smithsonian mathematical formulae

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. publications in the nature of meteorological,. physical,. and. handy books. of. mathematical. many. years a group of. information on geographical,. These include the. subjects.. 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, 1921).. The. many branches. present volume comprises the most important formulae of. 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 University of Chicago, contributed the section. equations.. The. on numerical solution. tables of elliptic functions were prepared. by. of differential. Col. R. L. Hippisley,. who has. contributed the. The compiler. Dr. Adams, and the Smithsonian Institution many physicists and mathematicians, especially to Dr. IJ. L.. Curtis and col-. C. B., under the direction of Sir George Greenhill, Bart.,. introduction to these tables. are indebted to. leagues of the Bureau of Standards, for advice, criticism, and cooperation in the preparation of this volume.. Charles D. Walcott, Secretary. May, ig22.. of the. Smithsonian Institution..

PREFACE The. original object of this collection of. some. together, compactly,. for the benefit of those. more. mathematical formulae was to bring. useful results of mathematical analysis. who regard mathematics. There are many such. itself. is. of the. not constantly using them, and to find them one. number. A. of. books which. may. and not as an end in remember, for one who. as a tool,. results that are difficult to is. obliged to look through a. not immediately be accessible.. collection of formulae, to. meet the object. largely a matter of individual selection;. of the present one,. for this reason this. volume. must be is. issued. an interleaved edition, so that additions, meeting individual needs, may be made, and be readily available for reference. It was not originally intended to include any tables of functions in this volume, but merely to give references to such tables. An exception was made, in. elliptic functions, calculated, on Sir George 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 list of useful tables of this. however, in favor of the tables of Greenhill's. new. plan,. kind, as well as of other volumes, having an object similar to this one,. is. appended.. 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 IntroShould the plan of. this collection. duction to the Tables of Elliptic Functions, I wish to express. And. I. wish also to record. my obligations. my. gratitude.. to the Secretary of the Smithsonian In-. and to Dr. C. G. Abbot, Assistant Secretary of the Institution, for the which they have met all my suggestions with regard to this volume.. stitution,. way. in. E. P. Princeton,. New. Jersey. Adams.

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. D. Bierens de Ha.a.n: E.. Jahnke and. F.. Integrals,. Cambridge, 191 1.. Nouvelles Tables dTntegrales Definies, Leiden, 1867.. 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 Braunschweig,. 1. W. LiGOWSKi: Taschenbuch O. Th. Burklen:. der reinen und angewandten Mathematik,. 888-1 894. der Mathematik, Berlin, 1893.. Formelsammlung. und. Repetitorium. der. Mathematik,. Berlin, 1922.. F.. Auerbach: 1909.. Taschenbuch fur Mathematiker und Physiker,. Leipzig, 1909.. i.. Jahrgang,.



CONTENTS PAGE Symbols I.. viii. Algebra. i. Geometry. 29. III.. Trigonometry. 61. IV.. Vector Analysis. 91. II.. V. Curvilinear Coordinates VI. VII. VIII.. 99. Infinite Series. 109. Special Applications of Analysis. 145. Differential Equations. 162. IX. Differential Equations (Continued). 191. X. Numerical Solution of Differential Equations. 220. XI. Elliptic Functions. 243. Introduction by Sir George Greenhill, F.R.S. Tables of Elliptic Functions, by Col. R. L. Hippisley. Index. 245. ..... 259. 311. vii.

SYMBOLS Whenever used. logarithm.. log. To. find the. nl. Factorial,. !. common logio a. = 0.43429. log a. = 2.30259. where n. Does not. > <. Greater than.. is. understood.. .. .. •. log a.. .. .. .. logio a.. an integer denotes. is. 1.2.3.4. ^. Equivalent notation 4:. the Naperian "logarithm. logarithm to base lo. equal.. Less than.. ^. Greater than, or equal. :^. Less than, or equal. CO. Binomial. —. Approaches. I. —. ^' ,. ^'. '. '. d{Xi, X2. a. 1. See 1.51-. coefficient.. Determinant where aik. flffc I. '. \. to.. to.. the element in the. is. Functional determinant.. iih.. row and ^th column,. See 1.37.. ). Absolute value of. a.. a. If. quantity. a real. is. its. "numerical value,. I. without regard to \. sign.. a] = modulus of a =. +V —. i. The imaginary =. ^. Sign of summation,. i.e.,. If. a. is. + Va^ +. a complex quantity, a =. a+. i^,. jS^,. i.. ^dk =. ai. +. 02. +. 03. +. •. .. •. •. +. On.. k=n. II. Product,. i.e.,. I. I (i. +. kx). =. (i. vui. +. x){i. +. 2x){i. +. 3^). .. .. .. .. (i. +. nx)..

I. 1.00. ALGEBRA. Algebraic Identities.. 1.. a". -. b". =. (a. -. b){a"--^. +. a^'-b. 2.. a". ±. Z>". =. (a. +. 6)(a"-i. -. a"-'^b. + a'^'W + + a"-^^- -. « odd: upper n even: lower 3.. (a;. +. ai){x. +. 0-2). Pi =. Ol. +. (x-. 02. Pk = sum of P„ = aiaoQs .. (a2. +. &2)(a2. +. i82). (a2. -. 62)(a2. _. ^2). +. Qn). = X". ab""--. + ±. b""'^).. fi''-^.. sign. sign.. + Pix"-i + P2a;"-2 + + Pn-lX + Pn.. +an,. all .. +. +. =F ab"-'-. .. the products of the a's taken ^ at a time. .. o„.. = (aa ^ (^ct. T ^. ± ^>a)2. ^ ^^)2, + ^^ + ^yy ^. 6/3)2. +. (a (3. J^)2. _. (^^. + 52 + ,2)(a2 + /32 y) = (^c^ Q^y _ ^,)2 + (,a _ ^^)2 + (aiS - a6)2. (a2 + b' + c2 + ^^)(a2 + |82 + y + 52) ^ (^ct + 6/3 + C7 + rfS)^ + (fljS - ba + c5 - dyf + (^7 - 66 - ca + d^Y + (a5 + 67 - CjS - c?a)2. {ac - bdY + (ai + bcY = {ac + bdY + {ad - be)-, (a + b)(b + c)(c + a) = (a + b + c){ab + be + ca) -abc. (a + b){b + c){c + a) = a^{b + c) + b"{c + a) + c\a + b) + 2abc. (a + b)(b + c){c + a) = bc(b + c) + ca{c + a) + ab{a + 6) + 2abc. 3(a + b){b + c){c + a) = {a + b + cY- {a' + b' + c'). (b - a){c - a){c - b) = a\c - b) + b~(a - c) + c-{b - a). (6 - a)(c - a)(c - b) = a{F- - c") + 6(^2 - a") + cia" - 6^). (6 - a){c - a){c - b) = bc{c - b) + ca{a - c) + ab{b - a). (a - bY +{b- cY + (c - aY = 2[(a - b){a - c) + {b - a){b - c) + (c- a)(c- 6)]. a3(52 _ ^2) _^ ^3(^2 _ ^2) ^ ^3(^2 _ ^2) = (^ _ J) (^ _ ^ (^ _ (-)(a^ + 6c + CO) (o + 6 + c)(a2 + 62 + c2) = 6c(6 + c) + ca(c + o) + ab{a + 6) + a^ + 6^ + c^. (a + 6 + c)(6c + CO + a6) = 0^(6 + c) + b'{c + a) + c2(a b) + 7,abc. {b + c- d){c + a - b){a + b - c) = a^ib + c) + b\c + c) + c2(a + 6) -(a3 + 63 + c3 + 2o6c). (^2. _|.. ). -\-.

