions. The Smithsonian Mathematical Tables: Hyperbolic Functions, contain the most complete five-place tables of Hyperbolic Functions. Table I. The common logarithms (base 10) of sinh «, cosh u, tanh u, coth u: u = o.oooi to M = o.iooo interval o.oooi, u = o.ooi to M = 3.000 interval o.ooi, M = 3.00 to M = 6.00 interval o.oi. Table II. sinh u, cosh u, tanh u, coth u. Same ranges and intervals. Table III. sin u, cos u, logio sin u, logio cos u: u = O.OOOI to u = O.IOOO interval 0.0001, u = o.ioo to u = 1.600 interval o.ooi. Table IV. logioe" (7 places), e" and e"" (7 significant figures): M = o.ooi to M = 2.950 interval o.ooi, u = 3.00 to M = 6.00 interval o.oi, M = i.o to M = 100 interval i.o (9-10 figures). Table V. five-place table of natural logarithms, log u. u = 1.0 to u = 1000 interval i.o, u = 1000 to M = 10,000 varying intervals. Table VI. gd u (7 places) ; u expressed in radians, u = o.ooi to m = 3.000, interval o.ooi, and the corresponding angular measure, u = 3.00 to m = 6.00, interval o.oi. Table VII. gd~^u, to o'.oi, in terms of gd u in degrees and minutes from 0° i' to 89° 59'. Table VIII. Table for conversion of radians into angular measure. TRIGONOMETRY 8q Kennelly: Tables of Complex Hyperbolic and Circular Functions. Cambridge, Harvard University Press, 1914. The complex argument, x + iq = pe^. In the tables this is denoted pZd. p = Vy? + g2^ (■g.n 5 = qlx. Tables I, II, III give the hyperbolic sine, cosine and tangent of {p^b) expressed as rZ.y. 8 = 45° to 5 = 90° interval 1° p = o.oi to p = 3.0 interval o.i. Tables IV and V give — 5 — , — ^ — expressed as rZ 7, = pZ 5, p = 0.1 to p = 3.0 interval o.i, 5 = 45° to 5 = 90° interval 1°. Table VI gives sinh (pZ 45°), cosh (pZ 45°), tanh (pZ 45°), coth (pZ 45°), sech (pZ 45°), csch (pZ 45°) expressed a.5 rZy. p = o to p = 6.0 interval 0.1, p = 6.05 to p = 20.50 interval 0.05. Tables VII, VIII and IX give sinh (x + iq), cosh (x + iq), tanh {x + iq), expressed as m + iv: a; = o to a; = 3.95 interval 0.05, gf = o to g = 2.0 interval 0.05. Tables X, XI, XII give sinh (x + iq), cosh {x + iq), tanh {x + iq) expressed as r£y. a; = o to x = 3.95 interval 0.05, q = oto q = 2.0 interval 0.05. Table XIII gives sinh {/^-\- iq), cosh (4 + iq), tanh (4 + iq) expressed both asM + iv and r Z y : • q = o to q = 2.0 interval 0.05. Table XIV gives - and logio — X = 4.00 to a; = 10.00 interval o.oi. Table XV gives the real hyperbolic functions: sinh 6, cosh 6, tanh d, coth 6, sech 0, csch 0. - o to 0=2. 5 interval o.oi, = 2.5 to = 7.5 interval 0.1. go MATHEMATICAL FORMULAE AND ELLIPTIC FUNCTIONS Pernot and Woods: Logarithms of Hyperbolic Functions to 12 Significant Figures. Berkeley, University of California Press, 1918. Table I. logio sinh x, with the first three differences. X = .0000 to X = 2 018 nterval o.ooi. Table II. logio cosh x. X = 0.000 to x = 2.032 interval o.ooi. Table III. logio tanh x. X = 0.000 to X = 2.018 interval o.ooi. sinh X Table IV. logi Table V. logio X X = 0.00 to X = 0.506 interval o.ooi. tanh X X X = 0.000 to X = 0.506 interval o.ooi. Van Orstrand, Memoirs of the National Academy of Sciences, Vol. XIV, fifth memoir, Washington, 1921. Tables of — > e", e~'', e"'^, e~"'^, e^ifc, sin x, cos x, to 23-62 decimal places or ni significant figures. IV. VECTOR ANALYSIS 4.000 A vector A has components along the three rectangular axes, x, y, z : ■^ X, ^ yt '■^ Z' A = length of vector. A = ^yAjTAjTT}. Direction cosines of A, — p, — p, —r. AAA 4.001 Addition of vectors. A + B = C. C is a vector with components. ^ 1/ ~ yl y -f- i> y. 4.002 6 = angle between A and B. cos 6 C = VA-' + £- + 2.45 cos 6 A,Bx+ AyBy+ AzBz AB 4.003 If a, b, c are any three non-coplanar vectors of unit length, any vector^ R, may be expressed: R = aa + 6b + fC, where a, b, c are the lengths of the projections of R upon a, b, c respectively. 4.004 Scalar product of two vectors: 5AB = (AB) = AB are equivalent notations. ^^ AB = AB cos AB. 4.005 Vector product of two vectors: FAB = A X B = [AB] = C. C is a vector whose length is ^^ C = AB sin AB. The direction of C is perpendicular to both A and B such that a right-handed rotation about C through the angle AB turns A into B. 92 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS 4.006 i, j, k are three unit vectors perpendicular to each other. If their direc- tions coincide with the axes x, y, z of a rectangular system of coordinates: X = Axi + Ay) + Az)s.. 4.007 4.008 4.009 4.010 ii = P = jj = j^ = kk = k^ = I, ij ■= ji = jk = kj = ki = ik = o. AB = BA Fij = - V]i = k, Fjk = - Fkj = i, FM = - Fik = j. cos AB = AxBx-\- Ayl FBA = i j k Ji. X ^ y ^ B X By B FAB = = {A yB, - A,By)i-^{A^Bx- A xB.)j + U ^By - A yB^)^. 4.10 If A, B, C, are any three vectors: AFBC = BFCA = CFAB = Volume of parallelepipedon having A, B, C as edges 11 By B, FA(B + C) = FAB + FAC. F(A + B) (C + D) = FA(C + D) + FB(C + D). FAFBC = Bi-AC - C5AB. FAFBC + FBFCA + FCFAB = o. FABFCD = ACBD - BCAD. F(FAB-FCD) = C5(DFAB) - D5(CFAB) = C^CAFBD) - D5(AFBC) = B5(AFCD) - AS'(BFCD) = Bi'CCFDA) - AS(CFDB). 4.20 I. 2. VECTOR ANALYSIS (/AB = M B+ Bdk. dVAB = VXdB + FdAB = VkdR - VRdk. 93 4.21 I. V - •— • — dx dy 2. VA = div A dA: dA. dA, H +■ dx dy dz 4. FVA = curl A = rot A i j k dx dy dz ■ri. X -^ y -^ z vv 32 32 32 ^ dx^ ^ a/ ^ az2 4.22 I. curl grad = curl V<^ = V\7V4> = o. 2. 3- 4- 5- 6. div curl A = o. curl curl A = curP A = V div A - V^A. V'A = iV'A. + JV'^ y + kV'^ .. .„ . 5 A d , d AV = A:,-r-+Ay— + A^—- dx dy dz 94 MATHEMATICAL FORMULAE AND ELLIPTIC FUNCTIONS 4.23 VAB = grad AB = (AV)B + (BV)A -h I'.A curl B + V.B curl A. Vl'AB = div TAB = B curl A - A curl B. FV TAB = (BV)A - (AV)B + A div 5 - B div A. div (pA = (j) div A + AV<^. curl (j)A = ]'■ VA + (j) curl A = F -grad ^.A + 2{u,v,w), [ 3 = , [ z = r cos 6. 4- 5- 6. hi = I, h = -, ha = r sin 6 dSr = r^sind dd dcf), dSe = r sin d drd (j), dStj, = r dr d 6. dT = r^sind dr d d dcj). I02 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS curlr A = — t—r. I -^ / sin fl /(^ 1 ^ \ 8. \ curie A ■ >sin0ll(^^^^^^)- ab>c, u>v>w. 6 = u: Ellipsoid. 6 = v: Hyperboloid of one sheet. 6 = w: Hjrperboloid of two sheets. 2. CURVILINEAR COORDINATES ■ 2 ^ {a- + m) (g' + v) (a- + w) * (a2 - 62) (a2 _ c2) ' 2^ (6^ + u) jb'- + i>) {b^ + w) 2^ (c^ + m) (c^ + v) (c' + w) {a' - c') (62 - c2) ^^, 4(a' + m) (&^ + m) (c^ + m) (m — d) (m — w) ' {v — w) {v — u) ' , „ 4(a- + w) (62 + w) (r + w) ht = — ; 77 '■ — ^ (,W — Mj (to — V) 103 div A = . ^^'^ + -) ^^; + -^ ^f + "^ ^ ( v (. - .) (. - TO) a}) {u — V) {u — w) du\ ) + 2 V(a^ + ^) (^i^ + f) (c2+ w) a (j) — w) (m — d) + 2 V(a^ + to) (62 + w) {(:'■ + to) a 5- V'=4 (m — to) (w — to) V(a' + m) (6^ + u) (c^ + m) a aTO f -\/(to - Z)) (m - D) .-^l^j { \/{u - to) (j; - to) ^„ j. £(v(.^ + .)(6^ + «)(.^ + .)A) V(a^ + .)(^ + .)(&.^ + .) a (^v(^M:^yWT (m — z)) (m — to) + 4 + 4 (m — t)) (d — w) Via^ + w) (6' + w) (c^ + to) a aw ' .)(c^ + .)£) (o— w) (ii— to) ^ V V(a^ + to) (6^ + to) (c^ + to) -. OTO V oic 6. curL A = J) — TO / (a^ + v) (6^ + ;>) (c- + z>) a_ , ^^ U — V dv vA^j , /(a" + w) (F + to) (c^ + to) a / , , ' V ^ -T-iVv - WAy y u — w dw\ curl„ A = curl„ A — w . lia^ + w) (62 + w) (c2 +. to) a / ^ . \ V T~ V M - TO^„ V J) — TO aw \ / s/ {a? +u) (62 + |<) (c2 _(_ m) _a_ D — M du ( Vto - uAA 2 + m ) (6^ + u) (c^ + z<) _a_ / / \ w — u ou \ / I04 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS 5.23 Conical Coordinates. The three orthogonal surfaces are: the spheres, I. X^ + / + 2^ = 1^, the two cones: 2. yO, y^ ^ + + +• ip: — W v^ — a = o. + 6V ' m2(z^ — c') (w' — c') ^1 =1, /'2^ = m2(j;2 _ jy2) ' hi {l? - ft2) (c2 - »2) ^ 2 _ (6^ - -UPl {C^ - VP) u^{v^ — w^) ^ J- A I ^ ^ 2^ ^ , V(z^ - b') {c' -v") d I /-2 2 i + uijf' — w^y dw ' 8. m'^mV 3m/ M2(r - W^) 6!) \ 6z)/ u^{v' - -up) dw\ dwj - V(6' - w") (c^ - w^) £; f Va' - w' A^ I V(y - le;') (c' - w'') dAu I d /^^ \ 1 curU A curl„ A = - IT- (m^^ ) ; u du\ J Uy/^ _ \/(i'' -6') (c^ - 1?) dAu dv 5.30 Elliptic Cylinder Coordinates. The three orthogonal surfaces are: I. The elliptic cylinders: ^2 + ^ cV (?{i^ — i) = I. CURVILINEAR COORDINATES 105 2. The hyperbolic cylinders: ^^^ = x. cV c^(i — ■>?■) 3. The planes: z = w. 2C is the distance between the foci of the confocal ellipses and hj^erbolas: 4. X = CUV. 5. y = cV*?— I Vi — J)^' 9- i , . I dAz dA^ curlu A = — , -^ r— > c-s/u^ -v^ dv dz , . dAu I dAz curl„ A = 5.31 ParaboUc Cylinder Coordinates. The three orthogonal surfaces are the two parabohc cylinders: I. y^ = 4CUX + 4c V. 2. An 3- 1 the planes: y^ = —4CVX + 4C^i>' z = w. 4- S- X = c{v — u). y = 2cy/m. I M + W I U-\-V h^ u hi V M + J) I 5m V " 5w/ 9v \u dv/ J dz^ io6 9- MATHEMATICAL FORMULAE AND ELLIPTIC FUNCTIONS dA^ V dAy curl„A= V^7~ + V dv u + v dz curls A = u u + dAu . . / u dAz V dz \ u + v du' curl. A = ^ f A (J^ A _ e (J^A.) u + v[du\yu + v I d-uyi u + v I 5.40 Helical Coordinates. (Nicholson, Phil. Mag. 19, 77, 1910.) A cylinder of any cross-section is wound on a circular cyUnder in the form of a heHx of angle a. o = radius of circular cylinder on which the central line of the normal cross-sections of the helical cylinder Ues. The 2-axis is along the axis of the cylinder of radius a. ti = p and V = 4> are the polar coordinates in the plane of any normal section of the helical cylinder. 4> is measured from a line perpendicular to z and to the tangent to the cylinder. w = 6 = the twist in a plane perpendicular to z of the radius in that plane measured from a line para:llel to the «-axis: X = (a + p cos 4>) cos 6 + p sin a sin 6 sin (/>, y = (a + p cos , z = a 6 tan a + p cos a sin d). h hi = h I —J P a^ sec^ a + 2ap cos + p^{cos^

= c sinh u sin v. The three orthogonal surfaces are the ellipsoids and h5^erboloids of revolution, and the planes, 6: c^ cosh^ u c^ sinb? u c^ cos ^v c'- sin'= v = I. With cos M = X, cos j; = /x .• 4• z = c X //, p = cV(X2 - i) ( I- m')- X^-i fe2 = 1 — p/- c^ (X2- m') , ^3' = c2(X2 - i) (i - m') 5.52 Spheroidal Coordinates (Oblate Spheroids): I. p -\-iz = c cosh(M + iv). z = c sinh u sin v. P = C cosh M COS V. cosh M = X, COS V = p,. 3- 4- I - /x" cHX^ - m')' fe^ c2(X2 - p^) c2(X2 - i) (i - m') 5.53 Parabolic Coordinates: I. z + ip = c{u + ivY- z = c{u^ — 1?), p = 2CUV. 3. M^ = X, D^ = p.. With curvilinear coordinates, X, p, 6: Io8 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS c'y \ + II c'y \ + jj. acVX/i 5.54 Toroidal Coordinates: z + g + ip I. u + iv = log P = z - a + ip a sinh u 2. cosh M — COS V a sin V cosh M — COS V , , cosh M — COS V , cosh u — cos i; • 3- hi = lh = , ha = :-r a a smh u The three orthogonal surfaces are: (a) Anchor rings, whose axial circles have radii, a coth M, and 'whose cross-sections are circles of radii, a csch u; (b) Spheres, whose centers are on the axis of revolution at distances, ± a cot V, from the origin, whose radii are, a CSC V, and which accordingly have a common circle, p = a, z = o; (c) Planes through the axis, w = 6 = const. VI. INFINITE SERIES 6.00 An infinite series: CO S M„ = Ml + M2 + Ms + . . . . n=i is absolutely convergent if the series formed of the moduli of its terms: |Mi| + |m2| + 1m21+.... is convergent. A series which is convergent, but whose moduli do not form a convergent series, is conditionally convergent. TESTS FOR CONVERGENCE 6.011 Comparison test. The series 2m„ is absolutely convergent if ] m„ | is less than C | z)„ | where C is a number independent of n, and v„ is the wth term of another series which is known to be absolutely convergent. 6.012 Cauchy's test. If I Limit I , « ^ M„ I. It is divergent if, \an+i / n+1 lOg no MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS 6.020 Alternating series. A series of real terms, alternately positive and nega- tive, is convergent if a„+i^a„ and limit In = O. In such a series the sum of the first s terms differs from the sum of the series by a quantity less than the numerical value of the {s + i)rf term. limit 6.025 If n—^co Un+\ I, the series 2m„ will be absolutely convergent if there is a positive number c, independent of n, such that, limit n Un+l Un I = 6.030 The sum of an absolutely convergent series is not affected by changing the order in which the terms occur. 6.031 Two absolutely convergent series, 5' = Ml + M2 + Ma + T = Vi -[- V2 -\- Vi -\- may be multiplied together, and the sum of the products of their terms, written in any order, is ST, ST = UiVi + M2'Bl + UiPi + 6.032 An absolutely convergent power series may be differentiated or inte- grated term by term and the resulting series will be absolutely convergent and equal to the differential or integral of the sum of the given series. 6.040 Uniform Convergence. An infinite series of functions of x, Six) = Ui{x) + Ui(x) + Uaix) + is uniformly convergent within a certain region of the variable x if a finite number, N, can be found such that for all values of « ^N the absolute value of the remain- der, I R„ I after n terms is less than an assigned arbitrary small quantity e at all points within the given range. Example. The series. ^(i -1- x^)" is absolutely convergent for all real values of x. Its sum is i + x^ if x is not zero. If X is zero the sum is zero. The series is non-uniformly convergent in the neigh- borhood of X = o. INFINITE SERIES III 6.041 A uniformly convergent series is not necessarily absolutely convergent, nor is an absolutely convergent series necessarily uniformly convergent. 6.042 A sufficient, though not necessary, test for uniform convergence is as follows: If for all values of x within a certain region the moduli of the terms of the series, 6' = ui{x) + Uiix) + are less than the corresponding terms of a convergent series of positive terms, T ^ M1 + M2 + M3 + . . . . where Mn is independent of .t, then the series S is uniformly convergent in the given region. 6.043 A power series is uniformly convergent at all points within its circle of convergence. 6.044 A uniformly convergent series, >S = Ui{x) + U2{x) + may be integrated term by term, and, 00 X S dx = ^ funix) dx. «=i 6.045 A uniformly convergent series, 5 = u-i{x) + thix) + . . . . may be differentiated term by term, and if the resulting series is uniformly convergent, 6.100 Taylor's theorem. f{x + h)= fix) + ^,/'(x) + ^/'(x) + + ^J^"Kx) + Rn. 6.101 Lagrange's form for the remainder: 2?„ =/(«+.) (x + dh)- j—^; o<0{y). The expansion of f{y) in powers of x is: /(y) =/(z) + x{z)f'iz) + f^f^ [!<^(z)ir(2)] + + 5|Sc{ i . When X = I, the series converges for w> — i and diverges for w^ -i. It is abso- lutely convergent only for n>o. When X = —I it is absolutely convergent for n>o, and divergent for n I put X = 7 in 6.30, I. (i + x)" = I + - X - n(m — n) „ n(m — n) (2m — n) , _1 i. ^i _| i '—± '- vi 2 1^3 , ,* nim — n) {2m — n) \_{k — i)m — n\ ^ 2. (i + x)"' = 1 - X + X^ - X^ + X* - 3. (i + x)"^ = I -,2X + 3X^ - 4x^ + 5x* - . . . . + . i-i I-I-3. 4. Vi-|-x=i-i--x- — ■ x^ -\ 2 ^ A o ^ 2 2-4 2-4-6 2-4-6-8 5- V7T X 2 2-4 2-4-6 2,-4-o-8 6. (i + «)i = I + - X - ^x^ + ^^ x^ - ''/"S" x^ + ^ 3 3-6 3-6-9 3-6-9-I2 ' ^ 7 ,-6 3'6-Q 3-6-0-I2 3'0-9 3-I-I 8. (i +x)3 = I + ^x + ^x^-^ -x" , ^ ' 2 2-4 2-4-6 2-4-6-8 3:l:l:3^4_ 3-fi-3-5 ^^ 2-4-6-8-IO 9. (i + x)-S = i-^x + ^x2- ^^ x= + ^ ^ ' 2 2-4 2-4-0 10. (l + X)' = I + - X - -^ x^ + -^ x^ 4 32 128 77 2048 x*-|- . . II. (i+x)-i = i--x + -^x2-^x' + ^x*- ^ ' 4 32 128 2048 12. (1 -'- x}i = I + - X — — X^ -I : 5 25 125 21 62s x*-|- . Il8 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS 13. (i + x)-= = i--x+—x^- — x^ + -^x*-. . . S 25 125 62s 14. (i + x)^ = I + ^x - — x^ + -^ x^ 23S_ ^4 _^ _ _ _ 6 72 1296 31104 / \ 1 I 7 o 01 , 172Q , 15. (i + x)-i = 1- - X + ~ x^ - -^ x^ -\ '—^ X* - . . . . 6 72 1296 31104 6.350 wV JG 2,-Xi iX.Jv OiV r~ c\ -^ I. = — r- + — r::^ + -^-^ + -T-8 + • • • ■ \x^<-l\. I— a;i + xi+aifi + x*i+x'* -■ 2. = -2 + -4 + -5 + [X^i]- a;— i.T+ix^+ix^+i -■ 6.351 I. , 1 " f lx\ nin — x) (xY i + Vi + x} =3"{i + «(-j + ^^(-j « may be any real number. 2. ( X + V 1 + a;M = I + jf x^ H ^ — j -' X* H ^ -^ ^^ x« + . . . 6.352 If a is a positive integer: {a— I n=o 6.353 If a and h are positive integers, and a I • 2 .^ 22»(» )2 (2M +1) L J 6.42 , . , ,„ „ 2 a;* 2-43;' 2-4-6 .T^ 1. (sin-i xf = x^^ \ — ^ + . . . . 323-53 3-5-7 4 "Z/(2«+l)! («+!)'' L J »=o 2. (sin-i a;)= =a;' + |}3^(i+^)^ + ^3Y(i+|2+^)^'+.--- [^^^i]" * — I /ka—1 3. (tan--)-^!2(-^-^fj^n S 2^0 + /> — 2 A J. \ .^^ 2ka + p — a — 2 kn= I a = I \&a=i / (Schwatt, Phil. Mag. 31, p. 490, 1916). 4. Vi — a;^ sin^i x = a; 1 ■ x? — oc' + 3 3-5 3-5-7 = a; + Zj^ ^> (2«-l)!(2«+l)'' L J sin-i X 2 2-42-4-6, — ==: = X -\-- x' -\ XT -\ X' + ■s/i-x^ 3 3-S 3-5-7 S 2^"(w!)^ (2W+ l) = V-i!:Mi....i [■,.<,]. INFINITE SERIES 1 23 6.43 I. log smx = logx- ■{ ^ a;? + -^ X* + — !— xfi + ' 6 180 2835 ^°s^-SSt^"^" ^^<'^^]- 2. log cos a; = — x^ — ~x* — - x^ ~ yfi - 2 12 45 2520 2^"-' (2^" - i) 5„ ^^ w(2»)! |_ 4 J 3. log tan X = log X + - a;2 + -^ X* + -^ x« + 4^ «' + 3 90 283s 18900 ^^ n(2n)'. L 4 J 4. log cos X = — I sin^ X + - sin^ x + - sin^ x + . 2[ 2 3 = - S^^^"^- [-^<7] 6.44 I. log (i + x) = X — x^ + - x' — x^ + . '2 3 4 «=I -" {log (i + x)}'' see 7.369. 2. log (x + Vi + x^) = X x' H i^ x^ - "^ " ^ " -^ " ^ x' + 2-3 2-4-s 2-4.6-7 , "V / ^. (2«- i)! x^-'+i r ^ ^ "1 ^ 2^" '«! {n- i)I (2W+ i) L J 3. log (i + Vi + a;2) = log 2 H x^ - ^^-? x"* + ^"'^"^'^ x* - . . . 2-2 2-4-4 2-4-6-6 = Iog2-2(-z)"-j!Lfi)= ,^" [x^^xl. ^rf 2'"' ^w!(«— l)! 2W |_ J 124 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS 4. log {l + Vl + X^) =logx + + I ivt I I-I-3 I X 2 ■ 3 x' ■ 2 ■ 4 • 5 a;= 1 , I "^/ N (2»-l)! X-^n-l r 1 5. log X = (x - i) - - (x - 1)2 + - (x - 1)3 ■ 2 3 10. Vi + »^ log {x + Vi + x^) = X + - x^ x' + 1-2 I-2-4 3 3-5 3-S-7 00 X ■ log (x + Vi + x^) 2^3 ^ 2-4 _5 2 ■4- 6 12. log (x + Vi + x^) = h 1323-53 ^ x^ 2 X* 2 • 4 x^ ,1 X - I I /x - iV I /x - l\' 6. log X = — — + - + - + . . . . X 2\ X / i\ X J n=i 1 ( X — 1 1 fx — iV 1 fx — lY ] 7. l0gX=2|— +-(^+-(-^j+...} ^^ 2;^ + I V^ + 1/ L J «=o I H~ X I I r I 8. log =2i — a; ,1 I X 3. — < I - log (i + ic) -=- tan""^ X > = 2x[ ° Vx J I-2-3 3-4-S x^ ox II 1-3 I 3. cosh-> X = log 2X 2 2X^ 2-4 4X* = log 2X - / iJ^M *~^" h<^> I «=o CD III ^^ /p2n+l r 4. tanh-1 x = x + -x' + -x° + -x' + . . .= 7, x^< i 357 AJ2n+i L- X X 2 3X'' 2-4 5x^ . , , I I II i-^ I CD '(«!)2 (2W+ l) CSch-X = 2 (--)" 2^n.,,{ftL+l) -""" P>^' , , I , 2 I X^ I-^ X* 6. cosh~^ - = log ^ X X 22 2-44 = sech-i X = log - - "y 2n^M ^'" U'■ (-x)" 2„\ \2 ^" h'i ^mJ 2» + I ■[' INFINITE SERIES 1 29 6.490 CX) 1. 1 = "V^g-*(2« + l). 2 sinh X ^ad 2 cosh % ^ «=o CO 3. - (tanh :*;-!)= ^ (-i)« e-2«^ «=i CO 4. - -log tanh - = y^— ^ e-- (^«+"-. ^ 2 * 2 ^^ 2M + I 2M + I »=o 6.491 i.2«--=^ ^S-(-)' By means of this formula a slowly converging series may be transformed into a rapidly converging series. 6.496 ' I I I I / v9 "I / \9 \ /^ \9 "t" I. tan X = 2x ■ 8y (2« - i)V - CO ^^ (2« — i)^7r^ — 4*^ M= I 00 I 2X 2X 2X I '^^ 2X X TT^ — x^ {2iry — x^ (37r)^ — x^ . • . ■ ^ ^^nV — x^ TT -^TT 57r CO = V ( i)n-i 4(2w - i)7r «= I (2w — i)^7r^ — 40;^ .2 I 2X 2X 2X X TT^ — 1^ (27r)^ — x^ (sttY — X = i+y(_i)n-i_i£ X ^^ n^TV^ — y? M = I By replacing x by ix the corresponding series for the hyperbolic functions may be written. 130 MATHEMATICAL PORMULjE AND ELLIPTIC FUNCTIONS INFINITE PRODUCTS 6.50 I « = I CO 2. sinh^ = ^JJ{^+;^)- » = I 3. cos x= nG-(24'i)v)' n = o 00 4. cosh^= n(^+(2«ri)v)- 6.51 sin X TT a; I. =11 cos — ■ » = I 6.52 I I — a; » = o JJ(i+a;2„), [«= Bix+i,y) = -^Bix,y), X -t y B{x, 1 - x) = sin TTX 6.610 For X real and positive: 6.611 6.612 \P(x+i) = ^ + xP(x), \p{l - x) = \p(x) + IT cot TTX. '/'(i) = -7- 2 log 2, vKi) = -y, yp{2) = 1-7. ^(4) = 1+^ + ^-7- 6.613 di INFINITE SERIES 133 6.620 ^w = S^ (-!)" + n M=0 6.621 6.622 = M*(^)-*(f) I3{x+i) + l3ix)=l, P{x) + )8(i -:*;)= ^-^!^- "^^ ' sin ir:« /3(i) = log 2, Kl)=f 6.630 Gauss's 11 Function: k I. n(*,z) = ^^JJ^ »=i ■ z + n 2. Jl{k,Z+l)=Ii{k,z)- '+' • 3. n(.) = ^_i::^i*n(^,.). 4. n (z) = r(0 + i). S. n (-z) n (z - l) = TT CSC TTS. 6.n(i) = :v?. 6.631 If z is an integer, w, n (w) = «! DEFINITE INTEGRALS EXPRESSED AS INFINITE SERIES 00 6.700 Jl^''^'^^^^ k\{2k + l.) k = o jjSft+l ■Sr J:„24+l 2"X' •3-5. . . . (2A + 1) 134 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS Darling (Quarterly Journal, 49, p. 36, 1920) has obtained an approximation to this integral: — -^tan-i I e-^^ii + x'e-^f Fresnel's Integrals: 6.701 />s (.V. = 2 (i^yf^^ k = ^4i+l = cos (a;2) 2 (-1)*- 224-^4ifc+l 3-5 (4^ + 1) «= o CO 6.702 X^sin(.^).x = 2(^^TSr^--^-' = sin (a;2) ^ (-1) A = o o2«: Ak+l k\=o I-3-5. . . (4^+ l) 2ik+\^ik+S "• ' fj ^ 1.3.5 (4^ + 3) 6.703 r ^(fi = '^ (-i)"^-r M = O 6.704 ^^^r ^-(^-y- ,, (jfe - i) ! Jo I - xt" CO s (a + wft) (a + k6 + i) (a + «6 + 2) . . . (a + nb + k — i) [i>o, a;2^i]. (Special cases, 6.445 and 6.923). CD CO o AJ nl[n + y) ^J y{y + 1) . . . {y + n) n= o n=o 6.706 If the sum of the series, 00 fix) = 2j ^"^^ Co< ^ <='] » = o is known, then CO (a + nb) {a + nb + 1) {a + nb + 2) {a + nb + k - 1) U>>o] n = o INFINITE SERIES 135 00 CO 6.707 / /(x) >^ - sin nx-dx= - j (tt - /^ /(< + 2m!-) -dt. »= I »= o Example i. f{x) = e~*^ C)fe>o]. « = I Replacing ^ by -, and subtracting, 2 l + ^^S^-^)"; 27r k^ -I- ^2 g*7r _ g-Air n= I Example 2. With f{x) = e"^ cos /xx and e"'^ sin /xx. X '^^ J ^ 1 ^ I TTsinh zXx X^ + ^l^ ^ I X^ + (« - jLt)^ X^ + (W + ;U)^ I cosh 2XTr - COS 2fJiT » r 1 . yLi '^^ I n — fjL n + n [ _ TTsm 2/x7r X^ + M^ ^J I X^ + (» — /x)^ X^ + in + /i)^ I cosh 2X7r — cos 2/x7r 6.709 If the sum of the series, OD J{x) = ^ a„x", » = o is known, then N.N / e-'ty-H{t)di aa + aiy + 023'(3' + i) + a^yiy + i) (y + 2) + = J^o • 6.710 The complete elliptic integral of the first kind: n d_x r'^ dd ^^ J o V(i - x-") (i - kV) " 7o Vi - F- sin^ d i£ ,,. .-v^-j; ^^7r(£j^, j^/±W^/2 + (i_^)2^M + . , , 2 (j>-+(^:> 7r(i + k') f , yi/i-a-.s- ■ ■ (2;?- i) .^^\ 2-4-6 . . . . 2^ / 136 MATHEMATICAL PORMUL^ AND ELLIPTIC FUNCTIONS 6.711 The complete elliptic integral of the second kind: /v.- F sin2 d dd. £=EL_AV^_/il3Vi* 2 [ \2/ I \2-4/ 3 =f{-s(^^;f^TT^ 2W I + Vi - /^2 = !: ( I + ^'^fi + y/ i-3 — (2n-i) Y 1 2(1 + ^0 r ^ L4 + Z/12-4-6 . . . (2« + 2)r J n=i Fourier's series 6.800 If /(x) is uniformly convergent in the interval: -c,irx fix) = -bo + h cos h 02 cos h 03 cos ^^ h ■ . . . ■'2 c c c • TTX . 27rx . -ITX + ai sm 1-02 sm \- 03 sm 1- , c c c ,. I r~^%, s. rmrx _, Om= - I f{x) cos (^X, C u —c c I (''^%f V . WTTX , I ffm = - I /(a;J sm ■ dx. C U —c c 6.801 If f{x) is uniformly convergent in the interval: o — - — sin ^ IT A^ 2W - I 6.803 /(x) = mx, --^x^+- 4 4 ?fi 4 = — m\x 1, - ^ X < = m(x-c), — ^ X ^ — 4 4 2/ 4 ., , 2WC'^^ , ,„_, I /(x) = —7-7/ (-i)" 7 «sm •' TT^ ^^ {271 — ir 2(211 — i)7r -X. {271 — l)^ c n= I 6.804 /(x) = mx, - - < x < + - J V / ' 2 2 . c 3c = w(x-c), + - < X < — , 2 2 CO 6.806 xj — / , ■■-■ sij « — r c /(x) = - 0, - 56 ^ X < - 36, = 1{X + 2b), -2,b ^x ^-h, = a, - b ^x ^ +b, = -l{x-2b), b ^x ^ 36, = -a, 36 ^ X ^ 56. ,, . 8-\/2a f TTX I ^TTX I 77rx I 77rx fix) = — 5— \ cos — r cos =^^-T J COS ^—r + —, cos ^—r- + ....1 138 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS 6.806 fix)=-^x + b, -l^x^o, b , ^ ^ I = — -jX+ b, o^x kil. f{x) = — 7, 7 ■ ^ cos (2W + i) ^ • n=Q 6.807 fix)=^x, o^x^b, CO 2aZ^ ■^^ I . fiTb . nirx ■ sin — r— sm - ir''b{l - b) ^ n" I I n=l S(_i)n-i r -] sin MX — 7r — 7r — ^ o < X < 27r • 90 12 12 48 aW «' L J n = I CO _ „^„ 7r''x TT^x' TTx"* x^ _ "V;^ sin nx To < x < 2x1 90 36 48 240 ^J n^ \_ J 624 ^^ w M = I INFINITE SERIES 139 6.820 ic^ 3 TT n= I 6.821 ^ 5 / . :, COS — c ^ X ^ c • 3 TV^^ n~ c 11 n= I CD n = I 00 » = r CD 6.822 e- = ^ (^"^ - I) { ^72 - 2 (-^^'"" c-^= '^^^ ^^ ) [° < ^ < ^J" »= I 00 6.823 cos 2a; — (— — .vlsin 2a; + sin^ x log (Asin^ .t) = / , — 7-^^ r^ \2 / ^^ n(n + i) n= I [o ^ X ^ tt]. 6.824 sin ix — (tt — 2a;)sin^ .t — sin a; cos x- log (4sin^ x) 03 Ssin 2(»+ i)x V . . 1 7 — , — r — o ^ .r ^ TT • n{n+i) L J CO TT . '^ cos 2».V r 7r"| sin -V = 7 7 r-7 ; r O ^ .T ^ — • 4 ji^ (2W - l) (2M + l) L 2 ] 6.825 ^ 2 4 - _„_ r sm X 6.830 1—2?' COS .V + « = I 6.831 tan-i CO I — r cos .V n= 1 6.832 - tan-i 5- = 7. sm(2W - i)x K ^6 = — = 1. 01 73430620, 945 2 7 52 = — = 1.6449340668 5? = ^g^ = 1.0083492774 « TT TT* -^^ = ^i:^^ = 1.2020S69032 -^B = — = 1.0040773562, 54 = ^ = 1.0823232337 59 = ^ ^ = 1.0020083928, e •^° , „,fi„,,^^^, '5'io = 1.0009945751, ^' = ;^i:7;7s = ^•°36927755i 5n = 1.0004941886. ^^^^^ III V. ^. , I 3" 5" 7" i^ (2^ + 1)' TT Ml = .— ) 4 M2 = 0.9159656 . . . M4 = 0.98894455 . ■. . . Me = 0.99868522 .... A table of Un from n = i tow = 38to 18 decimal places is given by Glaisher, Messenger of Mathematics, 42, p. 49, 1913. 6.902 Bernoulli's Numbers. 00 TT^" I I I I "^C^ I j7~ -''» ~ j-2n "■" "Pn '' ,2n "^ .2n + •••• = ^^ j^2n' (2m)! I^" 2^" 32" 42^ *= I (22" - l)7r2" pi ,1, I, I _IJ_^ I _ "^ I 2(2«)! ^"^^n + -^+--^ + ^ + - • ■ • - ^ (2yfe+l)2»" k = O (,2n-l_,)^2. ^_I___I_ ^_^ _y(_,)n-li. (2»)! i^" 22" ^3^" 42"^'--- JLJ^ ' F" A = I •"" j2n gSn '3- 42"-- •• ■ B. = l. 53 = ^. 42 B2 — ' 30 30 INFINITE SERIES 141 R S R 3617 00 510 R 691 43867 2730 798 R 7 R 1 7461 1 6 330 6.903 Euler's Numbers 7r^"+' ^ III ■^'» ^ ,2n+l + e2n+l »2n+l + • . . - / j\—'^) 22n+2(2w)! 22"+! S^n+l 72™+!^ • • • _^V ^) (2/fe - l)2''+l ^ = I ■> £i = I, £4 = I385,' £2 =5, £5 = 50521, £3 = 61, £5 = 2702765. 6.904 2W(2W — l) 2«(2« — l) (2W — 2) (2« 3) „ -C.n j i^n-\ H 1 £n-2 — . . . . 2! 4! - + (-1)" = 0. 6.905 tUtLllA 5„ = (2« _ 0£^. - (^^ - X) (2. - 2) (2. - 3) £^^ 2W 3! , (2W - l) {in - 2) (2M - 3) (2W - 4) (2W - 5) , / N , -I : 7i ^"-3 - + (-i)"~^ 6.910 6.911 Sr = B = I r i Si =e, ■S's = 52e, S2 = 26, Si = 2036, Sz = se, Si = 8Tje, Si = ise, Sa = 41406. CO n= I I -i)' S.-1, 53 = 32 - 31"' 64 ' -5^2 - .^ ' ^4 = TT^ + 307r2 - 384 768 142 MATHEMATICAL FORMULAE AND ELLIPTIC FUNCTIONS 6.912 1. log 2 = /, n = I CO 2. — (log if = / , — K = I 6.913 CD I. 2log2-i^2) ,K4J-i) - 2. 3 (iog3_,) = 2,^7-^7 2 ° ^ _^^ ^(g;^^ — i) » = I CO 3- -3 + flog3+2log2 = 2) ^(36;.'-x) « = I 6.914 5.^ J; ^3-5 (2 >.,-i)V I » = I 2 ■4- 6 2« j in-\- r U2 = 0.9159656 .... (see 6.901) 5o = 2 log 2 — — M2, 5_i = I TT TT 51 4 5-2 2 I TT 2 s, = — (2M2 + 27r i) I 3' s. 10 I gir 4 s, = -^ (18M2 327r + 13)- s, 178 I 5_2 = ^ log 2 + i - ^ (2M2 + i), e I 10 •J_3 =■ ' 3 97r 1 II I / „ N 5-4=^^l0g2 + —- — (18^+13), 5 22S7r 5_e=^ log 2 +^^-^(50^^ + 43). 2257r 6 ■5'7 = -7^ (S0M2 + 43) - ^' i267r 7 When r is a negative even integer the value w = - is to be excluded in the summation. 6.915 I A _ ^"3-5 ■ • • . (2w - i) _ (in- i)! 2-4-6 . . . . 27J 2^"~'-nl{n — i)! 4 ^mJ "dn^ — I 2. I 4 .^^ 4«' M=I INFINITE SERIES I43 CO 3- --i = ^^n : — 00 4. log (i + \^) - 1 = 2 (- 1)" ^"i^- n=i . ' - V .1 ^ 4W + I ^" 2 ^ " (2W - l) (2W + 2) TT 2 AiJ \2n — l) (2W + 2) 00 7. ^-- 1 = ^{-^YA,?Un+T). 2 TT^ ^^ 4M + I (2W — l) (2W + 2) 6.916 If w is an integer, and n = m'vs, excluded from the summation: 2 — = J - — — — -• (wj even) 4w^ ^^ n^ — « »=i 6.917 CO S« — I I. I «=2 2 ^ll Affi — I 2 ^1/4^^ n=i ■^> I2W^ — — I (4w2 _ i)2" £_ i„„ I +y3 _ . , >;;^ / .x„ 2-4-6 2;? I S(-)"d^ n = i 6.919 \i^-'o,.)=^[nlo,(^^) n=i 6.920 I. e = I + -. + -. + ^ + . . . . = 2.71828. i! 2! 3! 144 MATHEMATICAL FOEMULiE AND ELLIPTIC FUNCTIONS I I . I I '■ «-'-7i + ;i-rr--- . = 0.36788. 3- ;(-;) = ■ + ^ + j,H-.. . . = 1.54308. 4-2(^-ej = ^+^. + i! + -- . . = 1.175201. 5. cosi = i-ij+i-,-.... = 0.54030. 6. sinI = I-L^.^_.... = 0.84147. 6.921 4 III • Q III '■ lo - ' - 3^ + 3^ - ? + • • • • 16 III 3- 7^ = ^-7^ + ?-? + 25 III 4- ^=^-? + ?-? + 6.922 ^^' J^J}^ = e-- + e-^- + e'^^- + . . . ; r(i) = 3.6256 2r7r- 6.923 (Special cases of 6.705) : I. 1 1 2 1- • • • = log 2 - -. I-2-3 3-4'5 S-6-7 2 2. 1 7 ... = - (i - log 2). I-2-3 3-4-S S-6-7 2 ' * ^ III 3 2- ^^4 + ^^6 + 67^8 + - •• =J-log2. s- 7^ + ii6 + 7i^ + --- =i(^-^°^3)- 6. \- 2 sH \- ■ • ■ • = TT log 2. 2-3-4 6-7-8 I0-II-I2 82° 7- 7:^ + ^:i^ + 7^8^ + --- = g(^ + i5i)~4^°^^- VII. SPECIAL APPLICATIONS OF ANALYSIS. 7.10 Indeterminate Forms. 7.101 -. If Trrr assumes the indeterminate value - for a; = a, the true value o F{x) o of the quotient may be found by replacing /(x) and F{x) by their developments in series, if valid for x = a. Example: [sin^ X "I I - cos xJi=o' . sm^ X ('-^■•■J (-l^--)' I — cos X X^ X* I x^ 7!~4!"'""" 2!~4!"'" Therefore, r sin^ X n 2. [sin^ X ~| I — cos xj 3 lx=0 7.102 L'Hospital's Rule. If /(a + h) and F(a + h) can be developed by Taylor's fix) Theorem (6.100) then the true value of frrr ior x = a is, r yx) m_ F'{a) provided that this has a definite value (o, finite, or infinite). If the ratio of the first derivatives is still indeterminate, the true value may be found by taking that of the ratio of the first one of the higher derivatives that is definite. f(x) 7.103 The true value of ^rrr ior x = a is the limit, for h = o, of F{x) p\ F''>1{a) where/ *^^ (a) and F («' (a) are the first of the higher derivatives of f{x) and F{x) fix) that do not vanish for x = a. The true value of i,, , for a; =a is o if p>q, °° if Fix) /"'"' (a) p ^ J 7.120 o X 0° . . li, ioT X = a, f(x) X i*'(:»:) takes the form 0X0°, this product may be written, M I Fix) which takes the form - (7.101). ™.,«^ -rr Lmiit ^, , , Limit _, > 7.130 00 _ 00 . If, f(x) = 00 and F(x) = 00 , x—^a X— > 00 /(x)-F(x)=/(x){i-^ If ,, - is different from unity the true value of fix) — F(x) for x = a is <» . If ,. , = + I, the expression has the indeterminate form 00 x o which may be treated by 7.120. 7.140 100,0°, 00°- If { F(x)}<-^''ns indeterminate in any of these forms for X = a, its true value may be found by finding the true value of the logarithm of the given expression. Example; I X = y; log y = —tan x-log x, 148 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS tan X-log X Hence, Jj;=0 logx cot X csc'x •sm X _ 1=0 'j-X tan x~\ .x) J 1=0 7.141 If /(x) and x approach <» together, and , , — approaches a definite hmit, then. Limitf X— >oc [{/wr^] Limit /(x + i) (i) 7.150 Differential Coefficients of the form -. In determining the differential coefficient -j- from an equation /(x, y) = o, by means of the formula, dy _ dx dx df dy it may happen that for a pair of values, x = a, y = b, satisfying /(x, y) = o, -r- takes the form -. dx o Writing -^ = y', and applying 7.102 to the quotient (i), a quadratic equation is obtained for determining y', giving, in general, two different determinate values. If y' is still indeterminate, apply 7.102 again, giving a cubic equation for deter- mining y'. This process may be continued until determinate values result. Example: fi.^! y) = (x^ + 3^)^ — c'^xy = o. / = - 4x(x^ + y^) — c^y 43' (x^ + ;/) — c^x For X = o, y = o, y' takes the value Applying 7.102, ■y = i2x^ + \'f + i^xy — c^)y' A,y\y? + 33'^) + 8x31 — i). 17. r (a + 6^") « + ^fos«J_^= gi (^>o). 10 II 12 13 ISO 7.172 MATHEMATICAL FORMULAE AND ELLIPTIC FUNCTIONS I. X sin - = C. - -. e 2 ' c\x- £ X I. 7.173 1. pi^-] = X. L » Jo 2. [^ = I. L a; Jo Ksin nxY^I -V-j Jo= '^'"- 7.174 1. [x^]o = I. 2. a;° + *T°8^ = e ^• 3. a;'°BT^-^' = e. 4. x" log- = o (ot^ i). L ^Jo 5. Qog cos a; • cot x^o = o. cot- X 8. sin- • log (a + ie^) = c .1. [(cos^+asin£)]^=e- 4. [sin~* X ■ cot A-]o = I- 4{<-i)ri 8. [a™ log a;3o = {m>6). [gi _ g-x _ 2X~\ _ I (e^-^y Jo " 3' 10. [«e*]o = •» • r e^ - e~^ "I [log (i + a:)Jo 6. log tan (- + -1 • cot :»; = i. 12. -pS— = L \4 2/ Jo L log tan a; Jo SPECIAL APPLICATIONS OF ANALYSIS 151 7.175 — I I V X 7rx~i Ji e ^ L c 2 cjc 2. [(tt - 2x)tan x]^ = 2. 6. [(a + 6e tani)7r-2i]^ = g2_ 2 4. [(6= - e-)tan — 1 = - e^ 8. [(tan x)''""^> = i- 7.18 Limiting Values of Sums. Limit/i* + 2* + 3* + . . . . + «*\ I .. , ^ I- jfc+i = r"; — if ^> - I. ooif ^< — I. Limit / 1 I I 2. — + r-r + r— 1 + ---- + - n-^ 00 \wa wa + & Ma + 26 • • • • na + { in - i)b) log (a + b) - log . ^^^ j^^^_ Limit / n — 1^ n — 2^ n — 2,^ 3- w-^oo\^i-.2-(w+ i) 2-3-(w+2) 3-4-(« + 3) (w — w^ "\ _ . w(m + i)-{n+n)l ~ '^ ~ °^ ^' Limit 4- W— >00 (-'v)^(--'vJ-(--'^^^- + a" + & = ; \ w / J I — a' if a is a positive proper fraction. ^ fi-^co\_\ n ^ n y n y n] if 6>o and a is a positive proper fraction. V i-w V 2-n V 3-M V 6 w-w. I — y/a if 6>o and o is a positive proper fraction. + 2Vb, 7. ^i^i^^li +I + I + . . . .+i_logK =7 = 0.5772157 - . M— >oo| 23 M J (6.602). 152 MATHEMATICAL FORMULAE AND ELLIPTIC FUNCTIONS 7.19 Limiting Values of Products Limit I. \ nl\ n+ ij\ n + 2/ \ 2W — i/J if c>o. Limit 2 \ naj y na -\- b}\ na + 2hj ' ' ' ' \ na + {n - 1)6/ J ii a, b, c are all positive. I Limitp{w(w + i) (w + 2) (w + « — i)}»1 _ 2 m + i{n - 1) J~ e' if m>o. Limitf/ 2^ / 4c\ / 6c\ / 2nc\] 3 7.20 Maxima and Minima. 7.201 Functions of One Variable, y = j{x) is a maximum or minimum for the values of x satisfying the equation, j'(x) = -^ — = o, provided that f{x) is continuous for these values of %. 7.202 If, for x = a, f{d) = o, y =/W is a maximum if /"(a)o. Example: («2 + ax + ,8)2' /'(a;) = o when x = ±-n/;8, 2*;' — b^x — 2aj3 /"(x) = (x2 + ax + 18)' For X = +V^, /"(«^) = — = — -^ Maximum, V^ (2 ViS + a)2 SPECIAL APPLICATIONS OF ANALYSIS 153 + 2 For a; = -V/S, J"{x) = ^ \//3 (2V|8-«)' Minimum, ymax ymin -~ I ' a 2V/3 7.203 If for X = a, f'{a) = o and f"{a) = o, in order to determine whether y = f(a) is a maximum or minimum it is necessary to form the higher differential coefficients, until one of even order is found which does not vanish for x = a. y = /(^) is a maximum or minimum according as the first of the differential coefficients, f"{a), f"{a), /^(ffl), of even order which does not vanish is negative .or positive. 7.210 Functions of Two Variables. F{x, y) is a maximum or minimum for the pair of values of x and y that satisfy the equations. and for which dF dF / d^F Y d^F dW \dx dy) Bx^ dy 2{xu X2, ,Xn)=0 1. <, ^(t>k{Xl, 0C2, ,Xn) = O, where k {I, m, n) = P + m^ + n^ - 1 = o. This is the same as finding the minima and maxima of FQ, m, n) = anP + a^m^ + asan^ + ^anlm + ichzmn + 2013^. Equation (2) gives: (fflii + X)Z + a^m + aizti = o, ajsl + (022 + X)W + 023« = o, aial + aizm + (033 + X)» = o. Multiplying these 3 equations by /, m, n respectively and adding, x = -4- y2 SPECIAL APPLICATIONS OF ANALYSIS Then by (i. 1.363) tlie 3 values of r are given by the 3 roots of flu — 5 a-a. Ol3 ays. 022 — 023 ai3 023 O33 — 1 155 7.30 Derivatives. 7.31 First Derivatives. dx'^ I. -, — = wx" . dx^ de" dx 4- -^ =x^(i +loga;). 5- 6. d loga a: _ I _ logg e dx X log a X d log X I 7. = 2a;'°s=:-i log X.* 8 (f.V (f(loga;)'' dx (log x) ^~' { I + log X ■ log log x} . {'i~] is a, polynomial ir 7.354 If <^(-7-) is a polynomial in -r-, 7.355 Euler's Theorem. If m is a homogeneous function of the wth degree of r variables, x\, x^, . . . Xr, / d d dY \ dxi dx2 oXtI where m may be any integer, including o. 7.36 Derivatives of Functions of Functions. 7.361 If /(x) ^F{y), and y = ^{x), '■ t^m = TT p'^y) + jr p"^y) + ^ p"'^y) + •••• + S"" ^'"'^^)' where 7.362 I '^ '' dx" W ic^" W *'"-' i! W (w - i) (w - 2) w(w - i) 2) (A , ^ x2"-2 2! W 2. (-i)'.-^e"^ = ^e^ff-)" +(«-!) ,, , ^ ' dx'' X" ( \x/ I ! \x/ + (« - i) (« - 2) j — 2! , . , , , , «(» — 1) (n — 2) faY'^ + (n-i){n- 2) (« - 3) -^ j^ '- y + IS8 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS 7.363 1. £^ Fix^) = (2»)"F'")(x2) + '"^" ~ ^-' (2j;)"-2 F("-i)(x2) n(n — i} (n — 2) (n — T.) , . , „, „, , „. + -^^ — — j — — — {2x) "-« F'"-2) (x^) , njn - i) in- 2) (w - 3) (w - 4) (w - 5) , «„_3u 2\ , 4 j {2X)'' " F^" '>{x') + . . . . d" „ . , , ^ . f m(« - i) n(n — i)(» — 2)(m — i) 2. ^„ e"^ = i2axye^^- { X + ^T(^ + -^ 21(4..^ w(w - i) (n - 2){n- 3) (w - 4) (w - 5) 1 31(40*2)3 .... I 3. £^„ (I + axr fxjfjL - i){iJ, - 2) . . . . (ix - n+ i)(2aa;)" J w(w - i ) (i + ax'') (i + ax^)"-!" \ I • (m - « + i) 4ax2 «(w — i){n — 2){n — 3) /i + ax^Y + ■ )V.... (ix 7.364 2!(/x — w + i)(|U — « + 2) \ 4ax^ £;!i (I - x^)--^ = (- i)"-i ^"3-5 ■ • • • ^^^-^^ sin (m cos-i x) + ■ {n + i)»(w - i)(w - 2) F'"-2)(\/^) 2! (2V^-)"+2 2. -r- (i + av«J = — ^'^-^^ ^^ — r [a? 7.365 1. :^ F(e-) = —,e-F'ie-) + —,e^-F"{e-) + — %3^f"'(e-) + . . . dx'^ I ! 2 ! 3 ! where 2. Et = k---Ak- 1)" + ^^^-^^ (k-2)--. . . . I ! 2! _^ I _ _ p^;. sin (2 tan-^e-^) ^ ^ ^^ sin (3 tan'^e"^) , sin (4 tan~V~'') ^^^^ cos (2 tan-'e-^) _|_ ^^^^^ cos (3 tan-'e"^) da;''i+e2^ V(i + e^^)^ V(i + e^^)^ p^,... cos (4 tan-'e-^) ^ SPECIAL APPLICATIONS OF ANALYSIS 159 n(n — i) 7.366 I. 1^ F{\og x)=^[ CoF('')(log x) - CiF(''-ii(log x) + CJP^^-^K^og x)- n 1 Co= I, Ci =1 + 2 + 3 + + (»'- i) n C2 = i-2 + i-3 + i-4+ + i-(«-i) + 2-3 + 2-4+ + 2-(« - l) + 3-4+ + 3-(«- i) + + {n — 2) (« — i) = n(n — i)(n — 2){^n — i) 24 2. Cj, = Cj: + nCk~i. 3- —n — («— i) — n Ci = Ci + nCk-i. n i Co = I Cj: = 0, 2 3 Ci = I Ci = 3 4 Ci = 6, 3 C2= 3 4 C2 = II 4 C3 = 6. Ck Co = — 2 -3 . Ci = 3 Ci = 6 Ci = 10, C2 = 7 C2 = 25 C2 = 6s, —2 -3 -4 C3 =15 C3 = 90 C3 = 350. 7.367 Table of Ci. n = - 4 -3 — 2 - I + I + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 Co = I I I I I I I I I I I I Ci = 10 6 3 I 3 6 10 IS 21 28 36 C2 = 65 25 7 2 II 35 85 175 322 546 C3 = 350 90 15 6 SO 225 735 i960 4536 Cj - I70I 7770 34IOS 145750 61x501 301 966 3025 9330 28501 31 63 127 225 511 24 274 120 1624 1764 720 6769 13132 13068 5040 22449 67284 118124 109584 40320 Cc - Cfi- Cv - Cs = l6o MATHEMATICAL FORMULAE AND ELLIPTIC FUNCTIONS 7.368 1. £;(l0g xy = ^ ~^]^~' { C^i/.(l0g X)-1 - C,^,P{P - l)(l0g x)P-2 + Cr^pip -i){p- 2) (log xy-^-. . . |, where ^ is a positive integer. If n