:. •. MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS. 2. + c){ - a + b + c){a - b + c){a ^ b - c) = 2{b''c^ + cV + a^i^) + b' + c*). {a + b + c + dy + (a + b - c - dy + {a + c - b - d)- + {a + d - b - cY = 4(a2 + ^2 + + d^). If /I = ca + 67 + Cj8 B = a^ + ba + cy C = ay + b^ + ca. 21.. +. {a. b. -{a'. 22.. C-'. 23.. (a. 24.. [a^. 25.. (a3. + 6 + c)(a + (3 + y)=A+B + C. + 62 + c2 - (a6 + bc + ca)'] [a^ + /^^ + +. 63. ^. c*. - 30k) (a^ +. +. j8^. 7^. -. y^. 3 a/37). -. +. {a(3. = ^^. +. 5'^. + 7a)]. I3y. - 3ABC.. +. ALGEBRAIC EQUATIONS 1.200. The. expression. =. f(x) is. an. 1.201. co-v". +. aix"~i. +. 02'^"~^. +. +. an-ix. +. an. integral rational function, or a polynomial, of the ;zth degree in x.. The equation. = o has n. f{x). roots which. may. be real or complex,. dis-. tinct or repeated.. 1.202. f(x). 1.203. equation f(x) = o are. If the roots of the. =. ao(x. Symmetric functions. -. Ci){x. -. Ci,. C2,. -. {x. C2). .. Cn,. .,. .. Cn). commore important. of the roots are expressions giving certain. Among. binations of the roots in terms of the coefficients.. the. are Ci. +. C1C2. +. C2-\-. +. C1C2C3. C1C3. +. +. CiC2Ci. .. .. .. +. .. C„. C\C2Cz. + .. C2C3. .. +. +. = ao. +. CoCi+. Cidd. = (-1)". Cn. +. +. Cn-iCn. =. Cn-2Cn-lCn. =. — a<2,. ao. — Oo. 1.204. Newton's Theorem.. roots of iix). =. If. si,. denotes the. sum. of the ^th. o,. + Cn Ci + C2 + + 51^0 = O 2^2 + SiOi + 5ocro = 303 + Sia2 + 52^1 + ssao = o 4O4 + SiOs + S2ai SiOi + 54^0 =. Sk. ==. Ifll. -\-. '. o. powers of. all. the.

•^3. = - -—.

:. :. MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS. 4 1.221. It follows. from 1.220 that J\h) = R.. =. evaluating J{x) for x. 1.222. To. = Ao{x - hy. A,{x - hy-^. +. + .... + An-i{x -h)+ An.. 1.220 form J{h) = An.. last. term of each. efficients. of. express J{x) in the form. J{x). By. This gives a convenient way. h.. Repeat this process with each quotient, and the sums will be a succeeding value of the series of co-. line of. An, An-1,. ,. Aq.. Example f{x). =. 2,X^. +. 2X'*. -. SX". +. 2X. -4. h = 2. -4. 2. O. _6. i6. 8. 16. 24. 50. 6. 28. 88. 224. 14. 32. 48. 100. 96 =. As.

ALGEBRA the upper sign being used if. The. they are to be increased.. j{±h) where. The. /'(.v). is. +. the. y4o.v". where the. +. may. may. To. j-;/'"(±//). /(.v),. —. +. =. J"{x), the second derivative,. etc.. also be written. +. .4i.v"-i. coefficients. be diminished.. derivative of. first. resulting equation. sign. resulting equation will be. + ^'r (±//) +. .v/(±/o. 5. and the lower. the roots are to be diminished. if. A.2X"-~. +. +. An-lX. -^. An =. be found by the method of 1.222. by h change the. increase the roots. o. the roots are to. if. sign of h.. MULTIPLE ROOTS 1.240. If c is. a multiple root of J{x) =. o, of. order m,. i.e... repeated. m. times,. then fix) c is also. of order. (x. m—. 2 of. -. common. R =. c)-Q;. m-. i. divisor.. If. F{x). =. have no multiple roots. will is. o. oi the first derived equation, f'(x). the second derived equation, f'{x). The equation /(x) = o. 1.241. no. =. a multiple root of order. the greatest. common. o, ii. and fix). =. o;. so on.. and/(x) have. divisor oi f{x). and. /'(a;),. f{x)/F{x) =/iGv), and/i(x) will have no multiple roots.. An. 1.250. equation of odd degree, n, has at least one real root whose sign. is. opposite to that of a„. 1.251. An. root. a„. if. equation of even degree, n, has one positive and one negative real is. negative.. The equation f(x) = o has. 1.252. as. many. 1.253. Descartes' Rule of Signs:. than. has changes of sign from. it. No. +. to. If f{x). = o. is. Xl. and x =. X2. equation can have more positive roots. - and from -. No equation can have more negative roots 1.254. between x = and X2.. real roots. as there are changes of sign in f(x) between Xi. to +, in the terms of /(.v). than there are changes of sign in f{-x).. put in the form. ^oCv - h)by 1.222, and Ao, Ai,. +. Ai{x ,. -. An. h)"-^. are. + An =. +. all positive,. h. is. o. an upper. limit of the. positive roots. If /(-a-). h. is. = o. is. put. in a similar form,. a lower limit of the negative roots.. and the. coefiicients are all positive,.

:. MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS. O. /(i/x) = o is put in a similar form, and the coefficients are a lower limit of the positive roots. And with /(- i/x) = o, h. If. A. is. all positive, is. an upper. limit of the negative roots.. 1.255. Sturm's Theorem. fix). = oox". +. Form. the functions:. +. aix-"~i. a-iX"-'. +. .. .. .. +. .. a„. =/(x) = naox""-^ + (n - i)aiX"-2 + Mx) = -R, in fix) = Q^Mx) + R, Mx) = -R2 in Mx) = Q2f2ix) + R2 /lOv). The number of real number of changes stituted for x is. minus the number. substituted for. may. roots of f{x). x.. = o between x =. Xi. of sign in the series f(x), fiix),. .. ,. .. and x =. Mx),. .. /i, /2,. .. .. .. +. ^2 is .. .. .. /3(x). - 2x^ - 3X- + I ox - 4 - 3x2 _ 2.V + 5 = 9X2 _ 27.V+ II = -8x - 3. /4(X). = -1433. /(x). =. X*. /i(x). =. 2X^. /2(x). equal to the. when. Xi is sub-. series. when. X2. numerical factors. be introduced or suppressed in order to remove fractional. Example. a^_j. same. of changes of sign in the. In forming the functions. .. coefficients..