- l)(l0g X)P-^ + H-(-l)^+lC„_,,^(^-l)(^-2) .... 2-1 [• 7.369 log (i + x) = Coa;p - Ci ;f— + C- - I < a; < + I. 7.37 Derivatives of Powers of Functions. Ji y = 4>{x). ^^ - ('"\ Jl- ^l1 ('"'\ ^ '^"y^ < ('^\ ^ '^"y^ _• die" °^^ " \i/ i-ydx''~ \2J 2-f rfx" \3/ 3-y^ dx" ~ ' ' ., d^f dx" 7.38 d"{a + bx)" dx" m(m - i)(w - 2) (m - \jt - ij) b''(a + bx)'"~". d"(a + bx)-^ , , w!J" O i 1 '- = ( _ T)" dx"" ^ ' (a + ^)a;)"+i y) = - y' where X is a function of x alone and F is a function of y alone. The solution is: 8.002 Linear equations of the form: dy dx Solution: Jx dx + (y dy = C. t form: + P{x)y = Qix). y = e~->^^^">'** I fQ{x)e~-^P<-'^'"' dx + c\- 8.003 Equations of the form dy ^^ + P{x)y = y-Q{x). Solution: _L_g-(n-i)fPMd^ + (w - i) re(.t-)e-"'-')/^W''Mx = C. 8.010 Homogeneous equations of the form: dy_ _ P{x, y) dx Q{x, yY where P{x, y) and Q(x, y) are homogeneous functions of x and y of the same degree. The change of variable: y = TO, gives the solution: DIFFERENTIAL EQUATIONS 163 8.011 Equations of the form: dy a'x + b'y + c' dx ax + by + c If ab' — a'b ^ o, the substitution X = x' + p, y = y' + q, ' where ap + bq + c = o, a'p + b'q + c' = o, renders the equation homogeneous, and it may be solved by 8.010. If ab' — a'b = o and b' 4= o, the change of variables to either x and 2 or 3; and 2 by means of z = ax + by, will make the variables separable (8.001). 8.020 Exact differential equations. The equation, P(x, y)dx + Q{x, y)dy = o, is exact 11, dQ_dP^_ dx dy The solution is: or JP{x, y)dx + f I Q{x, y) - ^ j^ P{x, y)dx \dy = C, jQix, y)dy + J I P{x, y) - ^^Jq{x, y)dy | dx = C. 8.030 Integrating factors. v{x, y) is an integrating factor of P{x, y) dx + Q{x, y) dy = o, if i ^^e) - 1 ^^^^- 8.031 If one only of the functions Px + Qy and Px - Qy is equal to o, the reciprocal of the other is an integrating factor of the differential equation. 8.032 Homogeneous equations. If neither Px + Qy nor Px - Qy is equal to o, „ „ is an integrating factor of the equation if it is homogeneous. 164 MATHEMATICAL FORMULAE AND ELLIPTIC FUNCTIONS 8.033 An equation of the form, P{x, y)y dx + Q{x, y)x dy = o, has an integrating factor: xP -yQ 8.034 If dP dQ is a function of x only, an integrating factor is gfF(x)dx_ 8.035 If dx dP is a function of y only. an integrating factor is efF{y)dy_ 8.036 If dP dy Qy dQ is a function of the product xy only, an integrating factor is gfF{xy)d{xy) ^ 8.037 If dx dy) ^ H Px + Qy is a function of the quotient - only, an integrating factor is e.fP (S)^(S)- 8.040 Ordinary differential equations of the first order and of degree higher than the first. Write: I -A General form of equation: fix, y, p) = o. DIFFERENTIAL EQUATIONS 165 8.041 The equation can be solved as an algebraic equation in p. It can be written {p-R,)(p-R,) (p-R„)=o. The differential equations: P = Riix, y), p = Riix, y), may be solved by the previous methods. Write the solutions: /i(-v, y, c) = o; fiix, y,c)=o; where c is the same arbitrary constant in each. The solution of the given differential equation is: Mx, y, c)Ji{x, y,c) f„{x, y, c) = o. 8.042 The equation can be solved for y: I. y=f(x,p). Differentiate with respect to x: It may be possible to integrate (2) regarded as an equation in the two variables X, p, giving a solution 3. {y, P, c) = o. Ehminate p between (i) and (3) and the result will be the solution of the given equation. 8.044 The equation does not contain x: f{y, p)=o. It may be solved for p, giving, dy dx p^y^' which can be integrated. 1 66 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS 8.045 The equation does not contain y: f{x, p) = o. It may be solved for p, giving, which can be integrated. It may be solved for .v, giving, X = F{p), which may be solved by 8.043. 8.050 Equations homogeneous in x and y. General form: P{P,%} = 0. (a) Solve for p and proceed as in 8.001 (b) Solve for -; , X y = xf(p). Differentiate with respect to x: dx _ fip)dp X p-fipy which may be integrated. 8.060 Clairaut's differential equation: 1. _ _ y = px+f{p), the solution is: y = ex +f(,c). The singular solution is obtained by eliminating p between (i) and 2. x + f'ip) = o. 8.061 The equation I. y = xfip) + 4>{p)- The solution is that of the linear equation of the first order: dx fip) cj>'ip) dp p-m'' p-f{py which may be solved by 8.002. Eliminating p between (i) and the solution of (2) gives the solution of the given equation. DIFFERENTIAL EQUATIONS 1 67 8.062 The equation: x^^[{Ai + Aix + AiX^ + ....+ ^ma;""-!) cos vx + (Bi + B2X + Bsx" + .... + S^a;"-!) sin J^.v) = gf ^{Cicos {vx - fi) + C2.V cos {vx- Bi) + + C^a;"-! cos {vx - d„,)] = e''"^{Ci sin {vx + 60 + C2X sin {vx + 62) + + C^x"'-'^ sin {vx + d,„)] 1 68 MATHEMATICAL PORMUL^ AND ELLIPTIC FUNCTIONS where X ± i/x is the repeated root and tan 6k = -r' The particular integral. d 8^ 8.110 The operator D stands for — > ^^ for ^> The differential equation 8.100 may be written: (I>» + aiD"-i + 02 D--^ + + a„)>' = fiD)y = F(x) _ V{x) ^~ fioy f{D) = (Z? - XO(Z? - X2) (D- X„), where Xi, X2, , X„ are determined as in 8.101. The particular integral is: y ^ g\ix I e(X2-Xi) I (^-c / eC^-^')'' dx / g-^n^^^ V{x)dx. 8.111 Tvjy- may be resolved into partial fractions: f{D) £» - Xi £» - X2 -D - X„ The particular integral is: y = Nie^'" I e-'''"'V {x)dx + A'^ae^" / e-'^''^V{x)dx + + NneK'' f e-'>'r>'V{x)dx THE PARTICULAR INTEGRAL IN SPECIAL CASES 8.120 V{x) = const. = c, c y = —■ an 8.121 V{x) is a rational integral function of x of the wth degree. Expand ■ . - in ascending powers of D, ending with D". Apply the operators D, D^, , Z>" to each term of V(x) separately and the particular integral will be the sum of the results of these operations. DIPFERENTIAL EQUATIONS 169 8.122 F(x) = ce*^ unless ^ is a root of /(D) = o. If A is a multiple root of order r of J{D) = o where /(D) = (Z) - ky^PiD). 8.123 F(a5) = c cos (/fex + a). If f^ is not a root of /(D) = o the particular integral is the real part of /,!(4 x+a) . fiik) If ik is a multiple root of order r of f{D) = o the particular integral is the real part of where /<''' {ik) is obtained by taking the rth derivative of f{D) with respect to D, and substituting ik for D. 8.124 V{x) = c sin (/fea; + a). If ik is not a root of /(D) = o the particular integral is the real part of _y-gHkx+a) If ik is a multiple root of order r of /(D) = o the particular integral is the real part of f\ik) 8.125 F(x) = ce*^-Z, where X is any function of x. If X is a rational integral function of x this may be evaluated by the method of 8.121. 8.126 F(x) = c cos {kx + a)-X, where X is any function of x. The particular integral is the real part of /(D ■+ ik) ^ 8.127 V{x) = c sin {kx -\- a)-X. The particular integral is the real part of f{D + ik) 170 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS 8.128 V{x) =ce^^ cos {kx + a). If (/3 + ik) is not a root of /(£>) = o the particular integral is the real part of /(|8 + ^k) If (j8 + ik) is a multiple root of order r of /(£>) = o the particular integral is the real part of where /("■' (j8 + ik) is formed as m 8.123. • 8.129 V = cd'' sin {kx + a). If (jS + «^) is not a root oif{D) = o the particular integral is the real part of IW+ikT'- If (/3 + ik) is a multiple root of order r of f(D) = o the particular integral is the real part of f'K^ + ik) 8.130 V{x) = x^X, where X is any function of x. I -,r ■, { d I 1 -, mini — i) „ [ d? i 1 ,^ The series must be extended to the {m + i)th term. 8.200 Homogeneous hnear equations. General form: ^" T^ + «i* 1^1 + •■■■ + an-ix j^ + any = Vix). dx^ ax" ^ ax Denote the operator: Jot x'"j^=d{d-i)id-2) (e-m + i). The differential equation may be written: F{d)-y= Vix). The complete solution is the sum of the complementary function, obtained by solving the equation with V{x) = o, and the particular integral. DIFFERENTIAL EQUATIONS I? I 8.201 The complementary function. y = cxx^^ + C2.v^^ + + c„a;'^", where Xi, X2, , X„ are the n roots of f (X) = o if the roots are all distinct. If Xfc is a multiple root of order r, the corresponding terms in the comple- mentary function are: /*{&! + hi log X + is (log xY^ . . . . + hr (log x)'-^]. If X = /x ± «V is a pair of complex roots, of order r, the corresponding terms in the complementary function are: .T*'{[^i + ^2 log X + Az (log xf + . . . . + Ar (log x)'-i] cos {v log x) + iBi + B2 log X + Bs (log xy+.... + 5, (log xy-^2 sin (i^ log x) } . 8.202 The particular integral. If F(0)= (0-X:)(0-X2) id-K), y = x^^ j x^''~^'^''-dx I x^~^'~^dx / x^"-^"-^~'^V(x)dx. 8.203 The operator _,„, may be resolved into partial fractions: F{d) ~ - Xi e - X2 + ^ - X„' y = Nix^' f x-'>''-W {x)dx + NiX^' I x-'>'^-Wix)dx + + NnX^'' I x-'^''-Wix)dx. The particular integral in special cases. 8.210 V(x) = ex'', unless ^ is a root of F{d) = o. If ife is a multiple root of order r oi F(d) = o. _ c (log xY ^~ FW(yfe) ' where F''''>{k) is obtained by taking the rth derivative of F{6) with respect to 6 and after differentiation substituting k for 6. MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS 172 8.211 where X is any function of x. V{x) = cx^X, ex" F{d + k) X. 8.220 The differential equation: w^ d^-'^y dy (a + bxY — ^ + (a + bxY'^ai j-^^ + + (a + te)a„_i -£ + any= V{x), dx" dx may be reduced to the homogeneous linear equation (8.200) by the change of variable , , z = a + ox. It may be reduced to a linear equation with constant coefficients by the change of variable: ^ , 8.230 The general Unear equation. General form: d-i d-^y + P„_.| + P„=F, dx" dx"" where Po, -Pi, , Pn, V are functions of x only. The complete solution is the sum of: (a) The complementary function, which is the general solution of the equation with F = o, and containing n arbitrary constants, and (b) The particular integral. 8.231 Complementary Function. If ji, y^, . . . . , y-n are n independent solu- tions of 8.230 with V = o, the complementary function is y = c^yi + Ciy2 + + c„y„. The conditions that yi, y2, . . . . , y„ be w independent solutions is that the determinant A 4^ o. A = When i^^o: d—^y, d"-'y2 d^'^'^yn dx"-'- dx"--^ ' * ' * * ■ dx"-"- d^-^yi d"-^y2 d^-^yn dx'^-^ dx''-'' • ' dx"-^ dyi dx dy2 dx dy„ ■ dx yi yi ■ . ■ yn A = Ce' -■^9-.'^. DIFFERENTIAL EQUATIONS 173 8.232 The particular integral. If A^ is the minor of , ^_'^ in A, the par- ticular integral is: y = y'Jp;A'^'+y'-JpA'^'+---- + y''Jp;A'^'- 8.233 If yi is one integral of the equation 8.230 with v = o, the substitution will result in a linear equation of order « — i. 8.234 If yi, ^2, , yn-i are n — i independent integrals of 8.230 with V = o the complete solution^is: n — I «— I _ y = ^.y Ckk + Cn ^j Jk I -^ ^ P'" dx k=i k=i where A is the determinant: A = dx"-^ dx"~^ d'^~^yi d''~^y2 d^'-^yr^i dyi dyt dx dx dyn- yi Ji dx yn-i and A* is the minor of -f—^ in A. SYMBOLIC METHODS 8.240 Denote the operators: dx d a X -r = "■ dx 8.241 If X is a function of .t: 1. (D - w)-i X = e™^ / g-"^ Xdx. 2. (D-m)-^o = ce""'. 3. {6 -my- X = x^ I x-"-i Xdx. 4. {6 - m)-^ o - ex"'. 174 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS 8.242 If F{D) is a polynomial in D, 1. f (Z')e™^ = e'^''F{m). 2. F(D)e'^^X = e'^'^FiD + m)X. 3. e'^^F{D)X = F{D - w)e'»^Z. 8.243 If F{e) is a polynomial in 6, 1. F{d)x'" = x'"F{m). 2. F{d)x"'X = x^F{e + m)X. 3. x^F{e)X = F{e - m)x^X. 8.244 x^-~^=e{d-i) {6-2) {6 - w fi). INTEGRATION IN SERIES 8.250 If a linear differential equation can be expressed in the symbolic form: lx'"Fie)+fie)2y=o, where F{9) and/(0) are polynomials in d, the substitution, y = ^ a„xP+'"^, n = o leads to the equations, ao/(p) = o, oo^(p) +fli/(p + OT) = o, aiF(p + w) + a2/(p + 2m) = o, a^Fip + 2m) + azfip + 3m) = o. 8.251 The equation /(P) = o, is the "indicial equation." If it is satisfied ao may be chosen arbitrarily, and the other coefficients are then determined. 8.252 An equation: y = o. may be reduced to the form 8.250, where, fid) = (t>(e -m)d{e-i) {6 -2) xe -m + 1). If the degree of the polynomial /is greater than that of F the series always con- verges; if the degree of / is less than that of F the series always diverges. DIFFERENTIAL EQUATIONS 175 ORDINARY DIFFERENTIAL EQUATIONS OF SPECIAL TYPES 8.300 d-y dx- where X is a function of x only. y = / — ^ / ""(.v -ty-' Tdt + ci.v"-' + ax--"- + . . . + c„-i x + 1 where T is the same function of t that X is of x. 8.301 dx" ' where F is a function of y only. If \p{y) = 2 J Ydy, the solution is: r dv 8.302 Put d-y /d"-'y\ dx- ~ \dx--')' d-'y^ dY^ dx--' ^' dx ^^^>' ^ + '^^ = //(F) = '^^^)' Y = {x + ci), and this equation may be solved by 8.300. Or the equation can be solved: r dY r dY fYdY y-J F{Y)J F{Y) J F{Yy where the integration is to be carried out from right to left and an arbitrary constant added after each integration. EUminating Y between this result and Y = (t>(x + ci) gives the solution. 8.303 d , dx y _p(d''-'y\ x- \dx--^/' 176 MATHEMATICAL FORMULiE AND ELLIPTIC FUNCTIONS Put which may be solved by 8.301. If the solution can be expressed: Y = 4>{x), n — 2 integrations wiU solve the given differential equation. Or putting yl^{y) = 2jYdy, r dv r dv r YdY ^J {ci+^iY)}iJ {ci+^(F)}i J {c,+ xl,{Y)}i' where the integration is to be carried out from right to left and an arbitrary constant added after each integration. The solution of the given differential equation is obtained by elimination between this result and Y = 4,{x). 8.304 Differential equations of the second order in which the independent variable does not appear. General type: / dy d^y\ ^V'di'd^^)=°- dy dp d^y ^^dx' ^d^^'ih^' A differential equation of the first order results: Put K''"l) If the solution of this equation is: the solution of the given equation is, 8.305 Differential equations of the second order in which the dependent variable does not appear. General type: F Put P = \ dx dx^l dy_ dp __d^y dx' dx dx^ DIFFERENTIAL EQUATIONS 177 A differential equation of the first order results: If the solution of this equation is: the solution of the given equation is: y = Ci + ff{x)dx. 8.306 Equations of an order higher than the second in which either the inde- pendent or the dependent variable does not appear. The substitution: dx ^' as in 8.304 and 8.305 will result in an equation of an order less by unity than the given equation. 8.307 Homogeneous differential equations. If y is assumed to be of dimensions n, X of dimensions i, -7- of dimensions (n — i), -7^ of dimensions (n — 2), ax ax-, then if every term has the same dimensions the equation is homogeneous. If the independent variable is changed to 6 and the dependent variable changed to z by the relations, X = ^, y = ze'^, the resulting equation wiU be one in which the independent variable does not appear and its order can be lowered by unity by 8.306. If y, -J-, -T~2' ■ • • • ^.re assumed all to be of the same dimensions, and the equation is homogeneous, the substitution: y ^ gfudx^ wiU result in an equation in u and x of an order less by unity than the given equation. 8.310 Exact differential equations. A linear differential equation: ^"dx-^^'^Ux'^-'^ + ^'rfx + ^'' = ^' where P, Po, -Pi, P^arefunctionsof a:is exact if: d_P, d^ d-Pr. ^'- dx^ dx^- + ^-')^^ = °- where, 178 MATHEMATICAL FORMULiE A^ID ELLIPTIC FUNCTIONS The first integral is: a -p E^ (Jn-2 - rr^2 " ^^ + ^^2 ' ^'~^'~ dx'^ dx'~ +^~'^ dx-i If the first integral is an exact differential equation the process may be con- tinued as long as the coeflScients of each successive integral satisfy the condition of integrabiHty: 8.311 Non-linear differential equations. A non-Hnear differential equation of the «th order: dy [dx"' dx^-"-' .^^,y,x] = o, d^v to be exact must contain -^ in the first degree only. Put dn-ly liny gp dx"-^ ~ ^' dx'^ " dx Integrate the equation on the assumption that p is the only variable and -~ its differential coefficient. Let the result be Vi. In V dx — dVi, , is dx ' dx"-^ the highest differential coefficient and it occurs in the first degree only. Repeat this process as often as may be necessary and the first integral of the exact dif- ferential equation will be Vi+V, + =c. If this process breaks down owing to the appearance of the highest differential coefficient in a higher degree than the first the given differential equation was not exact. DIFFERENTIAL EQUATIONS 1 79 8.312 General condition for an exact differential equation. Write: dx -^ dx^ -^ ■ ■ ■ dx"" ' In order that the differential equation: V{x, y, y'; y", , >-''•>) = o, be exact it is necessary and sufficient that dV d /dV\ 32 /dV\ dy dx \dy'J dx^ \dy". (_ \n il IIL\ _ *■ ^''" dx" Uj''"'/ ~ °' 8.400 Linear differential equations of the second order. General form: S + ^l+e--. where P, Q, R are, in general, functions of x. 8.401 If a solution of the equation with i? = o: y == w can be found, the complete solution of the given differential equation is: y = c-iw + Ciw fe-'^P''- ^ + w Ce'J'P''- ^ CwRe-^^'''' dx. 8.402 The general linear differential equation of the second order may be reduced to the form: — + Iv = Re^-^P^" dx'^ where: y = ve~^-^P'^^, 2 dx 4 8.403 The differential equation: d^v -r,dy _ by the change of independent variable to 2 = fe-fPi- dx, becomes: d^y „„, ^ + Qe^fP'i^y = o. By the change of independent variable. dz = Qe-fP'^" dx, it becomes: d f I dy] dziudz] +y = °- l8o MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS 8.404 Resolution of the operator. The differential equation: ^y dy may sometimes be solved by resolving the operator, into the product, d-" , d The solution of the differential equation reduces to the solution of dy , - fld^c r j^ + sy = Cie .) t The equations for determining p, r, q, s are: pr = u, dr qr + ps + p— = v, ds qs + P^^ = w. 8.410 Variation of parameters. The complete solution of the differential equation: is y = c,Mx) + c,f,{x) + ^J'^^Rik)/'^'''' \mx)MO - Mx)M^) I d^, where fi{x) and fiix) are two particular solutions of the differential equation with R = o, and are therefore connected by the relation dx dx C is an absolute constant depending upon the forms of /i and /z and may be taken as unity. 8.600 The differential equation: (02 + hx) ^ + («i + hx) -j^ + (flo + hax)y = o. .8.501 Let Special cases. 8.502 hi = bi = ba = o. The solution is: where: DIFFERENTIAL EQUATIONS IS I y-i = Cie^'" + de^'", Ai _ — fli rb-y/ffli' — 4ao g2 X2 202 8.503 D = o, h = o, where: y = e^'' I Ci + C2 I e-(*+2>^)^-"^^(fx 8.604 D = 0, J2 4= o: , oi ^1 \ ^0 k = — m = — A= — ^■ (h 202 Oi y = e^'' where C1 + C2 I e-(*+2'''^(a2 + b2x)"'dx , b\ Oibi — O1J2 k = - m = T-, , 02 bi and X is the common root of: (H>^ + fliX + flo = o, ^12X2 + &iX + &o = o. 8.505 B -^ o, bi. = b\ = o. If 17 =/(^) is the complete solution of: y = e-7( where -'-/(^1. 400^2 - Ol O ^0 N *1 a = 5 — P = — . X = 402 (h 202 8.510 The differential equation 8.500 under the condition D ^ o can always be reduced to the form: «l| + (* + ' + 8Tf + f* = °- 8.511 Denote the complete solution of 8.510: 8.512 &2 = 61 = o: y = e^x+{^+vx)i F{2(ju + vxf], where: \__^ _ gi^ - 4gog2 /wy 202 4^2^ V960V ' V = - ^4i° /> = ? = !• A. X. J («i + M' 182 MATHEMATICAL FORMULAE AND ELLIPTIC EXTNCTIONS 8.513 &2 = o, &i =)= o: where: ,. bo aibi — 202^0 o bi A = - 7- tti = 7 , Pi = — , Ol O2O1 fl!2 0260^ - aibobi + aab^ 261' I <1= ~- 8.514 62:1=0, 60 = -¥: where: -. Ji 4a 0^2^ — 2aibib2 + (hbi^ 2O2 62 _ 4go52^ — 2aibib2 + (hb-? ^^ ~ bi^ aj)2 — Oibi I 8.515 62 + o, Jo + , 402 x.^ f /3i(a2 + ^2x) I ^ ^^^^1 /32- I where a2 = 02, 182 = J2, /3i = 262A + 61 and X is one of the roots of 62X2 + biK + bo = o. a2\^ + fllX + Oo ffllZ'2 — 02^1 ^ 2b2\+b, ' ^~ k .2 8.520 The solution of 8.510 will be denoted: 2. F(p, q,-0= e^ F{q, p, © 3- Fiq,P,0 = e-^F{p,q,-0. 4. P{p,g,0^e-''-'F{i-q,i-p,^). 5- Fi-p,- q, Q = e+p+. i7(i + g, I + ;^, ^). fJm 6. F(^ + ^,^,^)=|_,^(^,^_^). 7- ^(i-, ? + «, 8 = ( - i)" e-« 1^ { e?F(i>, 9, ^ DIPFERENTIAL EQUATIONS 183 8.521 The function F{p, q, ^) can always be found if it is known for positive proper fractional values of p and q. 8.522 p and q positive improper fractions: p = m + r, q = n + s where m and n are positive integers and r and s positive proper fractions. 8.523 p and q both negative: p = — {m — I + r) q = — {n — 1 + s), /r (_ ^ + I _ ,, _ „ + I - ,, Q = ( - i)- ^-+"+^+-1 ^ l^e-? ^ I e« i?(,, ,, ^) |]. 8.524 p positive, q negative: p = m + r, q= — n + s, F(m + r,-n + s,^)=j^ ^^n+i-.-s _ /?(i _ 5, i - r, ©J • 8.525 p negative, q positive: p = — m + r, q = n + s, d^ r d"^ f 1 "1 F{-m + r,n + s,0 = (- i)-+ne-f^ l^^+i— ^ | e? F(i - 5, i - r, © | J- 8.530 If either p or q is zero the relation D = o is satisfied and the complete solution of the differential equation is given in 8.502, 3. 8.531 li p = m, a, positive integer: = Fim, q, a = ci ^ [^-^e-^f^^-^eid^] + c. ^ [^«e-«] ■ 8.532 If ^ = OT, a positive integer and both q and ^ are positive: (/) = F(m, q, ^) = ci I W^-^i - m)«-i e'^" du + de'^ / (^ + ")"'"' ^'~' ^"^^ du. 8.533 If 5 = «, a positive integer: <^ = F{p, n, = ^ie-« |S[^' = Fip,n,^) = ci I M^-i(i - m)"-! e-^" ^m + Cae-? / Ct + u)p-^u''-^ e'^'' du. 184 MATHEMATICAL FORMULAE AND ELLIPTIC PUNCTIONS 8.640 The general solution of equation 8.510 may be written: 4> = F{p, q, = CiM + C2N, q> o N Jo ? >0 ,, Tip)Tiq) f _P ^ . PiP + i) ^' pip + i){p+2) ^^ ^^ " T{s) r 'si\^ s{s+i) 2\ s{s + i){s+2) s\^ s = p + q, ^, _ Tiq)e-i \ ip-i)q (p-i)(p- 2)q{q + i) ^3 I i!$ 2!^' (p- i)(p- 2) (p-n-i){q)iq+i) (q + n - 2) + (w - i) !$"-! p(p - j)(p - 2) . . . . (p - n)q(q + i) (? + 2) . . . . (q + n - 1) where o < p < i and the real part of ^ is positive. THE COMPLETE SOLUTION OF EQUATION 8.510 IN SPECIAL CASES 8.550 p>o, q>o, real part of $>o: Pip, q, = ci I W-^i - u)'2-^e-i''du + de-^ / (^ + u) ^-^W-^e-^^du. 8.551 p>o, q>o, ^o: Fip,q,^) = ^'^~^'Uci f {i-u)-^u-ie-^''du + cze-i I m-p(i +M)-«e-?"i 8.553 po, go: F(i - r, I - q, ^) = ci / u-^{i - u)-^e-^''du + Cae-i / (i + u)-^u-^e-^''du, ^o, q = n + s, where « is a positive integer and 5 a proper fraction. ^>o: F(i - s, 1 - p, ^) = ci I M-»(i - uyvg-i^du + C2e-^ / (^ + uy^u-Pe-i^du, ^ dy is satisfied by the confluent hypergeometric function. The complete solution is: y = ciMia, 7, x) + c^^-'^M{a - 7 + i, 2 - 7, a;) = M{ol, 7, x), l86 MATHEMATICAL FOEMULjE AND ELLIPTIC FUNCTIONS where ,^, , ax aia + i) x^ a(a + i)(q; + 2) x' ^ ' "' ' 71 7(7 + i) 2! ^ 7(7 + i)(7 + 2) 3! The series is absolutely and uniformly convergent for all real and complex values of a, 7, X, except when 7 is a negative integer or zero. When 7 is a positive integer the complete solution of the differential equation is: y= I ci + C2 log X I M{a, Tj ^) + ^2 1 ^ (| " r - ^j a{a + i) ^(j, _J 1 I i\ 7(7 + 1) 2\\a a+i 7 7 + i~'^ 2/ a{a + i)(a + 2) ^ / J , i i _ J _ _i L_ _ - l\ 7(7 + i)(7 + 2) 3! \« a+i a+2 7 7 + 1 7 + 2 ^ 2~3/ + . . . 8.601 For large values of x the following asymptotic expansion may be used: M{q, 7, x) r(7) / ,.-.-ry L a{(x-7+i) i . a(a+i)(a-7+i)(a-7+2) I 1 "r(7-a) '^ ^^ \^ I ^+ 2! ^2---| I r(7),.,.„-T, I . (i-«)(7-«) i , ii-a)(2-a)(y-a)(j-a+i) I , 1 1 (a) [ I a; 2! x^ j 8.61 1. IfCa, 7, x) = e'M{y-a, y, -x). 2. x^-"' M{a -7 + 1, 2 - 7, s;) = e^^a;^-'"' if (i - ol, 2-7, -a;). 3. - M{a + I, 7 + I, a;) = M{a + i, 7, a;) - ilf (a, 7, x). 4. ailf (a + 1, y + 1, x) = {a - y)M{a, y -\- 1, x) + yM{a, 7, x). 5. (a + «)ikf (a + I, 7 + I, x) = (a - y)M{oL, 7 + i, x) + 7ilf (a + i, 7, x). 6. a7lf(Q: + i, 7, a;) = y{a + x)ilf (a, 7, x) - x{y - a)M(a, 7 + 1, x). 7. aM (a + I, 7, x) = {x+ 2a - y)M{a, 7, x) + (7 - oi)M{a - i, 7, x). 8. — — xM{a, 7 + I, x) = (x + 7 - 1)71/(0;, 7, x) + (i - y)M{a, 7-1, x). 8.62 2. (i - a) / M{a, y, x) dx = {1 - y)M{a - i, 7 - i, x) + (7 - i). DUTEEENTIAL EQUATIONS 187 8.631 SPECIAL DIFFERENTIAL EQUATIONS AND THEIR SOLUTIONS IN TERMS OF M{a, J, x) 8.630 j-^ + 2(^ + gx) — + 4«?'+ P"^ - fin^ + ^q.x{p + qm) } y = o, y = e-(2'+9"')^lf [a, -, -q{x - mYj. + {^P + }Yi+ {f-i' + l{yp + yt-^o.i)}y-o, y = g-(p+Ox jlf (q;^ Y, 2tx). 8.632 j^ + 2(/> .+ ?x) ^ + lq + c{i- 4a) + (p + qxY - c\x - w)M y = o, 8.633 -y = e-(p+')^ x ^ M (a, 7, 2tx). 8.634 d+\-ir- + ''' + '^^-'^''jrx _^ I ' 1 _|- ((j2 ^- 2J7 - 4ac) + 2a{b - c)x + h{h - 2c)o^ \y = o, y = g-o^-i''^' If fa, 7, co^). 8.635 + ^ I (^2 - /2)x2'- + r(^g + 7^ - 2a/)x'- + ^ ^^(7 - g)(2 - g - 7) U = o, y = e ' X' M\oL, y, —j- 8.640 Tables and graphs of the function Mia, 7, x) are given by Webb and Airey (Phil. Mag. 36, p. 129, 1918) for getting approximate numerical solu- 1 88 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS tions of any of these differential equations. The range in a; is i to lo; in a, +0.5 to +4.0 and -0.5 to -3.0; in 7, i to 7. For negative values of x the equations of 8.61 may be used. SPECIAL DIFFERENTIAL EQUATIONS 8.700 where X{x) is any function of x. The complete solution is: y = Cie"^ + Cae-"^ + - I X{^) sinh n{x - ^) d^. nj 8.701 S + .| + »-, = xw. The complete solution, satisfying the conditions: a; = o y = yo, dy , dx^y^' ,„ f , sin n'x I , k . ,\ y = g-i"^ i y„ ; \- yo Icos n X -\ 7 sm n xj n Jo n' = \Jn^ - -• where 8.702 g-fnx)dx j^ -/: fe-J'^^^^''''g{x)dx + Ci + Ci. 8.703 d^ dx" +f(y){^)\i(y)-o, efny)dy ^y +C2. 8.704 ^j {c^-2feJ''^y)'iyg{y)dy]^ /gfoWiv gy feJ-MiyJ^y) dy + Ci. 8.705 8.706 DIPFEKENTiAL EQUATIONS 1 89 fef Xy^iy dy = cxf e'f '^^^^'^ dx + a. 8.707 8.708 - + ia + bx)j-^ + abxy=o. y = e -""{ci + ca/e""-***' dx} ■ d^y , 1 sdy , y = e-»*{ci + c fx-^^" dx] eP'y a dy h ■ — - -\ -] ^^ = 0. dx^ X dx x^ I. (a-iy>Ab; \ = ^V{a-iy-4b a- 1 y = X 2 {CiX + C2X~^}- 2. (a - i)2<46; X = ^ V4* - (a - ly a — I y = X r{ci COS (X log x) + C2 sin (X log x)}- 3. {a - lY = 46 a — 1 y = a;~^"(ci + C2loga;). 8.709 P^+2bx^+ia + bVy = o. dx' ax I. ab, \= y/a-b, 8.710 y = e 2 (ci cos X:*; + Ca sin Xx). y = ci(a + bx) + C2 1 e^ - (a + *»:) //^^^ ^* | " igo MATHEMATICAL FORMULAE AND ELLIPTIC FUNCTIONS 8.711 8.712 8.713 dx'^ X dx -^ x' if .a y •= - \ I cos IJ.X + C2 sm fjLx -\ — ^ X [ (^ d*y , d^y d^y , dy , di^+^'^di^+'d^^+'^i+'^y- y = cie-'"='{pi sin (wix + aO + Wi cos (coix + ai) } + C2e""''"lP2 sin (waa; + a^) + 0:2 cos (waa; + 0:2)}, where: 4Wi^ = z + c-2d^+ 2Vz^ - 40 - 2 d\/z - c + d^, 4W2^ =Z + C-2(i2_ 2^32 _ 4a ^ 2 d\/z^^^T^, 2pi = d + Vz - c + d^, 2P2 = d - y/z- c-\- d^, and z is a root of s' — cz^ — 4(0 — bd)z + 4(oc — ad"^ — ¥) = o. (Kiebitz, Ann. d. Physik, 40, p. 138, 1913) IX. DIFFERENTIAL EQUATIONS {continued ) 9.00 Legendre's Equation: 9.001 If w is a positive integer one solution is the Legendre polynomial, or Zonal Harmonic, Pn{x): 2"(m!)^ \ 2(2W — l) 2-4- (2W — l)(2» — 3) • • • • J 9.002 If « is even the last term in the finite series in the brackets is: (w!)3 (-1) = ^!)M! 9.003 If n is odd the last term in the brackets is: («!)2(»-i)! (-1)" ([K«-i)]!)U2«-i)! 9.010 If » is a positive integer a second solution of Legendre's Equation is the infinite series: ^"^'^^" (2«+l)!l'' + 2(2« + 3) (w+l)(w+2)(w+3)(w + 4) ^_(„+5) , 1 . 2-4-(2« + 3)(2«+S) • . • -J 9.011 P,„(cos d) = (-i)":Sr, ( sin^" e - ^' sin^-^ 6 cos^ (w!)2p"' " 2! (2m) + ....+ (-1)"-^ — — YZX'^ ^— cos^" 6 9.012 P^n+i (cos 0) = (-i)" ^^f„ + y^' { sin^" cos - ^ sin2-2 cos' ^ T^ ^ ^ (2W+1)! J (Brodetsky: Mess, of Math. 42, p. 65, 191 2) 191 192 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS 9.02 Recurrence formulae for Pn{x): I. (« + i)i'7H-i + nPn-i = (2w + i)a;P„ (2« + i)P„ dPn+l dPn—l dx dx dPr^l __dP„ 3- ("+^)^" = ^^-^-rf7- r, dP^ dP^i 4- nPn = x-z J dx dx 5. (i - *') ^ = (« + i) (^^" - -P^+i)- 6. (l - ^) ^" = »(^n-l - ^^n). dP„ 7. (2« + l) (l - X^) -T^ = n{n + l) {Pn-l - Pn+l). 9.028 Recurrence formulae for Qn{x). These are the same as those for Pn{x). 9.030 Special Values. PoW = I, PiW = X, P2W = i(3*' - i), Pz{x) = hiSx^ - 2,x), Pi(x) = |(35»*-30x' + 3), PbW = 1(63^^ - 7°*' + iS«:)> ■PeW = W(23ix« - 315^;' + 105*;' - S), ■PvW = rE-(429x' - 693^5 + 31SX' - ssx), P&{x) = r2T(643S^ - i2oi2a;« + 6930a;* - 1260^2 + 35). 9.031 „ , s I , x+ 1 Qi{x) = -^x log |— ^ - I, Q^ix) = \p,{x)\og=^^-\x, (3a(x) = ^P3Wlog|±-;-^x'+^^- DIFFERENTIAL EQUATIONS 1 93 9.032 P,„(o) = (-x)'- "3-5---.(2.-i) 2 •4-6 .... 2W P2n+l(o) = O, -P«(l) = I, Pn{-X) = (-l)»P„(x). 9.033 If z = f cos 0: aPn(cos g) w + I / „ - — a = -^ Pi (cos C/)P„(cos 6*) - P„+i(cos d) = f^^JT^rl^"-^(^°^^)-^'^^(^°^^^/ 9.034 Rodrigues' Formula: ^ ' 2"w! (ix" ^ ^ 9.035 If z = r cos 9: 9.036 If w ^ « P„(cosg) = (^%-£(j 7/t P„(X)P„(X) = 2y ^^_, U+2m-2^+ir'^'^^^ ^^)' where: ^ _ I-3-5 . . . ■ (2>-- i) MEHLER S INTEGRALS 9.040 For all values of w: 2 /' g "cos (w + h)(t)d(t> ' P„(cos 9) =- I / / ^ ==^' ^^o V 2(C0S (p — cos (7) 9.041 If M is a positive integer: P„(cos 0} = - : ^J V2 (cos - )4>d^ + i)2] In' -{n- i)2] . .^^ 1 +(--+3) g:;:;gj p^x(cos0) + ^^^ + 7)[^2_(^+3)2]^„._(^ + 4).3 ^n+3(C0S 0) + . . . |- CO fl TT TT-y (4W- l) / I-3-5. ■ . (2n- l) Y„ /_, m »=I 00 . c-r, fl ^ 7rV< (4W+1) / I-3-5. . . . (2w- i) V„ /^, m 2. sin P = / 7 r-7 ; r 7 i^2n(C0S C/). 4 2 ^^ (2W — l) (2W + 2) \ 2-4-6 . . . . 2W / «=i CO ^ fl ir'V^ 2«(4w - i) /i-3-s . . . . (2« - i)Vr, / fl\ 3. cot = - 7 . -r^^*^ ^ . ^ ^2n-l(cOS Q). ^ 2 ^^ (2W - i) \ 2-4-6 . ... in I CO 4. CSC = ^ + ^ 2 (4^ + ^) ( "'•^•,--^^""^T i-2n(cOS e). ^ 2 2 iW V 2'4-6 . . . . 2fJ / 9.063 . d I + sm - ,^ I. log — ^ = 1+2 ^1 ^"^''"^ ^)- sin- n=i 2 ^- ^°^ ^% si^ g ^ "^°^ '''' 2 " ^°^ (' + ^''' 2) = 2 « ■^"^'^°' ^)- 9.064 K(k) and £(yfe) denote the complete elliptic integrals of the first and second kinds, and fe = sin 0: 196 MATHEMATICAL FOIIMUL^ AND ELLIPTIC FUNCTIONS ^ - Eik) = ^ + 2^j;(-z)n.x ^4^+^) .^3-5-^--(^^-^) )V,„(cosg). 8 4^^ (2»- l) (2W+2) V 2-4-6....2W / ^ (Hargreaves, Mess, of Math. 26, p. 89, 1897) 9.070 The differential equation: If w is a positive integer, and —i>x> + i, two solutions of this differential equation are the associated Legendre functions '■d'"Pnix) P„ (x) = (i - x^)" dx'" ' Qn{x) = (I-XO^ d^^' 9.071 If n, m, r are positive integers, and n>m, r>m: Pn (x) Pr (x) dx = o a r ^ n, 2 (n + m)\ .. , — 7 TT n r = n. 2W + I {n — m)\ 9.100 Bessel's Differential Equation: d'^y ^ T. dy ^ I v^' 9.101 One solution is: dx^ X dx \ x'^J k=o 9.102 A second independent solution when v is not an integer is: y = J-v{x). 9.103 11 V = n, an integer: 9.104 A second independent solution when y = w, an integer, is: 'n-F„(x) = 2/„(x)-log- - 2j ^i ^(-j k = o CO (see 6.61). DIFFERENTIAL EQUATIONS 197 9.105 For all values of v, whether integral or not: Yyix) = —. I cos virJJx) - J-vix)], sin ^TT V / J-v{x) = cos virJvix) — sin vitYv{x), F_v(x) = sin VTrJvix) + cos vkYv{x). 9.106 For V = n, an integer: F_„(x) = (-i)"F„W. 9.107 Cylinder Functions of the third kind, solutions of Bessel's differential equation: 1. Hy (x) = /„(x) + iVvix). 2. Hy ix) = Jv{x) — iYv{oo). 3. Zr-„ {x) = e""^' eI{x). ' „II , , .„II. ^ 4. H-y (x) = e-'"^'H^(a;). 9.110 Recurrence formulae satisfied by the functions Jv, F„, Hv, Hv, Cp represents any one of these functions. 1. Cv-.\{x) — Cv+l{x) = 2^-Cv{x). ax 2. C _i(x) + Cp+i{x) = — Cv{x). X 3. -j-Cv{x) = Cv-l{x) Cv{x). 4. ^C (x) = -C^ix) - C:,+i{x). 5. T- I ^Qi) y = .v''C„_i(a;). 6. ,"2 = - \ Cy+2{x) + Cp-iix) - 2Cp(x) \ ■ 9.111 I. Mx) ^^ - Y.(x) %^ = — •■ 2. J,+,{x)Y,{x) - Mx)Y,+^{x) = — • (toe CIX TvX TTX ASYMPTOTIC EXPANSIONS FOR LARGE VALUES OF X 9.120 _ 1. Jv{x) = y — i P (x) cos I .V TTJ - Qvix) sin ( X tt] 2. Yvix) = y • — s Pvix) sin (a; tt j + Qp(x) cos f x tt 198 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS 3. hI{x) = ^''^'"^'''V^ {^"^^^ + ^^"(^^ }' 4. dl{x) = e~''^'"^''Vi {^^(^) - ^"^""^ }' where ^"W -^-^ ^\ ^) (^2k)\ 26* x2* *=i yK»>/ - ^ (, i; (2jfe - i) ! 2«*^-3 a^*-i SPECIAL VALUES 9.130 I. /o(x) = i-^,y +^,y -(^y +•••• 3. ^FoW = (log f + 7) /o W + gj - (^. (^ + + (^(^ + i + ^)u = (log 1+7) -^0 W + 4 ] ^ -^2(x) - - /4(x) + ^ /e W- 4. Jf. W = (log ^^ + 7) MX) - ^ /o (X) - ^ I : - -^ (. + I) (^J 213! V 2 3/\2 = flog - + y)Mx) - -/o(x) + -^Ux) -^Mx) \ 2 I X 1-2 2-3 7 = 0.5772157 (6.602). 9.131 Limiting values for x = o: 2-3 3-4 + -^ /7(x) - /o(x) = I, /i(x) = o, I"o(x) = ^(log^+7), Fi(x) = - — . irx DIFFERENTIAL EQUATIONS I99 9.132 Limiting values for x = oo : cos [X j ■ sin /o(x) = 1=^' Yo{x) = — '-x) <" . Itvx sin [x ) cos [x ) /i(x) = :== — ' Fi(x) = 7= •• . /TTX j-KX VT VT where 9.140 Bessel's Addition Formula: «»+*)=(^)'2:<-)i:(^)'^«(')- 9.141 Multiplication formula: CO 9.142 00 sl{k - s)lT{v + k - s + i)T{iJ. + s + i)' 5 = 9.143 7- r ^ 7- ^ ^ _ V (-1)^ (ti + v + 2k\ M''+''+2* DEFINITE INT-EGRAL EXPRESSIONS FOE BESSEL'S FUNCTIONS 9.150 fx 2 /j<(x) = r / ^ cos (x sin (j)) COS^" (ji'dfj). 9.151 /a 2 v2/ /"^ 7y(x) = ;;^ ; / COS (x COS (j)) sin^" (j)-dcf). 200 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS 9.152 /,y ^'kV{v + -j 2 If n is an integer: 9.153 9.154 9.155 9.156 9.157 /2n W = — I COS (x sin )d(j) = - I ^ ■ (-l)" /"" 2(-l)" /"- /2n(x:) = -^^ — I COS ix COS d>) COS {2n)d4> = / ^ "K Jo IT 'J o /2r.+i W = — I sin (x sin d4> =— I ' ■ IT Jo TTjo (_l)n /''^ • 2(-l) J2n+i{x) = -^^ — ^ / sin {x COS ^) COS {211 + i)4>d(j) = TT Jo T^ T /'+" I /""" 2'n-J-w 2irJo IT Jo INTEGRAL PROPERTIES 9.160 If CvilJ'X) is any one of the particular integrals: Mixx), Y^ifxx), Hliixx), E^liiix), of the differential equation: d?"y ^ 1 dy ^ ( ^ V •i Cp {iJ,kx)Cv {fJi,ix)xdx Ax I iJ.iCv{lXkx)Cv'{lxix) - tXkCv{tJLix)Cv'{lXkx) \ \;iXk^ii.i. 9.161 If (Xk and Hi are two different roots of Cy(/x6) - o, J' Cv{iXkx)C^{iiix)x dx = — ; \ iXkC^{iXia)Cy'{iika) - iXiCv{lXha)C^'{yi.id) a Mfc - Ml I J 9.162 If ixk and yLi are two different roots of Cv'(iJ.a) ^ , I arid Cviixb) = o, / Cv{iJikx)Cv{lJiix)xdx = pCv(lJ'ka)Cv{iJLia). If M* = M;: / Cp{lXkx)Cv{iXix)xdx = i j b^C/^fXkb) - a?Cv'Km) - U - ^CvKlXkO) |. DIFFERENTIAL EQUATIONS 20I EXPANSIONS IN BESSEL S FUNCTIONS 9.170 Schlomilch's Expansion. Any function f{x) which has a continuous differential coeflBcient for all values of x in the closed range (o, ir) may be expanded in the series: where 9.171 where 9.172 where: f{x) = ao + ^j akJ oikx), «o = /(o) + - / u I ' f {usin 6)dddu, (Ik = — I ucosku I 'f'{u sin 6) dddu. 00 f{x) = aox" +^ ahJnictkx) o) s\r^^ (bdcb, T{v + I) \2j J 9.214 If X is large, to a first approximation: In (x) = (27rx cosh /3)-i e^ (cosh/3-^sinh/3)^ i!r„ (x) = 7r(27rX cosh /3)-*e-^ (cosh ^ - /3 sinh /3)^ » = X sinh ;8. ' 9.215 Ber and Bei Functions. ber X + i bei x = / (x\/z), ber X — i bei x = Io{ix\/t), , I fxY I /x" ber X = I - T-;^. {-] + (2!)n2/ ' (4!)'\2 beix = (?T-^f^T + ^ 2/ (30^27 (5 0^2 DIFFERENTIAL EQUATIONS 205 9.216 Ker and Kei Functions: ker X + i kei x = Ko^x^i), ker X — i kei x = Ko{ixy/i), ker X = ( log y] ber x -^ — bei x — -7—^ ( i + - ) ( - \ "x V 4 (2!)2V 2/V2, I / I I I \ /x keix=(log|-7)beix-^berx + gJ-^(i + i + i)(fJ + 9.220 The Bessel-Clifford Differential Equation: dcp = Ci{x). 4. I Ci{x sin^ <^) sin' cj) dcj) = C^^x). J J COS 2 A/ iC Ci(x sin^ (/)) sin 4> dcj> = -• ° X DIFFERENTIAL EQUATIONS 207 9.229 Many differential equations can be solved in a simpler form by the use of the C„ functions than by the use of Bessel's functions. (Greenhill, Phil. Mag. 38, p. 501, 1919) 9.240 The differential equation: dx^ X dx d^y 2 (w + i) dy dx^ with the change of variable: y = zx~"~^, becomes Bessel's equation 9.200. 9.241 Solutions of 9.240 are: I. y = X-"-^ Jn+i{x). 2. y = i«;~"~* F„+j(x). 3- y = ^-""^ K+ii'^)- 4. y = x---i K+i{x). 9.242 The change of variable: X = 2-\/Zj transforms equation 9.240 into the Bessel-Clifford differential equation 9.220. This leads to a general solution of 9.240: When n is an integer the equations of 9.225 may be employed. C (^ - ^^'^ ^^ "^ ^^ 4/ X ■2 sin {x + €) cos {x + e) (?)-■ 9.243 The solution of d^y 2(n + i) dy • dx^ X dx may be obtained from 9.242 by writing sinh and cosh for sin and cos respectively. 9.244 The differential equation 9.240 is also satisfied by the two independent functions (when n is an integer) : I ( \ - ( ^ AV ^^'^ ^ V ^ dxj X = I y (-1)* yp' •3-5 . . (2W+ l) ^ 2*^!(2»+-3) (2W+ 2/fe+ l) A = o 2o8 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS *■« = (-.;!)" cos X X ^ I-3-5. ■ ■ (2»- i) y< x^'' ^n + l ^ y I 2*;fe!(i _ 2m) (3 - 2W) . . . .{2k-2n- l) k= o 9.245 The general solution of 9.240 may be written: 9.246 Another particular solution of 9.240 is: y = Ux) = ^ _ i — j -^ = ^,(x)- ii/'„(x), . , , i'^e-^'' f w(« + 1) , (w - i)w(» + i) (« + 2) ■' ^ ' si;"+i [ 21X 2-4-(za;)2 I-2-3 + ■ — - 2-4-6 . . . . 2M(ia;)" ; san = - X-^r^x{x), 9.247 The functions \pn{x), '^n{x),fn{x) satisfy the same recurrence formulae: d\pn{x) dx x-'^j — ■ + (2W + i)i/'„(x) = 1^7,-l(x). 9.260 The differential equation: (i^-y n(n+ i) with the change of variable: y = mVx is transformed into Bessel's equation of order n + -. 9.261 Solutions of 9.260 are: I. » 2 \ xdxj x Cn(x) = (-l)n\/^ /__5(x) = X''+'(- - » 2 \ X £„(x) = C„(x) - i5„(x) = x"+i f- - :^) \ xdx/ I (i\"cosa; t/x/ X J/ iJbJv I jO 9.262 The functions ^^Cx), Cn(a;), £„(a;) satisfy the same recurrence formulae: dSn{x) 11 + I dx ■Snix) - Sn+l(.x). DIFFERENTIAL EQUATIONS 209 2. j^ = Sr^iix) - - 5„(x). 3. ^„-i-i(x) = Sn{x) - Sn-l{x). 9.30 The hypergeometric differential equation: 9.31 The equation 9.30 is satisfied by the hypergeometric series: ^ ' '^' ' 17 1-2 7(7 + l) ^ a(a + !)(« + 2) /3(^ + i) (/3 + 2) ^, 1.2-3 7(7 + i) (7 + 2) The series converges absolutely when xi. When a; = +1 it converges only when a + ^ — 7o. 9.421 . I Virt I ?l!±l , , I-3-S . . . (2« - i) ph= ^ 9.422 ^ I = o p^i = o P"i = o p^i = o 9.423 ■h-i ■VI S y IT I-3-S . . . {2n + i)y IT _ 2^./]_ 9.424 I V p" r(i + v) where v may have any real value, except a negative integer. (Conjectural.) 9.425 P I = e" p — a ^ I = - (e"' - i) p — a a 9.426 With p = aq^, o _Li / \ I'^-'l • • • (2W — l) -2n^ - n (at)- {2at)''VTrat ^ At I 212 MATHEMATICAL FORMULiE AND ELLIPTIC FUNCTIONS 9.427 y/itat 9.428 If z = 2^ at g-aa: = -^ / e-»2(fD - e~«"= = — = / e"° 9 9.43 Many examples of the use of this method are given by Heaviside: Electro- magnetic Theory, Vol. II. Bromwich, Proceedings Cambridge Philosophical Society, XX, p. 411, 1921, has justified its application by the method of contour integration and appUed it to the solution of a problem in the conduction of heat. 9.431 Herlitz, Arkiv for Matematik, Astronomi och Fysik, XIV, 1919, has shown that the same methods may be appUed to the more general partial differential equations of the type, a./S and the relations of 9.42 are valid. 9.44 Heaviside's Expansion Theorem. The operational solution of the differential equation of 9.41, or the more general equation, 9.431, satisfying the given boundary conditions, may be written in the form, where F(^) and A(^) are known functions of /> = — . Then Heaviside's dt Expansion Theorem is: F{6) 'V F[d) Mo A77X + S. A(o) ^ ^^a[^{a) where a is any root, except o, of A(^) = o, A'(^) denotes the first derivative of A(/') with respect to p, and the summation is to be taken over all the roots of A(^) = o. This solution reduces to m = o at / = o. Many appUcations of this expansion theorem are given by Heaviside, Electromagnetic Theory, II, and III; Electrical Papers, Vol. II. Herlitz, 9.431, has also applied this expansion theorem to the solution of the problem of the distribution of magnetic induction in cylinders and plates. 9.45 Bromwich's Expansion Theorem. Bromwich has extended Heaviside's Expansion Theorem as follows. If the operational solution of the partial differential equation of 9.41, obtained to satisfy the boundary conditions, is DrFFERENTIAL EQUATIONS 213 where G is a constant, then the solution of the differential equation is u = G[m + N. + ^J^^ gOit where N^ and Ni are defined by the expansion, ^ = No + N,p + N,p'' + . . .; a. is any root of A(/>) = o, A'(^) is the first derivative of A(^) with respect to p, and the summation is over all the roots, a. This solution reduces to m = o at t = o. Phil. Mag. 37, p. 407, 1919; Proceedings London Mathematical Society, 15, p. 401, 1916. 9.9 References to Bessel Functions. Nielsen: Handbuch der Theorie der Cylinder Funktioneh. Leipzig, 1904. The notation and definitions given by Nielsen have been adopted in the pres- ent collection of formulae. The only difference is that Nielsen uses an upper index, J"{x), to denote the order, where the more usual custom of writing Jn(x) is here employed. In place of Hi" and fl'2" used by Nielsen for the cylinder functions of the third kind, H„^ and HJ^ are employed in this collection. Gray and Mathews: Treatise on Bessel Functions. London, 1895.^ The Bessel Function of the second kind, F„(x), employed by Gray and Mathews is the function TT - Yn{x) + (log 2 - j)Jn{x), 2 of Nielsen. SchafheitHn: Die Theorie der Besselschen Funktionen. Leipzig, 1908. Schafheitlin defines the function of the second kind, Y„{x), in the same way as Nielsen, except that its sign is changed. Note. A Treatise on the Theory of Bessel Functions, by G. N. Watson, Cambridge University Press, 1922, has been brought out while this volume is in press. This Treatise gives by far the most complete account of the theory and properties of Bessel Functions that exists, and should become the standard work on the subject with respect to notation. A particularly valuable feature is the Collection of Tables of Bessel Functions at the end of the volume and the Bibliography, giving references to all the important works on the subject. 9.91 Tables of Legendre, Bessel and aUied functions. P„(x) (9.001). 1 A second edition of Gray and Mathews' Treatise, prepared by A. Gray and T. M. MacRobert, has been published (1922) while this volume is in press. The notation of the first edition has been altered in some respects. 214 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS B. A. Report, 1879, pp. 54-57. Integral values of n from i to 7 ; from x = o.oi to X = 1. 00, interval o.oi, 16 decimal places. Jahnke and Emde: Funktionentaf eln, p. 83 ; same to 4 decimal places. P„(cos d) Phil. Trans. Roy. Soc. London, 203, p. 100, 1904. Integral values of n from I to 20, from = o to 6 = 90, interval 5, 7 decimal places. Phil. Mag. 32, p. 512, 1891. Integral values of n from i to 7, = o to d = 90, interval i ; 4 decimal places. Reproduced in Jahnke and Emde, p. 85. Tallquist, Acta Soc. Sc. Fennicae, Helsingfors, 33, pp. 1-8. Integral values of n from i to 8; = o to = 90, interval i, 10 decimal places. Airey, Proc. Roy. Soc. London, 96, p. i, 1919. Tables by means of which zonal harmonics of high order may be calculated. Lodge, Phil. Trans. Roy. Soc. London, 203, 1904, p. 87. Integral values of n from i to 20; d = o to Q = 90, interval 5, 7 decimal places. Reprinted in Rayleigh, Collected Works, Volume V, p. 162. dPnjcos 6) dd Farr, Proc. Roy. Soc. London, 64, 199, 1899. Integral values of n from i to 7; 6 = o to 6 = go, interval i, 4 decimal places. Reproduced in Jahnke and Emde, p. 88. Joix), Mx) (9.101). Meissel's tables, x = o.oi to a; = 15.50, interval o.oi, to 12 decimal places, are given in Table I of Gray and Mathews' Treatise on Bessel's Functions. Aldis, Proc. Roy. Soc. London 66, 40, 1900. a; = o.i to x = 6.0, interval 0.1, 21 decimal places. Jahnke and Emde, Funktionentafeln, Table III. x = o.oi to x = 15.50, interval o.oi, 4 decimal places. Jn{x) (9.101). Gray and Mathews, Table II. Integral values of n from w = o to w = 60; integral values of x from a; = i to a; = 24, 18 decimal places. Jahnke and Emde, Table XXIII, same, to 4 significant figures. B. A. Report, 1915, p. 29; n = o to » = 13. X = 0.2 to 3; = 6.0 interval 0.2 6 decimal places, X = 6.0 to x = 16.0 interval 0.5 10 decimal places. Hague, Proc. London Physical Soc. 29, 211, 1916-17, gives graphs of /„(a;) for integral values of n from o to 12, and n = 18, x ranging from o to 17. - J Fo(x) = Go(x) ; - ^ 7i(x) = G,{x). B. A. Report, 1913, pp. 116-130. x = o.oi to x = 16.0, interval o.oi, 7 decimal places. DUTEEENTIAL EQUATIONS 215 B. A. Report, 1915, a; = 6.5 to a; = 15.5, interval 0.5, 10 decimal places. Aldis, Proc. Roy. See. London, 66, 40, 1900: x= 0.1 to a; = 6.0. Interval 0.1, 21 decimal places. Jahnke and Emde, Tables VII and VIII, functions denoted KoW and Ki(x), 31; = 0.1 to X = 6.0, interval o.i; x = o.oi to x = 0.99, interval o.oi; x = i.o to X = 10.3, interval o.i; 4 decimal places. - - F„(x) = G„(x). 2 B. A. Report, 1914, p. 83. Integral values of n from o to 13. x = o.oi to X = 6.0, interval 0.1; x = 6.0 to x = 16.0, interval 0.5; 5 decimal places. - Fo(x) + (log 2 - 7)/o(«), Denoted Y^(x) and Yi{x) — Y\{x) + (log 2 - 7)/i(x). respectively in the tables. B. A. Report, 1914, p. 76, x = 0.02 to x = iS-So, interval 0.02, 6 decimal places. B. A. Report, 1915, p. 33, x = 0.1 to x = 6.0, interval 0.1; x = 6.0 to X = iS-S, interval 0.5, 10 decimal places. Jahnke and Emde, Table VI, x = o.oi to x = i.oo, interval o.oi; x = i.o to X = 10.2, interval 0.1, 4 decimal places. Fo(x), ri(x). Denoted A''o(x) and Nx{x) respectively. Jahnke and Emde, Table IX, x = 0.1 to x = 10.2, interval 0.1, 4 decimal places. — F„(x) + (log 2-7) Jn{x). Denoted F„(x) in tables. 2 B. A. Report, 1915. Integral values of n from i to 13. x = 0.2 to x = 6.0, interval 0.2; x = 6.0 to x = 15.5, interval 0.5, 6 decimal places. /„ + j(x). Jahnke and Emde, Table II. Integral values of « from » = o to n = 6, and « = — I to w = — 7 ; -v = o to .V = 50, interval i .0, 4 figures. Ji(x), J-iix). Watson, Proc. Roy. Soc. London, 94, 204, 19 18. X = 0.05 to X = 2.00 interval 0.05, X = 2.0 to X = 8.0 interval 0.2, 4 decimal places. Jaia), Ja-l{a) Yaici), Ya-i{2, 12 decimal places. These tables are reproduced in Gray and Mathews, Table VI. Jahnke and Emde, Table XXIV; same ranges, to 4 places. 7o (xVi) = Z - iY, V2/1 (xVT) = Xi + iYi DIFFERENTIAL EQUATIONS 21 7 Aldis, Proc. Roy. Soc. London, 66, 142, 1900; a; = o.i to x = 6.0, interval 0.1, 21 decimal places. Jahnke and Emde, Tables XV and XVI, same range, to 4 places. • Gray and Mathews, Table IV; x = 0.2 to x = 6.0, interval 0.2, 9 decimal places. Ya{xy/~i) (9.104) Denoted NaixV'i) in table. H^(xVi), nlixVi). Jahnke and Emde, Tables XVII and XVIII; x = 0.2 to x = 6.0, interval 0.2, 4-7 figures. 'f Hliix) = K,(x), (9.212). TT I - - H^iix) = Zi(x), Aldis, Proc. Roy. Soc. London, 64, 219-223, 1899; x = o.i to x = 12.0, interval o.i, 21 decimal places. Jahnke and Emde, Table XIV; same, to 4 places. iHliix), -Hliix) (9.107). Jahnke and Emde, Table XIII; x = 0.12 to x = 6.0, interval 0.2, 4 figures. berx, ber'x, ^^^is). bei X, bei' x, B. A. Report, 1912; x = o.i to x = 10. o, interval o.i, 9 decimal places. Jahnke and Emde, Table XX; x = 0.5 to x = 6.0, interval 0.5, and x = 8, 10, 15, 20, 4 decimal places. kerx.ker'x, ^g^is). kei X, kei x, B. A. Report, 1915; x = 0.1 to x = lo.o, interval 0.1, 7-10 decimal places, ber^ X + bei^ x, ber'^ X + bei'^ x, ber X bei' x — bei x ber' x, and the corresponding ker and kei ber X ber' x + bei x bei' x, functions. B. A. Report, 1916; x = 0.2 to x = 10. o, interval 0.2, decimal places. S„{x), S'nix), log5n(x), log5'„(x), a(x), C'n{x), logC„(x), logC'„(x), (9.261). £„(x), £'„(x), log£„(x), log£'„(x), B. A. Report, 1916; integral values of n from o to 10, x = i.i to x = 1.9, interval 0.1, 7 decimal places. 2l8 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS z)(x) = :^n( -^)xU_5(;)= ^^f:::^xij. -1 = '^ ij - 2 V 4/ •\2/ 1.15407^ "Hz Table I of Jahnke and Emde gives these two functions to 3 decimal places for X = 0.2 to x = 8.0, interval 0.2, and x = 8.0 to x = 12.0, interval i.o. Roots of Ja{x) = o. Airey, Phil. Mag. 36, p. 241, 1918: First 40 roots (p) with corresponding values of /i(p), 7 decimal places. Jahnke and Emde, Table IV, same, to 4 decimal places. Roots of Ji(x) = o. Gray and Mathews, Table III, first 50 roots, with corresponding values of Joix), 16 decimal places. Airey, Phil. Mag. 36, p. 241: First 40 roots (r) with corresponding values of /oW, 7 decimal places. Jahnke and Emde, Table IV, same, to 4 decimal places. Roots of /„(») = o. B. A. Report, 1917, first 10 roots, to 6 figures, for the following integral values of n: o— 10, 15, 20, 30, 40, 50, 75, 100, 200, 300, 400, 500, 750, 1000. Jahnke and Emde, Table XXII, first 9 roots, 3 decimal places, integral values of n 0—9. Roots of: TV (log 2 — 7)/n(») H Yn{x) = o. Dcnoted Y„(x) = o in table. Airey: Proc. London Phys. Soc. 23, p. 219, 1910-11. First 40 roots for n = o, I, 2, 5 decimal places. Jahnke and Emde, Table X, first 4 roots for w = o, i. E decimal places. Roots of: , . ' Denoted No(x) and Ni(x) in tables. Fi(x) = o. Airey: 1. c. First 10 roots, 5 decimal places. Roots of: Jo{x) ± (log 2 - y)Jo{x) + — Yo(x) = o. Denoted Joix) ± Yoix) = o. Ji(x) + (log 2 - y)Jiix) + - Yi{x) = o. Denoted Ji{x) + Yi{x) = o. Joix) - 2(log 2 - y)Joix) H — Yoix) = o. Denoted Joix) - zYoix) = o. * TT io/o(»;) ± (log 2 - y)Joix) + — Yoix) = o. Denoted loJoix) ± Yoix) = o. DirFEKENTL4L EQUATIONS 219 Airey, 1. c. First xo roots, 5 decimal places. Roots of" Jn+l(x) In+i(x) _ Jn{x) 7„(x) Airey, 1. c. First 10 roots: n = o, 4. decimal places, w = i, 2, 3, 3 decimal places. Jahnke and Emde, Table XXV, first 5 roots for w = o, 3 for « = i, 2 for n = 2: 4. figures. Airey, 1. c. gives roots of some other equations involving Bessel's functions connected with the vibration of circular plates. Roots of; Jy(x)Yv{x) = Jv{kx)Yp{kx). Jahnke and Emde, Table XXVI, first 6 roots, 4 decimal places, for V = o, 1/2, I, 3/2, 2, s/2: k = 1.2, 1.5, 2.0. Table XXVIII, first root, multiplied by (^ — i) for^ = i, 1.2, 1.5, 2 — 11, iQ) 39> "^ • ^ same as above. Table XXIX, first 4 roots, multiplied by (k — i) for certain irrational values of k, and J' = o, I. X. NUMERICAL SOLUTION OF DIFFERENTIAL EQUATIONS By F. R. Moulton, Ph.D., Professor of Astronomy, University of Chicago; Research Associate of the Carnegie Institution of Washington. INTRODUCTION Differential equations are usually first encountered in the final chapter of a book on integral calculus. The methods which are there given for solving them are essentially the same as those employed in the calculus. Similar methods are used in the first special work on the subject. That is, numerous types of differential equations are given in which the variables can be separated by suitable devices; httle or nothing is said about the existence of solutions of other types, or about methods of finding the solutions. The false impression is often left that only exceptionally can differential equations be solved. What- ever satisfaction there may be in learning that some problems in geometry and physics lead to standard forms of differential equations is more than counter- balanced by the discovery that most practical problems do not lead to such forms. 10.01 The point of view adopted here and the methods which are developed can be best understood by considering first some simpler and better known mathematical theories. Suppose I. F{x) =«"-!- flix""' + -I- o„_iX -t- a„ = o is a polynomial equation in x having real coefficients a\, a^, . . . , a„. If n is I, 2, 3, or 4 the values of x which satisfy the equation can be expressed as explicit functions of the coefficients. If n is greater than 4, formulas for the solution can not in general be written down. Nevertheless, it is possible to prove that n solutions exist and that at least one of them is real if n is odd. If the coefficients are given numbers, there are straightforward, though somewhat laborious, methods of finding the solutions. That is, even though general formulas for the solutions are not known, yet it is possible both to prove the existence of the solutions and also to find them in any special numerical case. 10.02 Consider as another illustration the definite integral I. I = \ fix) dx, where /(x) is continuous ior a^x^b. If F(x) is such a function that NUMERICAL SOLUTION OF DIFFERENTIAL EQUATIONS 221 then I = F{b) - F{a). But suppose no F{x) can be found satisfying (2). It is nevertheless possible to prove that the integral I exists, and if the value of (x) is given for every value of x in the interval a ^ .v ^ b, it is possible to find the numerical value of / with any desired degree of approximation. That is, it is not necessary that the primitive of the integrand of a definite integral be known in order to prove the existence of the integral, or even to find its value in any particular example. 10.03 The facts are analogous in the case of differential equations. Those having numerical coeflacients and prescribed initial conditions can be solved regardless of whether or not their variables can be separated. They need to satisfy only mild conditions which are always fulfilled in physical problems. It is with a sense of reUef that one finds he can solve, numerically, any particular problem which can be expressed in terms of differential equations. 10.04 This chapter will contain an account of a method of solving ordinary differential equations which is applicable to a broad class including all those which arise in physical problems. A large amount of experience has shown that the method is very convenient in practice. It must be understood that there is for it an underlying logical basis, involving refinements of modern analysis, which fully justifies the procedure. In other words, it can be proved that the process is capable of furnishing the solution with any desired degree of accuracy. The proofs of these facts belong to the domain of pure analysis and will not be given here. 10.10 Simpson's Method of Computing Definite Integrals. The method of solving differential equations which will be given later involves the computation of definite integrals by a special process which will be developed in this and the following sections. Let t be the variable of inte- gration, and consider the definite integral nb 1. F = jMdt. This integral can be interpreted as the area between the i-axis and the curve y = f{t) and bounded by the ordinates t = a and t = b, figure I. Let to = a,tn = b, ji =f{ti), and divide the interval a ^ < $ 6 up into n equal parts, each of length h = {b — a) In. Then an approximate value of F is 2. Fo = h(yi + yz + . . . + 3'«)- This is the sum of rectangles whose ordinates, figure i, are yi, yi, . . . , y„. 10.11 A more nearly exact value can be obtained for the first two intervals, for example, by putting a curve of the second degree through the three points 222 MATHEMATICAL FOEMULiE AND ELLIPTIC FUNCTIONS yo, yi, ys, and finding tlie area between tiie /-axis and tliis curve and bounded by the ordinates ta and fe. The equation of the curve is I. y = an + ai{t - to) + aiit - toY, where the coeflScients ao, ai, and 02 are determined by the conditions that y shall equal yo, yi, and ya at t equal to to, ti and fe respectively; or yo = ao, yi = ao + ai{ti - ta) + (h{ti - 'o, yu Aijin = yn- yn-i, These are the first differences of the values of the function y for successive values of t. All the successive intervals for t are supposed to be equal. 10.21 In a similar way the second differences are defined by A23'2 = Ai^a - Ai^i, A2y3 = Ai3)3 - Aiya, A2y„ = Aiy„ - Ai3;„_i, 10.22 In a similar way third differences are defined by Asys = A2>'s - A2y2, A33'4 = A2)'4 - A23'3, As^n = A2y„ - A23'„ and obviously the process can be repeated as many times as may be desired. 10.23 The table of successive differences can be formed conveniently from the tabular values of the function and can be arranged in a table as follows: Table I y A13) Aay Aay yo yi Ai^i 3-2 Ai3'2 A23'2 ys Aij^s A2)'3 Asya In this table the numbers in each column are subtracted from those immediately below them and the remainders are placed in the next column to the right on the same Une as the minuends. Variations from this precise arrange- ment could be, and indeed often have been, adopted. 10.24 A very important advantage of a table of differences is that it is almost sure to reveal any errors that may have been committed in computing the y,. If a single yi has an error c, it follows from 10.20 that the first difference Aiyi will contain the error +€ and Aij/.-i-i will contain the error — e. But the second differences A2y,-, Ai^i+i, and A23'i+2 wiU contain the respective errors +e, — 2e, -f e. Similarly, the third differences Asy,-, A3yi+i, A3yi+2, and A3yi+3 wiU contain the respective errors -|-e, —36, +3e, — e. An error in a single y; afiects j + i differences of order _;', and the coefl&cients of the error are the binomial coeffi- cients with alternating signs. The algebraic sums of the errors in the affected 224 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS numbers in the various difference columns are zero. Now in such functions as ordinarily occur in practice the numerical values of the differences, if the intervals are not too great, decrease with rapidity and run smoothly. If an error is present, however, the differences of higher order become very irregular. 10.25 As an illustration, consider the function y = sir i ior t equal to io°, 15°, The following table gives the function and its successive differ- ences, expressed in terms of units of the fourth decimal:' Table II t sin t Ai sin t A2 sin / A3 sin t 10° 1736 IS 2588 852 20 3420 832 — 20 25 4226 806 -26 -6 30 5000 774 -32 -6 35 5736 736 -38 -6 40 6428 692 -44 -6 45 7071 643 -49 -5 SO 7660 589 -54 -5 55 8191 531 -58 -4 60 8660 469 -62 -4 65 9063 403 -66 -4 70 9397 334 -69 -3 Stippose, however, that an error of two units had been made in determining the sine of 45° and that 7073 had been taken in place of 7071. Then the part of the table adjacent to this number would have been the following: Table III t sin t Ai sin A2 sin t A3 sin t 25° 4226 30 5000 774 35 5736 736 -38 40 6428 692 -44 - 6 45 7073 64s -47 - 3 50 7660 587 -58 — II 55 8191 531 -56 + 2 60 8660 469 -62 - 6 65 9063 403 -66 - 4 The irregularity in the numbers of the last column shows the existence of an error, and, in fact, indicates its location. In the third differences four numbers • Often it is not necessary to carry along the decimal and zeros to the left of the first significant figure. NUMERICAL SOLUTION OF DIFFEIUENTIAL EQUATIONS 225 will be affected by an error in the value of the function. The erroneous numbers in the last column are clearly the second, third, fourth, and fifth. The algebraic sum of these four numbers equals the sum of the four correct numbers, or — 18. Their average is —4.5. Hence the central numbers are probably —5 and —4. Since the errors in these numbers are -36 and +36, it follows that e is probably + 2. The errors in the second and fifth numbers are +e and — e respectively. On making these corrections and working back to the first column, it is found that 7073 should be replaced by 7071. 10.30 Computation of Definite Integrals by Use of Difference Functions. Suppose the values of f{t) are known for / = tn-i, tn-i, in, and tn+i- Suppose it is desired to find the integral Jr'tn+i fit) dt. tn I. The coefficients bo, &i, ^2, and 63 of the polynomial can be determined, as above, so that the function shall take the same values B.sf{i) for t = tn-2, tn-i, fn, and tn+i- With this approximation to the function J{t), the integral becomes (since in+l — tn = h) Ibo + bl{t- tn) +bi{t- tnf + b,{t- tn)^! dt tn = klbo + -bih + - biW + - bzh^J 2 3 4 -■ The coefficients 60, bi, bi, and J3 will now be expressed in terms of jn+i, Aiyn+i, A23'n+i, and Asy^+i. It follows from (2) that ' yn-2 = ba - 2b-Ji + 462^^ - 8^)3^', jr^T. = bo - bxh + bih"^ - bs¥, yn = bo, yn+1 = bo + bji + b^ffi + 53^'- Then it follows from the rules for determining the difference functions that f Aiy„_i = bji - 2,b2h^ + ibjt^, S. \ Aiyn = bih - bih^ + bzh->, [ A,yn+i = b,h + b^h' + bzh^. ( A^yn = 2^2^^ - 6bzh^, \ A^yn+l = 252^^. 7. Asyn+i = 6b3h\ 226 MATHEMATICAL FORMULAE AND ELLIPTIC FUNCTIONS It follows from the last equations of these four sets of equations that bo = yn+i - Ai)/„+i, bih = Aiy„+i — Aj^n+i - ^ Asji^i, 62^^ = - A2y„+i, baP = T Aayn+i. Therefore the integral (3) becomes 9. In = Jm-I - -AiJI^l - — A2yr.+1 - — As^n+l - The coefiScients of the higher order terms A43i„+i and Asyn+i are and 48 respectively. 10.31 Obviously, if it were desired, the integral from tn-i to tn-\, or over any other part of this interval, could be computed by the same methods. For example, the integral from t^\ to t^ is = A|^y,H-i - ^Aiy,H.i + ^A2y„+i + ^A3y„+i + .... J. NUMERICAL ILLUSTRATIONS 10.32 Consider first the application of Simpson's method. Suppose it is required to finH J'^if r "]5s° sin tdt = — cos / = 0.3327. 2S° L J 25° On applying 10.12 with the numbers taken from Table I, it is found that II = —[^.4226 + 2.0000 + 1. 1472 + 2.5712 + 1. 4142 + 3.0640 + .819131, which becomes, on reducing 5° to radians, /i = 0.3327, agreeing to four places with the correct result. 10.33 On applying 10.11 (4) and omitting alternate entries in Table II, it is found that J "45° 10° sin t di = — 1^.4226 + 2.2944 + .7071] = 0.1992, 25' 3 which is also correct to four places. These formulas could hardly be surpassed in ease and convenience of application. NUMERICAL SOLUTION OF DIFFERENTIAL EQUATIONS 227 10.34 Now consider the application of 10.30 (9). As it stands it furnishes the integral over the single interval tn to t,^i. If it is desired to find the integral from tn to tn+m, the formula for doing so is obviously the sum of m formulas such as (9), the value of the subscript going from n + i to n -\- m + i, or In, m = k\ U„+l + + J-n+m+lj - " ( Al^^+l + + Al)l„+^ij - — ( A2y„+i +.... + A2yn+m+l] - — i^Byn+l +-... + Asyn+^M-l) + • • • " On applying this formula to the numbers of Table I, it is found that J"SS° sin tdt = S°[(.Sooo + .5736 + .6428 + .7071 + .7660 + .8191) - - (.0774 + .0736 + .0692 + .0643 + .0589 + .0531) H (.0032 + .0038 + .0044 + .0049 + .0054 + .0058) -| (.0006 + .0006 + .0006 + .0005 + .0005 + .0004)3 24 = 0.3327, agreeing to four places with the exact value. When a table of differences is at hand covering the desired range this method involves the simplest numerical operations. It must be noted, however, that some of the required differences necessitate a knowledge of the value of the function for earher values of the argument than the lower limit of the integral. 10.40 Reduced Form of the Differential Equations. Differential equations which arise from physical problems usually involve second derivatives. For example, the differential equation satisfied by the motion of a vibrating tuning fork has the form d'^x _ df ~ ~ ' where /fe is a constant depending on the tuning fork. 10.41 The differential equations for the motion of a body subject to gravity and a retardation which is proportional to its velocity are ' d^x dx df^~'^dt' d^y _ dy where c is a constant depending on the resisting medium and the mass and shape of the body, while g is the acceleration of gravity. 228 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS 10.42 The differential equations for the motion of a body moving subject to the law of gravitation are (m jQ ^ iy JO ^- -P^ df ~ r" df " r" r'i = x^ + y^ + 2^. 10.43 These examples illustrate sufficiently the types of differential equations which arise in practical problems. The number of the equations depends on the problem and may be small or great. In the problem of three bodies there are nine equations. The equations are usually not independent as is illustrated in 10.42, where each equation involves all three variables x, y, and z through r. On the other hand, equations 10.41 are mutually independent for the first does not involve y or its derivatives and the second does not involve x or its deriva- tives. The right members may involve x, y, and z as is the case in 10.42, or they may involve the first derivatives, as is the case in 10.41, or they may involve both the coordinates and their first derivatives. In some problems they also involve the independent variable t. 10.44 Hence physical problems usually lead to differential equations which are included in the form d^x If df ^ = / dx dy 'di'di' tl dx dy *' ^' di' di' ' where / and g are functions of the indicated arguments. Of course, the number of equations may be greater than two. 10.45 If we let , dx , dy "" ^Jv ^ ^di' equations 10.44 can be written in the form dx di = X dx' -^ =fix,y,x',y',t), dy , dy' dt g{x, y, x', y', t). NUMERICAL SOLUTION OF DIFFERENTIAL EQUATIONS 229 10.46 If we let x = xi, x' = Xi, y = xs, y' = Xi, equations 10.45 are included in the form iji — Jn\Xi, X2, . . . , Xny t). This is the final standard form to which it will be supposed the differential equations are reduced. 10.50 Definition of a Solution of Differential Equations. For simplicity in writing, suppose the difierential equations are two in number and write them in the form dy It g(x,y,i), where / and g are known functions of their arguments. Suppose x = a, y = b at i = o. Then X = (/)(/), y = ^(t), is the solution of (i) satisfying these initial conditions if 4> ''■nd \l/ ^re such functions that <^(o) =a, ^(o) = b, 3. f =/(<^.'A,/), di g{ = «+/ /(, ^, t) dt, ^ = b + Jji, xjy, i) dt. The diflSculty arises from the fact that and yp are not known in advance and the integrals on the right can not be formed. Since 4> and yp are the solution values of x and y, we may replace them by the latter in order to preserve the original notation, and we have NUMERICAL SOLUTION OF DIFFERENTIAL EQUATIONS 231 2. X = a + £f{x, y, t) dt, y = b + J g{x, y, t) dt. If X and y do not change rapidly in numerical value, then/(x, y, t) and g(x, y, t) will not in general change rapidly, and a first approximation to the values of x and y satisfying equations (2) is Jj{a, b, t) dt, Xi = a + yy = b + Jj{a, b, t) dt. at least for values of t near zero. Since a and b are constants, the integrands in (3) are known and the integrals can be computed. If the primitives can not be found the integrals can be computed by the methods of 10.1 or 10.3. After a first approximation has been found a second approximation is given by Xi= a -{- y2'=b+ I g(xi, yi, t) dt. K i/o The integrands are again known functions of t because Xi and yi were determined as functions of t by equations (3). Consequently x^ and y2 can be computed. The process can evidently be repeated as many times as is desired. The wth approximation is / f{xr^i, y„_i, t) dt, Xn = a + yn = b + Jji^n-l, y„_i, t) dt.' There is no difiiculty in carrying out the process, but the question arises whether it converges to the solution. The answer, first established by Picard, is that, as n increases, x„ and y„ tend toward the solution for all values of t for which all the approximations belong to those values of x, y, and t for which / and g have the properties of continuity with respect to / and differentiability with respect to X and y. If, for example, / = sm X and the value of .t„ tends towards zero for t = T, then the solution can not be extended beyond t = T. It is found in practice that the longer the interval over which the integration is extended in the successive approximations, the greater the number of approxi- mations which must be made in order to obtain a given degree of accuracy. In fact, it is preferable to take first a relatively short interval and to find the solution over this interval with the required accuracy, and then to continue from the end values of this interval over a new interval. This is what is done in actual work. The details of the most convenient methods of doing it will be explained in the succeeding sections. 232 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS I 10.7 The Step-by-Step Construction of the Solution. Suppose the differential equations are ^^^ = gi=c,y,t), with the initial conditions x = a, y = b a,t t = o. It is more difficult to start a solution than it is to continue one after the first few steps have been made. There- fore, it will be supposed in this section that the solution is well under way, and it will be shown how to continue it. Then the method of starting*a solution will be explained in the next section, and the whole process will be illustrated numerically in the following one. Suppose the values of x and y have been found for t = h, t2, . . . . , tn. Let them be respectively Xi, yi; x^, yi; . . .; x„, y„, care being taken not to confuse the subscripts with those used in section 10.6 in a different sense. Suppose the intervals ti — h, h — U, . . . , tn — t,^i are all equal to k and that it is desired to find the values of x and y at t^^i, where tn+i — tn = h. It follows from this notation and equations (2) of 10.6 that the desired quantities are Jr>tn+I f {x, y, t) dt, tn g {x, y, t) dt. U The values of x and y in the integrands are of course unknown. They can be found by successive approximations, and if the interval is short, as is supposed, the necessary approximations will be few in number. A fortunate circumstance makes it possible to reduce the number of approxi- mations. The values of x and y are known at < = /„, /„_i, i„_2, . . . From these values it is possible to determine in advance, by extrapolation, very close approxi- mations to X and y ior t = t„+i. The corresponding values of / and g can be computed because these functions are given in terms of x, y, and t. They are also given for t = tn, /n-i, Consequently, curves for / and g agreeing with their values at t = tn+i, tn, tn-i, .... can be constructed and the integrals (2) can be computed by the methods of 10.1 and 10.3. The method of extrapolating values of x„+i and y„+i must be given. Since the method is the same for both, consider only the former. Since, by hypothesis, X is known for t = tn, in^i, tn-2, .... the values of Xn, Ai.t„, A^Xn, and AsXn are known. If the interval h is not too -large the value of AaXn+i is very nearly equal to As.tn. As an approximation AaXr^-i may be taken equal to A^Xn, or perhaps a closer value may be determined from the way the third differences NUMERICAL SOLUTION OF DIFFERENTIAL EQUATIONS 233 Asaj^s, AiXn-2, AsXn-i, and AsXn vary. For example, in Table II it is easy to see that A3 sin 75° is almost certainly -3. It follows from 10.20, 1, 2 that 3- < AiJC^i = AzXn+i + AiX„, Xn+l = Z\iX,t_|_i + Xn. After the adopted value of AaX^i has been written in its column the successive entries to the left can be written down by simple additions to the respec- tive numbers on the line of 4. For example, it is found from Table II that A2 sin 75° = -72, Ai sin 75° = 262, sin 75° = 9659. This is, indeed, the correct value of sin 75° to four places. Now having extrapolated approximate values of x^^-i and y„+i it remains to compute / and g for x = Xr^i, y = y„+i, t = t,^i. The next step is to pass curves through the values of / and giort = /^i, tn, tn-i, .... and to compute the inte- grals (2). This is the precise problem that was solved in 10.30, the only difference being that in that section the integrand was designated by y. On applying equation lOiSO (9) to the computation of the integrals (2), the latter give Xr^i = Xn + h lfn+1 - "Ai/^i - — A^fr^i - — Aj/^i . . . ], y^-l = yn + h Ign+l - -Aign+l ~ — Ajg^i Asg^-l • • • ], where J /n+l = f\Xv+l, Jn+l, tv+l), The right members of (4) are known and therefore Xn+i and y,^i are determined. It will be recalled that/n+j and g,^i were computed from extrapolated values of x„+i and y^i, and hence are subject to some error. They should now be re- computed with the values of x„+i and yn+i furnished by (4) . Then more nearly correct values of the entire right members of (4) are at hand and the values of x„+i and y^^i should be corrected if necessary. If the interval h is small it will not generally be necessary to correct x^i and y„+i. But if they require correc- tions, then new values of /,j+i and gn+i should be computed. In practice it is advisable to take the interval h so small that one correction to /n+i and g,^! is sufl&cient. After x„+i and y,^i have been obtained, values of x and y at tn+2 can be found in precisely the same manner, and the process can be continued to t = t^,, tn+i, .... If the higher differences become large and irregular it is advisable to interpolate values at the mid-intervals of the last two steps and to continue with an interval half as great. On the other hand, if the higher differences become very small it is advisable to proceed with an interval twice as great as that used in the earlier part of the computation. The foregoing, expressed in words, seems rather complicated. As a matter of fact, it goes very simply in practice, as will be shown in section 10.9. 