ALGEBRA Form a. row by. third Oia2. ,n-2. —. cross-multiplication:. 00O3. diOA. Form. dide. Oo(i5. —. Clod?. Oi. a fourth row by operating on these last two rows by a similar cross-. Continue. multiplication.. number whose. — ai. ai. no terms left. The column gives the number of roots. this operation until there are. of variations of sign in the first. real parts are positive.. If there are. any equal. roots. some. of the subsidiary functions will vanish.. In place of one which vanishes write the differential. coeflEicient of the last one which does not vanish and proceed in the same way. At the left of each row is written the power of x corresponding to the first subsidiary function in that row. This power diminishes by 2 for each succeeding coefficient in the row.. to. Any row may be multiplied or divided by any positive quantity in order remove fractions.. DETERMINATION OF THE ROOTS OF AN EQUATION Newton's Method. If a root of the equation f(x) = o is known to lie between Xi and 0C2 its value can be found to any desired degree of approximation by Newton's method. This method can be applied to transcendental equations 1.260. as well as to algebraic equations. If b is. an approximate value. This process. may. b. —. yjT: =. c. —. -p-r\. =. of. a second approximation,. c is. c^. a root,. a third approximation.. is. be repeated indefinitely.. Horner's Method for approximating to the real roots of f{x) = Let pi be the first approximation, such that pi -\- x > c > pi, where. 1.261. o.. c is. condition. by. 1.231.. the. The equation can always be transformed into one in which this holds by multiplying or dividing the roots by some power of 10 Diminish the roots by pi by 1.233. In the transformed equation. root sought.. A(i{x. -. px)^. +. Ai{x. -. PiY-^. + ....+ An-i{x -. pi). +An. = o. put. h -~ 10. and diminish the roots by. bIx -Pi\. ^" An-l. />2/io, yielding. ^y + bJx -Pi10/ V. a second transformed equation. ^)«-i 10/. +. .. .. .. .. + Bn. =. o..

MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS. 8 If. Bn and Bn-i are Then take. same. of the. was taken too large and must be dimin-. sign p2. ished.. pz. _ Bn. lOO. The. and continue the operation. ^. 1.262. lOO. lO. This method determines approximate values of. Method.. Graeffe's. Bn-1. required root will be:. the roots of a numerical equation, complex as well as. all. Write the equation. real.. of the «th degree. =. f{x). —. ao.v"". +. ai.v"~^. —. a^x""^. .. .. .. ±. .. a„ = o.. The product. contains only even powers of. Ao =. ao2. Ai =. ai^. A% =. a^. Az =. az'. —. an equation. wth degree in. of the. x"^.. The. 2a (jO^. — —. + 200^4 + 20105 — — 203O5 + 202O6 -. = a^. yl4. The. It is. x.. determined by. coefficients are. 201(73. 202^4. 200^6 201O7. +. 200O8. roots of the equation. -. A^y"". +. ^ly"-^. -. Aiy""-^. .. .. .. .. ± ^„ =. o. Continuing. are the squares of the roots of the given equation.. this process. we. get an equation. -. i?0M". whose roots are the. 2'"th. +. i?lM"~^. powers. Then. >. >. C2. .. .. .. .. ±. /?„. = O. of the roots of the given equation.. Let the roots of the given equation be Ci. -. i?2M"~^. Cs. Ci,. c^,. .. .. >. >. .. .. ,. Put X = 2^. Suppose. Cn.. first. that. Cn. for large values of X,. approximately. may. they. If the roots are real. be determined by extracting the Xth roots of ± is determined by taking the sign which. Whether they are. these quantities.. equation f(x) =. satisfies the. o.. Suppose next that complex roots enter so that there are equalities among the absolute values of the roots. I. ci I. ^. C2 I. I. ^. 1. I. rs. I. Cp+i. Suppose that. ^ 1. ^. ^ Cp+2 I. I. I. ^. 0> .. I. .. Cp. ;. .. I. .. ^. c„ I. 1. I. >. Cp+i I. 1. ;.









:. :. ALGEBRA Then. X. if. is. large. enough so that Cp^. large. is. compared. to Cp^^, c-^, c-^. Cp^ approximately satisfy the equation:. RouP - RiuP-^ :„^. when X. is. large. .. .. .. ±. .. i^p. = o. approximately satisfy the equation:. +. i?p+2M»-^-2. _. ±Rn. ^. enough the given equation breaks down into a number is shown in the process of deriving the suc-. This stage. of simpler equations. cessive equations. -. RiUP-^. Rp+^U^^-P-^. RnU"' Therefore. +. when. the coefficients are obtained from those of. certain of. by squaring.. the preceding equation simply. 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.. Quadratic Equations.. 1.270. X-. The. +. 2ax. +. =. 6. 0.. roots are. = -a. Xi. oci -\-. xi. X1X2. b. —. b. -s/a?. > <. b. roots are real,. a^. b. roots are complex,. a?. =. b. roots are equal.. a?. If. + Va^ -. = —a — = — 2a = b.. Xi. Cubic equations.. 1.271. +. (i). x^. (2). x = y. (3). /. ax^. +. +. te. c. =. Substitute. -. -?>py. -. 2q. =. where. 3, = 20. Roots of li. p. >. =. --. ab. 2. 3. 27. (3) o, g. >. o, q^. >. cosh. p^. <f>. ,. a^. Vp^. —. c.. o..

MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS. 10. = 2\/^ cosh. yi. li. >. p. o, q. <. o, q". >. —. y2=. -. — + W^P. sinh. —. ys=. -. - - ^VIp. sinh. -. •. /»^. cosh. 4>. =. Vp'. li. p. yi. = — 2y/p cosh —. y2. = -. — + iVsP sinh -. ya. = -. -^. - zV3?. <o sinh. </). =. yi. 2^. y2= -. /?. sinh. — + iV. - 3P. — tV-. --Ti. =. 3^3. .. p>. o, q^. <. —. cosh. /. 2. li. -•. sinh. 1. 3p cosh. — 3. p^. =. cos. —~ Vp'. yi. = 2\/p. y2. =. cos. ^ + V3p /—-. -vi. .. 4). --•. 2. 1.272. sin^. 3. Biquadratic equations. a^x^. +. aix^. +. a«::t;2. +. os-'v. +. ^4. =. 0.. Substitute. X = y. y^^—^Ey-'+^Gy-^—.F^o do-. ao"^. ao*.

:. ALGEBRA. H. - a^ G = ao^az - 3aoai02 + 2ai^ F = ao^ai - 4ao^Oifl3 + taoa^a^ — 3^1* / = a 0^4 - 401^3 + 302^ F = a^H - sH' J = aoOia-i + 2^10203 — ^003- — ai~ai — A = /^ — 27/^ = the discriminant G2 + 4H' = ao'iHI - aoJ). anQ^. oz^. of the roots of the biquadratic:. Nature. A. =. = o. Equal roots are present. Two. roots only equal. Three roots are equal. Two Two. real. :. I =. H. = I = J =. not both zero. G =. o;. ao'I. — i2H^ =. o.. and two complex roots. Roots are either. H< H>. :. distinct pairs of equal roots:. Four roots equal. A<o A>o. / are J = o. / and. :. o and atfl o and ao^I. all real. or. — 12^- < — i2H^ >. all. complex. o. Roots. all real. o. Roots. all. complex.. DETERMINANTS 1.300. 1.301. A. A. columns 1.302. If. determinant of the «th order, with. an. an. an. ai„. a-2i. 0-22. a^z. a^n. flsi. ^32. <?33. Ozn. flnl. Cn2. n- elements, is written. I. «u-. I. ,. ii,. h. a,. determinant. is. not changed in value by writing rows for columns and. for rows.. two columns or two rows. sulting determinant. is. unchanged. of a. determinant are interchanged the. in value. but. 1.303. A. determinant vanishes. 1.304. If. each element of a row or a column. the determinant. itself is. if. it. multiplied. is. re-. of the opposite sign.. has two equal columns or two equal rows.. by that. is. multiplied. factor.. by the same. factor.

:. :. :. MATHEMATICAL FORMULA AND. 12. A. ELLIPTIC FUNCTIONS. is not changed in value if to each element of a row or added the corresponding element of another row or column multiphed by a common factor.. 1.305. column. 1.306 or. determinant. is. If. each element of the. lib.. more terms the determinant. row or column consists of the sum of two up into the sum of two or more dethe Ith row or column the separate terms of. splits. terminants having for elements of. the lib row or column of the given determinant.. 1.307. corresponding elements of two rows or columns of a determinant. If. have a constant 1.308. ratio the determinant vanishes.. If the ratio of the differences of. corresponding elements in the ^th and. qth rows or columns to the differences of corresponding elements in the rth. and. sth.. 1.309. rows or columns be constant the determinant vanishes.. p rows or columns of a determinant whose elements are rational become equal or proportional when .v = h, the determinant. If. integral functions of x is. divisible. by. (x. —. h) p~^.. MULTIPLICATION OF DETERMINANTS 1.320. Two. may. determinants of equal order. be multiplied together by the. scheme I. an. I. X. I. 6i,-. =. Cij. 1. 1. I. where Cij. 1.321. If the. of lower order. =. aabji. +. +. +. an. oin. (hi. 022. 02n. I. o. o. G. O. I. O. G. an. an. a\n. O. G. 021. a-22. a-in. a„i. an2. GIG. anr. Onl. o. 1.322. an. ani. ai„bjn-. two determinants to be multiplied are of unequal order the one can be raised to one of equal order by bordering it; i.e.. On. am. ai2bj2. The product. of. two determinants flin. X. o. .. may. .. .. be written. bn. ^1. bni. bnn.

ALGEBRA an.

MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS. 14 At,. is. the. minor. first. terminant of order n. and column which. may. It. i.. A. determinant. of the. —. corresponding to o,y and. is. a de-. A. by crossing out the row and multiplying by (-i)'^^^. be obtained from. intersect in aij,. 1.342. an. flli. A. A An +. +. Ai/. a^.. Ay.. +. .. .. .. .. a.-„. +. +. a.iAoj. +. O A A;„ = -. .. A. o. aniAnj =. if. i. .^ II. t. .. i. if. A. if. ^'. 4=. / -(. =J. 4^y. =. /. 1.343. is. d-A. dAj,.i. dAij. daijdau. dan. duki. the coefficient of anaki in the complete expansion of the determinant A.. may. It. be obtained from A, except for sign, by crossing out the rows and columns. which intersect. in aij. and. aui.. 1.344. Aij. I. I. X. Oii. I. 1. A.; I. The determinant. |. An. \. is. = A" = A"-.. I. the reciprocal determinant to A.. 1.345. d'A. Aij Ail. daijdau. Ahj Aki. \. I. ^. _dA. _c^ _ 3A " dan da^ dan dakj. 1.346. d'A A2daijdakidapq. Aij. Ail. Aig. Akj Apj. Aki. Akg Apg. A,i. 1.347 32. A daudaki. daijdaki. 1.348. If. A. =. o,. aA a^. _. daij ddki. 1.350. If. aij. =. aji. the. determinant. aA aA. _. dan dakj. In. a. symmetrical. a skew determinant.. In a skew. symmetrical.. is. determinant Aij = Aji. 1.351. If. Oij. = -aji. the. determinant. is. determinant Aij. = i-iy-'Aji..

:. ALGEBRA 1.352. If. =. Oij. and an =. -Gji,. I^. determinant. the. o,. a skew. is. symmetrical. determinant.. A A. 1.360. skew symmetrical determinant skew symmetrical determinant. A. of. even order. of. odd order vanishes.. a perfect square.. is. system of linear equations. + +. ai2X2. a2lXi. 022.^2. + +. UnlXi. +. aniXi. +. anXi. .. .. h. + +. ainXn. +. QnnXn = kn. (hnXn. = =. ^2. has a solution. hAu + hA2i+. A-Xi =. +. ^nA,. provided that. A 1.361. If. A. =. o,. but. the. all. .A... where. 5. may. 1.362. If ^1. and. A. if. =. =. +. The If. I. ^2. =. .. •. .. •. •. .. o,. &'-A. u. kr-. dassdari. = kn =. o,. i,. 2,. .. .. .. n). n.. .,. the linear equations are homogeneous,. o,. =. = 1,2,.... A~. n).. condition that n linear homogeneous equations in n variables shall. be consistent 1.364. 2. :^ o.. a,,I. minors are not. be any one of the integers. X". 1.363. =. first. is. that the determinant, A, shall vanish.. there are n. +. i. anXl (hlXl. Cin\Xl. ClXl. linear equations in. + + + +. n variables. fll2^2. +. +. ainXn. 0^X2. +. +. a2nXn. an2X2. + +. +. On„.T„. C2.r2. +f„.V„. the condition that this system shall be consistent. On. is. = =. ^1. ^2. = kn = ^„+i that the determinant-'.

:. MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS. i6. 1.370. Functional Determinants.. If. yi, yi,. .. .. .. .,. jn are n functions of. xi, x^,. .. ,Xn). yk =fk(Xi,X2, the determinant: dyy dyi. dyi. dv,. d(yi,. dxi dx2. dXr,. dxjl. d{xi, X2,. dy2 dy2. dy2_. dxi dx2. dXr,. dyn dVn. dVn. dx2. dXn. dxi is. Jn). >'2. ;Xn). the Jacobian,. 1.371 F{xi,. If. 3'2,. y\,. Xi,. ,. Xn). ,. are the partial derivatives of a function. >'n. :. Ji. =. dF -Q^_. {i. =. I, 2,. .. .. w). .,. .. the symmetrical determinant:. JdF. H is. aF. \dxi' dxo. dXi dXj. dF\ *. •. •. •. a(Xl, X2,. QxJ Xn). ,. the Hessian.. 1.372. If vi,. A'2,. ,. yn are given as implicit functions of. X2,. .Ti,. ,. Xn by the n equations Fiiyi, y2,. Fniyu. ,. Xn). = o. .,. Xn). = O. yn, Xi, x^,. ,. yn, Xl, Xo,. 3'2,. .. then. dhuvi, d(xu 1.373. If the. ,yn). X2,. .. .. .,. Xn). n functions. ^ (_. d{xx, xo,. yi, y^,. the Jacobian, /, vanishes; and. if. ,F„). y d(FuF2. ^. .. .. .,. .T„). ^. d(Fr,. d(yu. .. ..,F„). .. .. >'2,. .. ,. t„). yn are not independent of each other o the w functions yi, >'2, ., 3'„ are not ,. 7=. .. independent of each other but are connected by a relation F<<yx,. F.2, >'.,. ,. >'n). = O. .. ..