234 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS 10.8 The Start of the Construction of the Solution. Suppose the differential equations are again with the initial conditions x = a,y = h a,tt = o. Only the initial values of x and y are known. But it foUows from (i) that the rates of change of x and y dXt = o zxtf{a, h, o) andg [a, b , o) respectively. Consequently, first approximations to values of x and y a,t t = h = h are J xfil = a + hfia, b, o), ^' \y,m = b + hg{a,b,o). Now it follows from (i) that the rates of change of x and y &tx = Xi, y = 311, t = h are approximately /(xi'^', yfi\ k) and g{xi^^\ y/^', /i). These rates will be different from those at the beginning, and the average rates of change for the first interval will be nearly the average of the rates at the beginning and at the end of the interval. Therefore closer approximations than those given in (2) to the values of x and y a,tt = U are 3. / ^i^^' = a + hhl f{a, b, o) + /(*:«, yi'^, 0], \ yfi) = b + ^h [g(a, b, o) + g{xfi\ yP-\ /.)]. The process could be repeated on the first interval, but it is not advisable when the interval is taken as short as it should be. The rates of change at the beginning of the second interval are approximately /(xi*^', yi^^', ii) and g(xi<^', yi^^', /i) respectively. Consequently, first approxima- tions to the values of x and y a,tt = h, where U — ti = h, are 4. / x^'i) = xfi^ + hfixfi\ yP\ k), \yP = y,^^) + hg{xP,yP\k). With these values of x and y approximate values of /2 and g^ are computed. Since /o, go', /i, gi '^re known, it follows that A1/2, A1J2; A2/2, and A2g2 are also known. Hence equations (4) of 10.7, for » + i = 2, can be used, with the exception of the last terms in the right members, for the computation of Xi and y2. At this stage of work Xo = o, yo = b; Xi, yi; Xi, y^ are known, the first pair exactly and the last two pairs with considerable approximation. After f^ and gi have been computed, Xi and yi can be corrected by 10.31 for « = i. Then ap- proximate values of Xz and ys can be extrapolated by the method explained in the preceding section, after which approximate values of f, and gs can be com- puted. With these values and the corresponding difference functions, x^ and y2 can be corrected by using 10.31. Then after correcting all the corresponding differences of all the functions, the solution is fully started and proceeds by the method given in the preceding section. 10.9 Numerical Illustration. In this section a numerical problem will be treated which will illustrate both the steps which must be taken and also the method of NUMERICAL SOLUTION OF DIFFERENTIAL EQUATIONS 235 arranging the work. A convenient arrangement of the computation which pre- serves a complete record of all the numerical work is very important. Suppose the differential equation is = -(l + K^)x + 2kV, I. df dx X = o, - = I at i = o. The problem of the motion of a simple pendulum takes this form when expressed in suitable variables. This problem is chosen here because it has an actual physi- cal interpretation, because it can be integrated otherwise so as to express t in terms of x, and because it will illustrate sufficiently the processes which have been explained. Equation (i) will first be integrated so as to express t in terms of x. dx On multiplying both sides of (i) by 2 — and integrating, it is found that the integral which satisfies the initial conditions is (^)'= (i - x^) (i - «V). On separating the variables this equation gives ■ _ p d^ ^' " Jo V(l - X^) (l - kV) ■ Suppose K^ < I and that the upper limit x does not exceed unity. Then 4- / = I -I- - kV + I K-^x' + 4 /c^x" -1- . . . . Vi - K^x^ 2 8 16 where the right member is a converging series. On substituting (4) into (3) and integrating, it is found 'that 5. i = sin 1 X -I- \\_-xVi - x' + sin-i x^ + |[-«Vi - x^ - lx{i - x'^f- -I- IxVi - x^ -\- f sin-i «]«*-!- ]. When X = 1 this integral becomes T — — 2 -ij;'''+(^J"-(^j«'+ Equation (5) gives t for any value of x between -i and -t-i. But the problem is to determine x in terms of t. Of course, if a table is constructed giving t for many values of x, it may be used inversely to obtain the value of x corresponding to any value of t. The labor involved is very great. When k^ is given numerically it is simpler to compute the integral (3) by the method of 10.1 or 10.3. In mathematical terms, t is an elliptical integral of x of the first kind, and the inverse function, that is, x as a function of /, is the sine-ampUtude function, which has the real period 4T. 236 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS Suppose K^ = - and let y = -t-- Then equation (i) is equivalent to the two equations which are of the form 10.60 (i), where (f = y, = -^ x + x', and X = o, y = 1 at / = o. The first step is to determine the interval which is to be used in the start of the solution. No general rule can be given. The larger /o and go the smaller must the interval be taken. A fairly good rule is in general to take h so small that hfo and hgo shall not be greater than 1000 times the permissible error in the results. In the present instance we may take h = o.i. First approximations to x and ya.tt = o.i are found from the initial conditions and equations 10.8 (2) to be XiW = Q -\ I = o.iooo, ■ 10 yfi^ = I H o = 1. 0000. 10 It follows from (8) and these values of Xi and yi that r/(«iW,yi'i),A) = i.oooo, 1 10. ' < [ gixi^^\ yfi\ h) = -0.1490. Hence the more nearly correct values of Xi and yi, which are given by 10.8 (3), are Xi® = o H [1.0000 + 1.0000I = o.iooo, y/^' = I H [0.0000 — 0.1490] = 0.9925. Since in this particular problem x = fy it, it is not necessary to compute both / and g by the exact process explained in section 10.8, for after y has been determined x is given by the integral. It follows from (7), (8), (10), and (ii) that a first approximation to the value of y at i = ^2 = 0.2 is 12. y2''^ = .0025 .1490 = .9776. With the values of y at o, .1, .2 given by the initial conditions and in equations (9) and (12), the first trial y-table is constructed as follows: NUMERICAL SOLUTION OF DIFFERENTIAL EQUATIONS First Trial y-Tahle 237 t y A,y A2)' . I . 2 I . 0000 ■9925 .9776 -.0075 -.0149 -.0074 Since y = f it now follows from the first equations of (11) and 10.7 (4) for n = i that an approximate value of X2 is I 00 4 10 13 X2 (1) = . 1 I .9776 + - .0149 H .0074 2 12 .1986. With this value of Xi it is found from the second of (8) that gi = .2901. Then the first trial g-table constructed from the values of g at / = o, o.i, 0.2, is: First Trial g-Table t S Aig A^g .0000 .1 - . 1490 - . 1490 . 2 — . 2901 — . 1411 + .0079 Then the second equation of 10.7 (4) gives for w = i the more nearly correct value of y^, 14. y^ = .9925 + — I I .2901 H .1411 — — .0079 ^12 12 ] = ■ 9705- This value of yz should replace the last entry in the first trial y-tahle. When this is done it is found that Ai^j = -.0220, A^yi = -.0145. Then the first equa- tion of 10.7 (4) gives 15. a;2 = .1000 + — .9705 + - .0220 + — .0145 = .1983. The computation is now well started although Xi, yi, Xi, and y, are still subject to slight errors. The values of Xi and yi can be corrected by applying 10.31 for n = 1. It is necessary first to compute a more nearly correct value of g. by using the value of x^ given in (15). The result is g^ = -.2896, A,g2 = -.1406, A2g2 = +.0084. Then the second equation of 10.7 (4) gives 16. y2 = .9925 + — + - .1406 - — .0084 2 12 •9705, agreeing with (14). This value of 0-2 is therefore essentially correct. An applica- tion of 10.31 then gives 17- ^ Xi = .0000 -I I .9705 -I- - .0220 10 1 2 -^.0X4S] = ■0907. 238 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS after which it is found that gi = '-.1486, Aigi = -.i486. Now the first trial y-table can be corrected by using the value of ^2 given in (14). The result is: Second Trial y-Tahl e t y A,y Aay 1. 0000 .1 ■9925 -.0075 .2 ■9705 — .0220 -.0145 In order to correct Xj and y^ by the same method, which is the most convenient one to follow, it is necessary fii;st to obtain approximate values of gz and ys. The trial g-table can be corrected by computing g with the values of x given by (17) and (15). Then the line for g, can be extrapolated. The results are: Second Trial g-Table • t g A.g ^2g .0000 .1 - . i486 - . i486 .2 - . 2896 — .1410 + . 0076 •3 -.4230 -•1334 +.0076 Then the second equation of 10.7 (4) gives for « = 2, 18. ys = .9705 + ■ .4230 + - .1334 - — .0076 2 12 = .9348. When this is added to the second trial y-table, it is found that 19. ys = -9348, Aiya = -.0357, A^y, = -.0137, Asys = +.0008. Now «2 and y^ can be corrected by applying 10.31 to these numbers and those in the last line of the second trial g-table. The results are X2 = .0997 H 10 .9348 + 1 .0357 - -^ .0137 + i .0008 24 .1980, y, = .9925 + 10 .4230 + -.1334 + ^.0076] = . 9705- The preliminary work is finished and x and y have been determined for t = o, .1, and .2 with an error of probably not more than one unit in the last place. As the process is read over it may seem somewhat complicated, but this is largely because on the printed page preliminary values of the unknown quantities can not be erased and replaced by more nearly correct ones. As a matter of fact, the NUMERICAL SOLUTION OF DLFFERENTIAL EQUATIONS 239 first steps are very simple and can be carried out in practice in a few minutes if the chosen time-interval is not too great. The problem now reduces to simple routine. There are an x-table, a y-table (which in this problem serves also as an /-table), a g-table, and a schedule for computing g. It is advisable to use large sheets so that all the computations except the schedule for computing g can be kept side by side on the same sheet. The process consists of six steps: (i) Extrapolate a value of gn+i and its differences in the g-table; (2) compute y„^-i by the second equation of 10.7 (4); (3) enter the result in the y-table and write down the differences ; (4) use these results to compute x^+i by the first equation of 10.7 (4); (5) with this value of Xn+i compute g„+i by the g-computation schedule; and (6) correct the extrapolated value of gn+i in the g-table. Usually the correction to g^i will not be great enough to require a sensible correction to y„+i. But if a correction is required, it should, of course, be made. It follows from the integration formulas 10.7 (4) and the way that the difference functions are formed that an error e in g^i produces the error |Ae in yn+i, and the corresponding error in Xn+i is — h^e. It is never advisable to use so large 04 a value of h that the error in x„+i is appreciable. On the other hand, if the differ- ences in the g-table and the y-table become so sinall that the second differences are insensible the interval may be doubled. The following tables show the results of the computations in this problem reduced from five to four places. Final a;-Table t X Aia; A2X A3X 0000 .1 0997 0997 . . .2 1980 0983 — .0014 •3 2934 0954 — .0029 -.0015 •4 3847 0913 — . 0041 — .0012 ■S 4708 0861 — .0052 — .0011 .6 5508 0800 — .0061 — .0009 •7 6243 0735 — .0065 - . 0004 .8 6909 0666 — . 0069 — .0004 •9 7505 0596 — .0070 — . 0001 I.O 8030 0525 — .0071 — .0001 I.I 8486 0456 — .0069 -f .0002 1.2 8877 0391 -.0065 + . 0004 1-3 9205 0328 -.0063 + . 0002 1-4 9472 0267 — .0061 + . 0002 1-5 9682 0210 -.0057 + . 0004 1.6 9837 015s -■0055 + . 0002 1-7 9940 0103 — .0052 -t- . 0003 1.8 9993 0053 -.0050 -f-.0002 1.9 9995 0002 -.0051 — .0001 240 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS Final y-Table t y Aiy A^y ^zy I 0000 .1 9925 -.0075 .2 9705 — .0220 -.0145 •3 9352 -■0353 -0133 + .0012 •4 8882 -.0470 -.0117 + .0016 •5 8320 — -0562 — .0092 + .0025 .6 7687 -•0633 — .0071 + .0019 •7 7009 -.0678 -.0045 + .0016 .8 6308 — .0701 -.0023 + .0022 •9 5602 — .0706 — . 0005 + •0008 I.O 4906 — .0696 + .0010 + .0015 I.I 4231 -.0675 + .0021 + .0OII 1.2 3.S84 -.0647 + .0028 + . 0007 1-3 2968 — .0616 + .0031 + .0003 1-4 2382 -.0586 + .0030 — .0001 i-S 1824 -■0558 + .0028 — .0002 1.6 1290 -■0534 + .0024 — . 0004 1-7 077s -•0515 + .0019 — . 0005 1.8 0271 • -.0504 + .001 1 — . 0008 1.9 0230 — -osoi + .0003 -.0008 Final e-Schedule t .1 .2 •3 •4 ■5 .6 •7 .8 •9 log* 8.9989 9.2967 94675 9-5851 g.6728 9.7410 9-7954 9.8394 9-8753 log*^ 6.9967 7.8901 8.4025 8.7553 9.0184 9.2230 9.3862 9-S182 9-6259 IX .2992 •5941 .8802 1-1541 1.4124 1.6524 1.8729 2.0727 2.2515 -ix -.1496 -.2970 -.4401 --5770 -.7062 -.8262 -•9365 -1.0364 -1. 1257 x^ .0010 .0077 .0252 .0569 .1044 .1671 ■2434 .3298 ■4227 g -.i486 -.2893 -.4149 -.5201 -.6018 -.6591 -.6931 - .7066 - -7030 NUMERICAL SOLUTION OF DIFFERENTIAL EQUATIONS 241 Final g-Table t ? A,g A.^ As? .0000 .1 — i486 - . i486 .2 — 2893 - • 1407 + .0079 ■3 4149 -.1256 + .0151 + .0072 •4 5201 -.1052 + .0204 + •0053 •5 6018 -.0817 + •0235 + .0031 .6 6591 -•0573 + .0244 + . 0009 •7 6931 -.0340 + •0233 — . 001 I .8 7066 -•0135 + .0205 — .0028 ■9 7030 + ■ 0036 + .0171 -.0034 I.O — 6867 + .0163 + .0127 -.0044 I.I — 6618 + .0249 + .0086 — .0041 1.2 — 6320 + .0298 + .0049 -.0037 1-3 6008 + .0312 + .0014 -■003s 1.4 5710 + .0298 -.0014 — .0028 i-S 5447. + .0263 -•003s — .0021 1.6 5236 + .0211 — .0052 — .0017 1-7 5088 + .0148 -.0063 — .0011 1.8 501 1 + .0077 — .0071 — . 0008 1.9 5008 + .0003 -.0074 -.0003 Final g-Schedule — Continued 1.0 I.I 1.2 1-3 1-4 1-5 1.6 1-7 1.8 1-9 9.9047 9.9287 9.9483 9.9640 9.9764 9.9860 9.9929 9-9974 9.9997 9-9998 9-7141 9.7861 9.8449 9.8920 9.9292 9.9580 9.9787 9.9922 9-9991 9.9994 2.4090 2-S4S8 2.6631 2.761S 2.8416 2.9046 2.9511 2.9820 2,9979 2.9985 -1.204s -1.2729 -I-33I6' -1.3807 -1.4208 -1-4523 -1-4756 -1.4910 -1.4989 -1.4992 ■SI78 .6iii .6996 ■7799 .8498 .9076 •9520 .9822 .9978 .9984 - .6867 - .6618 - .6320 - .6008 - -5710 - -5447 - -5236 - .5088 - .5011 - .5008 242 MATHEMATICAL FORMULA AND ELLIPTICAL FUNCTIONS As has been remarked, large sheets should be used so that the x, y, and g-tables can be put side by side on one sheet. Then the /-column need be written but once for these three tables. The g-schedule, which is of a different type, should be on a separate sheet. The differential equation (i) has an integral which becomes for i^ = - 2 , dx and -rr^y. dt fi 4-'^v% --v.i = 21. y^-^r-x' X 2 4 and which may be used to check the computation because it must be satisfied at every step. It is found on trial that (21) is satisfied to within one unit in the fourth place by the results given in the foregoing tables for every value of t. The value of t for which x = 1 and 3/ = o is given by (6). When k^ = ^ it is found that T = 1.8541. It is found from the final x-table by interpolation based on first and second differences that x rises to its maximum unity for almost exactly this value of t; and, similarly, that y vanishes for this value of t. XI ELLIPTIC FUNCTIONS By Sir George Greenhill, F. R. S. INTRODUCTION TO THE TABLES OF ELLIPTIC FUNCTIONS By Sir George Greenhill /dx PM 4- N -v/X —7=' and more generally, I — ^_ dx V ^ J P + Q VX ' where M, N, P, Q are rational algebraical functions of x, can always be expressed by the elementary functions of analysis, the algebraical, circular, logarithmic or hyperbolic, so long as the degree of X does not exceed the second. But when X is of the third or fourth degree, new functions are required, called elliptic functions, because encountered first in the attempt at the rectification of an ellipse by means of an integral. To express an elliptic integral numerically, when required in an actual question of geometry, mechanics, or physics and electricity, the integral must be normaUsed to a standard form invented by Legendre before the Tables can be employed; and these Tables of the Elliptic Functions have been calculated as an extension of the usual tables of the logarithmic and circular functions of trigonometry. The reduction to a standard form of any assigned elliptic integral that arises is carried out in the procedure described in detail in a treatise on the elliptic functions. 11.1. Legendre's Standard Elliptic Integral of the First Kind (E. I. I) is r d(j) ^ f" dx ^^ " Jo Vi - K^ sin2 = sm am u abbreviated by Gudermann to, a; = sn M cos - A

„ ■ I I ddnu , -r-^ = - K^ sin d) cos d) or — -, — = - «■= sn m en m du du IT 11.11. The complete integral over the quadrant, o<4><—, o Jo A' = I = = k'. • k' is the comodulus to k, . K^ + k'' = I, and the coperiod, dcj) K', is, K' IT X v(] I — k'^ sin^ )■ 11.12. sn^ u + cn^ M = I - \ I , cn^ u + K^ sn^ M = I dn^ u — K^ cn^u = /c'^. sn o = o, en o = dn, ' 0=1. sn iiT = I, cnK = o, dm. K = k'. 11.13. Legendre has calculated for every degree of 6, the modular angle, K = sin 6 , the value of Fcj) for every degree in the quadrant of the amplitude (p, and tabulated them in his Table IX, Fonctions elhptiques, t. II, 90 X 90 = 8100 entries. But in this new arrangement of the Table, we take u = Fcj) as the independent variable of equal steps, and divide it into 90 degrees of a quadrant K, putting r° u = eK = — o K, r° = 9o°e. 90 As in the ordinary trigonometrical tables, the degrees of r run down the left of the page from 0° to 45°, and rise up again on the right from 45° to 90°. Then columns II, III, X, XI are the equivalent of Legendre's Table of F^ and (^, but rearranged so that F(/> proceeds by equal increments 1° in f°, and the incre- ments in , of 4>. Jacobi adopted the idea in his Fundamenta nova, and employs the elliptic functions sin 4> = sin am u, cos (/> = cos am u, Acj) = A am u, single-valued, uniform, periodic functions of the argument u, with (quarter) period K, as 4> grows from o to ^tt. Gudermann abbreviated this notation to the one employed usually today. 11.2. The E. I. I is encountered in its simplest form, not as the elliptic arc, but in the expression of the time in the pendulum motion of finite oscillation, unrestricted to the small invisible motion of elementary treatment. The compound pendulum, as of a clock, is replaced by its two equivalent particles, one at in the centre of suspension, and the other at the centre of oscillation, P; the particles are adjusted so as to have the same total weight as the pendulum, the same centre of gravity at G, and the same moment of inertia about G or 0; the two particles, LE rigidly connected, are then the kinetic equiva- lent of the compound pendulum and move in the same way in the same field of force (Maxwell, Matter and Motion, CXXI). Putting OP = I, called the simple equivalent pendulum length, and P starting from rest at -B, in Figure i, the parti- cle P will move in the circular arc BA 5'as if sliding down a smooth curve ; and P will acquire the same velocity as if it fell vertically KP = ND; this is all the djmamical theory required. (velocity of PY = 2g-KP, (velocity of Ny= 2g-ND- sinMOP = 2g-ND NP^_g2 OP^ ^2 ND-NA-NE, and with AD = k, AN = y, ND = h-y, AE= 2I, NE= 2I- y, I where F is a cubic in y. Then t is given by an elliptic integral of the form Fig. i I — ^- This integral is normahsed to Legendre's standard form of his E. I. I by putting y = h sin^ cj), making AOQ = (j), h - y = h cos^ , 2I - y = 2I {1 - K^ sin^ (j)), „ h AD . o .i^D K^ = — , = -r^ = sm^ AEB. 2I AE K is called the modulus, AEB the modular angle which Legendre denoted by 6; -\/(i - K^ sin^ 4>) he denoted by A0. 248 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS With g = Irfi, and reckoning the time t from A, this makes in Legendre's notation. Then the angle 4> is called the amplitude of nt, to be denoted am nt, the particle P starting up from A at time / = o; and with u = nt, AP AQ , AN ''''' = AB = AD '"'''^AD DQ , PK cn« = ^ cn^u = j^ dnM = ^ dn^« = ^ Velocity of P = n-AB-ca. u = ^BP-PB', with an oscillation beat of T seconds in M = eK, e = 2t/T. 11.21. The numerical values of sn, en, dn, tn («, k) are taken from a table to modulus K = sin (modular angle, 6) by means of the functions Dr, Ar, Br, Cr, in columns V, VI, VII, VIII, by the quotients, \/7/ sneK = j^ cneK = jr dneK C Vk' tn eif = 4 jD o o r = 90 e M = eK. These D, A, B, C are the Theta Functions of Jacobi, normalised, defined by D(.r) = Q-^, ^('^=HK' B{r) = ^(90° - r) C{r) = D{go° - r). They were calculated from the Fourier series of angles proceeding by multiples of r°, and powers of q as coefiicients, defined by Qu = I — 2q cos 2r + 2(^ cos \r — 29^ cos dr -\- . . . . Hu = 2q^ sin f — 2q^ sin T,r + 2^^ sin 5r — . . . . 11.3. The Elliptic Integral of the Second Kind (E. I. II) arose first historically in the rectification of the ellipse, hence the name. With BOP = 4> m Figure 2, the minor eccentric angle of P, and 5 the arc BP from 5 to P at x = a sin 0, y = b cos 4>, INTRODUCTION TO THE TABLES OF ELLIPTIC FUNCTIONS 249 -rr = Vffl^ cos^ (j) + b^ sin^ (j) = o,^{4>t <<)> to the modulus k, the eccentricity of the ellipse, o A(^ -(/(^ is denoted by £(^ in Legendre's notation of his standard E. I. II; it is tabulated in his Table IX alongside of F(f> for every degree of the modular angle 6, and to every degree in the quadrant of the amplitude cj). But it is not possible to make the inversion and express ^ as a single-valued function of E4>. Fig. 2 11.31. The E. I. n, E(f>, arises also in the expression of the time, t, in the oscil- lation of a particle, P, on the arc of a parabola, as Fcj) was required on the arc of a circle. Starting from B along the parabola BAB', Figure 3, and with AO = h, OB = b, BOQ = cf), AN = y = h cos^ ^,NP = x = b cos ^ and with OS = 2h = b tan a, OA' = SB = b sec a, the parabola cutting the horizontal at B at an angle a, the modular angle, BRA'B' is a semi-ellipse, with focus at S, and eccen- tricity K = sin a. (Velocity of Pf = -(f = Q}^ cos^ ) ( -3-) Fig. 3 ._,j 2gy = 2gh cos' (^ = a'(i — sin' a sin' 4>) cos' 4> \ = F' cos' <^, if V denotes the velocity of P at ^, and OA' = a. Then with 5 the elliptic arc BR, y^-'^^^ and so the point R moves round the ellipse with constant velocity V, and ac- companies the point P on the same vertical, oscillating on the parabola from B to B'. In the analogous case of the circular pendulum, the time t would be given by the arc of an Elastica, in Kirchhoff's Kinetic Analogue, and this can be placed as a bow on Figure 1, with the cord along AE and vertex at B. Legendre has shown also how in the oscillation of R on the semi-eUipse BRB' in a gravity field the time i is expressible by elliptic integrals, two of the first and two of the second kind, to complementary modulus^Fonctions. elUptiques„, I,, p. 183). 2SO MATHEMATICAL FOEMULyE AND ELLIPTIC FUNCTIONS 11.32. In these tables, £<^ is replaced by the columns IV, IX, of E{r) and G{r) = E{go — r), defined, in Jacobi's notation, by E(r) = zneK = E±-eE cit/--;'/-^^ G{r) = znii-e)K, r = g'oe. E B.w, ^, ru/ (i-^) This is the periodic part of E(j) after the secular term eE = r=u has been set A. aside, E denoting the complete E. I. II, E = E\'K = P''^4>■d4>. ' The function zn u, or Zu in Jacobi's notation, or E{r) in our notation, is calculated from the series, i^sinhwTT— ;f^ This completes the explanation of the twelve columns of the tables. 11.4. The Double Periodicity of the Elliptic Functions. This can be visualised in pendulum motion if gravity is supposed reversed suddenly at B (Figure i) the end of a swing; as if by the addition of a weight to bring the centre of gravity above 0, or by the movement of a weight, as in the metronome. The point P then oscillates on the arc BEB' , and beats the elliptic function to the complementary modulus k' , as if in imaginary time, to imaginary argument nti = fK'i: and it reaches P' on AX produced, where tan AEP' = tan AEB-cn {nt'i, k), or tan EAP' = tan EAB-ca {nt' , k'); or with nt' = w, DR' = DB- en {iv, k'), DR = DB-cn (v, k'), with DR-DR' = DB\ EP' crossing DB in R'. en {iv, k) = en {v, k') i sn {v, k'] en (zj, k') dn (zj, k') , . , tsn(v, k) . , ,, sn ( w, K) = 7 -^ = J tn \v, K) en {v, K) dn {iv, k) = en {v, k') sn {K' - v, k') where K' denotes the complementary (quarter) period to comodulus k'. If m, m' are any integers, positive or negative, including o, sn {u + /^mK + zm'iK') = snu en [u + 4mK + 2m' {K + iK')^ = en m dn {u + 2mK + ^m'iK') = dn w 11.41. The Addition Theorem of the Elliptic Functions. , ^ snucnvdnv±snvcnuia.u sn [u ± V) = I — K^ sn^ M sn^ V en (v ±u) = ^'^ ^ ^" ^ ^ sn w dn M sn ;) dn 7) * I — K^ sn^ M sn^ V dn (v ±u) = ^'^ ^ ^'^ ^ ^ ''^^ sn t< en M sn t) en ;> I — K^ sn^ u sn^ V INTRODUCTION TO THE TABLES OF ELLIPTIC FUNCTIONS 25 1 11.42. Coamplitude Formulas, with n = ± isT, sn (K - u) = J = sn (iiC + u) en {K - u) = —5 CTi(K +u) = 5 dnw \ I J ^^^ k' dx).(K - u) = T — = dn(K + u) an M K tn M \ ■ / k' tn M 11.43. Legendre's Addition Formula for his E. I. II, E4> = fAcj) ■ dcj) = ydn^ u-du, (^ = ydn u-du = am m. E(t> + E\}/ - E(T = K^ sin a / \ Ea - Ed - 2E\f/ = 1 ■ 2I ■ 2 / — -, = am (» - m) or, in Jacobi's notation, , , , , — 2 K^ sn D en 7J dn D sn^ M zn{v + u) + znvv — u) — 2 znv = 5 — 5 5 I — K^ sn-" u sn^ V 11.5. The Elliptic Integral of the Third Kind (E. I. Ill) is given by the next integration with respect to u, and introduces Jacobi's Theta Function, Qu, defined by, d log Qu „ f = Zu = znu du e p- = exp. fzn u-du. Integrating then with respect to u, ^ , ^ ■, rs , \ C— 2 k'^ sn D cn D dn D sn' u . loge (z- + m) - loge (. -u)-2uzr,v = J ^-K'sn^usn^v '^"' and this integral is Jacobi's standard form of the E. I. Ill, and is denoted by - 2 n (m, n) ; thus, -TT , . TK^sn wcni) dni; sn^M , ,, G (d — m) n (m, v) = / 5 — 5 5 du = uza.v + ^\og q , , — r • ^ ' ' J I - k' sn' M sn' D (3 (i) + m) Jacobi's Eta Function, Hv, is defined by Hi) .- e^=VKsn., and then d log Hi) en D dn D , , , , — 7 — = h zn D, denoted by zs v: dv sxiv 252 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS SO that J a I en J) dn D , du , = M V 11 (m, V) snj) 1 , (d - m) 2 ® 9 (j) H- m) This gives Legendre's standard E. I. Ill, M d(t> L I + n sin^ (j) Acf) where we put n = — k^sv?v = — k^ sin^ \p, the normaUsing muhipUer, M. The E. I. Ill arises in the dynamics of the gyroscope, top, spherical pendulum, and in Poinsot's herpolhode. It can be visualized in the soUd angle of a slant cone, or in the perimeter of the reciprocal cone, a sphero-conic, or in the mag- netic potential of the circular base. 11.61. We arrive here at the definitions of the functions in the tables. Jacobi's Qu and Hm -are normahsed by the divisors Go and UK, and with r = goe, D(r) denotes -^^, A{r) denotes -^5^ whUe B{r) = ^(90 - r), C^r) = D{go - r), and B{p) = A{qo) = D{p) = C{go) = I, C(o) = D(9o) = ^• Then in the former definitions, A(r) Ajgo) ^, 7v^ = 7v — r sn M = Vk sn eK D{r) D{go) Bir) B(o) TvT = Tu^ en M = en eK D{r) £»(o) CW _ C(o) _ dneg Dir)-D{o) ^''"- V7" Then, with u = eK, v = fK, r = gee, ^ = 90/, f \ TT ri^ , 1 1 Q if - e) K («,.)-«s:.n/K+-l06gii-p^ znfK = E is), zn (i -/) Z = £ (90 - 5) = G (s). INTRODUCTION TO THE TABLES OF ELLIPTIC FUNCTIONS 253 The Jacobian multiplication relations of his theta functions can then be rewritten D{r + s)D{r - s) = DhDh - tan^ OAhA^s, A{r + s)A{r - s) = AhD^s - D^Ah, B{r + s)B{r - s) = BhBh - AhAh. But unfortunately for the physical appHcations the number 5 proves usually to be imaginary or complex, and Jacobi's expression is useless; Legendre calls this the circular form of the E. I. Ill, the logarithmic or hyperbohc form corre- sponding to real 5. However, the complete E. I. Ill between the limits o «^ Si>S2>S3> - a, and normalised to a standard form of zero degree these differential elements are I II V^i - Si ds VS s — a ds Vsi - S3 Vs III i^^4 s- a Vs S denoting the value of 5 when s = s>Si ^ r -Vsi -_.3 ds ^ ^^_, JJ[^ ^ cn-V^^ - dn- V Js ^ys y s - S3 y s - S3 ». "V^i - ^3 ds VS 'Vsi - S3 ds (, _ e)K = rvV:Liiif = sn-V^^^ = cn-v/^^ = dn-^J '' ' ''■' ^ '' Jsx -y/S y S - S2 y S - S2 V 51-53-5.-52 indicating the substitutions, = sin^ cb = sn^ eK, = sin^ \j/ = sn^ (i — e)K. S - S3 5-52 In the next interval 5 is negative, and the comodulus k' is required. 5i>5>52 ^^, ^ r Wsi-j^ds ^ ^„_i^/KZZ = cn-^jm^ = dn-^v/^^H^ J V- S '^51-52 V 5i - 52 y Si- S3 f t\ T^t r V^l - 53 ds . /51 - 53 • 5 - 52 _. ■ /52 - — 53 -51 — 5 Vff- dn-iV 5-53 S is positive again in the next interval, and the modulus is k. S2>S>S3 t \ s- P'V^i - 53 ds Si - S3-S2- s . (i - e)K = I ;= = sn ly = en- V J' VS V 52- 53-51- 5 V , Si — 52-5 ■y/S ' 52 - 53-51 - 5 V 52 - 58-5i - 5 5i — 5 ,K = ^^AL^iii£ = sn- V'-^^ = en- V'-^^ = dn" V" Jh \/S ''52-53 V 52 - 53 V 5i \/S '52-53 V 52 - 53 V 5i - 53 indicating the substitutions, '-^^^ = A'xl/ = dn2 (: - e)K, '-^^ = sin^ cj) = sn^ eK 5i — 5 52 — 53 5 = 52 sin^ (t> + S3 cos' 4>. INTRODUCTION TO THE TABLES OF ELLIPTIC FUNCTIONS 255 5 is negative again in the last interval, and the modulus /c'. Js V- S y Si- s \ Si- s y si- Sri , ^ r y/s.-s^ds ^ ^^.Jsr-^ ^ cn-l/^EI = dn-v/^ J-co v^r^ y si-s y si- s y si- — s s 11.8. For the notation of the E. I. II and the various reductions, take the treatment given in the Trans. Am. Math. Soc, 1907, vol. 8, p. 450. The Jacobian Zeta Function and the Er, Gr of the Tables, are defined by the standard integral Psji-s^ds^ j ^^.^^^£^^ rdn'{eK)-d{eK)=EsimeK = eH + zneK, Jh y/si — Si \/S I/O Jo or, r.as^ d^ ^ r/^^, (/i^O -^(/i^O = E am/Z' = fW + zn JK', Jsi Vsi - 53 V - 2/ ^° Isz Vsi - 53 v - s where zn is Jacobi's Zeta Function, and H, H' the complete E. I. II to modulus K, k', defined by, H = fjAicj}, k) dcj} = J;dn2 {eK)-d{eK) H' = JJa{4>, k') dck = jMn^ UK')-d{fK'). The function zn u is derived by logarithmic differentiation of 9m, d log 0M . , zn M = — f , or concisely, Qu = exp. J zn u-du, and a function zs u is derived similarly from d log Hu ZSM = 7 du d log Gm d log sn m du du en M dn M = znu -\ SUM For the incomplete E. I. II in the regions, 00 >5>Si>52>5>53 and - Si- S3 s - S3 sn* eK = or > S — S3 Si — S3 256 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS — ^=^ — = = I — as = - ii - e)H + zs eK 5 V^l -S3VS J^ S- S3 V^ f '-'' A = K^ f^^^ ^^^^^' ds=-{i-e){H- k'^K) + 2S cK f '-'' A = f'l^^ y^EZ]^ ds={i- e){K -H)+ zseK the integrals being °o at the upper limit, 5 =" «= , or at the lower limit, s = S3 where e = o and zs eK = «> . So also, 's - 52 Vsi - S3 , _ p' '1 sy- s ds eff + zn eK ,51 5 - 53 -sjs 'J'^ ' V^i - 53 V^ (i - e)H -zn eZ / 5-5i V^i - S3 , _ r S2- s _ds_ ^e(H - /c'2 s - 53 a/5 '^ ~ "^ Vsi - S3 VS (i - e)(H - k'^K) + zn eK VS "' -' V^T^^ VS (i - e)(il - k'^K) - zn eisT - ^3 V^i - S3 , r s - S3 ds e{K - H) - zn eK J s - S3 ^ys «^ • VS J Vsi - S3 VS (i - e^^K -H)+zn eK Similarly, for the variable a in the regions Si>(r>S2>S3>(T> - 00 S negative, and 9 xzr/ Si — (T Si — S3 svrjK = or 5i — 52 51—0" Ph. o sx-Sx>S2>S3, put ^ - ^3 = ^^ (T - 53 = (5, - 53) sn^ ., K^ = ^-^^ bil H Si — ^3 ^ Si — S3 , ■\/si — Sgds , (7 = — 5 — (i — k' sn'' u sn^ v), = = du. VS = V^i — S3 (52 — S3} sn i; en n dn 11, making • / 4VS ds /"k^ sn w cni) dnt) sn^M , „/ v / 7= = / i — I 5 du = n(w, V). J s - a ^s J I - K^ sn^ M sn^ D But in the region, Cr>5i>52>5>58, , V , SX-S3 \ /^ , ,,cnDdnu s - 53 = (52 - 53) sn^ u, a - S3 = — - — , - v2/ = (5i - 53)8 > sn^ V 2 sn** n 5, ^„ cr — 5 = — 5 — (i — K^ sn^ u sn^ v), sn^ 2) making, en w dn w , /\\/Y. ds C sn?) cnt)dnz) tr — 5 V^ J I — K^ sn^ M sn'' d sn » In a dynamical appUcation the sequence is usually s>S]_>a>S'i.>s>S3 or S>Sx>Si>S>S3>(J, making S negative, and the E. I. Ill is then called circular; the parameter v is then imaginary, and the expression by the Theta function is illusory. The complete E. I. Ill, however, was shown by Legendre to be tractable and falls into four classes, lettered (V) im'), p. 138, {i'), {k'), pp. 133, 134 (Fonc- tions elliptiques, I). 5i>«7>52 s\ — a sn^ fK' cn^ fK' = dnV^' = Si - 52 (T - 52 Si - Si a - S3 5i - 53 2S8 MATHEMATICAL FORMLU^E AND ELLIPTIC FUNCTIONS oo>5>5iX^'"ij^-^ = AifK') = Mi -/) -KznfK' S2>5>53/ ^^^ -^ = BifK')=W+KznfK' ^ + 5 = |ir. S3>0'> - oo Si - 53 sn2 /ii:' = 5i — (T •' 5i - (T 5i - 0- CO >5 >.i r " ^T^^ = CifK') = K zs fK' - i7r(i - /) ^ '-f^ ^ = Z)(/r) = K zs fK' + ^TTf TABLES OF ELLIPTIC FUNCTIONS By Col. R. L. Hippisley 26o ELLIPTIC FUNCTION K = 1.5737921309, K' = 3.831742000, E = 1.5678090740, E' = 1.012663506, r F<^ <^ E(r) D(r) A(r) 0.00000 00000 0° 0' 0.00000 00000 I . 00000 00000 0.00000 00000 I 0.01748 65792 i 0.00006 64649 I .00000 05812 0.01745 23906 2 0.03497 31585 2 0.00013 28485 I . 00000 23240 0.03489 94650 3 0.05245 97377 3 0.00019 90699 I. 00000 52264 0.05233 59088 4 0.06994 63169 4 0.00026 50480 • I. 00000 92847 0.06975 64107 5 0.08743 28962 5 0.00033 07023 I. 0000 I 44942 0.08715 56642 6 0.10491 94754 6 0.00039 59525 1.00002 08483 0.10452 83693 7 0.12240 60546 7 0.00046 07190 1.00002 83393 0.12186 92343 8 0.13989 26338 8 0.00052 49226 I . 00003 69582 0.13917 29770 9 0.15737 92131 9 0.00058 84849 I . 00004 66945 o.r5643 43264 10 0.17486 57923 10 0.00065 13283 1.00005 75362 0.17364 80247 II 0.19235 23716 II 0.00071 33760 I . 00006 94702 0.19080 88283 12 0.20983 89508 12 0.00077 45523 1.00008 24819 0.20791 15101 13 0.22732 55300 13 0.00083 47824 1.00009 65555 0.22495 08603 14 0.24481 21092 14 2 0.00089 39929 I. 0001 I 16738 0.24192 16887 15 0.26229 86885 15 2 0.09095 2 II 14 I. 00012 78184 0.25881 88257 16 0.27978 52677 16 2 o.ooioo 90670 I. 00014 49696 0.27563 71244 17 0.29727 18469 17 2 0.00106 47903 I .00016 31066 0.29237 14618 18 0.31475 84262 18 2 o.ooiii 92132 I .00018 22072 0.30901 67404 19 0.33224 50054 19 2 0. 001 I 7 22694 I .00020 22482 0.32556 78900 20 0.34973 15846 20 2 0.00122 38941 I .00022 32051 0.34201 98690 21 0.36721 81639 21 2 0.00127 40244 1.00024 50525 0.35836 76658 22 0.38470 47431 22 2 0.00132 25992 I .00026 77636 0.37460 63009 23 0.40219 13223 23 2 0.00136 95594 I .00029 13109 0.39073 08277 24 0.41967 79016 24 2 0.00141 48476 I. 0003 I 56657 0.40673 63347 25 0.43716 44808 25 3 0.00145 84087 I .00034 07982 0.42261 79464 26 0.45465 10600 26 3 0.00150 01897 1.00036 66779 0.43837 08251 27 0.47213 76393 27 3 0.00154 01398 1.00039 32731 0.45399 01723 28 0.48962 42185 28 3 0.00157 82103 I .00042 05516 0.46947 12303 29 0.5071 1 07977 29 3 0.00161 43549 I .00044 84801 0.48480 92833 30 0.52459 73770 30 3 0.00164 85297 I .00047 70246 0-49999 96593 31 0.54208 39562 31 3 0.00168 06931 1.00050 61502 0-51503 77311 32 0-55957 05354 32 3 0.00171 08062 I 00053 58215 0.52991 89180 33 0.57705 71 147 33 3 0.00173 88322 I .00056 60024 0.54463 86870 34 0.59454 36939 34 3 0.00176 47373 1.00059 66561 0-55919 25543 35 ,0.61203 02731 35 3 0.00178 84901 1.00062 77451 0.57357 60867 36 0.62951 68524 36 3 0.00181 00617 I .00065 92318 0- 58778 49028 37 0.64700 34316 37 3 0.00182 94261 1.00069 10776 0.60181 46744 38 0.66449 00108 38 3 0.00184 65599 1.00072 32438 0.61566 11280 39 0.68197 65900 39 3 0.00186 14423 1.00075 56912 0.62932 00458 40 0.69946 31693 .40 3 0.00187 40556 1.00078 83803 0.64278 72670 41 0.71694 97485 41 4 0.00188 43845 1.00082 12712 0.65605 86895 42 0.73443 63278 42 4 0.00189 24166 1.00085 43239 0.66913 02706 43 0.75192 29070 43 4 0.00189 81424 1.00088 74981 0.68199 80287 44 0.76940 94862 44 4 0.00190 15552 1.00092 07533 0.69465 80439 45 78689 60655 45 4 0.00190 26510 1.00095 40492 0.70710 64600 90° r F;A ^ G(r) C(r) B(r) Smithsonian Tables TABLE = 5° 9=0.000476569916867, 9 = 0.9990468602, H(K) =0.2955029021 261 B(r) C(r) G(r) ^ Fi^ 90-r I . 00000 00000 I .00190 80984 0.00000 00000 90° 0' 1-57379 21309 90 0.99984 76949 I .00190 75172 0.00006 63384 89 1-55630 55517 89 0.99939 08259 I. 00190 57743 0.00013 -25961 88 I. 5388 I 89724 88 0.99862 95323 I. 00190 28720 0.00019 86928 87 1-52133 23932 87 0.99756 40458 I. 00189 88136 0.00026 45481 86 I -50384 58140 86 0.99619 46912 I. 00 I 89 36042 0.00033 00820 85 1.48635 92347 85 0.99452 18855 I. 00188 72501 0.00039 52149 84 1.46887 26555 84 0.99254 61382 I. 00187 97590 0.00045 98676 83 I. 45138 60763 83 0.99026 80513 I. 00187 11401 0.00052 39616 82 1.43389 94971 82 0.98768 83186 I .00186 14039 0.00058 74190 81 1.41641 29178 81 0.98480 77260 I .00185 05621 0.00065 01626 80 1.39892 63386 80 0.98162 71510 I. 00183 86282 0.00071 21163 79 I. 38143 97593 79 0.97814 75623 I. 00182 56165 0.00077 32046 78 1-36395 31801 78 0.97437 00200 1.00181 15429 0.00083 33534 77 I . 