7. :. :. ALGEBRA Covariant property.. 1.374. by a. = an. r]2,. +. ^1. and the functions. w. +. OisS. +. yi, j2,. Vn of. ,. ,. yn of. a(|l,. .. .. .,. .. ,. I2,. Xn are transformed. a^. |. For the Hessian,. To change. 1.380. /. new. variables, ,. n). ,. ,. Xn). ,. *'. '. 1. K. 1. xi, X2,. .. .. .. .,. Xn. }'2,. ,. when. Jy„. yn)dyidy2. yi, ya,. yn are given functions. ,. Xn'.. ^^y^'. Jf d{Xi,. J. is. .,. .. the variables in a multiple integral. i=r where F{x). .. .. '. = J-\ an. fFiyi,. of Xi, Xi,. yn). ,. dix,, X2,. W = H'\ an. /. =. 2,. the determinant or modulus of the transformation.. is. \. ,. djy,, yo,. U. r. I,. Xn become the functions. ^n:. Vn). ,. =. (i. ain^r^. :V2,. ,. or. where. Xi,. ^1, ^2,. d(Vi, V2,. '. •^. to. Xi, X2,. linear substitution Xi. rji,. I. the variables. If. y". >. Xi,. ,. the result of substituting. in /^(yi, y2,. ,. dx.. F{x)dx,dx, y^l Xn). X\, X2,. .. .. .. Xn for. .,. yi, y^,. .. .. .,. yn. yj.. PERMUTATIONS AND COMBINATIONS 1.400 n.. Given n different elements. Represent each by a number, of permutations of the n different elements is,. i, 2, 3,. ,. The number. „Pn = n\ e.g.,. w =. 3. :. (123),. 1.401 r. (132),. (213),. Given n different elements.. (231),. The number. {r<n), or the number of r-permutations, vPr. e.g.,. w =. 4, y. (312),. (321). =6. =. 3!. of permutations in groups of. is.. {n-r)l. = 3:. (123) (132) (124) (142) (134) (143) (234) (243) (231) (213) (214) (241) (341) (314). (3i2)(32i)(324)(342)(4i2)(42i)(43i)(4i3)(423)(432) = 24.

:. :. MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS. l8. Given. 1.402. n. m. divided into. The number. elements.. different. specified groups, with Xi,. +. tively, {xi-\- X2-\-. = n. Xm). X'l,. 17. e.g.,. m. 6,. =. 3, xi. =. 2, X2. =. 0:3. 3,. of. ways. a^^ in. they. can. be. each group respec-. is. I. Xm\. X\.\X2\. n =. ,. =. i. (12). (345). (6). (13). (245). (6). (23). (145). (6). (24). (135). (6). (34). (125). (6). (35). (124). (6). (45). (123). (6). (25). (234). (6). (14). (235). (6). (15). (234). (6). X. 6. = 60. Given n elements of which Xi are of one kind, X2 of a second kind, Xm of an mth. kind. The number of permutations is. 1.403. ,. w!. Xi. 1.404. Given w. among. m. +. X2. +Xm. +. The number. different elements.. specified groups,. when blank groups {m. +n—. =. of. n.. ways they can be permuted. are allowed,. is. i)\. (m-i)l e.g.,. w =. 3,. m. =. 2:. (123,0) (132,0) (213,0) (231,0) (312,0) (321,0) (12,3) (21,3) (13,2) (31,2) (23,1) (32,1) (1,23) (1,32) (2,31) (2, 13) (3, 12) (3,21) (0,123) (0,213) (0,132) (0,231). (0,312) (0,321). = 24. different elements.. The number. of. ways they can be permuted. 1.405. Given n. among. m specified groups, when blank groups are not allowed,. contains at least one element,. (n e.g.,. w =. 3,. w. =. so that each group. is. nl{n — i)! - m)lim -. i)!. 2. (l2,3)(2I,3)(l3,2)(3I,2)(23,l)(32,l)(l,23)(l,32)(2,3l)(2,I3)(3,I2)(3,2l). Given n. 1.406 into. m. n =. e.g.,. different elements.. specified groups. s,. m=. The number. when blank groups. of. are allowed. into. m. 12. is. 2:. (I23,0)(l2,3)(l3,2)(23,l)(l,23)(2,3l)(3,12)(0,123). 1.407. =. ways they can be combined. Given n similar elements. different groups. The number. when blank groups. of. =. 8. ways they can be combined. are allowed. is.

:. ALGEBRA (n. n =. e.g.,. Group. I. Group. 2. Group. 3. 1.408 into. m. 6,. w. =. +m—. ig. i)!. 3. 6554443333222221111110000000] 0102013021403125041326051423 1=28 0010210312041320514230615243J. Given n similar elements. The number of ways they can be combined when blank groups are not allowed, so that each group. different groups. shall contain at least. one element,. is. (n-j)l (m —. i)l(n. —. tn)l. BINOMIAL COEFFICIENTS 1.51. /n\ '•. •. \k). _ / 11 \ _ ^ ~ _ n{n - i)(n ~ [n -kj ' ". 2). "". k)\. W + U + ij^. \k. +. .. .. kl. i)'. "-,!-'"'. 3.. 5.. n\ _ " kl{n -. (:). = o. if. ,,. <. k.. ^•(^cr)-cr)---(^(;'::)-. «-(^G.!X)-G-X)---(^("r) "^+ .1/. II.. I. +. "-....+. (-:)"«. \2/. :J-(:J---(;;T. =0.. \ll. =(::)'. .. (n. -. k. +. i).

MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS. 20 1.52. Table. c) I. 2. c). of. Binomial Coefficients.. c). (:). c) G). c). C). Co). c) C).

ALGEBRA 3.. ^'f{x). = Ay(x. + h)-. =. 3/0. +. /(.v. A«/W = fix +. 4.. -. Aj(x).. +. 3/(-v. -. nh). 21. +. 2h). 3f{x. + n-. '^/(x. +. +. I/O. (-. +. - fix).. //). + IT^^h) -. !i^^L^/(.v. i)"/(^).. 1.812. = cA/(x). 1.. A[c/(.t)]. 2.. A[/i(x) +/2(x). 3.. A[/i(.v) -/^Or)]. '^'. a constant).. {c. + A/^Ov) + + /2(.v + h) AMx) = Mx)-AMx) +Mx)-AMx) + AMx)-AMx). .Mx) _ Mx)-AMx) -Mx)-AMx) f2{x)-Mx + h) f2{x). 1.813. The. +. =. ==. ]. /i(.v). a polynomial of the «th degree. nth. difference of. =. f(x). A/i(.i-). ^^{x). .. OoXn. +. +. aix"~^. +. +. an-ix. is. constant.. a„. A"/(x) = nlaoh^.. 1.82 A"'} (x. nin — i){n —. =. (x-b-n-. -b)(x-b-h)(x-b-2h). {x. —. —. b){x. —. b. 2. h) {x. —. {n. ). —. —. b. 2h). .. m+ .. .. (.r. ,. ih)}. i)/^'". —. b. — n —. m—. ih).. I. 2.. A^ (.V. +. b) (x -\-b. +. h). +b+. (.V. n(n. -\-. (n. i). (-1)'. +. (x 3.. A'^a^ = (a^. 4.. Alog/(.v) =. 5.. A". .. 6.. A. sin {ex. /. A'" cos {cx. —. +. b) (x. +b+. 2h). +. h) {x. .. .. .. + b^. {x. .. (n. 2). + b+2h). .. .. .. .. n -. i)"'a-^. log(i+M£)). d). ,N. =. + d) =. 2. (. / (. sm — .. 2. sin. 1. sm. r.v. +. /. c/A'". —I. (. cos. (. rf. +w. ,. ex. -\-. d. -\-. m. —— +. •. 1. ^^. eh. ih). + m — i)h"' (x + b+ n + m -. 7r\ • 1. ih). If.

MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS. 22. Newton's Interpolation Formula.. 1.83. /(•v). =. /(a). +. —^ A/(a) + (-v. -. o) (x. ^. '-. '-f^,. -a-h). {x. -a-. A'fia). 2h). ^. _,_. 3-. '''. (:v. -. a) ix. -a-h). {x. -. a) (x. -. a. -. {x. -. (x. h). a. -. - n-. a. -. where ^ has a value intermediate between the greatest and least. and. f(a. +. =. nh). f(a). + ^ A/(a) +. ^ +. +. nh). ^„ r/. ^^^^. of a, {a. +. Ay(a). nA"-'f{a). +. + ^±ZlUllz3l A3y(.) A"/(a).. Symbolicall). 1.832. ,9. 1.. A. 2.. f{a. = e''rx-i. +. nh). If Wo. = (i+A)«/UJ =. Wx =. 1.840. ih). x.. 1.831. 1.833. +. /(«), Ml. (i. +. The operator. A). = f{a 'Wo. =. +. h),. e. ""dx iiQ.. th. = J{a. +. 2/?.),. inverse to the difference, A,. .. is. .. .. .,. m^ = f{a. the sum, S.. +. a70,. ,. n. .,. nh),.

,. ALGEBRA Indefinite. 1.843 I.. S[(.f. -b){x-b-. y. ^^. -b){x-b-h). +. b. .. +. b). (.V. +. h). .. +. (cx. .. .. - IT^h)']. b. {x-b-. nh). +. {x. .. +. (^x. +. b)(x. =. c?). 2. +. b. h). -. 7+ ^^ sm —. +. C.. =^. + n-. b. ih). '-. sin -.-.- lex. 2j cos. -. {x. .. I (.V. (;j. 4.. .. i)h. -7-^7 - i)h. =. -b-2h). h){x. (.V-. ^. +. {n. 23. Sums.. .. .. .. .. {x. +. b. +. =-+C. n ih). fi^j. +. C.. 2. ^. = -. sin (ex +(/). + C.. -j^. 2 sin. — 2. 1.844. If /(.r). is. a polynomial of degree n,. 1.845. If /(x). is. a polynomial of degree n,. f{x). =. flo-v". +. + .... +. ai.v"-i. o„_iX. +. a„,. +. CnX. and S/(x) = F(x). + C, + Ci-t" +. F{x) = CoX"+l. CoX"-^. +. .. .. .. .. +. Cn+1,. where. +. (w. (w. + 1. {n -^. The. +. i)n(n. coefficient c„+i. ^-p. may. -. i) ^. i)hco. = Co. Irco. +. nhci. n(n. -. i)n. „ Pco. +. be taken. ,. ;. =. i) ,, h"Ci. arbitrarily.. fli. + ,. .. (n. -. ^. i)hc2. =. 02.

:. 24 1.850. :. MATHEMATICAL FORMULAE AND ELLIPTIC FUNCTIONS. From. Definite Sums.. the indefinite sum,. S/(.v)=FCt)+C, a. definite. sum. obtained by subtraction,. is. a+nh. y]f(x) = F{a. +. - F{a +. nh). mh).. a+mh 1.851 a+nh. 2^f{x) =f{a) +fia. +. = F{a. By means. nh). +. h). -. F{a).. many. of this formula. +f{a +. +. 2/O. .. .. .. n-. +/(a +. .. ih). sums may be evaluated.. finite. 1.852 a-\-nh. ^(x -h){x-h-h){x-b-2h) -. {a. ^. h. +. nh) (a. -. b. .. n -. +. .. .. .. ih). .. .. {x-h- Y^lh) .. (a. .. -. b. k. -. n - kh). +. + i)h (a-b)(a-b-h) .... (a-b-kh) ik + i)h {k. 1.853 a-\-nh. /j{x -. a)(x. -a-. h). .. .. .. -. a. -. (n. -. k). (x. .. ih). a. ^ n(n - i)(n -. 2). 1.854. If f(x). +. = Ao. f(x). Ai(x. -. + Am{x — a+nh. "^. r/ s 2jf(x) = Aon 4. .. a). +. n(n-i). i). .. [m 1.855. m. —. —. w(w -. .. f(x). =. —. -. c)(x. -. o -/?). + .... a. —. m. n{n — i){n. —. 2) ,,. h). ,. .. {x. —. + A2—. ,. -h. +. can be expressed. it. a. -. .. ih),. ~h^. (n- m). .. 1). a polynomial of degree (m. If f(x) is. i). A2{x. a)(x. + Ai^—^. .. .. ^^^^. a polynomial of degree. is. .. .. +. (k. -. or lower,. i). it. can be expressed. + Ai{x + mh) + Aiix + mh) (x + m - ih) {x+- 2h) + .... + A„^i{x + mh). yl. .. .. .. and,. Ao. f(x). 2jxix +. h)(x. +. 2h). .. .. .. {x. +. mh). mh. \. a{a. +. h). .. .. .. {a +-. m-. ih).









:. ALGEBRA. 25. I. (a. + nh). n. {a. -\-. .. {a-\-. m—. -{-. ih). Ai. {m —. i)h. a{a. \. h). -\-. .. .. m—. 2h). + nh). {a. .. {a. .. .. -}-. n. +m—. 2h). ". Am-l. I. I. [a. h 1.856. f. +. a. nh. + Ai{x + mh) + + A„,{x + mh). = ^0. f{x). m. a polynomial of degree. If f(x) is. .. can be expressed. it. + mh)(x + m (x + h). A2{x .. .. ih). +. .. ,. .. .. and,. ^A^j. fix). ^x{x +. h). .. .. .. (a-. +. I. mh \a(a +. tnh). h). .. .. .. .. (a. +m—. 11. I. +. (a. nh). (a. +m+. n —. ih). a+nh. +. ^'(--^t]+'^-Ea + nh h [a ^^x. +. j. a. where, j+nk a+nh. ^^. _. I. ^^ X. 1,86. Euler's. I. I. a. a. I. +. h. a. +. I. 2h. a. +. n — ih. Summation Formula. b. ^/(x) =. if'^dz + A, x=b. -^ d'"f(x + h. x. (w -. m\ w!0m(2), with h =. Ai = -^, A2k +. numbers. i. I,. is. =. o;. ^1 =. I. ^. ,. 2. ,. z). hdx"^. a. (w -. i)!. 2)!. the coefficients A2k are connected relation,. ^''-^ ^. -. |. the Bernoullian polynomial.. by the. (6.902), Bk,. =. A,h ^f'ib) -f{a). I. - fcfymiz)^ J°. +. {/(6) -f{a). A2 =. — 12 I. Ai = ^. ,. '^. (2)fe)!. I ,. 720. ^ ^6 =. I. 30240. with Bernoulli's.