34646 66009 77 0.97029 56747 I. 00 I 79 64246 0.00089 24894 76 2 I .32898 00217 76 0.96592 57675 I .00178 02800 0.00095 05409 75 2 1.31149 34424 75 0.96126 16296 I .00176 31288 o.ooioo 74371 74 2 I .29400 68632 74 0.95630 46817 I .00174 49918 0.00106 31089 73 2 1.27652 02840 73 0.95105 64338 I. 00172 58912 0. 001 I I 74885 72 2 1-25903 37047 72 0.94551 84846 I .00170 58502 0. 001 17 05097 71. 2 I. 24154 71255 71 0.93969 25209 I. 00 I 68 48932 0.00122 2IO81 70 2 I . 22406 05463 70 0.93358 03176 I .00166 30459 0.00127 22208 69 2 1.20657 39670 69 0.92718 37364 I. 00 I 64 03347 0.00132 07868 68 2 I. 18908 73878 68 0.92050 47258 1.00161 67874 0.00136 77470 67 2 I . 17160 08086 67 0.91354 53203 I. 00159 24327 O.OOI4I 30440 66 3 1.15411 42293 66 0.90630 76400 I. 00 I 56 73002 0.00145 66228 65 3 1 . I366i 76501 65 0.89879 38894 I. 00154 14205 0.00149 84301 64 3 1.11914 10709 64 0.89100 63574 1.00151 48252 0.00153 84I5I 63 3 I. 10165 44916 63 0.88294 74161 I. 00148 75467 0.00157 65289 62 3 I. 08416 79124 62 0.87461 95204 I .00145 96182 O.OOI6I 27250 61 3 1.06668 13332 61 0.86602 52071 I. 00143 10738 0.00164 69^92 60 3 I. 04919 47539 60 0.85716 70941 I. 00140 19481 0.00167 91897 59 3 I. 03170 81747 59 0.84804 78798 I .00137 22768 0.00170 93771 58 3 I. 01422 15955 58 0.83867 03419 I. 00 I 34 20959 0.00173 74846 57 3 0.99673 50162 57 0.82903 73370 1.00131 14423 0.00176 34776 56 3 0.97924 84370 56 0.81915 17995 I .00128 03532 0.00178 73244 55 3 0.96176 18578 55 0.80901 67404 I .00124 88666 0.00180 89958 54 3 0.94427 52785 54 0.79863 52473 I .00121 70208 0.00182 84651 53 3 0.92678 86993 53 0.78801 04823 I .00118 48546 0.00184 57085 52 3 0.90930 21201 52 0.77714 56818 I .00115 24072 0.00186 07047 51 3 0.89181 55409 51 0.76604 41556 l.ooiii 97181 0.00187 34353 50 i 0.87432 89616 50 0.75470 92851 I. 00108 68272 0.00188 38846 49 3 0.85684 23824 49 0.74314 45232 I. 00105 37745 0.00189 20395 48 3 0.83935 58031 48 0.73135 33926 I. 001 02 06003 0.00189 78900 47 3 0.82186 92239 47 0.71933 94850 1.00098 73450 0.00190 14287 46 4 0.80438 26447 46 0.70710 64600 1.00095 40492 0.00190 26510 45 4 0.78689 60655 45 r A(r) ' •• — - D(r) E(r) E(r) D(r) A(r) 0.00000 00000 0" 0' 0.00000 00000 I . 00000 00000 0.00000 00000 I 0.01758 71423 I 0.00026 61 187 I . 00000 23404 0.01745 21509 2 0.03517 42845 2 I 0.00053 19095 I. 00000 93587 0.03489 89861 3 0.05276 14268 3 I 0.00079 70448 1.00002 10463 0.05233 51918 4 0.07034 85691 4 2 0.00106 I 1979 I .00003 73890 0.06975 54570 5 0.08793 571 13 5 2 0.00132 40433 1.00005 83670 0.08715 44758 6 0.10552 28536 6 3 0.00158 52573 1.00008 39546 0.10452 69489 7 0.12310 99959 7 3 0.00184 45182 i.oooii 41206 0.12 I 86 75849 8 0.14069 71382 8 4 0.00210 15066 I .00014 88284 0.13917 11019 9 0.15828 42804 9 4 0.00235 59064 I .00018 80356 0.15643 22298 10 0.17587 14227 10 5 0.00260 74044 1.00023 16945 0.17364 57109 II 0.19345 85650 II 5 0.00285 56913 1.00027 97518 0.19080 63023 12 0.21 104 57072 12 5 0.00310 04619 1.00033 2I49I 0.20790 87771 13 0.22863 28495 13 6 0.00334 14153 I .00038 88224 0.22494 79261 14 0.24621 99918 14 6 0.00357 82555 1.00044 97028 0.24191 85595 15 0.26380 71340 15 7 0.00381 06920 I. 00051 47160 0.25881 55080 16 0.28139 42763 16 7 0.00403 84394 1.00058 37829 0.27563 36252 17 0.29898 14186 17 7 0.00426 12186 I .00065 68193 0.29236 77883 18 0.31656 85609 18 8 0.00447 87567 1.00073 37362 0.30901 29003 19 0.33415 57031 19 8 0.00469 07873 I .00081 44399 0.32556 38912 20 0.35174 28454 20 8 0.00489 705 II 1.00089 88322 0.34201 57197 21 0.36932 99877 21 9 0.00509 72961 I .00098 68100 0.35836 33745 22 0.38691 71299 22 9 0.00529 12778 I .00107 82664 0.37460 18764 23 0.40450 42722 23 9 0.00547 87596 I. 001 17 30898 0.39072 62791 24 0.42209 14145 24 10 0.00565 95131 1. 00127 1 1647 0.40673 16711 25 0.439B7 85568 25 10 0.00583 33185 I. 00137 23717 0.42261 31771 26 0.45726 56990 26 10 0.00599 99643 I. 00147 65874 0.43836 59597 27 0.47485 28413 27 II 0.00615 92485 I. 00158 36848 0.45398 52206 28 0.49243 99836 28 II 0.00631 09780 I. 00 I 69 35336 0.46946 62019 29 0.51002 71258 29 II 0.00645 49693 I .00180 59998 0.48480 41881 30 0.52761 42681 30 II 01100659 10484 0*00671 90513 I. 00192 09464 0.49999 45073 31 0.54520 14104 31 12 1.00203 82334 0.51503 25321 32 0.56278 85526 32 12 0.00683 88242 I. 00215 77178 0.52991 36820 33 0.58037 56949 33 12 0.00695 02232 1.00227 92542 0.54463 34239 34 0.59796 28372 34 12 0.00705 31150 I .00240 26944 0.55918 72740 35 0.61554 99795 35 12 0.00714 73769 1.00252 78880 0.57357 07990 36 0.63313 71217 36 13 0.00723 28968 I .00265 46826 0.58777 96173 37 0.65072 42640 37 13 0.00730 95735 I .00278 29236 0.60180 94008 38 0.66831 14063 38 13 0.00737 73166 I .00291 24548 0.61565 58756 39 0.68589 85485 39 13 0.00743 60469 1.00304 31183 0.62931 48239 40 0.70348 56908 40 13 0.00748 56962 I. 00317 47551 0.64278 20847 41 0.72107 28331 41 13 0.00752 62073 1.00330 72046 0.65605 35555 42 0.73865 99754 42 13 0.00755 75345 1.00344 03056 0,66912 51936 43 0.75624 71176 43 13 0.00757 96433 I 00357 .38959 0.68199 30169 44 0.77383 42599 44 13 0.00759 25102 1.00370 78127 0.69465 31055 45 90-r 0.79142 14022 45 13 0.00759 61235 1.00384 18928 0.70710 16026 pf i' G(r) C(r) B(r) Smithsonian Tables TABLE d = 10° q = 0.00191359459017, 0= 0.9961728108, HK = 0.418305976553 263 B(r) C(r) G(r) ^ ■Pxp 90-r I. 00000 00000 1.00768 37857 0.00000 00000 90° 0' 1.58284 28043 90 0.99984 76907 1.00768 14453 0.00026 40908 89 1.56525 56621 89 0.99939 08092 I .00767 44270 0.00052 78635 88 I 1.54766 85198 88 0.99862 94947 I . 00766 27394 0.00079 10004 87 I 1.53008 13775 87 0.99756 39792 1.00764 63966 0.00105 31846 86 2 I. 5 I 249 42353 86 0.99619 45873 1.00762 54187 0.00131 41001 85 2 1.49490 70930 85 0.99452 17362 1.00759 983 I I 0.00157 34327 84 3 I. 4773 I 99507 84 0. 99254 59357 1.00756 96650 0.00183 08697 83 3 1.45973 28084 83 0.99026 77878 1.00753 49572 0.00208 61008 82 4 I .44214 56662 82 0.98768 79866 1.00749 57500 0.00233 88183 81 4 1.42445 85239 81 0.98480 73181 1.00745 20912 0.00258 87173 80. 4 1.40697 13816 80 0.98162 66600 1.00740 40338 0.00283 54962 79 5 1.38938 42394 79 0.97814 69814 1.00735 16366 0.00307 88572 78 5 I. 37179 70971 78 0.97436 93426 1.00729 49632 0.00331 85063 77 6 1.35420 99548 77 0.97029 48945 I .00723 40828 0.00355 41538 76 6 1.33662 28125 76 0.96592 48785 I. 007 I 6 90696 0.00378 55150 75 7 I. 3 I 903 56703 75 0.96126 06262 I .00710 00027 0.00401 23098 74 7 I. 30144 85280 74 0.95630 35586 I . 00702 69663 0.00423 42636 73 7 1.28386 13857 73 0.95105 51861 I . 00695 00494 0.00445 1 1077 72 8 1.26627 42435 72 0.94551 71076 1.00686 93457 0.00466 25790 71 8 I .24868 71012 71 0.93969 10107 1.00678 49535 0.00486 84209 70 8 I. 23109 99589 70 0.93357 86703 1.00669 69756 0.00506 83836 69 9 1.21351 28167 69 0.92718 19488 1.00660 55192 0.00526 22237 68 9 I. 19592 56744 68 0.92050 27950 I. 0065 I 06958 0.00544 97055 67 9 1.17833 85321 67 0.91354 32440 I. 0064 I 26209 0.00563 06006 66 10 I. 16075 13898 66 0.90630 54160 I. 00631 14139 0.00580 46884 65 10 1.14316 42476 65 0.89879 15164 1.00620 71982 0.00597 I 756 I 64 10 I. 12557 71053 64 0.89100 38343 I. 00610 01007 0.00613 15997 63 II I. 10798 99630 63 0.88294 47424 1.00599 02520 0.00628 40232 62 II I .09040 28208 62 0.87461 66961 1.00587 77858 0.00642 8839« 61 II I. 0726 I 56785 61 0.86602 22325 1.00576 28392 0.00656 58716 60 12 1.05522 85362 60 0.85716 39703 1.00564 55522 0.00669 49498 59 12 1.03764 13940 59 0.84804 46080 I .00552 60678 0.00681 59154 58 12 1.02005 42517 58 0.83866 69240 1.00540 45314 0.00692 86187 57 12 1.00246 71094 57 0.82903 37754 1.00528 10912 0.00703 29201 56 12 0.98487 99671 56 0.81914 80969 I. 005 I 5 58975 0.00712 86900 55 12 0.96729 28249 55 0.80901 29003 1.00502 91030 0.00721 58089 54 13 0.94970 56826 54 0.79863 12733 I . 00490 08620 0.00729 41679 53 13 0.932 1 1 85403 53 0.78800 63786 1.00477 13308 0.00736 36683 52 13 0.91453 13981 52 0.77714 14532 1.00464 06672 0.00742 42224 51 13 0.89694 43558 51 0.76603 98071 1.00450 90305 0.00747 57531 50 13 0.87935 71135 50 0.75470 48222 1.00437 65809 0.00751 81941 49 13 0.86176 99712 49 0.74313 99518 1.00424 34799 0.00755 14902 48 13 0.84418 28290 48 0.73134 87191 I. 00410 98897 0.00757 55973 47 13 0.82659 56867 47 0.71933 47160 1.00397 59729 0.00759 04823 46 13 0.80900 85444 46 0.70710 16026 1.00384 18928 0.00759 61235 45 13 0.79142 14022 45 r A(r) D(r) E(r) Fct> ■* Smithsonian Tabues 264 ELLIPTIC FUNCTIO N K = 1.5981420021, K' = KV3 = 2.768063U64, E = 1 . 5141504939, E' = 1 . 076405113, r F<^ ^ E(r) D(r) A(r) 0.00000 00000 0' 0' 0.00000 00000 I . 00000 00000 0.00000 00000 I 0.01775 71334 I I 0.00059 97806 I. 00000 53258 0.01745 10959 2 o- 03551 42667 2 2 0.00119 88113 1.00002 12966 0.03489 68785 3 0.05327 14001 3 3 0.00179 63433 1.00004 78929 0.05233 20359 4 0.07102 85334 4 4 0.00239 16296 1.00008 50825 0.06975 12596 5 0.08878 56668 5 5 0.00298 39265 I .00013 28199 0.08714 92460 6 0. 10654 28002 6 6 0.00357 24940 I. 00019 10470 0.10452 06976 7 0.12429 99335 7 7 0.00415 65975 1.00025 96929 0.12186 03254 8 . 14205 70669 8 8 0.00473 55081 1.00033 86738 0.13916 28498 9 0.1 598 I 42002 9 9 0.00530 85039 1.00042 78937 0.15642 30024 10 0.17757 13336 10 10 0.00587 48710 1.00052 72438 0.17363 55278 II 0.19532 84669 II II 0.00643 39044 I .00063 66031 0.19079 51850 12 0.21308 56003 12 12 0.00698 49088 1.00075 58383 0.20789 67491 13 0.23084 27336 13 13 0.00752 71998 1.00088 48041 0.22493 50127 14 0.24859 98670 14 14 0.00806 01044 I. 00 I 02 33434 0.24190 47877 15 0.26635 70004 ■ 15 15 0.00858 29622 I .00117 12875 0.25880 09068 16 0.28411 41337 16 16 0.00909 51263 I .00132 84561 0.27561 82249 17 0.30187 12671 17 17 0-00959 59638 I. 00149 46577 0.29235 16211 18 0.31962 84004 18 18 0.01008 48569 I .00166 96898 0.30899 59997 19 0.33738 55338 19 18 0.01056 12037 I. 001 85 33392 0- 32554 62922 20 0.35514 26672 20 19 0.01102 44188 1.00204 53820 0-34199 74584 21 0.37289 98005 21 20 0.01147 39339 1.00224 55845 0.35834 44886 22 0.39065 69339 22 21 0.01190 91990 1.00245 37025 0.37458 24043 23 0.40841 40672 23 21 0.01232 96827 1.00266 94826 0.39070 62603 24 0.42617 12006 24 22 0.01273 48729 I .00289 26619 0.40671 I 1462 25 0.44392 83339 25 23 0.01312 42775 I .00312 29684 0.42259 21874 26 0.46168 54673 26 24 0.01349 74251 1.00336 01217 0.43834 45471 27 0.47944 26006 27 25 0.01385 38651 1.00360 38326 0.45396 34276 28 0.49719 97340 28 25 0.01419 31688 I 00385 38044 0.46944 40717 29 0.51495 68674 29 25 0.01451 49297 I. 00410 97324 0.48478 17640 30 0.53271 40007 30 26 0.01481 87635 1.00437 13049 0.49997 18327 31 0.55047 11341 31 26 0.01510 43095 I .00463 82031 0.51500 96510 32 0.56822 82674 32 27 0.01537 12298 I .00491 01019 0.52989 06380 33 0. 58598 54008 33 27 0.01561 92109 I. 00518 66701 0.54461 02607 34 0.60374 25341 34 28 0.01584 79628 1.00546 75706 0.55916 40350 35 0.62149 96675 35 28 0.01605 72204 1.00575 24612 0.57354 75273 36 0.63925 68009 36 28 0.01624 67429 I . 00604 09949 0.58775 63556 37 0.65701 39342 37 29 0.01641 63146 I .00633 28201 0.60178 61912 38 0.67477 10676 38 29 0.01656 57446 1.00662 75813 0.61563 27596 39 0.69252 82009 39 29 0.01669 48676 1.00692 49193 0.62929 1842 1 40 0.71028 53343 40 29 0.01680 35433 1.00722 44718 0.64275 92769 41 0.72804 24676 41 30 0.01689 16569 1.00752 58740 0.65603 09607 42 0.74579 96010 42 30 0.01695 91191 1.00782 87587 0.66910 28494 43 0. 76355 67344 43 30 0.01700 58662 I. 00813 27567 0.68197 09600 44 0.78131 38677 44 30 0.01703 18597 1.00843 74977 0.69463 13711 45 0.79907 lOOII 45 30 0.01703 70869 I .00874 26104 0.70708 02248 90-r Fi/' \^ G(r) C(r) B(r) Smithsonian Tables TABLE e = 15° g = . 004333420509983, 9 = 0. 9913331597, HK = , 5131518035 265 B(r) C(r) G(r) 4' Fi/' 90-r I. 00000 00000 I. 01748 52237 0.00000 00000 90° 0' I. 598 14 20021 90 0.99984 76723 I. 01747 98979 0.00058 94801 89 I 1.58038 48688 89 0.99939 07356 I. 01746 39271 0. 001 I 7 82606 88 2 1.56262 77354 88 0.99862 93293 I 01743 73307 0.00176 56424 87 3 1.54487 06021 87 0-99756 36857 I .01740 01412 0.00235 09281 86 4 1-52711 34687 86 0.99619 41297 I. 01735 24037 0.00293 34228 85 5 1-50935 63353 85 0.99452 10792 I. 01729 41766 0.00351 24342 84 6 1-49159 92020 84 0.99254 50444 I. 01722 55307 0.00408 72741 83 7 1.47384 20686 83 0.99026 66280 1.01714 65496 0.00465 72589 82 8 1.45608 49353 82 0.98768 65251 I. 01705 73297 0.00522 17102 81 9 1.43832 78019 81 0.98480 55225 I. 01695 79795 0.00577 99557 80 10 I .42057 06685 80 0.98162 44990 I .01684 86202 0.00633 13300 79 II I -40281 35352 79 0.97814 44248 I. 01672 93849 0.00687 51750 78 12 I -38505 64019 78 0.97436 63613 I .01660 04190 0.00741 08412 77 13 1.36729 92685 77 0.97029 14608 I .01646 18796 0.00793 76880 76 14 1.34954 21352 76 0.96592 09661 I 01631 39354 0.00845 50845 75 15 I 33178 50018 75 0.96125 62102 I .01615 67668 0.00896 24102 74 16 I .31402 78684 74 0.95629 86158 I. 01599 05651 0.00945 90560 73 17 1.29627 07351 73 0.95104 96947 I 01581 55329 0.00994 44245 72 18 I. 27851 36017 72 0.94551 10478 I. 01563 18834 0.01041 79308 71 18 1.26075 64684 71 0.93968 43642 I 01543 98405 0.01087 90033 70 19 1.24299 93350 70 0.93357 14207 I. 01523 96380 0. 01 132 70844 69 20 I .22524 22016 69 0.92717 40815 I. 01503 15198 0.01176 16310 68 20 1.20748 50683 68 0.92049 42975 1.01481 57396 0.01218 21151 67 21 I -18972 79349 67 0.91353 41057 I. 01459 25602 0.01258 80246 66 22 I .17197 08016 66 0.90629 56284 I. 01436 22536 0.01297 88640 65 23 I. I 542 I 36682 65 0.89878 10728 I .01412 51003 0.01335 41547 64 23 I. 13645 65348 64 0.89099 27303 I .01388 13892 0.01371 34359 63 24 1.11869 94015 63 0.88293 29756 I. 01363 14174 0.01405 62649 62. 25 I . 10094 22681 62 0.87460 42661 I 01337 54893 0.01438 22180 61 25 I. 08318 51348 61 0.86600 91414 1.01311 39167 0.01469 08906 60 26 I .06542 80014 60 0.85715 02219 I. 01284 70184 0.01498 18982 59 26 I .04767 08681 59 0.84803 02085 I. 01257 51195 0.01525 48767 58 27 I. 0299 I 37347 58 0.83865 18817 I. 01229 85512 0.01550 94825 57 27 1.01215 66014 57 0.82901 81005 I .01201 76507 0.01574 53939 56 28 0-99439 94680 56 0.81913 18020 I. 01 173 27599 0.01596 23105 55 28 0.97664 23346 55 0.80899 59997 I .01144 42262 0,01615 99545 54 28 0.95888 52013 54 0.79861 37836 I .01115 24009 0.01633 80704 53 29 0.94112 80679 53 0.78798 83184 I. 01085 76397 0.01649 64258 52. 29 0.92337 09346 52 0.77712 28430 I. 01056 03017 0.01663 48119 51 29 0.90561 38012 51 0.76602 06691 I .01026 07491 0.01675 30432 ■50 29 0.88785 66678 50 0.75468 51808 1.00995 93468 0.01685 09584 49 29 0.87009 95345 49 0.743 I I 98330 1.00965 64622 0.01692 84205 48 30 0.85234 2401 1 48 0.73132 81506 1.00935 24642 0.01698 53170 47 30 0.83458 52678 47 0.71931 37274 1.00904 77232 0.01702 15600 46 30 0.81682 81344 46 0.70708 02248 1.00874 26104 0.01703 70869 45 30 0.79907 lOOII 45 r A(r) D(r) E(r) F0 Smithsonian Tables 266 ' ELLIPTIC FUNCTIO K = 1.6200258991, K' =2.5046500790, E = 1.5237992053, E' = 1.118377738 r F<^ E(r) D(r) A(r) 0.00000 00000 0° 0' 0.00000 00000 I . 00000 00000 0.00000 00000 I 0.01800 02878 I 2 0.00106 89581 I .00000 96218 0.01744 81883 2 0.03600 05755 2 4 0.00213 65522 1.00003 84757 0.03489 10694 3 0.05400 08633 3 6 0.00320 14202 I . 00008 65263 0.05232 33377 4 0.07200 II5II 4 7 0.00426 22042 I. 00015 37152 0.06973 96909 5 0.09000 14388 5 9 0.00531 75519 I .00023 99605 0.08713 48313 6 0.10800 17266 6 II 0.00636 61189 1.00034 51572 0.10450 34678 7 0.12600 20144 7 13 0.00740 65708 I .00046 91770 0.12184 03169 8 0.14400 23021 8 15 0.00843 75848 I. 0006 I 18689 0.13914 01051 9 0.16200 25899 9 17 0.00945 78515 1.00077 30591 0.15639 75697 10 0.18000 28777 10 19 0.01046 60772 1.00095 25510 0.17360 74610 II 0.19800 31655 II 20 0. 01 146 09855 1.00115 01262 0.19076 45434 12 0.21600 34532 12 22 0.01244 13188 I. 00136 55438 0.20786 35973 13 0.23400 37410 13 24 0.01340 58406 I. 00159 85414 0.22489 94205 14 0.25200 40288 14 25 0.01435 33370 I .00184 88351 0.24186 68298 IS 0.27000 43165 15 27 0.01528 26180 1.00211 61200 0.25876 06626 i6 0.28800 46043 16 28 0.01619 25197 t 1.00240 00704 0.27557 57786 17 0.30600 48921 17 30 0.01708 19057 1.00270 03405 0.29230 70609 i8 0.32400 51799 18 32 0.01794 96683 I .00301 65642 0.30894 94182 19 0.34200 54676 19 33 0.01879 47304 1.00334 83565 0.32549 77855 20 0.36000 57554 20 35 0.01961 60466 1.00369 53131 0.34194 71266 21 0.37800 60431 21 36 0.02041 26046 I .00405 701 12 0.35829 24349 22 0.39600 63309 22 37 0.021 18 34268 1.00443 30101 0.37452 87349 23 0.41400 66187 23 39 0.02192 75711 1.00482 28518 0.39065 10844 24 0.43200 69064 24 40 0.02264 41321 1.00522 60614 0.40665 45753 25 0.45000 71942 25 41 0.02333 22426 1.00564 21475 0.42253 43354 26 0.46800 74820 26 42 0.02399 10740 I . 00607 06033 0.43828 55296 27 0.48600 77697 27 44 0.02461 98378 I. 0065 I 09067 0.45390 33618 28 0.50400 80575 28 45 0.02521 77862 I .00696 25213 0.46938 30761 29 0.52200 83453 29 46 0.02578 42130 I . 00742 48968 0.48471 99582 30 0.54000 86330 30 46 0.02631 84541 1.00789 74700 0.49990 93370 31 0.55800 89208 31 47 0.02681 98888 1.00837 96651 0.51494 65858 32 0.57600 92086 32 48 0.02728 79396 I .00887 08946 0.52982 71240 33 0.59400 94963 33 49 0.02772 20732 1.00937 05600 0.54454 64181 34 0.61200 97841 34 SO 0.02812 18009 I .00987 80525 0.55909 99835 35 0.63001 00719 35 50 0.02848 66791 I. 01 039 27539 0.57348 33858 36 0.64801 03597 36 SI 0.02881 63091 I .01091 40371 0.58769 22416 37 0.66601 06474 37 51 0.0291 1 03382 1.01144 12669 0.60172 22208 38 0.68401 09352 38 52 0.02936 84591 1.01197 38011 0.61556 90470 39 .0.70201 12230 39 52 0.02959 04103 I .01251 09908 0.62922 84994 40 0.72001 15107 40 53 0.02977 59763 I. 01305 21815 0.64269 64140 41 0.73801 17985 41 53 0.02992 49874 I -01359 67138 0.65596 86845 42 0.75601 20863 42 53 0. 03003 73198 I 01414 39245 0.66904 12642 43 0.77401 23740 43 53 0.0301 1 28953 I. 01469 31466 0.68191 01665 44 0.79201 26618 44 S3 0.03015 16811 I. 01524 37112 0.69457 14668 45 90-r 0.81001 29496 45 53 0.03015 36896 I. 01579 49474 0.70702 13033 Fi// -A G(r) C(r) B(r) s MITH6 ONIAN Tables TABLE e = 20° g = . 007774680416442, 0=0. 9844506465, HK = 5939185400 267 B(r) C(r) G(r) ^ Fi/' 90-r 90 I . 00000 00000 I. 03 I 58 99246 0.00000 00000 90° 0' 1.62002 58991 0.99984 76215 I. 03158 03027 0.00103 62474 89 2 1.60202 56113 89 0.99939 05327 I. 03 I 55 14488 0.00207 12902 88 4 1.58402 53236 88 0.99862 88734 I. 03150 33980 0.00310 39250 87 6 1.56602 50358 87 0.99756 28767 I. 03143 62088 0.00413 29509 86 7 1.54802 47480 86 0.99619 28686 I 03134 99632 0.00515 71704 85 9 1.53002 44603 85 0.99451 92682 I .03124 47661 0.00617 53910 84 II I. 51202 41725 84 0.99254 25876 1.03112 07458 0.00718 64259 83 13 1.49402 38847 83 0.99026 34315 1.03097 80534 0.00818 90957 82 15 1.47602 35970 82 0.98768 24970 I. 03081 68627 0.00918 22293 81 16 1.45802 33092 81 0.98480 05736 1.03063 73701 0.01016 46651 80 18 1.44002 30214 80 0.98161 85429 1.03043 97942 0. 01 1 13 52523 79 20 1.42202 27337 79 0.97813 73781 1.03022 43759 0.01209 28519 78 22 1.40402 24459 78 0.97435 81442 1.02999 13775 0.01303 63381 77 23 1.38602 21581 77 0.97028 19968 I .02974 10829 0.01396 45994 76 25 1.36802 18704 76 0.96591 01827 1.02947 37972 0.01487 65396 75 27 1.35002 15826 75 0.96124 40390 I. 02918 98458 0.01577 10793 74 28 1.33202 12948 74 0.95628 49924 1.02888 95748 0.01664 71568 73 30 I .31402 10070 73 0.95103 45595 I 02857 33501 0.01750 37292 72 31 I .29602 07193 72 0.94549 43456 1.02824 15568 0.01833 97739 71 33 1.27802 04315 71 0.93966 60449 I .02789 45992 0.01915 42895 70 34 1.26002 01437 70 0.93355 14391 i. 02753 28994 0.01994 62967 69 36 I .24201 98560 69 0.92715 23977 I. 02715 69001 0.02071 48399 68 37 I. 22401 95682 '68 0.92047 08768 1.02676 70574 0.02145 89881 67 38 I. 20601 92804 67 0.91350 89187 1.02636 38468 0.02217 78360 66 40 1.18801 89927 66 0.90626 86515 1.02594 77596 0.02287 05049 65 41 I . 17001 87049 65 0.89875 22880 1. 0255 1 93029 0.02353 61442 64 42 1.15201 84171 64 0.89096 21252 1.02507 89985 0.02417 39320 63 43 1.13401 81294 63 0.88290 05436 1.02462 73829 0.02478 30767 62 44 1.11601 78416 62 0.87457 00067 I .02416 50064 0.02536 28172 61 45 I. 09801 75538 61 0.86597 30595 1.02369 24323 0.02591 24248 60 46 I .08001 72661 60 0.85711 23285 1. 0232 1 02363 0.02643 12037 59 47 I .06201 69783 59 0.84799 05205 I .02271 90060 0.02691 84920 58 48 I. 0440 I 66905 58 0.83861 04218 I. 0222 I 93398 0.02737 36626 57 49 1.02601 64028 57 0.82897 48973 1.02171 18465 0.02779 61243 56 49 I. 00801 61 150 56 0.81908 68896 1.02119 71444 0.02818 53227 55 50 0.99001 58272 55 0.80894 94182 I . 02067 58606 0.02854 07409 54 51 0.97201 55395 54 0.79856 55784 I .02014 86302 0.02886 19001 53 51 0.95401 52517 53 0.78793 85407 1.01961 60955 0.02914 83611 52 52 0.93601 49639 52 0.77707 15491 I .01907 89054 0.02939 97245 51 52 0.91801 46761 51 0.76596 79209 I. 01853 77143 0.02961 56313 50 53 0.90001 43884 50 0.75463 10450 I. 01799 31816 0.02979 57642 49 53 0.88201 41006 49 0.74306 43814 I. 01 744 59707 0.02993 98477 48 53 0.86401 38129 48 0.73127 14598 I. 01689 67484 0.03004 76489 47 53 0.84601 35251 47 0.71925 58784 I. 01634 61837 0.0301 I 89783 46 53 0.82801 32373 46 0.70702 13033 I .01579 49474 0.03015 36896 45 53 0.81001 29496 45 r A(r) D(r) E(r) F(#> Smithsonian Tables 268 ELLIPTIC FUNCTION K = 1 . 6489952185, K' = 2 . 3087867982, E = 1 . 4981149284, E' = 1 , 1638279645, r F(^ 4> E(r) D(r) A(r)- 0.00000 00000 0' 0' 0.00000 00000 I . 00000 00000 0.00000 00000 I 0.01832 21691 I 3 0.00167 60815 i.ooooi 53565 0.01744 18591 2 0.03664 43382 2 6 0.00334 99667 I .00006 14074 0.03487 84245 3 0.05496 65073 3 9 0.00501 94629 I. 000 I 3 80964 0.05230 44041 4 0.07328 86764 4 12 0.00668 23842 1.00024 53303 0.06971 45088 S 0.09161 08455 5 15 0.00833 65551 1.00038 29783 0.08710 34544 6 0.10993 30145 6 18 0.00997 98139 1.00055 08728 . 10446 59627 7 0.12825 51836 7 21 0.01161 00163 1.00074 88092 0.12179 67635 8 0-14657 73527 8 24 0.01322 50382 1.00097 65463 0.13909 05958 9 0.16489 95218 9 26 0.01482 27797 I. 00123 38067 0.15634 22095 10 •0.18322 16909 10 29 0.01640 1 1677 I .00152 02770 0.17354 63669 II 0.20154 38600 II 32 0.01795 81596 I. 00183 56081 0. 19069 78446 12 0.21986 60291 12 35 0.01949 17458 I. 00217 94159 0.20779 14345 13 0.23818 81982 13 37 0.02099 99533 1.00255 12815 0.22482 19454 14 0.25651 03673 14 40 0.02248 08485 1.00295 07519 0.24178 42052 15 0.27483 25364 15 43 0.02393 25396 I 00337 73404 0.25867 30615 i6 0.29315 47055 16 45 0.02535 31798 1.00383 05272 0.27548 33838 17 0.31147 68746 17 48 0.02674 09700 1.00430 97603 0.29221 00649 i8 0.32979 90437 18 50 0.02809 41609 I. 0048 I 44557 0.30884 80221 19 0.34812 12128 19 53 0.02941 10555 1.00534 39986 0.32539 21991 20 0.36644 33819 20 56 0.03069 001 18 1.00589 77438 0.34183 75673 21 0.38476 55510 21 57 0.03192 94445, 1.00647 50167 0.35817 91274 22 0.40308 77201 22 59 0.03312 78272 I .00707 51140 0.37441 19107 23 0.42140 98892 24 I 0.03428 36945 1.00769 73046 0.39053 09808 24 0.43973 28582 25 3 0.03539 56434 1.00834 08304 0.40653 14352 25 0.45805 42273 26 5 0.03646 23352 I . 00900 49074 0.42240 84064 26 0.47637 63964 27 7 0.03748 24970 1.00968 87266 0.43815 70635 27 0.49469 85655 28 9 0.03845 49232 I. 01039 14548 0.45377 26140 28 0.51302 07346 29 II 0. 03937 84764 i.oiiii 22358 0.46925 03045 29 0.53134 29037 30 12 0.04025 20886 I .01185 OI9I6 0.48458 54231 30 0.54966 50728 31 14 0.04107 47627 I .01260 ^i?3i 0.49977 32999 31 0.56798 72419 32 15 0.04184 55726 I. 01337 40113 0.51480 93092 32 0.58630 941 10 33 16 0.04256 36643 I .01415 80186 0.52968 88703 33 0.60463 15801 34 18 0.04322 82564 I. 01495 54899 0.54440 74492 34 0.6229s 37492 35 19 0.04383 86406 I. 01576 54535 0.55896 05600 35 0.64127 59183 36 20 0.04439 41 82 I I .01658 69227 0-57334 37662 36 0.65959 80874 37 21 0.04489 43196 I .01741 88967 0.58755 26819 37 0.67792 02565 38 22 0.04533 85655 I. 01 826 03617 0.60158 29737 38 0.69624 24256 39 23 0.04572 65058 I .01911 02927 0.61543 0361 1 39 0.71456 45947 40 23 0.04605 78000 I .01996 76540 0.62909 06189 40 0.73288 67638 41 23 0.04633 21809 1.02083 14013 0.64255 95777 41 0.75120 89328 42 24 0.04654 94543 I .02170 04820 0.65583 31255 42 0.76953 11019 43 24 0.04670 94981 1.02257 38374 0.66890 72089 43 0.78785 32710 44 24 0.04681 22622 1.02345 04035 0.68177 78347 44 0.80617 54401 45 24 0.04685 77678 I .02432 91 122 0.69444 10704 45 0.82449 76092 46 24 0.04684 61065 I .02520 88930 0.70689 30463 90-r F^^ -A G(r) C(r) B(r) Smithsonian Tables TABLE e = 25° q = 0. 012294560527181, 9 0=0. 975410924642, HK = 0,666076159327 269 B(r) I . 00000 00000 0.99984 751 n 0.99939 00912 0.99862 78812 0.99756 I I 158 0.99619 01235 0,99451 53263 0.99253 72400 0.99025 64734 0.98767 37287 0.98478 98010 0.98160 55779 0.97812 20395 0.97434 02576 0.97026 13962 0.96588 67101 0.96121 75452 0.95625 53377 0.95100 16139 0.94545 79893 0.93962 61686 0.93350 79444 0.92710 51976 0.92041 98958 0.91345 40932 0.90620 99299 0.89868 96309 0.89089 55058 0.88282 99477 0.87449 54326 0.86589 45184 0.85702 98444 0.84790 41300 0.83852 01744 0.82888 08549 0.81898 91269 0.80884 80221 0.79846 06482 0.78783 01874 0.77695 98956 0.76585 31015 0.75451 32053 0.74294 36775 0.73 I 14 80583 0.71912 99561 0.70689 30463 C(r) 1.05041 79735 I .05040 26167 1.05035 65652 1.05027 98750 I. 0501 7 26395 1.05003 49895 I . 04986 70926 1.04966 91533 1.04944 14129 I .04918 41489 A(r) I .04889 76746 1.04858 23391 1.04823 85265 1.04786 66559 1.04746 71802 I .04704 05862 1.04658 73936 I. 04610 81546 1.04560 34530 1.04507 39038 1.04452 01522 I . 04394 28728 1.04334 27690 I .04272 05719 I . 04207 70396 1.04141 29561 1,04072 91305 I . 04002 63960 1.03930 56088 1.03856 76470 I. 03781 34098 1.03704 38161 1.03625 98035 1.03546 23272 1.03465 23588 1.03383 08852 1.03299 89073 1. 032 1 5 74386 I. 03 130 75044 1.03045 01401 1.02958 63905 I. 02871 73077 1.02784 39507 1.02696 73835 I .02608 86741 I .02520 88930 G(r) 0.00000 00000 0.00159 57045 0.00318 96046 0.00477 98977 0.00636 47840 D(r) 0.00794 2i 0.00951 II627 o. 01 106 90855 O.OI26I 44653 0-01414 55416 0.01566 05663 O.OI715 78054 0.01863 55407 0.02009 20712 0.02152 57149 0.02293 48102 0.02431 77177 0.02567 28218 0.02699 85322 0.02829 32857 0.02955 55477 0.03078 38140 0.03197 66123 0.03313 25038 0.03425 00853 o. 03532 79902 0.03636 48907 0.03735 94992 0.03831 05700 0.03921 69009 0.04007 73349 0.04089 07619 0.04165 61200 0.04237 23976 0.04303 86345 o. 04365 39236 0.04421 74127 0.04472 83056 0.04518 58637 o- 04558 94076 0.04593 83183 0.04623 20386 0.04647 00744 0.04665 I996I 0.04677 74393 0.04684 61065 E(r) 90° o' 89 3 88 6 87 9 86 12 85 15 84 17 83 20 82 23 81 26 80 29 79 31 78 34 77 37 76 39 75 42 74 44 75 47 72 49 71 52 70 54 69 56 68 58 68 o 67 2 66 4 65 6 64 8 63 10 62 II 61 13 60 14 59 16 58 17 57 18 56 19 55 20 54 21 53 22 52 22 51 23 50 24 49 24 48 24 47 24 46 24 ¥\j/ 1.64899-52185 1.63067 30494 1,61235 08803 1.59402 871 12 I 57570 65421 1-55738 43730 1.53906 22039 1.52074 00348 1,50241 78657 I , 48409 56966 1-46577 35275 1-44745 13584 I -42912 91893 I ,41080 70202 1-39248 485 II I .37416 26821 1-35584 05130 I -33751 83439 1.31919 61748 1.30087 40057 1-28255 18366 I .26422 96675 1.24590 74984 1-22758 53293 I ,20926 31602 I ,19094 09911 1.17261 88220 I. 15429 66529 I -13597 44838 I -11.765 23147 1.09933 01456 I .08100 79765 1.06268 58075 1-04436 36384 I .02604 14693 I .00771 93002 0-98939 71311 0.97107 49620 0-95275 27929 o.- 93443 06238 0.91610 84547 0.89778 62856 0.87946 41 165 0.861 14 19474 0.84281 97783 0.82449 76092 7 E(r) D(r) A(r) 0.00000 00000 0° 0' 0.00000 00000 1 . 00000 00000 0,00000 00000 I 0.01923 60575 I 6 0.00332 09329 1.00003 I 945 I 0,01740 91115 2 0.03847 21150 2 12 0.00663 71847 1,00012 77415 0,03481 29991 3 0.05770 81725 3 18 0.00994 40836 I .00028 72724 0,05220 64403 4 0.07694 42300 4 24 0.01323 69759 I. 0005 I 03436 0,06958 42154 5 0.09618 0287s 5 30 0.01651 12357 1.00079 66833 0,08694 I 1086 6 0.11541 63450 6 36 0.01976 22733 1.00114 59427 0,10427 19100 7 0.13465 24025 7 42 0.02298 55446 I. 00155 76965 0,12157 14162 8 0.15388 84600 8 48 0.02617 65594 1.00203 14429 0.13883 44322 9 0.17312 45176 9 54 0.02933 08900 1.00256 66050 0.15605 57726 10 0.19236 05751 II 0.03244 41797 I. 00316 25308 0.17323 02632 II 0.21159 66326 12 5 0.03551 21508 I. 0038 I 84944 0.19035 27418 12 0.23083 26901 13 II 0.03853 06122 1-00453 36968 0.20741 80603 13 0.25006 87476 14 16 0.04149 54668 1.00530 72668 0.22442 10857 14 0.26930 48051 15 22 0.04440 27192 I .00613 82620 0.24135 67013 i^ 0.28854 08626 16 27 0.04724 84818 1.00702 56701 0.25821 98088 i6 0.30777 69201 17 32 0.05002 89819 1.00796 84103 0.27500 53288 17 0.32701 29776 18 37 0.05274 05671 1.00896 53340 0,29170 82026 i8 0.34624 90351 19 42 0.05537 97118 I ,01001 52268 0,30832 33939 19 0.36548 50926 20 47 0.05794 30217 i,oiiii 68099 0,32484 58897 20 0.38472 11501 21 52 0.06042 72392 I .01226 87413 0.34127 07019 21 0.40395 72077 22 56 0.06282 92476 I. 01346 96177 0,35759 28687 22 0.42319 32652 24 0.06514 60751 1.01471 79763 0,37380 74559 23 0.44242 93227 25 5 0.06737 48988 1.01601 22964 0.38990 95585 24 0.46166 53802 26 9 0.06951 30473 I. 01735 10012 0,40589 43019 25 0.48090 14377 27 13 0.07155 80036 1,01873 24599 0.42175 68435 26 0.50013 74952 28 16 0.07350 74079 1,02015 49897 0-43749 23737 27 0.51937 35527 29 20 0.07535 9058.8 I ,02161 68576 0.45309 61179 28 0.53860 96102 30 23 0.07711 09151 I ,02311 62828 0.46856 33375 29 0-55784 56677 31 27 0.07876 10969 1,02465 14386 0.48388 93314 30 0.57708 17252 32 30 0.08030 78862 I ,02622 04548 0.49906 94371 31 0.59631 77827 33 32 0.08174 97274 I ,02782 14201 0.51409 90330 32 0.61555 38402 34 35 0.08308 52267 1,02945 23841 0.52897 35386 33 0.63478 98977 35 37 0.08431 31523 1,03111 13599 0.54368 84170 34 0.65402 59552 36 40 0.08543 24331 1,03279 63263 0.55823 91754 35 0.67326 20128 37 42 0.08644 21580 1-03450 52308 0.57262 13672 36 0.69249 80703 38 43 0.08734 15741 1.03623 59914 0.58683 05928 37 0.71173 41278 39 45 0.08813 00853 1,03798 64996 0,60086 25017 38 0.73097 01853 40 46 0.08880 72502 I 03975 46228 0,61471 27930 39 0.75020 62428 41 48 0.08937 27798 1,04153 82068 0.62837 72177 40 0.76944 23003 42 49 0.08982 65352 1-04333 50787 0.64185 15792 41 0.78867 83578 43 49 0.09016 85246 1,04514 30495 0-65513 17355 42 0.80791 44153 44 50 0,09039 89009 1,04695 99164 0.66821 35999 43 0.82715 04728 45 SO 0,09051 79579 1,04878 34660 0.68109 31428 44 0.84638 65303 46 51 0.09052 61280 I. 05061 14765 0.69376 63926 45 0.86562 25878 47 51 0.09042 39779 I .05244 17208 0,70622 94378 90-r F;^ i G(r) C(r) B(r) Smithsonian Tables TABLE = 35° 9 = 0.024915062523981, 6 = 0.9501706456, HK = 0. 7950876364 273 B(r) C(r) G(r) f ¥4^ 90-r 90 I . 00000 00000 I. 10488 66859 0.00000 00000 90° 0' I. 73124 51757 0.99984 69394 I . 10485 47369 0.00300 62320 89 6 I .71200 91181 89 0.99938 78065 I. 10475 89287 0.00600 93218 88 12 I . 69277 30606 88 0.99862 27471 I. 10459 93781 0.00900 61288 87 17 1.67353 70031 87 0.99755 20048 I. 10437 62795 0.01199 35156 86 23 1.65430 09456 86 0.99617 59200 I . 10408 99048 0.01496 83495 85 29 1.63506 48881 85 0.99449 49305 I . 10374 06029 0.01792 75043 84 35 1. 61582 88306 84 0.99250 95707 I. 10332 87996 0.02086 78620 83 40 1.59659.27731 83 0.99022 04719 I. 10285 49965 0.02378 63141 82 46 1.57735 67156 82 0.98762 83615 1.10231 9771 1 0.02667 97640 . 81 51 I. 55812 06581 81 0.98473 40633 1.10172 37756 0.02954 51279 80 57 1.53888 46006 80 0.98153 84966 I .10106 77362 0.03237 93372 80 2 I. 51964 85431 79 0.97804 26763 I. 10035 24524 0. 035 I 7 93404 79 8 I. 50041 24856 78 0.97424 77II7 1.09957 87957 0.03794 21046 78 13 I .48117 64281 77 0.97015 48073 1.09874 77089 0.04066 46178 77 19 I. 46194 03706 76 0.96576 52612 I .09786 02047 0. 04334 38907 76 24 1.44270 43130 75 0.96108 04649 I. 0969 I 73646 0.04597 69592 75 29 1.42346 82555 74 0.95610 19028 1.09592 03375 0.04856 08861 74 34 I .40423 21980 73 0.95083 II5I6 1.09487 03382 0.05109 27637 73 38 1.38499 61405 72 0.94526 98796 1.09376 86463 0.05356 97I6I 72 43 1.36576 00830 71 0.93941 98461 I. 09261 66042 0.05598 89014 71 48 1.34652 40255 70 0.93328 29005 1.09141 56156 0.05834 75147 70 52 I .32728 79680 69 0.92686 09817 I. 09016 71440 0.06064 27902 69 56 1.30805 19105 68 0.92015 6II73 1.08887 27107 0.06287 20041 69 I I. 28881 58530 67 0.9I3I7 04228 1.08753 38930 0.06503 24775 68 5 1.26957 97955 66 0.90590 61007 I. 08615 23221 0.06712 15792 67 9 1.25034 37380 65 0.89836 54396 1.08472 96815 0.06913 67285 66 12 I. 231 10 76805 64 0.89055 08135 1.08326 77048 0.07107 53988 65 16 I. 21 187 16230 63 0.88246 46805 I. 08176 81732 0.07293 51200 64 19 I. 19263 55655 62 0.87410 95823 I .08023 29140 0.07471 34824 63 23 I .17339 95080 61 0.86548 81427 1.07866 37978 0.07640 81398 62 26 1.15416 34504 60 0.85660 30670 1.07706 27365 0.07801 68127- 61 29 I. 13492 73929 59 0.84745 71408 1.07543 16809 0.07953 72924 60 31 I. I 1569 13354 58 0.83805 32290 1.07377 26184 0.08096 7^440 59 34 1.09645 52779 57 0.82839 42745 1.07208 75705 0.08230 52102 58 36 I. 07721 92204 56 0.81848 32973 1.07037 85902 0.08354 86152 57 39 1.05798 31629 55 0.80832 33933 1.06864 77599. 0.08469 57684 56 41 1.03874 71054 54 0.79791 77333 1.06689 71884 0.08574 48680 55 43 1.01951 10479 53 0.78726 95615 I. 065 I 2 90086 0.08669 42053 54 44 1.00027 49904 52 0.77638 21945 I 06334 53750 0.08754 21680 53 46 0.98103 89329 51 0.76525 90201 I. 06 I 54 84606 0.08828 72448 52 48 0.96180 28754 50 0.75390 34961 1.05974 04548 6.08892 80287 51 49 0.94256 68179 49 0.74231 91490 1.05792 35605 0.08946 32214 50 49 0.92333 07604 48 0.73050 95727 1.05609 99913 0.08989.16370 49 50 0.90409 47028 47 0.71847 84273 1.05427 19690 0.09021 22056 48 50 0.88485 86453 46 0.70622 94378 1.05244 17208 0.09042 39779 47 51 0.85562 25878 45 A(r) D(r) E(r) F(p r Smithsonian Tables 274 ELLIPTIC FUNCTIO N K = 1.7867691349, K' = 1.9356810960, E = 1. 3931402485, E' = 1. 3055390943, r o F0 4> E(r) r)(r) A(r) 0.00000 00000 0° 0' 0.00000 00000 I . 00000 00000 0.00000 00000 I 0.01985 29904 I 8 0.00437 25767 J .00004 34107 0.01737 52657 2 0.03970 59807 2 16 0.00873 86910 I. 0001 7 35897 0.03474 53796 3 0.05955 89712 3 24 0.01309 18945 1.00039 03787 0.05210 51913 4 0.07941 19615 4 32 0.01742 57681 1.00069 35136 0.06944 95525 5 0.09926 49519 5 41 0.02173 39351 I .00108 26253 0.08677 33185 6 0.11911 79423 6 49 0.02601 00761 I. 00155 72398 0.10407 13496 7 0.13897 09327 7 57 0.03024 79420 1. 002 1 1 67791 0.12133 85117 8 0.15882 39231 9 5 0.03444 13683 I .00276 05620 0.13856 96780 9 0.17867 69135 10 13 0.03858 42875 I . 00348 78042 0.15575 97300 10 0.19852 99039 II 21 0.04267 07422 I . 00429 76203 0.17290 35587 II 0.21838 28943 12 28 0.04669 48973 I. 005 I 8 90239 0.18999 60657 12 0.23823 58847 13 36 0.05065 10519 I. 006 I 6 09295 0.20703 21648 13 0.25808 88751 14 43 0-05453 36499 I. 00721 21534 0.22400 67828 14 0.27794 18655 15 51 0.05833 72913 1.00834 14154 0.24091 48609 15 0.29779 48558 16 58 0.06205 67422 1.00954 73402 0.25775 13559 i6 0.31764 78462 18 5 0.06568 69435 I .01082 84592 0.27451 12417 17 0.33750 08366 19 12 0.06922 30203 I .01218 32120 0.291 18 95099 i8 0.35735 38270 20 18 0.07266 02895 I. 01360 99487 0.30778 11718 19 0.37720 68174 21 25 0.07599 42673 1.01510 69318 0.32428 12593 20 0.39705 98078 22 31 0.07922 06754 I. 01 667 23379 0.34068 48260 21 0.41691 27981 23 37 0.08233 54475 I. 01830 42606 0.35698 69491 22 0.43676 57885 24 42 0.08533 47336 I .02000 07123 0.37318 27300 23 0.45661 87789 25 48 0.08821 49046 I .02175 96267 0.38926 72959 24 0.47647 17693 26 53 0.09097 25564 1.02357 88616 0.40523 58014 25 0.49632 47597 27 59 0.09360 45123 I .02545 62012 0.42108 34293 26 0.51617 77501 29 4 0.09610 78252 1.02738 93589 0.43680 53924 27 0.53603 07405 30 8 0.09847 97792 1.02937 59801 0.45239 69344 28 0.55588 37309 31 13 0.10071 78905 1.03141 36450 0.46785 33318 29 0.57573 '672 12 32 17 0.10281 99075 1.03349 98717 0.48316 98948 30 o. 59558 97 1 16 33 22 0.10478 38101 1.03563 21191 0.49834 .19688 31 0.61544 27020 34 25 0.10660 78092 1.03780 77899 0.51336 49360 32 0.63529 56924 35 28 . 10829 03444 I .04002 42340 0.52823 42166 33 0.65514 86828 36 31 0. 