MATHEMATICAL FORMULAE AND ELLIPTIC FUNCTIONS. 26 1.861 h. ,-^{/"'w-r'w|+5^{/'w-/"W/-.. 1.862. 2^. /*. ,. ^. I. I. f/Wx. I. ^'''^r. 2. 12. dX. 720. C.r'*. I. d^Ux. 30240. rfx". ,. SPECIAL FINITE SERIES. difference,. I. n the number. the last term, and s. = a. I. = a. s. + +. {a. +. d). (n. —. i) 8. = -[2a. +. (w. +. -. sum, a the. If 5 is the. Arithmetical progressions.. 1.871. {a. +. 26). +. .. .. term, 5 the. first. common. of terms,. .. .. [o. +. (». -. i)5]. i)5]. 2. Geometrical progressions.. 1.872. 5. = a. + /)" />. If /><!,. CO,. -. a/j2. +. ^. a/>"-^. I. I. s. i-p. Harmonical progressions,. 1.873 if. n =. +. c/?. a,. the reciprocals, i/o, i/b, i/c, i/d,. b,c,d,. .. .. .. .. .... form. form an harmonical progression an arithmetical progression.. 1.874.. a;. =. «. I. "^ Xv-v^^. o. 2.. = w(«. +. (2«. i). +. i). I. "^^ n^ TjX* =. w"*. ,. 4.. 6. =. .ii^. 5. 1. n^. n. 3. 30. 1. 2.

ALGEBRA 1.875 X. =. n. *. =. I. 27. In general,. W'--. Bi,. B2,. B3,. .... are. Bernoulli's. coefficients (1.51); the series. term in. n^. if. k. is. numbers. (6.902),. ends with the term in n. (^j. if. k. is. the. are. odd.. 1.876 I. I. I. 11° I. I. 1234. .. (h. ?i{n. 7=. Euler's constant. 02. =. I. 12. 1.877. +. i). {n. +. 2). = 0.5772156649. .. .. .. 02. I. 2)1. binomial. even, and with the. n{n. +. I).

MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS. 28 1.879. Formula.. Stirling's. =. log (w!). o<^<i. The. coefficients. +. log y/i-K. Ak. w. {. + -j. log. —w. w. are given in 1.86.. 1.88 1.. i. 2.. I. +. +. i!. 2-2!. +. •2-3. +. +. 3-3!. .. .. +. 2-3-4+3-4-5. .. .-\-n-n\= (w+i)!. .. .. .. + «(•«+. .. i). +. (w. = -»(«+. 2). (w. i). +. 2). («. 4 3.. I-2-3 .. .. .. .. .. r. .. .. (w. .. +. 2-3-4. +r-. .. _ m(« 4.. 5-. +. I-^. i'-?. +. =. 2(/>. (/». I. -. i). i). ^^^[^/'g. -. K^ +. ^ 6.. +. +. +. 3(/?. («. i). I. o. c(a. ^)(6. ^ (6. +. I. —. w(w. +. i). (». +. 2). +. (/>. -. 2). (3/». +. 39. i). •. •. +. i). a)o(o. +. •. .. i). .. .. + +. .. .. +. (^>. .. .. -. 2W. .. .. .. +. w(/>. -. 2).. .. .. .. (6. i). .. .. .. (a. +. r). +. w -. l). i)].. + w-i) + - i) ;/. w). (a. +. (/>. i). .. (w. +. 2). - 2w +. ^i^^. •. +. (g. 5(6 I. .. I. +. 2). l)(3^. i). 2). r+. i). +1). +. +. -. (w. +. ^ I. +. +. i). = -w(w. -. (9. +. (r. .. .. i). w —. i). 6. g. -. I. +. I. —. a. -. w) (g. -. «). + 3)..

,. GEOMETRY. II.. Transformation of coordinates in a plane.. 2.00. Change. 2.001. Let x, y he a, system of rectangular or oblique coorReferred to x, y the coordinates of the new origin O' referred to a parallel system of coordinates with origin at 0'. of origin.. dinates with origin at 0. are a,. b.. Then. the coordinates are. x', y'.. X =. x'. y =. y'. 2.002. Origin unchanged.. Let. be the angle between the x. OJ. from the. j8. x-axis.. -. coordinates.. y axes measured counter-clockwise from make an angle a with the x-axis and the. a;'-axis. with the x-axis.. All angles are. measured counter-clockwise. Then :». sin CO. y sin. CO co'. 2.003. ObUque. Directions of axes changed.. Let the. the X- to the y-axis.. /-axis an angle. +a + b.. Rectangular axes.. = = =. -. x' sin {co. x' sin o: i8. -. a). +. y' sin (o). -. /3). y' sin /3. -J-. a.. Let both new and old axes be rectangular, the new. axes being turned through an angle d with respect to the old axes.. 2. 2.010 let. 2. Polar coordinates.. X =. x' COS B. —. y =. x' sin. d. -\-. y' sin. d. y' cos d.. Let the y-axis make an angle. co. with the. the X-axis be the initial line for a system of polar coordinates. are measured in a counter-clockwise direction from the. _. r sin (co. -. B). sin CO. y = 2.011. Then. If the X,. B. sin r. -.. sm. CO. co. =. X =. r cos. B. y =. r sin B.. y axes are rectangular,. 29. —. a;-axis.. r, B.. a;-axis. and. All angles.

.. MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS. 30. 2.020. Transformation of coordinates in three dimensions.. 2.021. Change. of origin.. Let. dinates with origin at O.. 0' are. a, b, c.. Then. x, y, z. x, y, z. the coordinates of the. +. X =. x'. y =. y+. b. +. c. =. z'. a. Transformation from one to another rectangular system.. The two systems. are x, y, z. and. x' y' z'. Referred to. .t,. Referred to. x, y, z. the direction cosines of y' are h, m^, fh. Referred to. x, y, z. the direction cosines of. y, z. The two systems. origin. z'.. z. changed.. new. referred to a parallel system of coordinates with origin at. O' the coordinates are x' , y' ,. 2.022. be a system of rectangular or oblique coor-. Referred to. the direction cosines of x' are. are connected. z'. /i,. are h,. by the scheme. :. Wi, wi. nts,. Us. Origin un-.