10983 00821 1.04227 87515 0.54294 52702 34 0.67500 16732 37 34 0.1 1 122 59132 1.04456 85961 0.55749 35973 35 0.69485 46636 38 37 0.1 1247 69491 1.04689 09786 0.57187 47405 36 0.71470 76540 39 39 0.11358 25187 1.04924 30699 0.58608 42864 37 0.73456 06443 40 41 0.1 1454 21645 I. 05 I 62 20047 0.6001 1 78665 38 0.75441 36347 41 42 0.1 1535 56375 1.05402 48851 0.61397 11590 39 0.77426 66251 42 44 0.1 1602 28932 1.05644 87839 0.62763 98902 40 0.79411 96155 43 46 0.1 1654 40861 1.05889 07481 0.641 I I 98356 41 0.81397 26059 44 46 0.1 1 69 1 95649 I. 06134 78029 0.65440 68220 42 0.83382 55963 45 47 0.11714 98662 I. 0638 I 69550 0.66749 67282 43 0.85367 85867 46 47 0.1 1723 57096 1.06629 51962 0.68038 54871 44 0.87353 15771 47 48 0.11717 79914 1.06877 95074 0.69306 90869 45 90-r 0.89338 45674 48 48 0.1 1697 77784 I. 07126 68617 0.70554 35725 Fi/- "A G(r) C(r) B(r) Smithsonian Tables TABLE e = 40° q = 0.033265256695577, 9 = 0. 9334719356, 275 HK = 0. 8550825245 B(r) ^ C(r) G(r) ^ Fi/' 90-r 90 I. 00000 00000 1.14254 42177 0.00000 00000 90° 0' 1.78676 91349 0,99984 63487 I. 14250 07942 0.00382 84907 89 8 I. 76691 61445 89 0.99938 54451 I . 14237 05769 0.00765 31872 88 15 1.74706 31541 88 0.99861 74408 1.14215 37243 0. 01 147 02963 87 23 \.-]2Tzi 01637 87 0-99754 25881 I .14185 05008 0.01527 60269 86 30 1-70735 71733 86 0.99616 12401 I . 14146 12760 0.01906 65913 85 38 1.68750 41829 85 0.99447 38506 I . 14098 65243 0.02283 82057 84 46 I .66765 I I 926 84 0.99248 09734 I . 14042 68243 0.02658 70918 83 53 I .64779 82022 83 0.99018 32628 1,13978 28584 0.03030 94781 83 I I .62794 521 18 82 0.98758 14726 I. 13905 54113 0.03400 16009 82 8 1.60809 22214 81 0.98467 64560 I. I 3824 53698 o- 03765 97054 81 16 1.58823 92310 80 0.98146 91652 I. 13735 37211 0.04128 00477 80 23 I . 56838 62406 79 0.97796 06509 I. 13638 15521 0.04485 88958 79 30 1-54853 32502 78 0.97415 20616 I -13533 00476 0.04839 25314 78 37 1.52868 02598 77 0.97004 46432 I . 13420 04893 0.05187 72514 77 44 I . 50882 72694 76 0.96563 97386 I . 13299 42539 0-05530 93702 76 51 1.48897 42791 75 0.96093 87866 I .13171 28116 0.05868 52206 75 57 I. 46912 12887 74 0.95594 33213 I. I 3035 77242 0.06200 I 1573 75 4 I .44926 82983 73 0.95065 49716 1,12893 06433 0-06525 35577 74 10 I. 42941 53079 72 0- 94507 54604 1,12743 33082 0.06843 88251 73 17 1.40956 23175 71 0.93920 66032 1,12586 75438 0-07155 33910 72 23 I 38970 93271 70 0. 93305 03082 1,12423 52584 0-07459 37177 71 29 1-36985 63367 69 0.92660 85744 1,12253 84414 0- 07755 6301 I 70 34 1-35000 33463 68 0.91988 34913 1,12077 ,91607 0.08043 76736 69 40 I.330IS 03560 67 0.91287 72377 I. I I 895 95604 0-08323 44077 68 45 I. 31029 73656 66 0.90559 20807 1.11708 18582 0.08594 3II88 67 51 1.29044 43752 65 0.89803 03745 1.11514 83422 0.08856 04692 66 56 1.27059 13848 64 0.89019 45598 1.11316 13690 0.09108 3I7I4 66 1-25073 83944 63 0.88208 7I6I8 1.11112 33599 0-09350 79923 65 5 I .23088 54040 62 0.87371 07901 I . 10903 67986 0-09583 17573 64 9 I.2II03 24136 61 0.86506 81367 I. 10690 42279 0.09805 13545 63 14 I.I9II7 94233 60 0.85616 I975I I . 10472 82465 O.IOOI6 37391 62 18 I.I7I32 64329 59 0.84699 51593 1.10251 15061 O.I02I6 59383 61 21 1-15147 34425 58 0.83757 06220 I . 10025 67080 0.10405 50557 60 25 I -I3I62 04521 57 0.82789 13739 1.09796 65999 0.10582 82770 59 28 I . III76 74617 56 0.81796 05020 1.09564 39724 0.10748 28746 58 32 I.09I9I 44713 55 0.80778 I 1684 1.09329 16556 O.I090I 62132 57 34 1.07206 14809 54 0.79735 66091 I. 09091 25160 O.I 1042 57553 56 37 1.05220 84905 53 0.78669 01322 1.08850 94525 0.1 1 170 90668 55 39 I -03235 55001 52 0.77578 5II73 I. 08608 53932 0.1 1286 38228 54 42 I. 01250 25098 51 0.76464 50133 1.08364 32917 0.1 1388 78137 53 44 0-99264 95194 50 0.75327 33376 1.08118 61237 0.1 1477 8951 I 52 45 0.97279 65290 49 0.74167 36742 I. 0787 I 68830 0-I1553 52736 51 46 0.95294 35386 48 0.72984 96728 1.07623 85782 0.11615 49535 50 46 0.93309 05482 47 0.71780 50468 I -07375 42288 0.1 1663 63025 49 47 0.91323 75578 46 0.70554 35725 I. 07126 68617 0.1 1697 77784 48 48 0.89338 45674 45 A(r) D(r) E(r) ^ F<|) r Smithsonian Tables 270 ELLIPTIC FUNCTION K = K' = 1. 8540746773, E = E' = 1. 3506438810, r F0 4> E(r) D(r) A(r) 0.00000 00000 0° 0' 0.00000 00000 I. 00000 00000 0.00000 00000 I 0.02060 08297 I II 0.00559 22185 I .00005 761 14 0.01732 23240 2 0.04120 16595 2 22 0.01117 56998 1.00023 03752 0.03463 96092 3 0.06180 24892 3 32 0.01674 17286 I. 00051 80814 0.05194 68175 4 0.08240 33190 4 43 02228 16343 I .00092 03796 0.06923 89126 5 0.10300 41487 5 54 0.02778 68124 I. 00143 67802 0.08651 0861 I 6 0.12360 49785 7 4 0.03324 87460 1.00206 66547 0. 10375 76329 7 0. 14420 58082 8 15 0.03865 90273 I .00280 92364 0. 12097 42023 8 0.16480 66380 9 25 0.04400 93780 1.00366 36213 0.13815 55494 9 0.18540 74677 10 36 0.04929 16689 I . 00462 87696 0.15529 66598 10 0.20600 82975 II 46 0.05449 79400 1.00570 35065 0.17239 25270 II 0.22660 91272 12 56 0.05962 04166 1.00688 65237 0.18943 81524 12 0.24720 99570 14 6 0.06465 15306 I .00817 63813 0.20642 85463 13 0.26781 07867 15 15 0.06958 39334 1.00957 I 509 I 0.22335 87294 14 0.28841 16165 16 25 0.07441 05129 I. 01 107 02088 0.24022 37330 15 0.30901 24462 17 34 0.07912 44078 I .01267 06562 0.25701 86008 16 0.32961 32760 18 43 0.08371 90207 I. 01437 09030 0.27373 83893 17 0.35021 41057 19 52 0.08818 80301 1.01616 88793 0.29037 81691 18 0.37081 49355 21 I 0.09252 54012 I. 01 806 23965 0- 30693 30262 19 0.39141 57652 22 9 0.09672 53955 I .02004 91494 0- 32339 80622 20 0.41201 65950 23 17 0.10078 25794 I. 02212 67193 0.33976 83967 21 0.43261 74247 24 25 0.10469 18308 1.02429 25769 0.35603 91671 22 0.45321 82545 25 33 0.10844 83455 1.02654 40853 0.37220 55308 23 0.47381 90842 26 40 0.1 1204 76417 1.02887 85035 0.38826 26656 24 0.49441 99139 27 47 0. II 548 55630 I. 03 I 29 29893 0.40420 57714 25 0.51502 07437 28 54 0.11875 82813 1.03378 46028 0.42003 007 I I 26 0.53562 15734 30 0.^186 22978 1.03635 03103 0-43573 08120 27 0.55622 24032 31 6 0.12479 44425 I . 03898 69880 0.45130 32670 28 0.57682 32329 32 12 0.12755 18736 I .04169 14251 0.46674 27359 29 0.59742 40627 33 17 0.13013 20757 1 .044/16 03288 0.48204 45468 30 0.61802 48924 34 22 0.13253 28561 1.04729 03271 0.49720 40572 31 ' 0.63862 57222 35 27 0.13475 23413 I 05017 79739 0.51221 66556 32 0.65922 65519 36 32 0.13678 89725 I 05311 97528 0.52707 77628 33 0.67982 73817 37 36 0.13864 14993 1.05611 20812 0.54178 28334 34 0.70042 821 14 38 39 0.14030 89744 I 05915 13149 0- 55632 73569 35 0.72102 90412 39 43 0. 141 79 07457 1.06223 37524 0.57070 68597 36 0.74162 98709 40 46 0.14308 64509 I 06535 56397 0.58491 69061 37 0.76223 07007 41 48 0. 14419 60059 I. 06851 31742 0- 59895 31001 38 0.78283 15304 42 51 0.145 1 1 96000 I. 07170 25103 0.6I28I 10868 39 0.80343 23602 43 54 0.14585 76849 I 07491 97630 0.62648 65539 40 0.82403 31899 44 54 0.14641 09671 I. 07816 10137 0- 63997 52334 41 0.84463 40197 45 55 0.14678 03964 I. 08142 23139 0.65327 29030 42 0.86523 48494 46- 56 0.14696 71583 1.08469 96910 0.66637 53880 43 0.88583 56792 47 57 0.14697 26631 1.08798 91523 0.67927 85625 44 0.90643 65089 48 57 0- 14679 85365 I .09128 66907 0.69197 83514 45 0.92703 73387 49 57 0. 14644 66094 1.09458 82886 0.70447 07318 90-r Fi^ 4^ G(r) C(r) B(r), Smithsonian Tables TABLE d = 45° q = e-^ = 0. 04321391826377, 0=0. 9135791382, HK:^= 0. 9135791382 277 B(r) C(r) G(r) 1^ Fi/- 90-r 90 I . 00000 00000 I. 18920 71150 0.00000 00000 90° 0' 1.85407 46773 0-99984 54246 1.18914 94665 0.00470 60108 89 10 1-83347 38476 89 0-99938 I75I4 I. 18897 65912 0.00940 76502 88 20 1.81287 30178 88 0.99860 91406 I. I 8868 87000 0.01410 05467 87 30 1.79227 21881 87 0.99752 78584 I. 18828 61440 0.01878 03289 86 40 I -77167 13583 86 0.99613 82775 1. 1 8776 94140 0.02344 26255 85 49 I -75107 05286 85 0.99444 08767 1.18713 91403 0.02808 30653 84 59 I . 73046 96988 84 0.99243 62407 I .18639 60914 0.03269 72774 84 9 1.70986 88691 83 0.99012 50593 1. 18554 1 1736 0.03728 08916 83 18 1.68926 80393 82 0.98750 81276 I. 18457 54293 0.04182 95382 82 28 1.66866 72096 81 0.98458 63450 I. I 8350 00363 0.04633 88487 81 37 1.64806 63798 80 0.98136 07I5I I. I 823 I 63059 0.05080 44575 80 47 1.62746 55501 79 0.97783 23446 1.18102 56817 0.05522 19994 79 56 I . 60686 47203 78 0.97400 24430 1. 1 7962 97376 0-05958 71139 79 5 1.58626 38906 77 0.96987 23216 1.17813 01756 0.06389 54439 78 14 I -56566 30608 76 0.96544 33929 I . 17652 88244 0.06814 26379 77 23 1-54506 2231 1 75 0.96071 71696 1.17482 76366 0.07232 43506 76 32 1.52446 14013 74- 0-95569 52639 I. I 7302 86866 0.07643 62449 75 40 1.50386 05716 73 0-95037 93863 I . 17113 41680 0.08047 39933 74 48 1.48325 97418 72 0-94477 13447 1.16914 63907 0.08443 32799 73 57 I .46265 89121 71 0-93887 30433 I. I 6706 77783 0.08830 98027 73 5 1.44205 80823 70 0.93268 64814 I . 16490 08653 0.09209 92756 72 13 1.42145 72526 69 0.92621 37526 I. 16264 82937 0-09579 74315 71 20 1 .40085 64228 68 0.91945 70430 1.16031 2809? 0.09940 00252 70 27 1-38025 55931 67 0.91241 .86305 1-15789 72608 0. 10290 28362 69 34 I -35965 47634 66 0.90510 08831 I . 15540 45920 0.10630 16727 68 41 1-33905 39336 65 0.89750 62579 I. 15283 78419 0.10959 23752 67 48 1.31845 31039 64 0.88963 72995 I . 15020 01398 0.1 1277 08206 66 54 1.29785 22741 63 0.88149 66386 I -14749 4701 I 0.11583 29266 66 1.27725 14444 62 0.87308 69906 I. 14472 48239 0. II 877 46567 65 6 1.25665 06146 61 0.86441 I I 542 I .14189 38846 0.12159 20252 64 II 1.23604 97849 60 0.85547 20099 I - 13900 53339 0.12428 I 1025 63 16 1.21544 89551 59 0.84627 25182 I . 13606 26928 0.12683 80211 62 21 1.19484 81254 58 0.83681 57184 I. I 3306 95480 0.12925 89815 61 26 1.17424 72956 57 0.82710 47269 I . 13002 95477 0.13154 02588 60 30 I -15364 64659 56 0.81714 27355 I. 12694 63970 0.13367 82099 59 34 I- 13304 56361 55 0.80693 30099 I. 12382 38537 0.13566 92789 58 38 1 .11244 48064 54 0.79647 88881 1,12066 57231 0.13751 00077 57 42 1.09184 39766 53 0.78578 37785 1. 1 1747 S8542 0.13919 70407 56 45 I. 07124 31469 52 0.77485 I 1587 I. I 1425 81342 0.14072 71344 55 47 1.05064 23171 51 0.76368 45735 i.iiioi 64844 0.14209 71663 54 50 1.03004 14874 50 0.75228 76332 I . 10775 48548 0.14330 41415 53 52 I .00944 06576 49 0.74066 40121 I. 10447 72199 0-14434 52037 52 53 0.98883 98279 48 0.72881 74469 1.10118 75735 0.14521 76436 51 55 0.96823 89981 47 0.71675 17348 1.09788 99237 0.14591 89078 50 56 0.94763 81684 46 0.70447 07318 1.09458 82886 0.14644 66094 49 57 0.92703 73387 45 r A(r) D(r) E(r) 4> F<^ Smithsonian Tables 278 ELLIPTIC FUNCTION K = 1.9355810960, K' = 1 . 7867691349, E = 1.3055390943, E' = 1.3931402485, r F0 4> E(r) D(r) A(r) 0.00000 00000 0" 0' 0.00000 00000 I . 00000 00000 0.00000 00000 I 0.02150 64566 I 14 0.00699 85212 I .00007 52700 0.01724 17831 2 0.04301 29132 2 28 0-01398 53763 1.00030 09884 0.03447 86990 3 0.06451 93699 3 41 0.02094 89334 I . 00067 68809 0.05170 58810 4 0.08602 58265 4 55 0.02787 76288 I .00120 24903 0.06891 84630 5 0.10753 22831 6 9 0.03476 00006 I. 00187 71775 0.0861 I 15805 6 0.12903 87397 7 22 0.04158 42717 I .00270 01222 0.10328 03705 7 0.15054 51963 8 36 0.04834 06320 1.00367 03237 0.12041 99725 8 0.17205 16530 9 49 0.05501 67694 I . 00478 66023 0-13752 55283 9 0- 19355 81096 II 3 0.06160 24003 I . 00604 76005 0.15459 21831 10 0.21506 45662 12 16 0.06808 70479 1.00745 17850 0.17161 50856 II 0.23657 10228 13 28 0.07446 05194 I . 00899 74482 0.18858 93888 12 0.25807 74795 14 41 0.08071 29320 I .01068 27105 0.20551 02505 13 0.27958 39361 15 53 0.08683 47367 I. 01250 55225 0.22237 28335 14 0.30109 03927 17 6 0.09281 67403 I. 01446 36673 0.23917 23067 15 0.32259 68493 18 18 0.09865 01256 I. 01655 47635 0.25590 38457 16 0.34410 33059 19 29 0.10432 64694 I .01877 62678 0.27256 26330 17 0.36560 97626 20 40 0.10983 77593 I .02112 54784 0.28914 38591 18 0.38711 62192 21 51 0. 11517 64068 I -02359 95379 0.30564 27234 19 0.40862 26758 23 2 0.12033 52604 I. 02619 54370 0.32205 44344 20 0.43012 91324 24 13 0.12530 76146 I. 02891 00179 0-33837 421 10 21 0.45163 55891 25 22 0.13008 72182 I -03173 99787 0-35459 72832 22 0.47314 20457 26 31 0.13466 82799 1.03468 18764 0.37071 88930 23 0.49464 85023 27 41 0.13904 54724 1.03773 21323 0.38673 42953 24 0-51615 49589 28 50 0-1432 I 39340 1.04088 70352 0.40263 87589 25 0.53766 14155 29 59 0. 14716 92687 I .04414 27466 0.41842 75678 26 0.55916 78722 31 6 0-15090 75443 1.04749 53052 0.43409 60218 27 0.58067 43288 32 14 0.15442 52892 1.05094 06315 0.44963 94381 28 0.60218 07854 33 21 0-I5771 94871 1-05447 45329 0.46505 31522 29 0.62368 72420 34 29 0.16078 75703 I .05809 27090 0.48033 25191 30 0.64519 36987 35 36 0.16362 74123 I. 06179 07561 0.49547 29148 31 0.66670 01553 36 41 0.16623 73178 1.06556 41737 0.51046 97376 32 0.68820 66119 37 46 0.16861 60131 I . 06940 83686 0.52531 84091 33 0.70971 30685 38 51 0. 17076 26341 I. 07331 86617 0.54001 43761 34 0.73121 95251 39 56 0.17267 67142 I .07729 02929 0-55455 31119 35 0-75272 59818 41 I 0.17435 81713 1.08131 84270 0.56893 01 177 36 0.77423 24384 42 4 0.17580 72936 1.08539 81601 0.58314 09242 37 0.79573 88950 43 7 0.17702 47258 1-08952 45247 0.59718 10935 38 0.81724 53516 44 9 0.17801 14536 1.09369 24965 0.61 104 62201 39 0.83875 18083 45 12 0.17876 87890 I .09789 70001 0.62473 19335 40 0.86025 82649 46 15 0.17929 83544 1.10213 29153 0.63823 38991 41 0.88176 47215 47 15 0.17960 20675 I. I 0639 50831 0.65154 78204 42 0.90327 11781 48 16 0.17968 21252 I .11067 83124 0.66466 94406 43 0.92477 76347 49 16 0.17954 09878 I. I 1497 73861 0-67759 45449 44 0.94628 40914 50 17 0.17918 13641 I. I 1928 70673 0.69031 89618 45 0.96779 05480 51 17 0.17860 61952 1.12360 21058 0.70283 85652 90-r Fi^ ^ G(r) C(r) B(r) Smithsonian Tables TABLE e = 50° q = 0. 055019933698829, 60= . 8899784604, HK = 0. 9715669451 279 B(r) C(r) G(r) 4^ Fi/' 90-r I . 00000 ooooo 1.24728 65857 0.00000 ooooo 90° 0' 1.93558 10960 90 0.99984 40186 I. 24721 12154 0.00561 92362 89 12 I. 9 I 407 46394 89 0.99937 6I3I9 1.24698 51964 0. 01 123 36482 88 25 1.89256 81828 88 0.99859 65127 1.24660 88048 0.01683 84106 87 37 I. 87106 17261 87 0.99750 54487 1.24608 24999 0.02242 89646 86 50 1.84955 52695 86 0.99610.33424 1.24540 69243 0.02799 96670 86 2 1.82804 88129 85 0.99439 07108 1.24458 29027 0.03354.64884 85 14 1.80654 23563 84 0.99236 81849 I. 24361 14410 0.03906 43123 84 26 1.78503 58997 83 0.99003 65093 1.24249 37250 0.04454 82835 83 39 1.76352 94430 82 0.98739 65416 I. 24123 11192 0.04999 35367 82 51 1.74202 29864 81 0.98444 92517 1.23982 51648 0.05539 5I96I 82 3 I. 7205 I 65298 80 0.98II9 57210 1.23827 75779 0.06074 83740 81 14 I .69901 00732 79 0.97763 7I4I7 1.23659 02476 0.06604 81700 80 26 1.67750 36165 78 0.97377 48160 1.23476 52334 0.07128 96708 79 37 1.65599 71599 77 0.96961 01546 I .23280 47629 0.07646 79497 78 49 1.63449 07033 76 0.96514 46762 I. 23071 12287 0.08157 80662 78 I .61298 42467 75 0.96038 00059 1.22848 71860 0.08661 50665 77 10 I. 59147 77901 74 0.95531 78745 I. 22613 53491 0.09157 39836 76 21 1.56997 13334 73 0.94996 01 167 1.22365 85882 0.09644 98379 75 31 1.54846 48768 72 0.94430 86698 I. 22 105 99257 0.10123 76383 74 42 1.52695 84202 71 0.93836 55727 I. 21834 25328 0.10593 23833 73 52 1.50545 19636 70 0.93213 29639 I. 21550 97252 0.1 1052 90627 73 I 1.48394 55069 69 0.92561 30802 I. 21256 49596 0.1 I 502 26595 72 II 1.46243 90503 68 0.91880 82552 I. 2095 I 18289 0.11940 81521 71 20 1.44093 25937 67 0.9II72 09173 1.20635 40582 0.12368 05174 70 30 I. 41942 61371 66 0.90435 35883 1.20309 54999 0.12783 47335 69 39 I. 39791 96805 65 0.89670 88815 I. 19974 01294 0.13186 57834 68 47 I. 37641 32238 64 0.88878 94998 I . 19629 20396 0.13576 86595 67 55 1.35490 67672 63 0.88059 82341 I. 19275 54368 0.13953 83674 67 2 1.33340 03106 62 0.87213 79612 1.18913 46345 0.14316 99314 66 10 I. 31 189 38540 61 0.86341 16420 I. I 8543 40490 0.14665 83999 65 18 1.29038 73973 60 0.85442 23195 1.18165 81935 0.14999 88516 64 24 1.26888 09407 59 0.84517 3II66 1.17781 16727 0.15318 64017 63 30 1.24737 44841 58 0.83566 72345 I. 17389 91774 0.1 562 I 62095 62 36 1.22586 80275 57 0.82590 79506 I. I 6992 54783 0.15908 34859 61 42 1.20436 15709 56 0.81589 86I6I I. I 6589 54205 0.16178 35017 60 48 I. 18285 51142 55 0.80564 26543 1.16181 39175 0.16431 15964 59 52 1.16134 86576 54 0.79514 35583 1. 1 5768 59453 0.16666 31878 58 56 I .13984 22010 53 0.78440 48891 I. I 535 I 65361 0.16883 37818 58 1. 1 1833 57444 52 0.77343 02735 1. 1493 1 07723 0.1 708 I 89832 57 4 1,09682 92877 51 0.76222 34019 I . 14507 37802 0.1 7261 45069 56 8 1.07532 28311 50 0.75078 80264 I. I 408 I 07240 0.17421 61892 55 10 I. 05381 63745 49 48 0.73912 79584 I. 13652 67992 0.17562 00006 54 12 1.03230 99179 0.72724 70671 I. 13222 72263 0.17682 20583 53 13 I. 01080 34613 47 0,71514 92767 1.12791 72446 0.17781 86395 52 15 0.98929 70046 46 0.70283 85652 I. 12360 21058 0.17860 61952 51 17 0.96779 05480 45 A(r) D(r) E(r) 4> F<^ r Smithsonian Tables 28o ELLIPTIC FUNCTION K = 2.0347163122, £' = 1.7312451757, E = 1.2586796248, E' = 1.4322909693, r F<^ E(r) D(r) A(r) 0.00000 00000 0' 0' 0.00000 00000 I . 00000 00000 0.00000 00000 I 0.02260 79479 I 18 0.00862 00346 I . 00009 74600 0.01712 13223 2 0.04521 58958 2 35 0.01722 45749 I .00038 97217 0.03423 80342 • 3 0.06782 38437 3 53 0.02579 81795 1.00087 64305 0.05134 55249 4 0.09043 17916 5 10 0.03432 55123 I. 00155 69957 0.06843 91832 5 0.1 1303 97395 6 28 0.04279 13942 1.00243 05914 0.08551 43971 6 0.13564 76875 7 45 0.05118 08539 1.00349 61575 0.10256 65538 7 0.15825 56354 9 2 0.05947 91769 1.00475 24006 0.11959 10390 8 0.18086 35833 10 19 0.06767 19530 I .00619 77962 0.13658 32373 9 0.20347 15312 II 36 0.07574 51216 1.00783 05901 0.15353 85318 10 0.22607 94791 12 52 0.08368 50144 1.00964 88003 0.17045 23039 II 0.24868 74270 14 9 0.09147 83960 I .01165 02201 0.1 873 I 99332 12. 0.27129 53749 15 25 0.09911 25013 I. 01383 24199 0,20413 67975 13' 0-29390 33229 16 40 0.10657 50694 1.01619 27508 0.22089 82730 14 0.31651 12708 17 56 011385 43755 I. 01872 83473 0.23759 97340 15 0.33911 92187 19 II 0.12093 92580 I. 02143 61311 0.25423 65532 16 0.36172 71666 20 25 0.12781 91435 I. 02431 28147 0.27080 41017 17 0.38433 51145 21 40 . 13448 40670 1.02735 49050 0.28729 77496 18 0.40694 30624 22 54 0. 14092 46901 1.03055 87080 0.30371 28656 19 0.42955 10103 24 7 0.14713 23140 1.03392 03331 0.32004 48178 20 0.45215 89583 25 20 0.15309 88906 1.03743 56974 0.33628 89743 21 - 0.47476 69062 26 33 0.1 588 I 70288 1.04110 05314 0.35244 07031 22 0.49737 48541 27 45 0.16427 99989 I. 0449 I 03831 0.36849 53729 23 0.51998 28020 28 56 0.16948 17327 1.04886 06244 0.38444 83538 24 0.54259 07499 30 8 0.1 7441 68208 1.05294 64558 0.40029 50181 25 0.56519 86978 31 18 0.17908 05075 I. 05716 29130 0.41603 07408 26 0.58780 66457 32 28 0.18346 86827 I. 06150 48720 0.43165 09003 27 0.61041 45937 33 38 0.18757 78710 1.06596 70560 0.44715 08801 28 0.63302 25418 34 46 0. 19140 52188 1.07054 40415 0.46253 60691 29 0.65563 04895 35 55 0.19494 84794 1.07523 02647 0.47777 18627 30 0.67823 84374 37 3 0.19820 59959 I .08002 00285 0.49288 36645 31 0.70084 63853 38 10 0.20117 66827 I .08490 75092 0.50785 68872 32 0.72345 43332 39 16 0.20386 00053 1.08988 67634 0.52268 69541 33 0.74606 2281 1 40 23 0.20625 59591 1.09495 17358 0.53736 93004 34 0.76867 02290 41 28 0.20836 50468 I . 10009 62656 0.55189 93747 35 0.79127 81769 42 33 0.21018 82554 1.10531 40947 0.56627 26408 36 0.81388 61249 43 38 0.21 172 70324 1.11059 88749 0,58048 45794 37 0.83649 40728 44 41 0.21298 32611 1. 1 1594 41760 0.59453 06894 38 0.85910 20207 45 45 0.21395 92364 1.12134 34929 0,60840 64905 39 0.88170 99686 46 48 0.21465 76400 I. I 2679 02542 0,62210 75244 40 0.90431 79165 47 50 0.21508 15155 I. 13227 78297 0,63562 93'57i 41 0.92692 58644 48 51 0.21523 42440 I. 13779 95386 0,64896 75812 42 0.94953 38123 49 53 0.21511 95200 I. I 4334 86579 0.66211 78175 43 0.97214 17602 50 53 0.21474 13276 I . I 489 I 84299 0.67507 57177 44 0.99474 97081 51 53 0.21410 39170 I. 15450 2071 I 0.68783 69663 45 I. 01735 76561 52 52 0.21321 17818 I . 16009 27802 0,70039 72833 90-r Fi/' 4^ G(r) C(r) B(r) • Smithsonian Tables TABLE e = 55° g =0. 069042299609032, 9 0=0. 8619608462, HK = 1. 0300875730 281 B(r) C(r) G(r) i' Fi^ 90 -r 90 I . 00000 00000 1.32039 64540 0.00000 00000 90° 0' 2.03471 53122 0.99984 19155 1.32029 87371 0.00654 66917 89 15 2.01210 73643 89 0.99936 77261 1.32000 57060 0.01308 82806 88 31 1.98949 94164 88 0.99857 76238 1.31951 77192 0.01961 96606 87 46 1.96689 14685 87 0.99747 19280 I. 31883 53734 0.02613 57182 87 I 1.94428 35205 86 0.99605 10861 I. 31795 95033 0.03263 13295 86 17 I. 92167 55726 85 0.99431 56720 I. 31689 11801 0.03910 13564 85 32 I . 89906 76247 84 0.99226 63864 1.31563 17106 0.04554 06434 84 47 1.87645 96768 83 0.98990 40553 1.31418 26349 0.05194 40144 84 2 1.85385 17289 82 0.98722 96302 I. 31254 57253 0.05830 62693 83 17 I. 83124 37810 81 0.98424 41 861 I. 31072 29838 0.06462 21812 82 32 1.80863 58331 80 0.98094 89213 I. 3087 I 66392 0.07088 64934 81 46 1.78602 78851 79 0.97734 51558 1.30652 91449 0.07709 39167 81 I I. 76341 99372 78 0.97343 43300 I. 30416 31759 0.08323 91270 80 15 I. 7408 I 19893 77 0.96921 80039 I. 30162 16250 0.08931 67629 79 29 I. 7 I 820 40414 76 0.96469 78546 1.29890 75994 0.09532 14240 78 43 1.69559 60935 75 0.95987 56758 1.29602 44173 0. 10124 76688 77 56 1.67298 81456 74 0.95475 33753 1.29297 56032 0.10709 00133 77 10 1.65038 01977 73 0.94933 29736 1.28976 48840 0. 1 1284 29301 76 23 . 1.62777 22497 72 0.94361 66021 1.28639 61840 0.1 I 850 08473 75 35 I. 60516 43018 71 0.93760 65006 1.28287 36204 0.12405 81487 74 48 1.58255 63539 70 0.93130 50161 I .27920 14980 0.12950 91731 74 1.55994 84060 69 0.92471 45998 1.27538 43041 ' 0.13484 82153 73 12 1.53734 04581 68 0.91783 78055 I .27142 67027 0.14006 95267 72 23 1.51473 25102 67 0.91067 72870 1.26733 35291 0.14516 73172 71 35 I. 492 I 2 45623 66 0.90323 57961 I. 26310 97835 0.1 5013 57566 70 46 I. 46951 66144 65 0.89551 61797 1.25876 06253 0.15496 89777 69 56 1.44690 86665 64 0.88752 13778 1.25429 13663 0.15966 10790 69 7 1.42430 07185 63 0.87925 44206 I . 24970 74646 0.16420 61290 68 16 I. 40169 27706 62 0.87071 84265 I. 24501 45176 0.16859 81701 67 26 1.37908 48227 61 0.86191 65988 I. 2402 I 82552 0.17283 12244 66 35 1.35647 68748 60 0.85285 22237 1.23532 45329 0.17689 92991 65 43 1.33386 89269 59 0.84352 86672 1.23033 93242 0.18079 63935 64 51 I .31126 09790 58 0.83394 93726 1.22526 87137 0.1 845 I 65064 63 59 1.28865 303 1 1 57 0.8241 I 78578 I. 2201 I 88895 0.18805 36444 63 6 1.26604 50832 56 0.81403 77126 I. 21489 61356 0.19140 18312 62 12 1.24343 71353 55 0.80371 25960 I . 20960 68240 0.19455 51177 61 19 I .22082 91873 54 0.79314 62334 1.20425 74072 0.19750 75927 60 24 I . 19822 12394 53 0.782-54 24136 I. 19885 44102 0.20025 33955 59 30 1.17561 32915 52 0.77130 49868 I . 19340 44225 0.20278 67279 58 35 I. 15300 53436 51 0.76003 78612 0.74854 50007 1.18791 40899 I. 18239 01066 0.20510 18688 0.20719 31885 57 56 39 42 I . 13039 73957 I . 10778 94478 1. 085 1 8 14999 1.06257 35519 1.03996 56041 50 49 48 47 46 0.73683 04220 0.72489 81922 0.71275 24260 I. I 7683 92068 1.17126 81567 I . 16568 37461 0.20905 51650 0.21068 24001 0.21206 96376 55 54 53 46 48 50 0.70039 72833 I. 16009 27802 0.21321 17818 52 52 I. 01735 76561 45 r A(r) D(r) E(r) * F<^ Smithsonian Tables 282 ELLIPTIC FUNCTION K = 2. 1565156475, K' = 1. 6857503548, E = 1. 211056028, E' = 1.4674622093, r F E(r) D(r) A(r) 0.00000 00000 0° 0' 0.00000 00000 I . 00000 00000 0.00000 00000 I 0.02396 12850 I 22 0.01050 21636 I. 00012 58452 0.01694 24822 2 0.04792 25699 2 45 0.02098 36904 1.00050 32288 0.03388 07351 3 0.07188 38549 4 7 0.03142 40274 1.00113 16945 0.05081 05279 4 0.09584 51399 5 29 0.04180 27880 I. 00201 04822 0.06772 76275 5 0.1 1980 64248 6 51 0.05209 98337 I. 00313 85295 0.08462 77970 6 0.14376 77098 8 13 0.06229 53533 I 00451 44723 0. 10150 67944 7 0.16772 89948 9 35 0.07236 99392 I .00613 66468 0.1 1836 03717 8 0.19169 02798 10 56 0.08230 46606 1.00800 309 I I 0.13518 42734 9 0.21565 15647 12 17 0.09208 I 1326 i.oioii 15480 0.15197 42358 lO 0.23961 28497 13 38 0.10168 15801 I. 01245 94672 0.16872 59855 II 0.26357 41347 14 58 0.1 I 108 88976 I .01504 40088 0.18543 52386 12 0.28753 54197 16 18 0.12028 67034 I. 01786 20463 0.20209 76999 13 0.31 149 67046 17 38 0.12925 93879 I. 02091 01701 0.21870 90619 14 0.33545 79896 18 57 0.13799 21563 I .02418 46923 0.23526 50037 15 0.35941 92746 20 16 0.14647 10652 1.02768 16504 0.25176 11911 16 0.38338 05595 21 35 0.15468 30530 I. 03139 68120 0.26819 32750 17 0.40734 18445 22 53 0.16261 59647 I 03532 56803 0.28455 68916 18 0.43130 31295 24 10 0.17025 85702 1.03946 34991 0.30084 76617 19 0.45526 44145 25 26 0.17760 05773 1.04380 52583 0.31706 11903 20 0.47922 56994 26 42 0.18463 26382 1.04834 57003 0.33319 30665 21 0.50318 69844 27 58 0.19134 63517 I 05307 93260 0.34923 88634 22 0.52714 82694 29 13 0.19773 42593 . I .05800 04010 0.36519 41381 23 0.551 10 95544 30 27 0.20378 98371 I. 063 10 29632 0.38105 44318 24 0.57507 08393 31 41 0.20950 74827 I .06838 08291 0.39681 52701 25 0.59903 21243 32 54 0.21488 24988 1.07382 76019 0.41247 21633 26 0.62299 34093 34 7 0.21991 10718 1.07943 66784 0.42802 06069 27 0.64695 46942 35 18 0.22459 02484 1.08520 12575 0.44345 60826 28 0.67091 59792 36 29 0.22891 79082 I .09111 43480 0.45877 40585 29 0.69487 72642 37 39 0.23289 27342 I. 09716 87771 0.47396 99905 30 0.71883 85492 38 49 0.23651 41807 I .10335 71989 0.48903 93230 31 0.74279 98341 39 58 0.23978 24399 I . 10967 21031 0. 50397 74905 32 0.76676 11191 41 6 0.24269 84060 I .11610 58243 0.51877 99184 33 0.79072 24041 42 13 0.24526 36394 I. 12265 05510 0.53344 20249 34 0.81468 36890 43 20 0.24748 03283 I. 12929 83350 0.54795 92224 35 0.83864 49740 44 26 0,24935 12513 I .13604 I 1010 0.56232 69191 36 0.86260 62590 45 31 0.25087 97387 1.14287 06563 0.57654 05212 37 0.88656 75440 46 35 0.25206 96336 I. 14977 87007 0. 59059 54347 38 0.91052 88289 47 39 0.25292 52540 I. 15675 68364 0.60448 70673 39 0.93449 01 139 48 42 0.25345 13545 1.16379 65783 0.61821 08313 40 0.95845 13989 49 44 0. 25365 30884 1.17088 93642 0.63176 21451 41 0.98241 26838 50 45 0.25353 59713 1.17802 65652 0.64513 64364 42 1.00637 39688 51 46 0.25310 58450 I. 18519 94959 0.65832 91446 43 I 03033 52538 52 46 0.25236 88429 1.19239 94253 0.67133 57232 44 1.05429 65388 53 45 0.25133 13558 1.19961 75873 0.68415 16433 45 90-1 1.07825 78237 ■ 54 44 0.25000 00000 1.20684 51910 0.69677 23959 Fi/- i^ ^ G(r) C(r) B(r) Smith soNiAN Tables • J TABLE e = 60° g = 0. 085795733702195, 9 = 0. 8285168980, HK = 1. 0903895688 283 B(r) C(r) G(r) i' Fi/' 90-r 90 I . 00000 00000 1.41421 35624 0.00000 00000 90° 0' 2. I 565 I 56475 0.99983 87925 I .41408 70799 0.00746 45017 89 19 2.13255 43625 89 0.99935 52434 I. 41370 77878 0.01492 38646 88 38 2.10859 30775 88 0.99854 95732 I. 41307 61515 0.02237 29430 87 57 2.08463 17926 87 0.99742 21491 I .41219 29466 0.02980 65777 87 16 2.06067 05076 86 0.99597 34843 I. 41 105 92570 0.03721 95889 86 35 2.03670 92226 85 0.99420 42378 1.40967 64744 0.04460 67701 85 53 2.01274 79377 84 0.99211 52135 I .40804 62958 0.05196 28815 85 II 1.98878 66527 83 0.98970 73588 I .40617 07222 0.05928 26440 84 29 1.96482 53677 82 0.98698 17641 1.40405 20551 0.06656 07336 83 47 I . 94086 40827 81 0.98393 96610 I .40169 28947 0.07379 17757 83 5 I. 91690 27978 80 0.98058 24210 1.39909 61356 0.08097 03401 82 23 1.89294 15128 79 0.97691 15541 1.39626 49639 0.08809 09364 81 41 1.86898 02278 78 0.97292 87065 1.39320 28531 0.09514 80095 80 58 I. 84501 89429 77 0.96863 56591 1-38991 35592 0.10213 59353 80 15 I. 82 105 76579 76 0.96403 43250 1.38640 1 1 169 0.10904 90175 79 32 1.79709 63729 75 0.95912 67478 1.38266 98339 0.1 1588 14840 78 49 I -77313 50879 74 0.95391 50985 1.37872 42853 0.12262 74837 78 5 1-74917 38030 73 0.94840 16738 1.37456 93090 0.12928 10844 77 21 I. 72521 25180 72 0.94258 88926 1.37020 99983 0.13583 62697 76 37 I. 70125 12330 71 0.93647 92941 1.36565 16965 0.14228 69378 75 53 1.67728 99480 70 0. 93007 55342 1.36089 99899 0.14862 68991 75 8 1.65332 86631 69 0.92338 03829 1-35596 07006 0.15484 98749 74 23 1.62936 73781 68 0.91639 67210 1.35083 98797 0.16094 94967 73 37 1.60540 60931 67 0.90912 75372 1-34554 37995 0.1 669 I 93054 72 51 I. 58144 48082 66 0.90157 59245 1.34007 89457 0.17275 27505 72 5 1-55748 35232 65 0.89374 50771 1.33445 20094 0.17844 31913 71 18 1-53352 22382 64 0.88563 82868 1.32866 98789 0.18398 38964 70 30 1.50956 09532 63 0.87725 89396 1.32273 96308 0.18936 80462 69 42 1.48559 96683 62 0.86861 05122 I. 31666 85215 0.19458 87340 68 54 I. 46163 83833 61 0.85969 65682 I. 31046 39783 0.19963 89691 68 5 1.43767 70983 60 0.85052 07549 1.30413 35898 0.20451 16802 67 16 1.41371 58134 59 0.84108 67990 1.29768 50969 0.20919 97204 66 26 1-38975 45284 58 0.83139 85036 I. 291 12 63832 0.21369 58722 65 36 1-36579 32434 57 0.82145 97438 1.28446 54650 0.21799 28546 64 45 I. 34183 19584 56 0.81 127 44636 1. 27771 04815 0.22208 33313 63 53 I. 31787 06735 55 0.80084 66719 1.27086 96850 0.22595 99196 63 I 1.29390 93885 54 0.79018 04386 1.26395 14305 0.22961 52018 62 9 1.26994 81035 53 0.77927 98915 1.25696 41655 0.23304 17372 61 15 1.24598 68185 52 0.76814 92120 I. 24991 64194 0.23623 20761 60 21 1.22202 55336 51 0.75679 26317 I. 2428 I 67937 0.23917 87758 59 27 I. 19806 42486 50 0.74521 44290 1.23567 39504 0.24187 44177 58 32 i.17410 29636 49 48 0.73341 89253 1.22849 66025 0.24431 16265 57 36 I . 15014 16787 i3.-]2l/^l 04816 1.22129 35025 0.24648 30908 56 39 1.12618 03937 47 0.70919 34952 1. 2 1 407 34320 0.24838 15864 55 42 1.10221 .91087 46 0.69677 23959 1.20684 51910 0.25000 00000 54 44 1.07825 78237 45 r A(r) D(r) E(r) <^ F<^ . Smithsonian Tables 284 ELLIPTIC FUNCTION K = 2. '3087867982, K' = 1. 6489952185, E = 1. 1638279645, E' = 1. 4981149284, r F(> E(r) D(r) A(r) 0.00000 00000 0° 0' 0.00000 00000 I . 00000 00000 0.00000 00000 I 0.02565 31866 I 28 0.01271 71437 I .00016 31607 0.01667 62945 2 0.05130 63733 2 56 0.02540 65870 I .00065 24464 0.03334 89266 3 0.07695 95599 4 24 0.03804 07622 I .00146 72698 0.05001 42309 4 0.10261 27466 5 52 0.05059 23651 I .00260 66524 0.06666 85367 5 0.12826 59332 7 20 0.06303 44839 I . 00406 92257 0.08330 81651 6 0.15391 91199 8 47 0.07534 07235 1.00585 32333 0.09992 94260 7 0.17957 23085 10 14 0.08748 53252 1.00795 65320 0.1 1652 86159 8 0.20522 54932 II 41 0.09944 32800 I. 01037 65954 0.13310 20150 9 0.23087 86798 13 8 0.11119 04341 1.01311 05159 0.14964 58850 10 0.25653 18665 14 34 0.12270 35875 1.01615 50083 0.16615 64662 II 0.28218 50531 16 0.13396 05824 I. 01950 64139 0.18262 99754 12 0.30783 82398 17 25 0.14494 03827 I .02316 07042 0.19906 26038 13 0.33349 14264 18 50 0.15562 31436 I .02711 34860 0.21545 05144 14 0.35914 46131 20 14 0.16599 02705 I .03136 00060 0.23178 98405 15 0.38479 77997 21 38 0.17602 44678 1.03589 51569 0.24807 66833 16 0.41045 09864 23 I 0.18570 97766 I. 0407 I 34825 0.26430 71 105 17 0.43610 41730 24 23 0.19503 16024 I .04580 91848 0.28047 71545 18 0.46175 73596 25 44 0.20397 67323 1.05117 61304 0.29658 281 10 19 0.48741 05463 27 4 0.21253 33427 1.05680 78572 0.31260 00376 20. 0.51306 37329 28 24 0.22069 09968 1.06269 75825 0.32858 47528 21 0.53871 69196 29 43 0.22844 06338 1.06883 82109 0.34447 28350 22 0.56437 01062 31 I 0.23577 45496 1.07522 23418 0.36028 01217 23 0.59002 32929 32 19 0.24268 63696 I .08184 22789 0.37600 24088 24 0.61567 64795 33 36 0.24917 10151 1.08869 00386 0.39163 54503 25 0.64132 96662 34 52 0.25522 46626 I 09575 73598 0.40717 49584 26 0.66698 28528 36 7 0.26084 46988 I 10303 57129 0.42261 66028 27 0.69263 60395 37 21 0.26602 96698 1.11051 63106 0.43795 601 17 28 0.71828 92261 38 34 0.27077 92271 I .11819 01175 0.45318 87717 29 0.74394 24127 39 46 0.27509 40704 I. 12604 78613 0.46831 04285 30 0.76959 55994 40 58 ■ 0.27897 58872 I. 13408 00433 0.48331 64880 31 0.79524 87860 42 9 0.28242 72920 I . 14227 69496 0.49820 24170 32 0.82090 19727 43 18 0.28545 17629 I. 15062 86634 0.51296 36449 33 0.84655 51593 44 26 0.28805 35786 I 15912 50752 0.52759 55647 34 0.87220 83460 45 34 0,29023 77551 I. 16775 58964 0.54209 35352 35 0.89786 15326 46 41 0.29200 99830 I. I 765 I 06705 0.55645 28823 36 0.92351 47193 47 47 0. 29337 65659 I. 18537 87860 0.57066 89018 37 0,94916 79059 48 52 0.29434 43597 I . 19434 94887 0.58473 68614 38 0.97482 10926 49 56 0.29492 07141 I. 20341 18951 0.59865 20033 39 I .00047 42792 50 59 0.2951 1 34159 1. 2 1 255 50050 0.61240 95465 40 I. 02612 74659 52 I 0.29493 06347 I. 22176 77148 0.62600 46907 41 I. 05178 06525 53 2 0.29438 08705 I. 23103 88308 0.63943 26185 42 1-07743 38392 54 2 0.29347 29047 1.24035 70830 0.65268 84992 43 I . 10308 70258 55 I 0.29221 57532 I. 24971 11383 0.66576 74922 44 I. 12874 02125 56 0.29061 86227 1.25908 96145 0.67866 47507 45 I 15439 33991 56 58 0.28869 08691 1.26848 10938 0.69137 54254 90-r Fi/' ^ G(r) C(r) B(r) Smithsonian Tables lABLE e = 65° g = 0. 106054020185994, = 0. 7881449667, HK = 1. 1541701350 28s B(r) C(r) G(r) <^ Fi/- 90-r 90 I . 00000 00000 1.53824 62687 0.00000 00000 90° 0' 2.30878 67982 0.99983 4I4I2 1.53808 15440 0.00834 87781 88 23 2.28313 36115 89 0.99933 66526 1.53758 75740 0.01669 26008 88 46 2.25748 04249 88 0.99850 77970 1.53676 49688 0.02502 65041 88 9 2.23182 72382 87 0.99734 80125 1-53561 47447 0.03334 55075 87 32 2.20617 40516 86 0.99585 79109 1-53413 83232 0.04164 46052 86 54 2 . 18052 08649 85 0.99403 82778 1-53233 75281 0.04991 87582 86 16 2.15486 76783 84 0.99189 00707 I -53021 45843 0.05816 28855 85 38 2.12921 44916 83 0.98941 44182 1.52777 21 140 0.06637 18564 85 2.10356 13050 82 0.98661 26176 I -52501 31340 0.07454 04819 84 22 2.07790 81184 81 0.98348 61339 I. 52194 10514 0.08266 35068 83 44 2.05225 49317 80 0.98003 65970 I -51855 96596 0.09073 56016 83 6 2.02660 I 745 I 79 0.97626 57996 I. 51487 31329 0.09875 13547 82 27 2.00094 85584 78 0.97217 56947 1,51088 60218 0.10670 52642 81 48 1-97529 53718 77 0.96776 83924 1 . 50660 32466 - 0.1 1459 17308 81 9 1.94964 21851 76 0.96304 61576 I .50203 00916 0.12240 50500 80 30 1.92398 89985 75 0.95801 14060 I. 49717 21977 0.13013 94047 79 50 1-89833 581 18 74 0.95266 67013 1.49203 55559 0.13778 88583 79 10 I .87268 26251 73 0.94701 475II 1.48662 64993 0.14534 73477 78 30 1.84702 94385 72 0.94105 84035 1.48095 16947 0.15280 86769 77 49 I. 82137 62519 71 0.93480 06429 I. 47501 81348 0.16016 65105 77- 8 1-79572 30652 70 0.92824 45859 1.46883 31288 0.1 6741 43683 76 26 I .77006 98786 69 0.92139 34772 1.46240 42933 0.17454 56190 75 44 I. 74441 66919 68 0.91425 06851 1.45573 95424 0.18155 34763 75 2 I -71876 35053 67 0.90681 96968 1.44884 70781 0.18843 09933 74 19 1,69311 03186 66 0.89910 41 140 1-44173 53793 0.19517 10594 73 36 1.66745 71320 65 0.891 10 76479 I. 43441 31916 0.20176 63966 72 52 I. 64 I 80 39453 64 0.88283 41 144 1.42688 95162 0.20820 95570 72 8 1.61615 07587 63 0.87428 74294 1.41917 35981 0.21449 29211 71 23 1-59049 75721 62 0.86547 16034 1.41127 49149 0.22060 86968 70 37 1.56484 43854 61 0.85639 07366 1.40320 31647 0.22654 89197 69 51 I. 53919 I 1988 60 0.84704 90138 1.39496 82541 0.23230 54536 69 4 I -51353 80121 59 0.83745 06991 1.38658 02852 0.23786 99932 68 17 1.48788 48255 58 0.82760 01310 1.37804 95440 0.24323 40676 67 29 1.46223 16388 57 0.81750 I7I68 1.36938 64865 0.24838 90447 66 41 1.43657 84522 56 0.80715 99276 1.36060 1 7261 0.25332 61379 65 .52 I. 41092 52655 55 0.79657 92934 I. 35 I 70 60205 0.25803 64133 65 2 1.38527 20789 54 0.78576 43973 I. 34271 02582 0.26251 08001 64 II I. 35961 88922 53 0.77471 98708 1.33362 54449 0.26674 01012 63 20 1-33396 57055 52 0.76345 03889 1.32446 26900 0.27071 50065 62 28 I. 30831 25189 51 0.75196 06646 1.31523 31927 0.27442 61086 61 35 1.28265 93322 50 0.74025 54443 1.30594 82284 0.27786 39198 60 41 1.25700 61456 49 0. 72833 95027 I. 29661 91348 0.28101 88920 59 46 I -23135 29589 48 0.71621 76383 1.28725 72976 0.28388 14388 58 51 1.20569 97723 47 0.70389 46686 1.27787 41372 0.28644 19600 57 55 I. 18004 65856 46 0,69137 54254 1.26848 10938 0.28869 08691 56 58 I -15439 33991 45 r A(r) D(r) E(r) F E(r) D(r) A(r 0.00000 00000 0° 0' 0.00000 00000 1 . 00000 00000 0.00000 00000 I 0.03075 62572 I 46 0.01878 71553 I .00028 90226 0.01564 67728 2 0.06151 25143 3 37 0.03752 01201 I. 001 15 57568 0.03129 2071 I 3 0.09226 87715 5 17 0.05614 50985 1.00259 92025 0.04693 44040 4 0.12302 50287 7 2 0.07460 90790 I. 00461 76935 0.06257 22754 5 0.15378 12859 8 47 0.09286 02109 1.00720 88997 0.07820 41558 6 0.18453 75430 10 31 0.1 1084 81632 I. 01036 98288 0.09382 84843 7 0.21529 38002 12 15 0. 12852 44620 I. 01409 68295 0.10944 36574 8 0.24605 00574 13 58 0. 14584 27986 I. 01 838 55946 0.12504 80220 9 0.27680 63145 15 40 0.16275 93073 1.02323 I 1658 0.14063 98665 10 0.30756 25717 17 22 0.17923 28093 1.02862 79374 0.15621 74137 II 0.33831 88289 19 3 0.19522 50184 I .03456 96626 0.17177 88130 12 0.36907 50860 20 43 0.21070 07095 I. 04 I 04 94593 0.18732 21327 13 0.39983 13432 22 22 0.22562 78479 I .04805 98163 0.20284 53538 14 0.43058 76004 23 59 0.23997 76797 1.05559 26010 0.21834 63622 15 0.46134 38576 25 36 0.25372 47838 1.06363 90673 0.23382 29430 i6 0.49210 01147 27 12 0.26684 70884 I .07218 98642 0.24927 27739 17 0.52285 63719 28 46 0.27932 58519 I. 08 I 23 50446 0.26469 34194 i8 0.55361 26291 30 19 0.29114 56129 1.09076 40755 0.28008 23255 19 0.58436 88862 31 50 0.30229 41 1 10 I. 10076 58484 0.29543 68145 20 0.61512 51434 33 21 0.31276 21816 I .11122 86903 0.31075 40803 21 0.64588 14006 34 50 0.32254 36297 I. 122 14 03756 0.32603 I I 842 22 0.67663 76577 36 17 0.33163 50828 I. 13348 81382 0.34126 50509 23 0.70739 39149 37 43 0.34003 58309 I. 14525 86847 0.35645 24653 24 0.73815 01721 39 8 o. 34774 76532 I. 15743 82078 0.37159 00694 25 0.76890 64293 40 31 0.35477 46364 I. I 7001 24008 0.38667 43599 26 0.79966 26864 41 52 0.36112 29881 I. I 8296 64722 0.40170 16862 27 0.83041 89436 43 12 0.36680 08467 I. 19628 51612 0.41666 82489 28 0,86117 52008 44 31 0.37181 80918 1.20995 27538 0.43157 00988 29 0.89193 14579 45 48 0.37618 61563 1.22395 30995 0.44640 31361 30 0.92268 77151 47 3 0.37991 78428 1.23826 96285 0.46116 31110 31 0.95344 39723 48 18 0.38302 71460 1.25288 53692 0.47584 56238 32 0.98420 02294 49 30 0.38552 90817 I .26778 29672 0.49044 61259 33 I .01495 64866 SO 41 0.38743 95246 1.28294 47038 0.50495 99214 34 1.04571 27438 51 51 0.38877 50552 1.29835 25154 0.51938 21695 35 I .07646 90010 52 59 038955 28159 I. 31398 80140 0. 53370 78866 36 I. 10722 52581 54 5 0.38979 03785 1.32983 25072 0.54793 19494 37 I. 13798 15153 55 10 0.38950 56204 1.34586 70195 0.56204 90989 38 I. 16873 77725 56 14 0.38871 66125 1.36207 23140 0.57605 39442 39 I . 19949 40296 57 16 0.38744 15171 1.37842 89138 0.58994 09669 40 I . 23025 02868 58 17 0.38569 84955 I. 39491 71251 0.60370 45267 41 I .26100 65440 59 17 0.38350 56260 1.41151 70596 0.61733 88663 42 I .29176 28011 60 15 0.38088 08305 1.42820 86579 0.63083 81 179 43 1. 3225 1 90583 61 12 0.37784 18107 1-44497 17132 0.64419 63092 •44 1.35327 53155 62 8 0.37440 59923 I. 461 78 58952 0.65740 73705 45 90-r 1.38403 15727 63 2 0.37059 04774 1.47863 07744 0.67046 51423 » H ^ G(r) C(r) B(r) Smithsonian Tables TABLE B = 75° q = 0. 163033534821580, 9 0= 0. 6753457533, HK = 1. 3046678096 289 B(r) C(r) G(r) ^ Fi/' 90-r I . 00000 00000 1.96563 05108 0.00000 00000 90° 0' 2.76806 31454 90 0.99981 60886 1.96533 12951 0.00989 91720 89 33 2.73730 68882 89 0.99926 44975 1.96443 40309 0.01979 47043 89 5 2.70655 06310 88 0.99834 56552 1.96293 98674 0.02968 29453 88 38 2.67579 43738 87 0.99706 02753 1.96085 07176 0.03956 02195 88 10 2.64503 81 167 86 0.99540 93546 I. 95816 92561 0.04942 28154 87 43 2.61428 18595 85 0.99339 41714 1.95489 89147 0.05926 69738 87 15 2.58352 56023 84 0.99101 62829 I. 95104 38778 0.06908 88752 86 47 2.55276 93451 83 0.98827 75221 1.94660 90763 0.07888 46278 86 19 2.52201 30880 82 0.98517 99940 I .94160 01803 0.08865 02550 85 51 2.49125 68308 81 0.98172 60720 1.93602 35909 0.09838 16828 85 22 2.46050 05736 80 0.97791 83923 I . 92988 64309 0. 10807 47268 84 54 2.42974 43165 79 0.97375 98498 I. 92319 65349 0.1 I 772 50798 84 25 2.39898 80593 78 0.96925 35914 1.91596 24373 0.12732 82981 83 55 2.36823 I 802 I 77 0.96440 30106 I. 908 I 9 33609 0.13687 97883 83 26 2.33747 55450 76 0.95921 17405 I . 89989 92030 0.14637 47936 82 56 2.30671 92878 75 0.95368 36468 I .89109 05214 0.15580 83802 82 25 2 27596 30306 74 0.94782 28200 I. 88177 85195 0.16517 54225 81 55 2.24520 67734 73 0.94163 35686 I. 87197 50301 0.17447 05894 81 24 2.21445 05163 72 0.93512 04092 I. 86169 24991 0.18368 83293 80 52 2.18369 42591 71 0.92828 80593 1.85094 39670 0.19282 28550 80 20 2.15293 80019 70 0.92114 14274 1.83974 30516 0.20186 81293 79 48 2.12218 17448 69 0.91368 56040 I. 82810 39279 0.21081 78488 79 15 2.09142 54876 68 0.90592 58521 I. 81604 13089 0.21966 54291 78 41 2.06066 92304 67 0.89786 75972 1.80357 04247 0.22840 39887 78 7 2.02991 29733 66 0.88951 64174 1.79070 70015 0.23702 63334 77 32 I. 99915 67161 65 0.88087 80328 1.77746 72401 0.24552 49406 76 56 I .96840 04589 64 0.87195 82952 I . 76386 77929 0.25389 19433 76 20 1.93764 42017 63 0.86276 31773 1.74992 57419 0.26211 91147 75 43 I .90688 79446 62 0.85329 87622 1.73565 85746 0.27019 78524 75 6 I. 87613 16874 61 0.84357 12322 I. 72108 41609 0.27811 91636 74 27 1.84537 54302 60 0.83358 68580 1.70622 07286 0.28587 36500 73 48 1.81461 91731 59 0.82335 19876 I. 69 I 08 68389 0.29345 14936 73 8 1.78386 29159 58 0.81287 30353 1.67570 13618 0.30084 24433 72 28 I. 753 10 66587 57 0.80215 64710 1.66008 34507 0.30803 58026 71 46 1.72235 04016 56 0.79120 88085 1.64425 25175 0.31502 04176 71 4 I. 69159 41444 55 0.78003 65955 1.62822 82065 0.32178 46673 70 20 1.66083 78872 54 0.76864 64021 I. 61203 03692 0.32831 64547 69 36 1.63008 16300 53 0.75704 48103 1.59567 90385 0.33460 32006 68 50 1.59932 53729 52 0.74523 84036 1.57919 44025 0.34063 18384 68 4 1.56856 91157 51 0.73323 37566 1.56259 67789 0.34638 88130 67 16 I. 53781 28585 50 0.72103 74248 1.54590 65890 0.35186 00808 66 28 1.50705 66014 49 0.70865 59347 I. 52914 43320 0.35703 I I 148 65 38 1.47630 03442 48 0.69609 57739 1.51233 05588 0.36188 69115 64 47 1.44554 40870 47 0.68336 33823 1.49548 58469 0.36641 20039 63 55 I. 41478 78299 46. 0.67046 51423 1.47863 07744 0.37059 04774 63 2 1.38403 15727 45 A(r) D(r) E(r) F0 r Smithsonian Tables 290 ELLIPTIC FUNCTION K = 3. 1533852519, K' = 1. 5828428043, E = 1. 0401143957, E' = 1. 5588871966, r ¥4) E(r) D(r) A(r) 0.00000 00000 0° 0' 0.00000 00000 I . 00000 00000 0.00000 00000 I 0.03503 76139 2 0,02346 68886 I. 00041 13182 0.01460 06854 2 0.07007 52278 4 I 0.04685 05457 I. 00164 48264 0,02920 20956 3 0.10511 28417 6 I 0.07006 85417 1.00369 91860 0,04380 49412 4 0,14015 04556 8 0.09304 00333 I .00657 21668 0,05840 99043 5 0.17518 80695 9 59 0.1 1568 65173 I .01026 06485 0,07301 76251 6 0.21022 56835 II 58 0.13793 25365 I. 01476 06225 0,08762 86871 7 0.24526 32974 13 55 0.15970 63263 I .02006 71948 0,10224 36040 8 0.28030 091 13 15 52 0.18094 03901 I. 0261 7 45886 0.1 I 686 28061 9 0.31533 85252 17 47 0.20157 19949 1.03307 61484 0,13148 66263 10 0.35037 61391 19 41 0.22154 35813 1.04076 43440 0,14611 52882 II 0.38541 37530 21 34 0.24080 30831 1.04923 07759 0,16074 88922 12 0.42045 13669 23 26 0.25930 41559 1.05846 61800 0.17538 74040 13 0.45548 89808 25 16 0,27700 63163 1,06846 04345 0. 19003 06422 14 0.49052 65947 27 4 0.29387 49943 1.07920 25667 0.20467 82669 15 0.52556 42086 28 51 0.30988 15035 I .09068 07598 0.21932 97686 i6 0.56060 18226 30 36 0.32500 29380 I . 10288 23622 0.23398 44577 17 0.59563 94365 32 20 0.33922 20017 I "579 38955 0.24864 14540 i8 0.63067 70504 34 I 0.35252 67798 I . 12940 10647 0.26329 96779 19 0.66571 46643 35 41 0.36491 046x8 I. 14368 87684 0-27795 78408 20 0.70075 22782 37 18 0.37637 10249 I . 15864 IIIOI 0,29261 44375 21 0.73578 98921 38 54 0.38691 08879 I. 17424 14105 0,30726 77376 22 0.77082 75060 40 28 0.39653 65430 I. 19047 22196 0.32191 57797 23 0.80586 51199 41 59 0.40525 81757 I 20731 53312 0.33655 63638 24 0.84090 27338 43 29 0.41308 92784 1.22475 17970 0,35118 70467 -25 0.87594 03477 44 56 0,42004 62655 1.24276 1942 1 0-36580 51367 26 0.91097 79617 46 22 0,42614 80965 1,26132 53814 0,38040 76896 27 0.94601 55756 47 45 0.43141 59095 1.28042 10369 0-39499 15050 28 0.98105 31895 49 7 0.43587 26721 1.30002 71557 0-40955 31244 29 1 .01609 08034 50 26 0.43954 28505 1,32012 13294 0,42408 88287 30 1.05112 84173 51 44 0.44245 21005 1.34068 05139 0,43859 46375 31 I. 08616 60312 52 59 0.44462 69813 I. 36168 10508 0.45306 63090 32 1.12120 36451 54 12 0,44609 46931 1.38309 86893 0-46749 93405 33 I. 15624 12590 55 24 0,44688 28394 I . 40490 86089 0.48188 89699 34 1.19127 88729 56 33 0,44701 92128 1.42708 54443 0.49623 01775 35 I. 2263 I 64868 57 41 0.44653 16053 1.44960 33094 0.51051 76900 36 I. 26135 41008 58 47 0.44544 76404 1.47243 58241 0-52474 59832 37 1.29639 17147 59 51 0.44379 46284 1-49555 61410 0- 53890 92878 38 I. 33142 93286 60 53 0.44159 94403 I. 51893 69731 0-55300 15938 39 1.36646 69425 61 54 0.43888 84024 1-54255 06233 0.56701 66575 40 I. 40150 45564 62 53 0.43568 72080 1.56636 90138 0.58094 80084 41 1.43654 21703 63 50 0.43202 08450 1-59036 37173 0.59478 89567 42 1-47157 97842 64 45 0.42791 35381 1. 61450 -59885 0.60853 26019 43 1.50661 73981 65 39 0.42338 87053 1.63876 67967 0.62217 18423 44 I. 54165 50120 66 32 0.41846 89243 I. 663 I I 68595 0-63569 93846 45 90-r 1.57669 26259 67 23 0.41317 59112 1.68752 66770 0.64910 77548 F^ \^ G(r) C(r) B(r) Smithsonian Tables TABLE e = 80° 291 1 =v. ^uoouaYoszc 0965, e = 0. 59 0423578356, HK = 1.406061468420 B(r) C(r) G(r) ^ F^ 90-1 1 I . 00000 00000 2.39974 38370 0,00000 00000 90° 0' 3.15338 52519 90 0.99979 75549 2.39930 24464 0,01049 98939 89 39 3.11834 76380 89 0-999 I 9 04200 2.39797 88675 0,02099 72691 89 18 3,08331 00241 88 0.99817 91961 2.39577 48778 0,03148 95952 88 57 3,04827 24102 87 0.99676 48832 2.39269 34364 0.04197 43187 88 36 3,01323 47963 86 0.99494 88778 2,38873 86793 0,05244 88508 88 15 2,97819 71823 85 0.99273 29703 2.38391 59122 0.06291 05559 87 54 2.94315 95684 84 0.9901 1 93406 2.37823 16019 0.07335 67394 87 32 2,90812 19545 83 0.98711 05534 2.37169 33654 0,08378 46353 87 II 2,87308 43406 82 0.98370 95524 2.36430 99572 0.09419 13935 86 49 2.83804 67267 81 0.97991 96536 2.35609 12550 0.10457 40674 86 27 2,80300 91128 80 0-97574 45380 2.34704 82431 0.1 1492 96001 86 4 2,76797 14989 79 0.97118 82434 2-33719 29943 0.12525 48110 85 42 2.73293 38850 78 0.96625 51552 2.32653 86504 0.13554 63814 85 19 2,69789 62711 77 0.96094 99971 2.31509 94002 0.14580 08404 84 56 2,66285 86572 76 0.95527 78200 2.30289 04563 0.1 560 I 45490 84 32 2,62782 10432 75 0.94924 39913 2.28992 80308 0,16618 36848 84 8 2.59278 34293 74 0.94285 41832 2.27622 93087 0,17630 42256 83 44 2.55774 58154 73 0-9361 I 43595 2.26181 24201 0,18637 19320 83 19 2,52270 82015 72 0.92903 07633 2,24669 641 12 0.19638 23298 82 54 2.48767 05876 71 0.92160 99031 2.23090 12139 0.20633 06915 82 28 2.45263 29137 70 0,91385 85385 2.21444 76139 0.21621 20167 82 I 2.41759 53578 69 o- 90578 36660 2.19735 72184 0.22602 10124 81 35 2.38255 77459 68 0.89739 25035 2.17965 24214 0. 23575 20713 81 7 2,34752 01320 67 0.88869 24749 2.16135 63692 0.24539 92508 80 39 2,31248 25181 66 0.87969 I 1946 2.14249 29245 0.25495 62494 80 10 2,27744 49041 65 0.87039 645 I I 2. 12308 66296 0,26441 63838 79 41 2,24240 72902 64 0.86081 61906 2. 10316 26690 0,27377 25638 79 II 2,20736 96763 63 0.85095 85006 2.08274 68307 0,28301 72673 78 40 2.17233 20624 62 0.84083 15928 2.06186 54682 0,29214 25142 78 8 2.13729 44485 61 0.83044 37863 2.04054 54606 0,30113 98388 77 35 2.10225 68346 60 0.81980 34906 2.01881 41730 0,31000 02630 77 2 2.06721 92207 59 0.80891 91886 1.99669 94165 0,31871 42670 76 28 2.03218 16068 58 0.79779 94194 1.97422 94075 0,32727 17611 75 52 I -99714 39929 57 0.78645 27612 1,95143 27275 0,33566 20561 75 16 I .96210 63790 56 0.77488 78149 1.92833 82823 0.34387 38337 74 39 1.92706 87650 55 0.76311 31867 1,90497 526H 0.35189 51171 74 I 1.89203 11511 54 0.75113 74717 1,88137 30959 0,35971 32414 73 21 I ■ 85699 35372 53 0.73896 92379 1.85756 142 10 0,36731 48250 72 41 I. 82195 59233 52 0.72661 70097 1.83357 00328 0.37468 57413 71 59 I. 78691 83094 51 0.71408 92524 I . 80942 88493 0,38181 10919 71 16 I. 75 I 88 06955 50 0-70139 43563 I. 78516 78703 0,38867 51812 70 32 I. 71684 30816 49 0.68854 06225 I. 76081 71386 0,39526 14938 69 47 I. 68180 54677 48 0-67553 62475 I , 73640 67003 0,40155 26735 69 1.64676 78538 47 0.66238 9309s I. 71 196 65668 0. 40753 05071 68 12 1.61173 02399 46 0.64910 77548 1.68752 66770 0,41317 59II2 67 23 1.57669 26259 45 r A(r) D(r) E(r) 4> F<^ Smithsonian Tables 292 ELLIPTIC FUNCTION K = 3. 2553029421, K' = 1. 5805409339, E = 1. 033789462, E' = 1. 5611417453, r F 90-r 90 I . 00000 00000 2.52833 01251 0.00000 00000 90° 0' 3.25530 29421 0.99979 22836 2.52784 54320 0.01060 10292 89 41 3.21913 29095 89 0.99916 93515 2.52639 20136 0.02119 97963 89 21 3.18296 28768 88 0.99813 18540 2.52397 18509 0.03179 40278 89 2 3.14679 28441 87 0.99668 08734 2.52058 82420 0.04238 14278 88 42 3. I 1062 281 14 86 0.99481 79213 2.51624 57960 0.05295 96662 88 22 3.07445 27787 85 0.99254 49353 2.51095 04254 0.06352 63677 88 2 3.03828 27460 84 0.98986 42745 2.50470 93354 0.07407 90993 87 42 3. 002 1 1 27133 83 0.98677 87139 2-49753 10120 0.08461 53590 87 22 2.96594 26806 82 0.98329 14382 2.48942 52067 0.09513 25631 87 2 2.92977 26479 81 0.97940 60344 2.48040 29203 0.10562 80337 86 41 2.89360 26152 80 0.97512 64836 2.47047 63835 0.1 1 609 89854 86 20 2.85743 25825 79 0.97045 71520 2.45965 90364 0.12654 25123 85 59 2.82126 25499 78 0.96540 27806 2.44796 55051 0.13695 55734 85 38 2.78509 25172 77 0.95996 84748 2.43541 15773 0.14733 49785 85 16 2.74892 24845 76 0.95415 96925 2.42201 41749 0.15767 73727 84 54 2.71275 24518 75 0.94798 22318 2.40779 13262 0.16797 92208 84 32 2.67658 24191 74 0.94144 22181 2.39276 21349 0.17823 67907 84 9 2.64041 23864 73 0.93454 60898 2.37694 67487 0.18844 61360 83 45 2.60424 23537 72 0.92730 05843 2.36036 63252 0.19860 30778 83 21 2.56807 23210 71 0.91971 27230 2.34304 29976 0.20870 31860 82 57 2.53190 22883 70 0.91 178 97950 2.32499 98377 0.21874 17592 82 32 2.49573 22556 69 0.90353 93417 2.30626 08184 0.22871 38038 82 7 2.45956 22230 68 0.89496 91397 2.28685 07750 0.23861 40125 81 41 2.42339 21903 67 0.88608 71836 2.26679 53647 0.24843 67407 81 14 2.38722 21576 66 0.87690 16690 2.24612 10260 0.25817 59833 80 47 2.35105 21249 65 0.86742 09743 2.22485 49364 0.26782 53494 80 19 2.31488 20922 64 0.85765 36425 2.20302 49697 0.27737 80358 79 50 2.27871 20595 63 0.84760 83633 2.18065 96524 0.28682 68004 79 20 2.24254 20268 62 0.83729 39541 2.15778 81197 0.29616 39332 78 50 2.20637 19941 61 0.82671 93416 2 . 13444 00706 0.30538 12272 78 19 2.17020 19614 60 0.81589 35429 2. I 1064 57227 0.31446 99478 77 47 2.13403 19287 59 0.80482 56467 2.08643 57672 0.32342 08014 77 14 2.09786 18960 58 0.79352 47945 2.06184 13229 0.33222 39026 76 40 2.06169 18634 57 0.78200 01623 2.03689 38902 0.34086 87415 76 5 2.02552 18307 56 0.77026 0941 I 2.01162 53056 Q. 34934 41494 75 29 1.98935 17980 55 0.75831 63194 1.98606 76958 0.35763 82644 74 53 I. 95318 17653 54 0.74617 54642 1.96025 34320 0.36573 84971 74 14 1.91701 17326 53 0.73384 75039 I. 9342 I 50843 0.37363 14953 73 35 I .88084 16999 52 0.72134 15096 1.90798 53771 0.38130 31100 72 55 1.84467 16672 51 0.70866 64787 I. 88159 71433 0.38873 83616 72 13 1.80850 16345 50 0.69583 I3I78 1.85508 32817 0.39592 14068 71 30 1.77233 16018 49 . Q 0.68284 48256 1.82847 67117 0.40283 55079 70 46 I. 73616 15691 48 0.66971 56781 0.65645 24120 1.80181 03311 0.40946 30040 70 I 1.69999 15365 47 I. 7751 I 69734 0.41578 52846 69 14 1.66382 15038 46 0.64306 34108 1.74842 93662 0.42178 27675 68 25 1.62765 14711 45 A(r) D(r) E(r) F0 r Smithsonian Tables 294 ELLIPTIC FUNCTION K = 3. 3698680267, K' = 1. 5784865777, E = 1. 027843620, E' = 1. 5629622295, r V E(r) D(r) A(r) 0.00000 00000 0° 0' 0.00000 00000 I . 00000 00000 0.00000 00000 I 0.03744 29781 2 9 0.02600 53438 1.00048 71379 0.01396 87846 2 0.07488 59561 4 17 0.05190 80180 I .00194 80481 0.02793 96081 3 0.1 1232 89342 6 26 0.07760 64875 1.00438 12208 0.04191 44920 4 0.14977 19123 8 35 0.10300 14601 1.00778 41400 0-05589 54231 5 0.18721 48904 10 40 0.12799 69416 1.01215 32844 0.06988 43359 6 0.22465 78684 12 46 0.15250 12188 I .01748 41292 0.08388 30956 7 0.26210 08465 14 51 0.17642 77402 1.02377 1 1470 0-09789 34813 8 0.29954 38246 16 55 0.19969 58914 I. 03100 78103 0.11191 71690 9 0.33698 68027 18 58 0.22223 16400 1. 039 1 8. 65941 0-12595 57152 10 0.37442 97807 20 59 0.24396 80481 I .04829 89781 0.14001 05412 II 0.41187 27588 22 58 0.26484 56468 I 05833 54510 0.15408 29167 12 0.44931 57369 24 56 0.28481 26740 1.06928 55135 0.16817 39451 13 0.48675 87150 26 52 0.30382 51779 1.08113 76835 0.18228 45483 14 0.52420 16930 28 46 0.32184 69961 I 09387 95005 19641 54524 15 0.56164 4671 1 30 38 0.33884 96193 I 10749 75312 0.21056 71740 i6 0.59908 76492 32 28 0.35481 19530 1.12197 73762 0.22474 00071 17 0. 63653 06273 34 16 0.36971 99918 I 13730 36763 0.23893 40100 i8 0.67397 36053 36 2 0.38356 64197 I. 15346 01207 0.25314 89941 19 0.71141 65834 37 46 0.39635 01539 I. I 7042 94549 0.26738 45123 20 0.74885 95615 39 27 0.40807 58450 I . 18819 34902 0.28163 98484 21 0.78630 25396 41 6 0.41875 33497 1.20673 31 139 0.29591 40077 22 0.82374 55176 42 42 0.42839 71871 I .22602 82998 0.31020 57076 23 0.86118 84957 44 16 0.43702 59916 I .24605 81209 0-32451 33701 24 0.89863 14738 45 48 0.44466 19725 1.26680 07616 0-33883 51142 25 0.93607 44519 47 18 0.45133 03888 1.28823 35321 0.35316 87494 26 0.97351 74299 48 45 0.45705 90462 I. 31033 28836 0-36751 17704 27 I .01096 04080 SO 10 0.46187 78212 I 33307 44242 0.38186 13526 28 1.04840 33861 51 32 0.46581 82181 1.35643 29365 0.39621 43484 29 I .08584 63641 52 52 0.46891 29597 1.38038 23962 0.41056 72843 30 I. 12328 93422 54 10 0.471 19 56148 1.40489 59917 0.42491 63594 31 I. 16073 23203 55 26 0.47270 02620 1.42994 61457 0.43925 74448 32 1.19817 52984 56 39 0.47346 1 1908 I -45550 45373 0.45358 60835 33 I. 23561 82764 57 50 0.47351 26377 I. 48154 21259 0.46789 74917 34 1.27306 12545 59 0.47288 85574 1.50802 91764 0.48218 65611 35 I. 31050 42326 60 7 0.47162 24256 1-53493 52855 0.49644 78621 36 1.34794 72107 61 12 0.46974 70729 I .56222 94100 0.51067 56480 37 1-38539 01887 62 15 0.46729 45464 1.58987 98960 0.52486 38600 38 1.42283 31668 63 16 0.46429 59969 I. 6 I 785 45092 0.53900 61335 39 I .46027 61449 64 15 0.46078 15892 I .64612 04680 0-55309 58052 40 I. 49771 91230 65 12 0.45678 04338 I . 67464 44762 0-56712 59210 41 I. 53516 21010 66 7 0.45232 05363 I -70339 27583 0.58108 92454 42 1.57260 50791 67 I 0.44742 87637 I -73233 10960 0.59497 82708 43 I .61004 80572 67 53 0.44213 08242 I. 76142 48657 0.60878 52287 44 1.64749 10353 68 44 0.43645 12599 1.79063 90777 0.62250 21016 45 1.68493 40133 69 32 0.43041 34495 I .81993 84164 0.63612 06349 90-r Fi^ i^ G(r) C(r) B(r) Smithsonian Tables TABLE e = 82° g = 0.22 9667169881194, = 0. 6464169465, HK = 1. 4676481002 295 B(r) C(r) G(r) i F\j/ 90- 90 89 88 87 86 r I . 00000 00000 0.99978 62II2 0.99914 50809 0.99807 73170 0.99658 40972 2.68054 03437 2.68000 36787 2.67839 44283 2.67571 48255 2.67196 85860 0.00000 00000 0.01069 49135 0.02138 78301 0.03207 67423 0.04275 96209 90° 0' 89 42 89 24 89 6 88 48 3 36986 80267 3.33242 50486 3.29498 20705 3.25753 90925 3.22009 61 144 0.99466 70666 0.99232 83334 0.98957 04645 0.98639 64786 2.66716 09043 2.66129 84418 2.65438 93156 2.64644 30842 0.05343 44040 0.06409 89867 0.07475 12085 0.08538 88428 88 30 88 12 87 53 87 35 3.18265 31363 3.14521 01582 3.10776 71802 3.07032 42021 85 84 83 82 0.90200 98400 2.63747 07296 0.09600 95847 87 16 3.03288 12240 81 0.97881 44497 0.97441 46367 2.62748 46381 2.61649 85778 0.10661 10385 0.11719 07054 86 57 86 37 2.99543 82459 2-95799 52679 80 70 0.96961 51474 2.60452 76741 0.12774 59701 86 18 2.9205s 22808 78 0.96442 1 1348 2.59158 83828 0.13827 40870 85 58 ^ KJKJ ^ ^ '-f -^^^ 2.88310 93II7 77 0.95883 81466 2 57769 84606 0.14877 21662 85 38 2.84566 63336 76 0.95287 21117 2.56287 69342 0.15923 71580 85 17 2.80822 33556 7^ 0.94652 93269 2.54714 40664 0.16966 58376 84 56 2.77078 03775 / 74 0.93981 64421 2.53052 13208 0.18005 47885 84 35 2-73333 73994 73 0.93274 04449 2.51303 13248 0. 19040 03849 84 13 2.69589 44213 72 0.92530 86446 2.49469 78294 0.20069 87739 83 51 2.65845 14433 71 0.91752 86553 2.47554 56695 0.21094 58556 83 28 2.62100 84652 70 0.90940 83786 2.45560 07207 0.221 13 72633 83 5 2.58356 54871 69 0.90095 59853 2.43488 98556 0.23126 83422 82 41 2.54612 25090 68 0.89217 98975 2.41344 08985 0.24133 41265 82 16 2.50867 95310 67 0.88308 87690 2.39128 25787 0.25132 93157 81 51 2.47123 65529 66 0.87369 14660 2.36844 44831 0.26124 82501 81 25 2-43379 35748 65 0.86399 70475 2.34495 70070 0.27108 48837 80 59 2 39635 05967 64 0.85401 47452 2.32085 13053 0.28083 27574 80 32 2-35890 76187 63 0.84375 39427 2.29615 92414 0.29048 49692 80 4 2.32146 46406 62 0.83322 41555 2.27091 33365 0.30003 41444 79 35 2.28402 16625 61 0.82243 50100 2.24514 67182 0.30947 24031 79 5 2.24657 86844 60 0.81139 62227 2.21889 30687 0.31879 13276 78 35 2.20913 57064 59 0.8001 I 75795 2.19218 65719 0.32798 19272 78 4 2.17169 27283 58 0.78860 89149 2.16506 18621 0.33703 46027 77 31 2 . 13424 97502 57 0.77688 009 1 1 2.13755 39706 0.34593 91087 76 58 2.09680 67721 56 0.76494 09778 2.10969 82742 0.35468 45152 76 23 2-05936 37941 55 0.75280 14315 2.08153 04423 0.36325 91686 75 48 2.02192 08160 54 0.74047 12755 2.05308 63856 0.37165 06505 75 II 1.98447 78379 53 0.72796 02805 2.02440 22044 0.37984 57377 74 34 1.94703 48599 52 0.71527 81443 I. 99551 41373 0.38783 03601 73 55 1.90959 18818 51 0.70243 44736 1.96645 851 15 0.39558 95596 73 14 I. 872 14 89037 50 0.68943 87648 1.93727 16923 0.40310 74491 72 33 1.83470 59256 49 0.67630 03866 1.90799 00345 0.41036 71725 71 50 1.79726 29476 48 0.66302 85617 1.87864 98345 0.41735 08655 71 6 I. 7598 I 99695 47 0.64963 23506 I . 84928 72824 0.42403 96200 70 20 1.72237 69914 46 0.63612 06349 I .81993 84164 0.43041 34495 69 32 1.68493 40133 45 A(r) D(r) E(r) 4> F E(r) D(r) A(r) 0.00000 00000 0' ' 0' 0.00000 00000 I . 00000 00000 0.00000 00000 I 0.03889 35833 2 14 0.02751 52459 1.00053 54142 0.01357 81428 2 0.07778 71666 4 27 0.05491 49171 I. 00214 11230 0,02715 91294 3 0.1 1 668 07500 6 40 0.08208 48196 I. 0048 I 55243 0,04074 57840 4 0.15557 43333 8 53 0.1 089 I 34862 I 00855 59486 0.05434 08922 5 0.19446 79166 II 4 0.13529 34531 I. 01335 86590 0.06794 71815 6 0.23336 14999 13 15 0.16112 24388 I.01921 88518 0.08156 73027 7 0.27225 50833 15 25 0.18630 43989 I. 02613 06577 0.09520 38101 8 0.31114 86665 17 33 0.21075 04315 I .03408 71422 0.10885 91438 9 0.35004 22499 19 40 0.23437 95237 1.04308 03072 0.12253 561 I I 10 0.38893 58332 21 45 0.25711 91248 I. 05310 10924 0.13623 53681 II 0.42782 94166 23 48 0.27890 55463 I. 06413 93774 0.14996 04030 12 0.46672 29999 25 50 0.29968 41874 I. 0761 8 39836 0.16371 25182 13 0.50561 65832 27 50 0.31940 95974 I .08922 26769 0.17749 33 141 14 0.54451 01665 29 47 0.33804 53836 I .10324 21710 0.19130 41733 15 0.58340 37499 31 42 0.35556 39822 I . 11822 81308 0.20514 62446 16 0.62229 73332 33 35 0.37194 63079 1.13416 51764 0.21902 04287 17 0.66119 09165 35 26 0.38718 13038 1.15103 68883 0.23292 73637 18 0.70008 44998 37 14 0.40126 54102 I. 16882 58124 0.24686 74120 19 0.73897 80832 38 59 0.41420 19722 I. I 875 I 34668 0.26084 06476 20 0.77787 16665 40 42 0.42600 06064 1.20708 03483 0.27484 68440 21 0.81676 52498 42 23 0.43667 65427 1.22750 59404 0.28888 54637 22 0.85565 88331 44 I 0.44624 99581 I ,24876 87226 0.30295 56475 23 0.89455 24165 45 37 0.45474 53170 1.27084 61798 0.31705 62057 24 0-93344 59998 47 10 0.46219 07281 I. 29371 48135 0.33118 56095 25 0.97233 95831 48 40 0.46861 73287 I -31735 01537 0-34534 19839 26 I. 01 123 31664 50 8 0.47405 87042 1.34172 67728 0.35952 31012 27 I. 05012 67498 51 33 0.47855 03463 I. 3668 I 82994 0.37372 63757 28 1.08902 03331 52 56 0.48212 91569 1-39259 74348 0.38794 88593 29 1.12791 39164 54 17 0.48483 29959 I. 41903 59703 0.40218 72381 30 I . 16680 74997 55 35 0.48670 02770 I ,44610 48057 0.41643 78306 31 I .20570 10830 56 50 0.48776 96093 1-47377 39701 0,43069 65861 32 1,24459 46664 58 4 0.48807 94838 I. 50201 26433 0-44495 90849 33 1.28348 82497 59 14 0.48766 80032 1.53078 91792 0.45922 05390 34 1.32238 18330 60 23 0.48657 26520 1,56007 11317 0.47347 57948 35 I. 36127 54163 61 30 0.48483 01039 1,58982 52804 0.48771 93356 36 I. 40016 89997 62 34 0.48247 60647 1,62001 76598 0.50194 52865 " 37 1.43906 25830 63 36 0-47954 51456 1,65061 35895 0.51614 74196 38 1.47795 61663 64 36 0.47607 07644 I. 68157 77058 0.53031 91603 39 I. 5 I 684 97496 65 35 0.47208 50753 I. 71287 39955 0.54445 35952 40 1.55574 33330 66 31 0.46761 89121 1.74446 58318 0.55854 34803 41 1.59463 69163 67 25 0.46270 1 762 1 I. 77631 601 10 0.57258 12511 42 1.63353 04996 68 18 0.45736 17475 1.80838 67918 0.58655 90333 43 I . 67242 40829 69 9 0.45162 56249 1.84063 99362 0.60046 86540 44 I. 71 131 76663 69 58 0.44551 87962 1.87303 67513 0.61430 16549 45 1. 7502 1 12496 70 45 0.43906 53283 1.90553 81344 0.62804 93057 90-r F\l/ ^ G(r) C(r) B(r) TABLE e = 83° 9 = 0. 242912974306665, 00 = 0. 5211317465, HK = 1. 4872214813 297 B(r) C(r) G(r) 4^ Fi^ 90-r 90 I . 00000 00000 2.86452 59727 0,00000 00000 90° 0' 3.50042 24992 0.99977 91249 2.86392 54580 0,01078 10889 89 44 3.46152 89158 89 0.999 I I 67583 2.86212 47652 0.02156 04536 89 27 3.42263 53325 88 0.99801 36755 2.85912 64461 0.03233 63597 89 II 3.38374 17492 87 0.99647 I I 670 2.85493 47485 0,04310 70526 88 55 3.34484 81659 86 0.99449 10345 2.84955 56077 0.05387 07471 88 38 3.30595 45826 85 0.99207 55874 2.84299 66356 0.06462 56168 88 21 3.26706 09992 84 0.98922 76367 2.83526 71062 0.07536 97836 88 5 3.22816 74159 83 0.98595 04884 2.82637 79377 0.08610 13069 87 48 3.18927 38326 82 0.98224 79350 2.81634 16722 0,09681 81718 87 30 3.15038 02493 81 0.97812 42473 2.80517 24517 0,10751 82779 87 13 3.11148 66659 80 0.97358 41628 2.79288 59919 0.11819 94268 86 55 3.07259 30826 79 0.96863 28755 2.77949 95523 0.12885 93097 86 37 3.03369 94993 78 0.56327 60226 2.76503 19042 0.13949 54938 86 19 2.99480 59160 77 0.95751 96711 2.74950 32957 0.15010 54088 86 I 2.95591 23326 76 0.95137 03036 2.73293 54142 0.16068 63318 85 42 2.91701 87493 75 0.94483 48022 2.71535 13465 0.17123 '53724 85 23 2,87812 51660 74 0.93792 04329 2.69677 55363 0.18174 94560 85 3 2.83923 15827 73 0.93063 48276 2.67723 37397 0.19222 53067 84 43 2.80033 79993 72 0.92298 59663 2.65675 29786 0.20265 94294 84 22 2,76144 44160 71 0.91498 21585 2.63536 14921 0,21304 80901 84 I 2,72255 08327 70 0.90663 20234 2.61308 86858 0,22338 72956 83 39 2.68365 72494 69 0.89794 44698 2.58996 50797 0.23367 27719 83 17 2.64476 36660 68 0.88892 86753 2.56602 22548 0.24389 99414 82 54 2.60587 00827 67 0.87959 40653 2.54129 27973 0.25406 38981 82 31 2.56697 64994 66 0.86995 02909 2.51581 02430 0,26415 93822 82 7 2.52808 29161 65 0.86000 72069 2.48960 90190 0.27418 07525 81 42 2,48918 93327 64 0.84977 48495 2.46272 43859 0.28412 19576 .81 16 2,45029 57494 63 0.83926 34134 2.43519 23782 0.29397 65053 . 80 50 2,41140 21661 62 0.82848 32287 2.40704 97447 0.30373 74301 80 23 2.37250 85828 61 0.81744 47382 2.37833 38874 0.31339 72593 79 55 2.33361 49994 60 0.80615 84738 2.34908 28015 0.32294 79773 79 26 2.29472 14161 59 0.79463 50337 2.31933 50143 0.33238 09873 78 56 2.25582 78328 58 0.78288 50590 2.28912 95239 0,34168 70724 78 26 2.21693 42495 57 0.77091 92109 2.25850 57383 0.35085 63539 77 54 2.17804 06662 56 0.75874 81476 2.22750 34151 0.35987 82486 77 21 2.13914 70828 55 0.74638 25018 2.19616 26008 0.36874 14237 76 47 2.10025 34995 54 0.73383 28587 2.16452 35708 0.37743 37507 76 12 2.06135 99162 53 1 \J\J »-' XJ 9 0.72UO 97334 2.13262 67708 0,38594 22578 75 36 2,02246 63329 52 0.70822 35503 2.10051 27578 0.39425 30813 74 58 1.98357 27495 51 0.69518 46210 0.68200 31247 2.06822 21426 0.4023s 14155 74 20 I .94467 91662 50 2.03579 55331 0.41022 14630 73 40 1.90578 55829 49 0.66868 90878 2.00327 34790 0.41784 63843 72 58 1.86689 19996 1.82799 84162 1,78910 48329 48 0.65525 23646 0.64170 26188 I .97069 64170 1,93810 46179 0,42520 82479 0,43228 79822 72 16 71 31 47 46 0.62804 93057 1.90553 81344 0.43906 53283 70 45 1,75021 12496 45 A(r) D(r) E(r) 4> F<^ r Smithsonian Tables 298 ELLIPTIC FUNCTION K = 3. 6518559695, K' = 1. 5751136078, E = 1. 017236918, E' = 1. 5664967878, r F0 4> E(r) D(r) A(r) 0.00000 00000 0' 0' 0.00000 00000 I . 00000 00000 0,00000 00000 I 0.04057 61774 2 I 0.02925 15342 1.00059 38572 0.01311 92586 2 0.08115 23549 4 29 0.05837 13484 I .00237 48641 0.02624 22974 3 0.12172 85323 6 55 0.08722 94380 1.00534 13262 0,03937 28749 4 0.16230 47098 9 16 0.11569 91812 I .00949 04192 0,05251 47063 5 0.20288 08872 II 33 0.14365 89152 1.01481 81886 0,06567 14426 6 0.24345 70646 13 49 0.17099 33783 1.02131 95491 0,07884 66485 7 0.28403 32421 16 4 0-19759 49853 1.02898 82841 0,09204 37819 8 0.32460 94195 18 17 0-22336 49075 I. 0378 I 70450 0.10526 61731 9 0.36518 55969 20 29 0.24821 39381 1.04779 73504 0,11851 70041 10 0.40576 17744 22 39 0.27206 31341 I. 05891 95857 0.13179 92889 II 0. 44633 79518 24 46 0.29484 42309 1.07117 30024 0.145 I I 58534 12 0.48691 41293 26 52 0.31649 98365 1.08454 57174 0,15846 93168 13 0.52749 03067 28 56 0-33698 34175 1.09902 47131 0.17186 20726 14 0.56805 64841 30 58 0.35625 90959 1. 1 1459 58374 0.18529 62711 ': 15 0.60864 26616 32 55 0.37430 12782 1.13124 38038 0.19877 38016 16 0.64921 88390 34 51 0,39109 41430 I. 14895 21925 0,21229 62758 17 0.68979 50165 36 44 0.40663 10147 I. 16770 34514 0.22586 50123 18 0-73037 1 1939 38 36 0.42091 36481 I. 18747 88983 0.23948 10211 19 0.77094 73713 40 24 0-43395 14533 1.20825 87235 0.25314 49894 20 0.81152 35488 42 9 0.44576 06829 1.23002 19929 0.26685 72683 21 0.85209 97262 43 51 0-45636 36044 I . 25274 66524 0.28061 78600 22 0.89267 59037 45 31 0.46578 76783 1.27640 95335 0.29442 64067 23 0.93325 2081 1 47 8 0.47406 47564 1.30098 63590 0.30828 21794 24 0.97382 82585 48 42 0.48123 03147 1.32645 17509 0,32218 40690 25 I .01440 44360 50 13 0.48732 27312 1.35277 92393 0.33613 05773 26 1.05498 06134 51 42 0.49238 26159 1-37994 1272 1 0,35011 98097 27 I 09555 67908 53 8 . 0.49645 21966 I .40790 92268 0,36414 94689 28 1.13613 29683 54 31 0.49957 47663 1.43665 34239 0.37821 68497 29 I. 17670 91457 55 51 0.50179 41897 I .46614 31412 0.39231 88350 30 I. 21728 53232 57 9 0.50315 44701 1.49634 66307 0.40645 18927 31 1.25786 15006 58 25 0.50369 93739 1.52723 I 1369 0.42061 20743 32 1.29843 76780 59 38 0.50347 21104 1.55876 29167 0.43479 50141 33 I. 33901 38555 60 48 0.50251 50624 I .59090 72622 0.44899 59303 34 1-37959 00329 61 56 0.50086 95651 1.62362 85241 0.46320 96265 35 I .42016 62104 63 2 0.49857 57270 1.65689 01387 0.47743 04952 36 1.46074 23878 64 5 0.49567 22903 1.69065 46558 0,49165 25218 37 1.50131 85652 65 7 0.49219 65260 1.72488 37696 0,50586 92908 38 I. 541 89 47427 66 6 0.48818 41583 1-75953 83514 0,52007 39919 39 I .58247 09201 67 3 0.48366 93168 1-79457 84847 0.53425 94285 40 1.62304 70975 67 58 0.47868 45099 1.82996 35024 0,54841 80268 41 1.66362 32750 68 51 0.47326 06189 1.86565 20265 0.56254 18461 42 I. 70419 94524 69 42 0.46742 69071 I .90160 20099 0.57662 25903 43 1-74477 56299 70 31 0.46121 10428 1.93777 07807 0- 59065 16209 44. 1.78535 18073 71 19 0.45463 91336 1. 9741 1 50881 0.60461 99704 45 90-r 1.82592 79847 72 5 0.44773 57684 2.01059 11517 0-61851 83573 FiA ^ G(r) C(r) B(r) Smithsonian Tables TABLE e = 84' 9 299 = 0.25794019576 3337, GO = 0.4929628191, HK = 1.5205617314 B(r) C(r) G(r) ^ Fi/- 90-r 90 I . 00000 00000 3.09301 99213 0.00000 00000 90° 0' 3.65185 59695 0.99977 07150 3.09233 85676 0.01085 90483 89 45 3. 61 127 97920 89 0.99908 31458 3.09029 54977 0.02171 66503 89 31 3.57070 36146 88 0.99793 81489 3.08689 36827 0.03257 13506 89 16 3.53012 74372 87 0.99633 71496 3.08213 80679 0.04342 16747 89 I 3.48955 12597 86 0.99428 21381 3^.07603 55627 0.05426 61204 88 47 3.44897 50823 85 0.99177 56649 3.06859 50269 0.06510 31473 88 32 3 . 40839 89048 84 0.98882 08340 3.05982 72527 0.07593 11673 88 17 3.36782 27274 83 0.98542 12955 3.04974 49431 0.08674 85345 88 2 3.32724 65500 82 0.98158 12363 3.03836 26866 0.09755 35344 87 46 3.28667 03725 81 0- 97730 53698 3.02569 69280 0.10834 43731 87 30 3.24609 41951 80 0.97259 89240 3. 01 I 76 59358 0.11911 91660 87 14 3.20551 80177 79 0.96746 76286 2.99658 97659 0.12987 59255 86 58 3.16494 18402 78 0.96191 77007 2.98019 02223 0.14061 25487 86 42 3.12436 56628 77 0.95595 58299 2.96259 08137 0.15132 68040 86 25 3.08378 94853 76 0.94958 91609 2.94381 67083 0.16201 63172 86 8 3.04321 33079 75 0.94282 52769 2.92389 46843 0.17267 85562 85 50 3.00263 71305 74 0.93567 21802 2.90285 30783 0.18331 08161 85 32 2.96206 09530 73 0.92813 82732 2.88072 17308 0.19391 02013 85 14 2.92148 47756 72 0.92023 23376 2.85753 19293 0.20447 36088 84 55 2.88090 85981 71 0.91196 35133 2.83331 63492 0.21499 77081 84 36 2.84033 24207 70 0.90334 12763 2.80810 89917 0.22547 89218 84 16 2.79975 62433 69 0.89437 54154 2.78194 51210 0.23591 34034 83 55 2.75918 00658 68 0.88507 60096 2.75486 I 1988 0.24629 70143 83 34 2.71860 38884 67 0.87545 34034 2.72689 48173 0.25662 52995- 83 13 2.67802 77109 66 0.86551 81826 2.69808 46313 0.26689 34606 82 51 2.63745 15335 65 0.85528 11491 2.66847 02880 0.27709 63287 82 28 2.59687 53561 64 0.84475 32958 2.63809 23575 0.28722 83335 82 4 2.55629 91786 63 0.83394 57809 2.60699 22604 0.29728 34722 81 39 2.51572 30012 62 0.82286 99019 2.57521 21966 0.30725 52753 81 14 2.47514 68238 61 0.81 153 70701 2.54279 50725 0.31713 67705 80 48 2.43457 06463 60 0.79995 87840 2.50978 44281 0.32692 04449 80 21 2.39399 44689 59 0.78814 66036 2.47622 43648 0.33659 82039 79 53 2.35341 82914 58 0.77611 21247 2.44215 94723 0.34616 13287 79 24 2.31284 21140 57 0.76386 69524 2.40763 47564 0.35560 04313 78 54 2.27226 59366 56 0.75142 26764 2.37269 55671 0.36490 54063 78 23 2.23168 97591 2.19111 35817 55 0.73879 08451 0.72598 29409 0.71301 03561 0.69988 43682 2.33738 75276 0.37406 53814 77 51 54 2.30175 64635 2.26584 83337 2.22970 91619 0.38306 86651 0.39190 26919 0.40055 39659 77 76 76 18 44 8 2.15053 74042 2.10996 12268 2.06938 50494 53 52 51 0.68661 61 1 72 0.67321 65825 0.65969 65607 0.64606 66446 0.63233 72022 2.19338 49695 2.15692 17102 2.12036 52053 2.08376 10820 2.04715 47117 0.40900 80023 0.41724 92673 0.42526 11165 0.43302 57335 0.44052 40667 75 74 74 73 72 31 53 13 32 49 2.02880 88719 1.98823 26945 1.94765 65171 1.90708 03396 1.86650 41622 50 49 48 47 46 0.61851 83573 2.01059 11517 0.44773 57684 72 5 1.82592 79847 45 r A(r) D(r) E(r) <#> F<^ Smithsonian Tables 300 ELLIPTIC FUNCTION K = 3. 8317419998, K' = 1. 5737921309, E = 1, 0126635062, E' = 1. 6678090740, r F<^ E(r) D(r) A(r) 0.00000 00000 0° 0' 0,00000 00000 I . 00000 00000 0,00000 00000 I 0.04257 49111 2 26 0,03129 75841 I . 00066 67396 0,01256 98450 2 0.08514 98222 4 52 0.06244 25476 I . 00266 63652 0,02514 45765 3 0.12772 47333 7 18 0,09328 44601 1.00599 70974 0.03772 90570 4 0. 17029 96444 9 43 0.12367 72052 1,01065 59692 0,05032 81006 5 0.21287 45555 12 6 0.15348 09749 I ,01663 88247 0,06294 64495 6 o- 25544 94667 14 29 0. 18256 40780 1,02394 03165 0.07558 87497 7 0.29802 43778 16 50 0.21080 45154 1.03255 39030 0,08825 95281 8 0- 34059 92889 19 9 0.23809 12866 1,04247 18453 0.10096 31685 9 0.38317 42000 21 26 0.26432 54039 1.05368 52030 0,11370 38895 10 0.42574 91111 23 42 0.28942 06026 I ,06618 38299 0,12648 57214 II 0,46832 40222 25 55 0.31330 37505 I, 07995 63700 0,13931 24846 12 0.51089 89333 28 5 0.33591 49667 1,09499 02519 0,15218 77682 13 0-55347 38444 30 13 0.35720 74739 I .11127 16844 0.16511 49087 14 0.59604 87555 32 18 0.37714 72117 1,12878 56513 0,17809 69700 15 0.63862 36666 34 21 0.39571 22464 1,14751 59063 0,19113 67239 i6 0.68119 85777 36 20 0.41289 20138 I , 16744 49685 0,20423 66315 17 0.72377 34889 38 17 0.42868 64336 1,18855 41 178 0,21739 88246 i8 0.76634 84000 40 II 0,44310 49337 I. 2 I 082 33907 0,23062 50891 19 0.80892 331H 42 I 0,45616 54173 1,23423 15771 0.24391 68485 20 0.85149 82222 43 49 0,46789 32075, 1,25875 62174 0.25727 51484 21 0.89407 31333 45 33 0.47831 99952 1,28437 36007 0,27070 06428 22 0,93664 8o/|,i^ 47 15 0.48748 28142 1.31105 87634 0,28419 35800 23 0.97922 29555 48 53 0.49542 30625 1.33878 54900 0.29775 37910 24 I. 02 I 79 78666 50 28 0,50218 55842 1,36752 63142 0,31138 06778 25 1.06437 27777 52 0,50781 78217 1.39725 25218 0.32507 32040 26 I. 10694 76888 53 29 0,51236 90454 1.42793 41552 0.33882 98857 27 I. 14952 25999 54 56 0,51588 96635 1.45954 00195 0.35264 87839 28 I. 19209 751 10 56 19 0,51843 06138 1,49203 76904 0.36652 74982 29 1.23467 24222 57 39 0,52004 28338 1.52539 35243 0,38046 31619 30 1.27724 73333 58 59 0,52077 68087 1.55957 26706 0.39445 24378 31 I. 3 I 982 22444 60 12 0,52068 21896 1.59453 90851 0.40849 15164 32 1,36239 71555 61 24 0,51980 74799 1.63025 55479 0,42257 61 140 33 1.40497 20666 62 34 0,51819 97811 1.66668 36814 0,43670 14735 34 1.44754 69777 63 41 0.51590 45944 1.70378 39728 0,45086 23658 35 I. 49012 18888 64 46 0,51296 56697 1.74151 57980 0.46505 30926 36 1,53269 67999 65 48 0,50942 48984 1.77983 74487 0.47926 74909 37 1.57527 17110 66 48 0,50532 22421 1,81870 61627 0.49349 89386 38 1,61784 66221 67 46 0,50069 56936 1,85807 81564 0.50774 03615 39 1,66042 15332 68 41 0,49558 12646 1,89790 86607 0.52198 42419 40 1,70299 64444 69 35 0.49001 29952 1,93815 19599 0,53622 26281 41 1.74557 13555 70 26 0.48402 29824 1,97876 14331 0.55044 71457 42 I. 78814 62666 71 16 0,47764 14227 2,01968 95998 0,56464 90099 43 1,83072 1 1777 72 3 0,47089 66670 2,06088 81669 0.57881 90394 44 1,87329 60888 72 49 0.46381 52836 2,10230 80805 0,59294 76712 45 1,91587 09999 73 33 0.45642 21286 2,14389 95792 0,60702 49768 90-r Ff ^ G(r) C(r) B(r) Smithsonian Tables TABLE e = 85° 9=0. 275179804873563, 0= 0. 4510905222, HK = 1. 5588714533 301 B(r) C(r) G(r) 4^ Fi/- 90-r I . 00000 00000 3.38728 70037 o.aoooo 00000 90° 0' 3.83174 19998 90 0.99976 05041 3.38649 90904 0.01092 82185 89 47 3.78916 70887 89 0.99904 23353 3.38413 65337 0.02185 52713 89 34 3.74659 21776 88 0.99784 64504 3.38020 28815 0.03277 99847 89 22 3.70401 72665 87 0.99617 44409 3.37470 40379 0.04370 I 1679 89 9 3.66144 23554 86 0.99402 85290 3.36764 82512 0.05461 76051 88 56 3.61886 74443 85 0.99141 15622 3.35904 60961 0.06552 80467 88 43 3.57629 25331 84 0.98832 70058 3.34891 04507 0.07643 12000 88 29 3.53371 76220 83 0.98477 89335 3-33725 64694 0.08732 57205 88 16 3.49114 27109 82 0.98077 20177 3.32410 15504 0.09821 02023 88 2 3.44856 77998 81 0.97631 15168 3.30946 52989 0.10908 31677 87 49 3.40599 28887 80 0.97140 32619 3.29336 94854 0.1 1994 30573 87 35 3.36341 79776 79 0.96605 36420 3.27583 79999 0.13078 82183 87 20 3.32084 30665 78 0.96026 95874 3.25689 68018 0.14161 68937 87 6 3.27826 81554 77 0- 95405 85520 3.23657 38654 0.15242 72092 86 51 3.23569 32443 76 0.94742 84947 3.21489 91220 0.16321 71605 86 35 3.19311 83332 75 0.94038 78585 3.19190 43978 0.17398 45990 86 20 3.15054 34221 74 0.93294 55499 3.16762 33486 0. 18472 72171 86 4 3.10796 85109 73 0.92511 09158 3.14209 13909 0.19544 25321 85 48 3 06539 35998 72 0.91689 37204 3 -11534 56304 0.20612 78689 85 31 3.02281 86887 71 0.90830 41205 3.08742 47870 0.21678 03419 85 13 2.98024 37776 70 0.89935 26403 3.05836 91 177 0.22739 68349 84 55 2.93766 88665 69 0.89005 01452 3.02822 03368 0.23797 39802 84 37 2.89509 39554 68 0.88040 78152 2.99702 15345 0.24850 81357 84 18 2.85251 90443 67 0.87043 71 170 2.96481 70925 0.25899 53603 83 58 2.80994 41332 66 0.86014 97763 2.93165 25995 0.26943 13876 •83 38 2.76736 92221 65 0- 84955 77491 2.89757 47641 0.27981 15977 83 17 2.72479 431 10 64 0.83867 31932 2.86263 13272 0.29013 09871 82 55 2.68221 93999 63 0.82750 84383 2.82687 09732 0.30038 41353 82 33 2.63964 44888 62 0.81607 59576 2.79034 32412 0.31056 51708 82 10 2.59706 95776 61 0.80438 83372 2.75309 84351 0.32066 77330 81 46 2.55449 46665 60 0.79245 82474 2.71518 75345 0.33068 49323 81 21 2. 51 191 97554 59 0.78029 84129 2.67666 21047 0.34060 93073 80 55 2.46934 48443 58 0.76792 15834 2.63757 42081 0.35043 27789 80 28 2.42676 99332 57 0.75534 05043 2.59797 63158 0.36014 66018 80 2.38419 50221 56 0.74256 78883 2.55792 12198 0.36974 13124 79 31 2.34162 OHIO 55 0.72961 63864 0.71649 85603 0.70322 68545 0.68981 35699 2.51746 19471 2.47665 16742 2.43554 36438 2.39419 10827 0.37920 66740 0.38853 16185 0.39770 41848 0.40671 14546 79 78 77 77 2 30 58 24 2.29904 51999 2.25647 02888 2.21389 53777 2.17132 04666 54 53 52 51 0.67627 08370 0.66261 05910 0.64884 45467 0.63498 41750 0.62104 06800 2.35264 71220 2.31096 47190 2.26919 65819 2.22739 50955 2.18561 22515 0.41553 94843 0.42417 32345 0.43259 64967 0.44079 18172 0.44874 04204 76 76 75 74 74 50 13 35 56 16 2.12874 55554 2.08617 06443 2.04359 57332 2.00102 08221 1.95844 591 10 50 49 48 47 46 0.60702 49768 2.14389 95792 0.45642 21286 73 33 I. 9 I 587 09999 45 r A(r) D(r) E(r) <^ F<^ Smithsonian Tables 302 ELLIPTIC FUNCTION K = 4. 0527581695, K' = 1. 5727124350, E = 1. 0086479569, E' = 1. 5688837196, r ¥ E(r) D(r) A(r) 0.00000 00000 0° 0' 0.00000 00000 I . 00000 00000 0.00000 00000 I 0.04503 06463 2 35 0.03379 31823 I .00076 14948 0.01189 42847 2 0.09006 12927 5 9 0.06740 53633 1.00304 53671 0.02379 47903 3 0.13509 19390 7 43 0.10065 84494 1.00684 97794 0.03570 77106 4 0.18012 25853 10 16 0.13338 00630 1.01217 16668 0.04763 91855 5 0.22515 32316 12 48 0.16540 61602 I. 01900 67332 0.05959 52742 6 0.27018 38780 15 18 0.19658 33739 1.02734 94459 0.07158 19286 7 0.31521 45243 17 46 0.22677 10168 I. 03719 30291 0.08360 49670 8 0.36024 51706 20 13 0.25584 26948 1.04852 94558 0.09567 00478 9 0.40527 58170 22 37 0.28368 75021 I. 06 I 34 94387 0.10778 26441 10 0.45030 64633 24 58 0.3 102 1 07894 1 .07564 24197 0. 11994 80182 II 0.49533 71096 27 18 0.33533 45137 I. 09 I 39 65585 0.13217 11972 12 0.54036 77559 29 34 0.35899 71966 I . 10859 87206 0.14445 69485 13 0.58539 84023 31 47 0.381 15 35291 I. 12723 44637 0.15680 97563 14 0.63042 90486 33 57 0.40177 36714 I . 14728 80243 0.16923 37988 15 0.67545 96949 36 4 0.42084 23033 I . X6874 23039 0. 18173 29260 16 0.72049 03413 38 8 0.43835 74800 1.19157 88539 0.1 943 I 06384 17 0.76552 09876 40 8 0.45432 93515 I. 21577 78616 0.20697 00661 18 0.81055 16339 42 S 0.46877 87966 1.24131 81358 0.21971 39498 19 0.85558 22802 43 58 0.48173 60209 I. 268 I 7 70925 0.23254 46217 20 0.90061 29266 45 53 0.49323 91602 1.29633 07415 0.24546 39877 21 0.94564 35729 47 35 0.50333 29227 1.32575 36734 0.25847 35115 22 0.99067 42192 49 18 0.51206 72988 I. 35641 90478 0.27157 41984 23 1.03570 48656 50 57 0.51949 63591 1.38829 85826 0.28476 6581 I 24 1.08073 551 19 52 33 0.52567 71528 I. 42136 25446 0.29805 07071 25 I. 12576 61582 54 6 0.53066 87177 1.45557 97413 0.31142 61261 26 I. I 7079 68045 55 36 0.53453 12033 I. 49091 75157 0.32489 18800 27 1. 21582 74509 57 2 0.53732 51072 1.52734 17416 0.33844 64932 28 1.26085 80972 58 25 0.53911 06227 I. 56481 68225 0.35208 79650 29 1.30588 87435 59 45 0.53994 70893 1.60330 56919 0.36581 37630 30 I. 35091 93898 61 2 0.53989 25408 I .64276 98172 0.37962 08180 31 1-39595 00362 62 16 0.53900 33421 I. 68316 92055 0.39350 55205 32 I .44098 06825 63 28 0.53733 39051 1.72446 24133 0.40746 37182 33 I. 48601 13288 64 36 0.53493 64751 1.76660 65590 0.42149 07161 34 I. 53104 19752 65 42 0.53186 09786 1.80955 73388 0.43558 12766 35 1.57607 26215 66 45 0.52815 49246 1.85326 90463 0.44972 96226 36 I. 621 10 32678 67 46 0. 52386 33506 1.89769 45959 0.46392 94409 37 I. 66613 39141 68 44 0.51902 88062 1.94278 55494 0.47817 38881 38 I. 711 16 45605 69 40 0.51369 13678 I .98849 21476 0.49245 55978 39 I. 75619 52068 70 33 0.50788 86793 2.03476 33449 0.50676 66888 40 I. 80122 58531 71 25 0.50165 60117 2.08154 68491 0.52109 87757 41 1.84625 64995 72 14 0.49502 63387 2. 12878 91642 0.53544 29804 42 I. 89128 71458 73 2 0.48803 04242 2.17643 56384 0.54978 99455 43 I. 93631 77921 73 47 0.48069 69176 2.22443 05163 0.56412 98491 44 I. 98134 84385 74 31 0.47305 24550 2.27271 69945 0.57845 24208 ' 45 90-r 2.02637 90848 75 12 0.46512 17631 2.32123 72832 0.59274 69597 Fi// 4^ G(r) C(r) B(r) Smithsonian Tables Table = 86° 9=0. 295488385558687, 00=0. 4242361430, HK = 1 . 6043008048 303 B(r) C(r) G(r) ^ FiA 90-r I . 00000 00000 3.78623 65254 0.00000 00000 90° 0' 4.05275 81695 90 0.99974 76964 3.78529 99318 0.01098 79345 89 49 4.00772 75232 89 0.99899 1 1477 3.78249 16163 0.02197 49829 89 38 3.96269 68769 88 0.99773 14382 377781 59714 0.03296 02520 89 28 3.91766 62306 87 0-99597 03726 3.77128 03065 0.04394 28343 89 17 3.87263 55842 86 0,99371 04703 3.76289 48312 0.05492 18007 89 6 3.82760 49379 85 0.99095 49588 3.75267 26317 0.06589 61931 88 54 3.78257 42916 84 0.98770 77652 3.74062 96405 0.07686 50165 88 43 3.73754 36452 83 0.98397 35058 3.72678 46000 0.08782 72314 88 32 3.69251 29989 82 0.97975 74732 3.71115 90191 0.09878 17452 88 20 3.64748 23526 81 0.97506 56227 3.69377 7124S 0.10972 74034 88 8 3.60245 17063 80 0.96990 45558 3.67466 58061 0.12066 29807 87 56 3.55742 10599 79 0.96428 15032 3 65385 45535 0.13158 71709 87 44 3.51239 04136 78 Q. 95820 43054 3.63137 53926 0.14249 85767 87 32 3.46735 97673 77 0.95168 I39I4 3.50726 281 14 0.15339 56986 87 19 3.42232 91209 76 0.94472 17573 3 58155 36840 0.16427 69227 87 5 3.37729 84746 75 0.93733 49419 3.55428 71880 0.17514 05085 86 52 3.33226 78283 74 0.92953 IOOI7 3.52550 47184 0.18598 45746 86 38 3.28723 71820 •73 0.92132 04850 3 49524 97967 0.19680 70842 86 24 3.24220 65356 72 0.9I27I 44039 3.46356 79762 0.20760 58292 86 9 3 19717 58893 71 0.90372 42062 3.43050 67437 0.21837 84126 85 54 3.15214 52430 70 0.89436 17453 3. 3961 I 54178 0.22912 22300 85 38 3.10711 45967 69 0.88463 92502 3.36044 50445 0.23983 44495 85 22 3.06208 39503 68 0.87456 92937 3.32354 82896 0.25051 19896 85 5 3 01705 33040 67 0.86416 47610 3.28547 93300 0.261 15 14957 84 48 2.97202 26577 66 0.85343 88167 3.24629 37417 0.27174 93142 84 30 2.92699 201 13 65 0.84240 48716 3.20604 83874. 0.28230 14649 84 II 2.88196 13650 64 0.83107 65499 3.16480 13024 0.29280 36106 83 52 2.83693 07187 63 0.81946 76545 3.12261 15798 0.30325 10250 83 32 2.79190 00724 62 0.80759 21336 3 07953 92551 0.31363 85568 83 II 2.74686 94260 61 . 79546 40466 0.78309 75297 3 03564 51912 2.99099 09630 0.32396 05923 0.33421 10135 82 49 82 26 2.70183 87797 2.65680 81334 2.61177 74870 60 59 58 0.77050 67624 2.94563 87432 0.34438 31544 82 3 0.75770 59335 2.89965 I 1884 0.35446 97527 81 39 2.56674 68407 57 56 0.74470 92077 2.85309 13269 0.36446 28984 81 13 2.52I7I 61944 0.73153 06927 0.71818 44065 0.70468 42455 0.69104 39537 0.67727 70914 2.80602 24483 0.37435 39786 80 47 2 47668 55480 55 2.75850 79940 2.71061 14508 2.66239 62465 2.61392 56481 0.38413 36176 0.39379 16142 0.40331 68729 0.41269 73321 80 19 79 50 79 20 78 49 2.43165 49017 2.38662 42554 2.34159 36091 2.29656 29627 54 53 52 51 0.66339 70061 0.64941 68038 0.63534 93209 0.62120 70978 0.60700 23531 2.56526 26633 2.51646 99446 2.46760 96971 2.41S74 35896 2.36993 26700 0.42191 98869 0.43097 03076 0.43983 31542 0.44849 16855 0.45692 77651 78 17 77 43 77 8 76 31 75 52 2.25153 23164 2.20650 I67OI 2.I6147 10238 2. 1 1644 03774 2.07140 9731 1 50 49 48 47 46 0.59274 69597 2.32123 72S32 0.46512 17631 75 12 2.02637 90848 45 A(r) D{r) E(r) ^ Fc^ r Smithsonian Tables 304 ELLIPTIC FUNCTION K = 4. 3386539760, K' = 1 5718736105, E = 1.0052585872, E' = 1.5697201504 I, r F 4> E(r) D(r) A(r) o 0.00000 00000 0° 0' 0.00000 00000 1 . 00000 00000 0.00000 00000 I 0.04820 72664 2 46 0.03700 05198 1.00089 26934 0.01102 97158 2 0.09641 45328 5 31 0.07377 86246 1.00357 01695 0.02206 73089 3 0.14462 17992 8 15 o.iioii 59944 1.00803 06141 0.03312 06260 4 0.19282 90656 10 59 0.14580 23384 I. 01427 09982 0.04419 74541 5 0.24103 63320 13 41 0.18063 90239 1.02228 70707 0.05530 54893 6 0.28924 35984 16 21 0.21444 22668 1.03207 33471 0.06645 23081 7 0.33745 08648 18 59 0.24704 57854 1.04362 30963 0.07764 53371 8 0.38565 81312 21 34 0.27830 28485 1.05692 83239 0.08889 18239 9 0.43386 53976 24 7 0.30808 76822 I. 07197 97531 0.10019 88085 10 0.48207 26640 26 37 0.33629 62369 1.08876 68032 0.11157 30946 II 0.53027 99304 29 3 0.36284 63422 I. 10727 75652 0.12302 12218 12 0.57848 71968 31 27 0.38767 73064 I . 12749 87762 0.13454 94383 13 0.62669 44632 33 46 0.41074 90335 1.14941 57909 0.14616 36738 H 0.67490 17296 36 2 0.43204 07437 1.17301 25520 0.15786 95139 IS 0.72310 89960 38 14 0.45154 93887 I. 19827 15591 0.16967 21746 i6 0.77131 62624 40 23 0.46928 78534 I. 22517 38362 0.18157 64776 17 0.81952 35288 42 27 0.48528 30289 1.25369 88987 0.19358 68272 i8 0.86773 07952 44 28 0.49957 38349 1.28382 47193 0.20570 71870 19 0.91593 80616 46 24 0.51220 92565 1.31552 76945 0.21794 10587 20 0.96414 53280 48 16 0.52324 64512 1.34878 26100 0.23029 14612 21 I. 01235 25944 50 5 0.53274 89656 1.38356 26077 0.24276 091 1 1 22 1.06055 98608 51 50 0.54078 50933 I. 41983 91529 0.25535 14044 23 I. 10876 71272 53 30 0.54742 63924 1.45758 20021 0.26806 43994 24 I. I 5697 43936 55 7 0.55274 63730 1.49675 91734 0.28090 08008 25 I. 20518 16600 56 40 0.55681 93566 1-53733 69175 0.29386 09452 26 1.25338 89264 58 10 0.55971 95044 1.57927 96919 0.30694 45879 27 I. 30159 61928 59 36 0.56152 00057 I. 62255 01370 0.32015 08913 28 1.34980 34592 60 58 0.56229 24153 I. 66710 90551 0.33347 84147 29 I. 39801 07256 62 17 0.56210 61265 1.71291 53925 0.34692 51057 3° I. 4462 I 79920 63 33 0.56102 79658 I .75992 62260 0.36048 82928 31 1.49442 52584 64 46 0.55912 18929 1.80809 67519 0.37416 46804 32 1.54263 25248 65 55 0.55644 87947 1.85738 02804 0.38795 03444 33 I 59083 97912 67 2 0.55306 63561 1.90772 82336 0.46184 07305 34 1.63904 70676 68 6 0.54902 89975 I .95909 01488 0.41583 06538 35 1.68725 43240 69 7 0.54438 78661 2. 01 141 36867 0.42991 42995 36 1-73546 15904 70 5 0.53919 08711 2.06464 46451 0.44408 52267 37 1.78366 88568 71 I 0.53348 27539 2. I 1872 69773 0.45833 63730 38 I. 83187 61232 71 54 0.52730 51847 2.17360 28173 0.47266 00609 39 1.88008 33896 72 45 0.52069 68791 2.22921 25107 0.48704 80065 40 1.92829 06560 73 34 0.51369 37297 2.28549 46508 0.50149 13298 41 1.97649 79224 74 20 0.50632 89466 2.34238 61220 0.51598 05665 42 2.02470 51888 75 5 0.49863 32034 2.39982 21493 0.53050 56822 43 2.07291 24552 75 47 0.49063 47860 2.45773 63538 0-54505 60878 44 2.12111 97216 76 58 0.48235 9741 1 2.51606 08149 0.55962 06569 45 2.16932 69880 77 7 0.47383 20219 2.57472 61393 0.57418 77451 90-r F^ ^ G(r) C(r) B(r) Smithsonian Tables TABLE, e = 87° «- = 0. 320400337134867, 9 = 0. 3802048484, HK = 1. 6608093163 305 B(r) C(r) G(r) ^ F^ 90-r I . 00000 00000 4-37119 23556 0.00000 00000 90° 0' 4-33865 39760 90 0.99973 08085 4.37002 95871 0.01103 73956 89 51 4.29044 67096 89 0.99892 36540 4.36654 32014 0.02207 41777 89 43 4.24223 94432 88 0-99757 97949 4 36073 89539 0.03310 97273 89 34 4.19403 21768 87 0.99570 13248 4.35262 64203 0-04414 34137 89 25 4.14582 49104 86 0.99329 I I 666 4.34221 89731 0- 055 1 7 45893 89 16 4.09761 76440 85 0.99035 30638 4-32953 37471 0.06620 25830 89 7 4.04941 03776 84 0.98689 15704 4-31459 15972 0.07722 66944 88 58 4.00120 31112 83 0.98291 20378 4.29741 70454 0.08824 61873 88 49 3-95299 58448 82 0.97842 05999 4.27803 82196 0.09926 02826 88 39 3-90478 85784 81 0.97342 41557 4.25648 67836 0.11026 81515 88 30 3.85658 13120 80 0.96793 03503 4.23279 78580 0.12126 89076 88 20 3.80837 40456 79 0.96194 75529 4.20700 99336 0.13226 15989 88 10 3.76016 67792 78 0.95548 48341 4-17916 47765 0.14324 51989 88 3-71195 95128 77 0.94855 19406 4-14930 73254 0.1 542 I 85972 87 49 3.66375 22464 76 0.941 I 5 92676 4. I I 748 55826 0.16518 05896 87 38 3.61554 49800 75 0.93331 78308 4.08375 04971 0.17612 98666 87 27 3-56733 77136 74 0.92503 92359 4.04815 58427 0.18706 50017 87 16 3.51913 04472 73 0.91633 56463 4.01075 80891 0.19798 44386 87 4 3.47092 31808 72 0.90721 97509 3.97161 62682 0.20888 64763 86 51 3.42271 59144 71 0.89770 47288 3.93079 18356 0.21976 92546 86 38 3.37450 86480 70 0.88780 42140 3.88834 85274 0.23063 07363 86 25 3.32630 13816 69 0.87753 22590 3.84435 22135 0.24146 86896 86 II 3.27809 41 152 68 0.86690 32971 3.79887 07472 0.25228 06673 85 57 3.22988 68488 67 0.85593 21039 3-75197 38123 0.26306 39853 85 42 3.18167 95824 66 0.84463 37589 3-70373 27678 0.27381 56982 85 27 3-13347 23160 65 0.83302 36055 3.65422 04910 0.28453 25731 85 II 3.08526 50496 64 0.82111 72113 3.60351 12193 0.29521 10610 84 54 3-03705 77832 63 0.80893 03281 3.55168 03915 0.30584 72655 84 37 2.98885 05168 62 0.79647 88516 3.49880 44891 0.31643 69081 84 19 2.94064 32504 61 0.78377 87810 3.44496 08773 0.32697 5291 I 84 2.89243 59840 60 0.77084 61787 3.39022 76481 0.33745 72566 83 40 2.84422 87176 59 0.75769 71307 3-33468 34641 0.34787 71421 83 19 2.79602 14512 58 0.74434 77069 3.27840 74042 0.35822 87319 82 57 2.74781 41848 57 0.73081 39218 3.22147 88118 0.36850 52042 82 35 2.69960 69184 56 0.71711 16962 3.16397 71463 0.37869 90740 82 II 2-65139 96520 55 0.70325 68193 0.68926 491 16 0.67515 13887 0.66093 14267 3.10598 I 837 I 3.04757 21420 2.98882 70090 2.92982 49435 0.38880 21304 0.39880 53693 0.40869 89202 0.41847 19672 81 81 80 80 47 21 54 26 2.60319 23856 2.55498 51 192 2.50677 78528 2.45857 05864 54 53 52 51 0.64661 99275 0.63223 14865 0.61778 03606 0.60328 04384 0.58874 52 1 10 2.87064 38790 2. 81 136 10542 2.75205 28945 2.69279 48995 2.63366 15364 0.428 1 1 26638 0.43760 80415 0.44694 391 1 1 0.45610 47583 0.46507 363 I I 79 79 78 78 77 56 25 53 19 44 2.41036 33200 2.36215 60536 2.31394 87872 2.26574 15208 2.21753 42544 50 49 48 47 46 0.57418 77451 2.57472 61393 0.47383 20219 77 7 2.16932 69880 45 r A(r) D(r) E(r) F0 Smithsonian Tables 3o6 ELLIPTIC FUNCTION K = 4. 7427172653, K' = 1. 5712749524, E = 1. 0025840855, E' = 1. 5703179199, r F<^ E(r) D(r) A(r) 0.00000 00000 0° 0' 0,00000 00000 I . 00000 00000 0.00000 00000 I 0.05269 68585 3 I 0.04150 83698 I. 00 I 09 49202 0.00984 61866 2 0.10539 37170 6 2 0.08272 60369 1.00437 91719 0.01970 23988 3 0.15809 05755 9 I 0.12336 86879 1.00985 12249 0.02957 86287 4 0.21078 74340 II 59 0.16316 44916 I .01750 85180 0.03948 48012 5 0.26348 42925 14 56 0.20185 96235 1.02734 74434 0-04943 07415 6 0.31618 11510 17 49 0.23922 29917 1.03936 33238 0.05942 61408 7 0.36887 80095 20 40 0.27504 99964 I 05355 03843 0.06948 05245 8 0.42157 48680 23 28 0.30916 52198 1.06990 17180 0.07960 32187 9 0.47427 17265 26 13 0.34142 40166 1.08840 92458 0.08980 33 I 81 10 0.52696 85850 28 53 0.37171 30376 I. 10906 36709 0.10008 96542 II 0.57966 54435 31 30 0.39994 97772 1.13185 44282 0.1 1047 07636 12 0.63236 23020 34 2 0.42608 12751 I. 15676 96284 0.12095 48573 13 0.68505 91605 36 30 0.45008 21300 I. I 8379 59985 0.13154 97896 14 0.73775 60190 38 53 0.47195 19964 1.21291 88175 0.14226 30292 15 0.79045 28775 41 12 0.49171 27333 I. 24412 18489 0.15310 16293 i6 0.84314 97360 43 26 0.50940 53625 1.27738 72698 0.16407 21997 17 0.89584 65946 45 35 0.52508 69758 I. 31269 55975 0.17518 08788 i8 0.94854 34531 47 40 0.53882 77072 1.35002 56142 0-18643 33074 19 I. 00124 03116 49 40 0.55070 78595 1-38935 42896 0.19783 46027 20 I 05393 7 I 701 51 34 0.56081 52531 1.43065 67027 0.20938 93338 21 I. 10663 40286 53 25 0.56924 28378 1-47390 59633 0.221 10 14976 22 I 15933 08871 55 II 0.57608 65921 I -51907 31337 0.23297 44971 23 I. 21202 77456 56 52 0.58144 37172 I. 56612 71505 0.24501 1 1 193 24 I .26472 46041 58 29 0.58541 1 1 188 I. 61503 47485 0.25721 35159 25 I. 3 I 742 14626 60 2 0.58808 41618 1.66576 03865 0.26958 31846 26 1.37011 83211 61 31 0.58955 56773 I. 71826 61750 0.28212 09517 27 I. 42281 51796 62 55 0.58991 51945 I. 7725 I 18082 0.29482 69565 28 1-47551 20381 64 16 0.58924 83721 1.82845 44989 0.30770 06377 29 1.52820 88966 65 33 0.58763 66017 1.88604 89185 0.32074 07202 3° I 58090 57551 66 46 0.58515 67551 1.94524 71416 0-33394 52050 31 1.63360 26136 67 56 0.58188 10541 2.00599 85969 0.34731 13599 32 1.68629 94721 69 3 0.57787 70364 2.06825 00238 0-36083 57125 33 I ■ 73899 63306 70 6 0.57320 76019 2.13194 54360 0.37451 40449 34 I. 79169 31891 71 7 0.56793 1 1 188 2 . 19702 60925 0.38834 13902 35 1.84439 00476 72 4 0.56210 15757 2.26343 04764 0.40231 20314 36 1.89708 69061 72 59 0.55576 87678 2.33109 42822 0.41641 95021 37 1.94978 37646 73 51 0.54897 85058 2-39995 04116 0.43065 65890 38 2.00248 06231 74 41 0.54177 28388 2.46992 89791 0.44501 53371 39 2.05517 74816 75 28 0.53419 02851 2-54095 73266 0.45948 70563 40 2.10787 43401 76 12 0.52626 60647 2.612.96 00482 0.47406 2331 I 41 2.16057 I 1986 76 55 0.51803 23296 2.68585 90255 0.48873 10316 42 2.21326 80571 77 35 0.50951 83887 2-75957 34731 0.50348 23272 43 2.26596 49156 78 14 0.50075 09241 2.83401 99954 0.51830 47025 44 2.31866 17741 78 50 0.49175 41985 2. 909 I I 26530 0.53318 59750 45 2.37135 86326 79 25 0.48255 02516 2.98476 30422 0.54811 33155 90-r Fi 4^ G(r) C(r) B(r) Smithsonian Tables TABLE d = 88° 9=0. 353165648296037, 9 0=0. 3246110213, HK = 1. 7370861537 307 B(r) C(r) G(r) ^ F^ 90-r I . 00000 00000 5 35291 58734 0.00000 00000 90° 0' 4.74271 72653 90 0.99970 65254 5-35135 39870 0. 01 107 55804 89 54 4.69002 04068 89 0.99882 66090 5.34667 I I 120 0.02215 08037 89 47 4 63732 35483 88 0.99736 17711 5-33887 55928 0.03322 53090 89 41 4.58462 66898 87 0.99531 45401 5.32798 13106 0.04429 87274 89 35 4-53192 98313 86 0.99268 84456 5.31400 76445 0.05537 06778 89 28 4-47923 29728 85 0.98948 80069 5.29697 94165 0.06644 07630 89 21 4.42653 61 143 84 0.98571 87199 5.27692 68222 0.07750 85650 89 15 4-37383 92558 83 0.98138 70401 5 25388 53459 0.08857 36405 89 8 4.32114 23973 82 0.97650 03636 5.22789 56618 0.09963 55161 89 I 4.26844 55388 81 0.97106 70046 5.19900 35203 0.1 1069 36828 88 54 4.21574 86803 80 0.96509 61704 5.16725 96214 0.12174 75905 88 46 4.16305 18218 79 0.95859 79343 5. 1327 I 94744 0.13279 66420 88 39 4. I I 035 49633 78 0.95158 32050 5-09544 32457 0.14384 01862 88 31 4.05765 81048 77 0.94406 36948 5-05549 55939 0.15487 75112 88 23 4.00496 12463 76 0.93605 18846 5.01294 54947 0.16590 78361 88 15 3.95226 43878 75 0.92756 09875 4.96786 60538 0.17693 03026 88 6 3-89956 75293 74 0.91860 49094 4.92033 43119 0.18794 39654 87 58 3.84687 06707 73 0.90919 82095 4.87043 10392 0.19894 77822 87 48 3.79417 38122 72 0.89935 60570 4.81824 05226 0.20994 06015 87 39 3.74147 69537 71 0.88909 41880 4-76385 03454 0.22092 1 1 507 87 29 3.68878 00952 70 0.87842 88604 4-70735 I 1607 0.23188 80216 87 18 3.63608 32367 69 0.86737 68071 4.64883 64589 0.24283 96552 87 8 3-58338 63782 68 0.85595 51894 4.58840 23314 0.25377 43247 86 56 3-53068 95197 67 0.84418 15481 4.52614 72300 0.26469 01 166 86 45 3.47799 26612 66 0.83207 37552 4.46217 17234 0.27558 49098 86 32 3.42529 58027 65 0.81964 99644 4.39657 82526 0.28645 63526 86 19 3-37259 89442 64 0.80692 85610 4.32947 08849 0.29730 18370 86 6 3.31990 20857 63 0.79392 81 128 4.26095 50677 0.3081 I 8471 I 85 52 3.26720 52272 62 0.78066 73195 4-19113 73836 0.31890 30470 85 37 3.21450 83687 61 0.76716 49636 4.12012 53075 0.32965 20072 85 21 3.16181 15102 60 0.75343 98604 4.04802 69653 0.34036 14062 85 5 3.10911 46517 59 0.73951 08099 3-97495 08972 0.35102 68681 84 48 3.05641 77932 58 0.72539 65478 3.90100 58247 0.36164 35409 84 29 3.00372 09347 57 0.7IIII 56987 3.82630 04227 0.37220 60448 84 10 2.95102 40762 56 0.69668 67291 3-75094 30973 0.38270 84160 83 51 2.89832 72177 55 0.68212 79026 3.67504 17706 0.39314 40446 83 30 2.84563 03592 54 66745 72351 3.59870 36716 0.40350 56060 83 8 2.79293 35007 53 0.65269 24519 0.63785 09470 3.52203 51359 3.44514 14133 0.41378 49862 0.42397 31992 82 44 82 20 2.74023 66422 2.68753 97837 52 51 0.62294 97425 0.60800 54504 0.59303 42368 0.57805 17864 0.56307 32704 3.36812 64840 3.29109 28843 3.21414 15421 3.13737 16225 3.06088 03834 0.43406 02965 0./14403 52686 0.45388 59368 0.46359 88357 0.47315 90851 81 55 81 28 80 59 80 29 79 58 2.63484 29252 2.58214 60667 2.52944 92081 2.47675 23496 2.42405 5491 I 50 49 48 47 46 0.5481 I 33155 2.98476 30422 0.48255 02516 79 25 2.37135 86326 45 r ' A(r) D(r) E(r) 4> F0 Smithsonian Tables 3o8 ELLIPTIC FUNCTION K = 5. 4349098296, K' = 1. 5709159581, E = 1. 0007515777, E' = 1. 5706767091, r F<^ E(r) D(r) A(r) 0.00000 00000 0° 0' 0.00000 00000 I . OpOOO 00000 0.00000 00000 I 0.06038 78870 3 27 0.04919 51488 I .00148 76066 0.00797 98676 2 0.12077 57740 6 54 0.09795 31901 1.00595 04088 0.01597 27570 3 0.18116 36610 10 19 0.14584 95983 I. 01 338 83449 0.02399 16544 4 0.24155 15480 13 42 0.19248 42494 I .02380 12862 0.03204 94760 5 0.30193 94350 17 3 0.23749 17959 I. 037 I 8 89963 0.04015 90322 6 0.36232 73220 20 19 0.28055 00559 1.05355 10766 0.04833 29925 N7 0.42271 52090 23 32 0.32138 60670 1.07288 68948 0.05658 38508 8 0.48310 30960 26 40 0.35977 96610 I. 095 I 9 55002 0.06492 38899 9 0.54349 09830 29 43 0.39556 46136 I. 12047 55228 0.07336 51472 10 0.60387 88700 32 40 0.42862 75917 I. 14872 50597 0.08191 93794 II 0.66426 67569 35 32 0.45890 52450 I. I 7994 15472 0.09059 80283 12 0.72465 46439 38 18 0.48637 98590 1.21412 16208 0.09941 21860 13 0.78504 25309 40 58 0.51 107 40138 I. 25126 09628 0.10837 25614 14 0.84543 04179 43 32 0.53304 46717 I. 29135 41391 0.1 1 748 94454 15 0.90581 83049 45 59 0.55237 70723 1.33439 44250 0.12677 26784 i6 0.96620 61919 48 20 0.56917 87466 1.38037 36227 0.13623 16162 17 I .02659 40789 50 35 0.58357 38857 1.42928 18693 0.14587 50978 i8 1.08698 19659 52 44 0.59569 82320 1. 48 1 10 74384 0.15571 14129 19 I . 14736 98529 54 47 0.60569 45851 1.53583 65353 0.16574 82707 20 1.20775 77399 56 43 0.61370 89715 1.59345 30865 0.17599 27682 21 I .26814 56269 58 35 0.61988 74725 1.65393 85266 0.18645 13603 22 I 32853 35139 60 20 0.62437 36797 I. 71727 15815 0.19712 98307 23 1.38892 14009 62 0.62730 67243 1.78342 80514 0.20803 32624 24 I .44930 92879 63 35 0.62881 98144 1.85238 05926 0.21916 60113 25 1.50969 71749 65 5 0.62903 92100 1.92409 85022 0.23053 16788 26 I 57008 50619 66 30 0.62808 35657 1.99854 75042 0.24213 30872 27 I . 63047 29489 67 51 0.62606 35735 2.07568 95405 0.25397 22556 28 I . 69086 08359 69 7 0,62308 18462 2.15548 25676 0.26605 03772 29 I. 75124 87229 70 19 0.61923 29878 2.23788 03597 0.27836 77989 3° 1.81163 66099 71 27 0.61460 38040 2.32283 23203 0.29092 40017 31 I . 87202 44969 72 31 0.60927 36149 2.41028 33038 0.30371 75832 32 I. 93241 23839 73 32 0.60331 46378 2.50017 34479 0.31674 62424 33 I .99280 02709 74 29 0.59679 24144 2.59243 80185 0.33000 67656 34 2.05318 81579 75 23 0.58976 62623 2.68700 72681 0.34349 50157 35 2. I I 357 60449 76 14 0.58228 97341 2.78380 63098 0.35720 59222 36 2.17396 39318 77 2 0.57441 10737 2.88275 50068 0.37113 34754 37 2.23435 18188 77 48 0.56617 36598 2.98376 78796 0.38527 072 1 1 38 2.29473 97058 78 31 0.55761 64315 3.08675 40315 0.39960 97596 39 2.35512 75928 79 II 0.54877 42910 3.19161 70942 0.41414 17461 40 2.41551 54798 79 49 0.53967 84809 3.29825 51932 0.42885 68946 41 2.47590 33668 80 25 0.53035 69362 3.40656 09346 0.44374 44843 42 2.53629 12538 80 58 0.52083 46089 3.51642 14148 0.45879 28694 43 2.59667 91408 81 30 0.51113 37664 3.62771 82525 0.47398 94906 44 2.65706 70278 82 0.50127 42646 3.74032 76441 0.48932 08915 45 2.71745 49148 82 28 0.49127 37968 3.85412 04436 0.50477 27366 90-r Ftp \^ G(r) C(r) B(r) Smith SOMAN Tables TABLE g = 89° g = 0.403309306338378, 6 0=. 0. 2457332317, HK = 1. 8599580878 309 I . 00000 00000 0.99966 43156 0.99865 79343 0.99698 28696 0.99464 24694 0.99164 140^2 0.98798 56557 0.98368 24869 0.97874 04272 0.97316 92390 0.96697 98856 0.96018 44944 0.95279 63165 0.94482 96828 0.93629 99559 0.92722 34802 0.91761 75278 0.90750 02426 0.89689 05812 0.88580 82522 0.87427 36532 0.86230 78063 0.84993 22921 0.83716 91826 0.82404 09732 0.81057 05141 0.79678 09414 0.78269 56083 0.76833 80165 0-75373 17477 0.73890 03962 0.72386 75024 0.70865 64877 0.69329 05904 0.67779 28032 0.66218 58136 0.64649 19448 0.63073 30999 0.61493 07081 0.59910 56732 0.58327 83254 0.56746 83750 0.55169 48696 0.53597 61539 0.52032 98326 0.50477 27366 A(r) 7 56958 97180 7.56705 29325 7-55944 77064 7-54678 94142 7-52910 36233 7.50642 60102 7.47880 22428 7.44628 78301 7.40894 79407 7-36685 71893 7.32009 93943 7.26876 73054 7.21296 23044 7.15279 40797 7.08838 02759 7.01984 61207 6 94732 40301 6-87095 31948 6.79087 91481 6.70725 33191 6.62023 25717 ■6.52997 87323 6.43665 81080 6.34044 09975 6.24150 I 1966 6.14001 55012 6.03616 32083 5.93012 56192 5.82208 55452 5.71222 68183 5.60073 38100 5-48779 09576 5-37358 23026 5.25829 10413 5.14209 90885 5.02518 66588 4-90773 1 863 1 4.78991 03252 4.67189 48167 4.55385 49133 4-43595 66732 4.31836 23371 4.20123 00521 4.08471 36196 3.96896 22668 3.85412 04436 0.00000 00000 o. 01 1 10 10463 0.02220 19579 0-03330 25985 0.04440 28272 o 05550 24979 0.06660 14556 0.07769.95354 0.08879 65593 0.09989 23340 O.I 1098 66481 0.12207 92686 0.I33I6 99380 0.14425 83704 0.15534 42469 0.16642 721 18 0.17750 68667 0.18858 27648 0.19965 44048 0.21072 12232 0.22178 25863 0.23283 77807 0.24388 60035 0.25492 63501 0.26595 78012 0.27697 92084 0.28798 92768 0.29898 65471 0.30996 93739 0.32093 59022 0.33188 40408 0.34281 14317 0.35371 54168 0.36459 29992 0.37544 08012 D(r) 0.38625 50154 o. 39703 13507 0.40776 49715 0.41845 04298 0.42908 15883 o. 43965 15347 0.45015 24856 0.46057 56791 0.47091 12546 0.48II4 8II89 0.49127 37968 E(r) 90" o' 89 56 89 53 89 49 89 45 89 42 89 38 89 34 89 30 89 26 89 22 89 17 89 13 88 58 88 53 88 47 88 41 88 35 88 29 88 22 88 15 88 7 87 59 87 87 87 87 87 87 86 86 86 86 85 85 85 85 51 42 33 23 12 I 50 37 24 10 55 40 23 6 47 84 27 84 6 83 44 83 20 82 55 82 28 5.43490 98296 5.37452 19426 5-31413 40556 5-25374 61686 5-19335 82816 5.13297 03946 5.07258 25077 5.01219 46207 4.95180 67337 4.89141 88467 4.83103 09597 4.77064 30727 4.71025 51857 4.64986 72987 4.58947 94117 4.52909 15247 4.46870 36377 4.40831 57507 4 34792 78637 4.28753 99767 4.22715 20897 4.16676 42027 4.10637 63157 4.04598 84287 3.98560 05417 3.92521 26547 3.86482 47677 3.80443 68807 3.74404 89937 3.68366 I 1067 3.62327 32197 3.56288 53328 3.50249 74458 3.44210 95588 3.38172 16718 90 3-32133 37848 3.26094 58978 3.20055 80108 3.14017 01238 3.07978 22368 3-01939 43498 2.95900 64628 2.89861 85758 2.83823 06888 2.77784 28018 '2.71745 49148 F 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 Smithsonian Tables INDEX The numbers A PAGE Absolute convergence 109 Addition formulas, Elliptic Functions 250 Algebraic equations 2 Algebraic identities i Alternating series .' no Archimedes, spiral of 52 Area of polygon 36 Arithmetical progressions 26 Asymptotes to plane curves 40 Axial vector 9S B Ber and Bei functions 204 BernouUian numbers 25 polynomial 140 Bessel functions 196 addition formula 199 multiplication formula 199 references 213 Bessel-Clifford differential equation . . 205 Beta functions 132 Binomial coefiScients 19 Binormal 59 Biquadratic equations 10 Bromwich's expansion theorem 212 C Cassinoid 53 Catenary S^ Cauchy's test 109 Center of curvature, plane curves 39 surfaces S^ Change of variables in multiple inte- grals 17 Characteristic of surface S^ Chord of curvature, plane curves 39 Circle of curvature 39 Circular functions, see Trigonometry Cissoid 53 Clairaut's differential equation i66 Coefficients, binomial 19 Combinations i7 Comparison test i°9 31 refer to pages. PAGE Complementary function 167 Concavity and convexity of plane curves 38, 42 Conchoid 53 Conditional convergence 109 Confluent hypergeometric function 185 Conical coordinates 104 Consistency of linear equations 15 Convergence of binomial series 117 tests for infinite series 109 Covariant property 17 Cubic equations '. 9 Curl 93 Curvature, plane curves 38 space curves 58 Curves, plane ; 36 space 57 Curvilinear coordinates 99 Curvilinear coordinates, surfaces of revolution io6 Cycloid 51 Cylindrical coordinates 32, 102 Cylinder functions, see Bessel functions 197 D d'Alembert's Test 109 Definite integrals, computation by dif- ference functions 225 Simpson's method 221 expressed as infinite series 134 de Moivre's theorem 66 Derivatives 155 of definite integrals 156 of implicit functions 161 Descartes' rule of signs s Determinants 11 Difference functions 222 Differential equations 162 numerical solution 220 Differentiation of determinants 13 Discriminant of biquadratic equa- tion II Divergence 93 Double periodicity of elliptic functions 250 1 312 INDEX E PAGE Ellipse 46 Ellipsoidal coordinates 102 Elliptic cylinder coordinates 104 Elliptic integrals, first kind 245 second kind 248 third kind 251 Elliptic integral expansions 135, 19s Envelope 40 Envelope of surfaces 56 Epicycloid 52 Equations, algebraic 2 transcendental, roots of 84 Equiangular spiral 53 Eta functions 251 Euler's constant 27 summation formula 25 transformation formula 113 theorem for homogeneous functions. 157 Eulerian angles 32 Evolute 39 Exact differential equations 163, 177 Expansion of determinants 13 Expansion theorem, Bromwich's 212 Heaviside's 212 PAGE Homogeneous differential equations 162, 166, 177 Homogeneous linear equations 15 Homer's method 7 I'Hospital's rule 145 Hyperbola 48 Hyperbolic functions 71 spiral 52 Hypergeometric differential equation 209 series 209 Hypergeometric function, confluent. . . 185 Hypocycloid 52 Identities, algebraic i Implicit functions, derivatives of 161 Indeterminate forms 145 Indicial equation 174 Infinite products 130 series 109 Integrating factors 163 Interpolation formula, Newton's 22 Intrinsic equation of plane curves. ... 44 Involute of plane curves 39 Finite differences and sums Finite products of circular functions . Finite series, special Fourier's series Fresnel's integrals Functional determinants 20 84 26 136 134 16 Jacobian i6 K Ker and Kei functions 205 Kummer's transformation 114 Gamma function Gauss's n function theorem Geometrical progressions. Gradient of vector Graeffe's method Green's theorem Gregory's series Gudermannian H Harmonical progressions Harmonics, zonal Heaviside's operational methods. expansion theorem Helical coordinates Hessian 131 133 95 26 93 8 95 122 76 26 191 210 212 106 16 Lagrange's theorem 112 Laplace's integrals 193 Latus rectum, ellipse 48 hyperbola 49 parabola 46 Leclert's transformation 115 Legendre's equation 191 Leibnitz's theorem 157 Lemniscate 53 Limiting values of products 152 sums 151 Linear equations 15 Linear vector function 96 Lituus S3 Logarithmic spiral S3 M Maclaurin's theorem 112 Markoff's transformation formula. ... 113 INDEX 313 PAGE Maxima and minima 152 Mehler's integrals 193 Minor of determinant 14 Multinomial theorem 120 Multiplication of determinants 12 Multiple roots of algebraic equations . . s N Neoid 53 Neumann's expansion, zonal har- monics 194 Newton's interpolation formula 22 method for roots of equations 7 theorem on roots of algebraic equa- tions.; 2 Normal to plane curves 36 Numbers, Bernoulli's 140 Euler's 141 Numerical series 140 Numerical solution of differential equa- tions 220 O Oblate spheroidal coordinates 107 Operational methods 210 Orthogonal curvilinear coordinates. . . . 100 P n function. Gauss's 133 Parabola 45 ParaboKc coordinates 107 Parabolic cylinder coordinates 105 ParaboKc spiral 53 ParaUelepipedon, volume of 92 Partial fractions 20 Particular integral 167 Pedal curves 4° Pendulum 247 Permutations and combinations 17 Plane S3 Plane curves 36 polar coordinates -41 Plane geometry 34 Points of inflexion 39> 42 Polar coordinates 32, loi Plane curves 4i Polar subtangent 37 subnormal 37 normal 37 tangent 37 Polar vector 95 PAGE ■ Polynomial 2 Bernoullian 25 series 119 Principal normal to space curves 58 Products, finite of circular functions. . 84 limiting values of 152 of two series no Progressions 26 Prolate spheroidal coordinates 107 Q Quadratic equations 9 Quadriplanar coordinates 33 R Raabe's test 109 Radius of curvature, plane curves. . . 38, 42 space curves 58 surfaces 55 Radius of torsion 59 Reciprocal determinants 14 Resolution into partial fractions 20 Reversion of series 116 Rodrigues' formula 193 Roots of algebraic equations 2 transcendental equations 84 Rot 93 Routh's rule 6 S Scalar product 91 Schlomilch's expansion, Bessel func- tions 201 Series, finite, circular functions 81 infinite 109 special finite 26 numerical 140 of Bessel functions 201 h}^ergeometric 209 of zonal harmonics i94 Simpson's method 221 Singular points 41 Skew determinants 14 Skew-S)Tnmetrical determinants 15 Solid geometry S3 Space curves 57 Spherical polar coordinates loi Spherical triangles 78 Spheroidal coordinates 107 Spiral of Archimedes 52 Stirling's formula 28 314 INDEX PAGE Stokes's theorem 95' Sturm's theorem 6 Subnormal 36 Sub tangent 36 Sums, limiting values of 151 Summation formula, Euler's 25 Surfaces 55 Symbolic form of infinite series 112 Symbolic methods in differential equa- tions 173 Symmetrical determinants 14 Symmetric functions of roots of algebraic equations 2 T Tables, binomial coefficients 20 hyperbolic functions ' 72 trigonometric functions 62 Tangent to plane curves 36 Taylor's theorem iii Theta function 248, 251 Toroidal coordinates 108 Tractrix 53 Transtendental equations, roots of 84 Transformation of coordinates 29 determinants 12 PAGE equations 4 infinite series 113 Triangles, solution of plane 77 spherical 78 Trigonometry 61 Trilinear coordinates 33 Trochoid 51 U Uniform convergence no Unit vector 92 V Variation of parameters 180 Vectors, axial 95 polar 95 functions, linear 96 Vector product gi W Witch of Agnesi 53 Z Zeta function 255 Zonal harmonics 191