GEOMETRY COSffi2. 2.024. +m -. h. -. «3. I. +. 31. B. cos^. k - nh -. I. /i. + W2 -. 7 fii. -. I. Transformation from a rectangular to an oblique system,. x\. tangular system:. x, y,. y' , z' oblique system.. = h = mi. cos xy'. = h. cos xz. =. cos yx'. cos yy'. cos ys'. = Ms. cos zx'. =. cos zy'. = m2 = iH. cos xx'. til. X =. /iX'. y = mix' z. =. nix'. cos y'z' cos 2'x'. cos. x')''. + + +. = hh = hh = ^1/2. ^2)''. nhy' tiiy'. + + +. IsZ'. Wsz' W32'. + W2W3 + ;Z2W3 + W3W1 + nmi + minh + Wi'«^. = + m-^ + + m^ + n^ = h^ + W3^ + W3^ =. /i^. li. cos 22 '. fix'. 1 I I. Transformation from one to another oblique system. cos xx'. = h. =. Iz. W3.

MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS. 32 2.026. Transformation from one to another oblique system.. If Hx, fly, Hz are the normals to the olanes normals to the planes y'z', z'x', x'y',. X cos xux. =. x' cos x'nx. y cos yriy. =. x' cos x'ny. =. x' cos .r'wz. z cos. = X. cos xwx'. = X. cos xw v'. = X. cos xnz'. the. cos z'nx. cos z'uy. cos. z'fiz.. .. -{-. >». j,'.. Transformation from rectangular to spherical polar coordinates.. makes an angle 6 with the on the x-y plane makes an angle with the x-axis.. r,. of r. y 'w y. n/. + y cos >'«i' + z cos znj z cos zw + cos yuy + y cos jWz' + z cos zwz'.. x' cos .t'«x'. cos z'uz. xy and Wx', Uy',. + y' cos y'nx + z' + y' cos y'uy + z' + y' cos )''wz + z'. y' cos z'. 2.030. zwz. yz, zx,. the radius vector to a point. x =. r sin. 6 cos. y =. r sin. 6 sin. =. r"^. (j>. x"^. +^+. B = cos~^. cj). z^. V + -v^. z. =. r cos. 6. .. =. (b. z-axis, the projection. = >'". +. z^. y tan-i ,. a:. 2.031. Transformation from rectangular to cylindrical coordinates.. p, the perpendicular. from the. z-axis to. a point makes an angle 6 with the. x-z plane.. X = p cos. p = V.T-. y = p sin. 2.032. =. +. tan-i. y^. I. ^. 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. projection of Oz' perpendicular to Oz. is. to the east.. Angles. if. Oz. is. vertical,. directed to the south, then. z'Oz. =. ^K = yVK. =. B, </>,. \p.. and the. OK is directed.

:. GEOMETRY The. direction cosines of the two systems of axes are given. scheme. 33. by the following.

:. :. :. :. :. :. :. MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS. 34. ^"i, y>i, X3, Xi denote the distances of a point P from the four sides of a tetrahedron (the tetrahedron of reference); k, mi, m; h, fth, fh'i h, ms, nz; and I4, nii, Hi the direction cosines of the normals to the planes Xi = o, X2 = o, x^ = o,. Xa. = o with. respect to a rectangular system of coordinates x, y,. di the distances of these. Xi. X2 (i). Xi Xi s\, 52,. and. V. 53,. its. and 54 are the areas volume. 3F = By means. X = AiXi y = A2X1. =. Z. any. of. +. X^J^. A3X1. -\-. +. + + +. B1X2. + B2X2 + B3X2. dx, d^, ds,. di di. ds d\.. .T353. +. .T454.. determined. + Di,. CiXz. + A, + D3.. C2.V3. C3.T3. surface,. =. F(x,y,z). may. and. 4 faces of the tetrahedron of reference. of the. .Ti5i. — — — —. + triiy + WiZ + nhy + fhz + mzy + naZ + m4y + n\Z. = hx = hx = ^3^ = hx. of the first 3 equations of (i) x, y, z are. The equation. z;. 4 planes from the origin of coordinates. o,. be written in the homogeneous form. Fi. \AyXi. +. B1X2. +. C1X3. + -p. M2:>^1. +. B2X2. +. C2X3. +. A 3X1 +. B3X2. +. C3X3. + -y. (51.V1. -|7 (SiXi. (siXi. +. S2X2. +. S3X3. +. SiXi). ,. +. S2X2. +. S3X3. +. S4Xa). \,. +. S2X2. +. S3X3. +. SiXi). PLANE GEOMETRY 2.100. The equation. 2.101. If. of a line. Ax + By + C = angles p. p. is. makes with. the. .v-. If. a' and. (3'. a+. are the angles the line. y cos. The equation. of a line. may. y = ax a b. a and. +. (3.. makes with the. p = y cos a' — X cos 2.103. upon the Hne, and. and j-axes p = X cos. 2.102. o.. the perpendicular from the origin. x-. and. >'-axes. j3'.. be written b.. = tangent of angle the line makes with = intercept of the y-axis by the line.. the. .r-axis,. (3. the.

:. :. :. :. GEOMETRY 2.104. The two. 35. lines:. y =. QiX^. bi,. +. bo,. y = aox intersect at the point bi. X =. Gl. 2.105. — —. bi,. Ol. between the two. If (^ is the angle. lines. o = ±. +. I. 2.106. Equations of two parallel. Ax By + Ax + By -\-. -\-. -{-. •. 2.104 02. C1C2. +. b\,. \y = ax +. b^.. +. bu. o. -=. = o C2 = o. ax. =-. y. {. Co = o. Equations of two perpendicular Hnes. Ax + By Bx - Ay. a^bi. O2. lines. Ci. -{-. 2.107. -. (?i. ,. tan. — —. a\bi. y =. Oo. :. Ci. y = ax. (. or. \ \. X — + h. ,. -. y =. a. {. 2.108. Equation of. A {x 2.109. through. line. Equation. -{-. through. of line. Ax B(x 2.110. Xi, yi. Ax + By + C = - xi) B{y - yi) =. Equation. xi). of. -{-. By. - A(y -. Hne through y. -. y\. or or. C =. yi). or. = o. or. a 1. -. —. tan. <b. r. :. a tan. 2.112. Perpendicular distance from the point. ^ p =. 2.113. Axx. ,. (-v. through the two points,. Equation. Ax. -^. +. By ^ C = o B\\. .Vi,. or. =. Polar equation of the line y = ax -^b\. a - ay. b cos. =. sin {d. where tan. a=. a.. -. .Vi,. or. +C. r. -. >'i. (/). +. b,. =. •. with the. line. ,. x{).. and. yi,. yi to. p =. x^, yt. :. the line. y = ax .. '. X\),. to the line. y = ax y. b,. -\-. = a{x -. yi. making an angle. +. 2.111. of line. y = ax. -. y. and perpendicular. o. .Vi, 3-1. =. parallel to the line. o. Xi, yi. -^. and. o. '^'1 '. + -. b, 0-^1. -. ^ •. ,. y = ax. -{. b:.

MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS. 36 2.114. perpendicular to the line from the origin, makes an angle. If p, the. j3. with the axis: /». Area Jn = A.. 2.130 Xn,. 2. A =. r cos (0. -. i3).. polygon whose vertices are at. of. yi.{xn. =. -. X'z). +. ytixi. -. xs). +. ysix-z. -. ^-4). Xx,. ju. +. X2,. ji;. +. yn{xn-i. -. Xi).. PLANE CURVES 2.200. The equation. of a plane curve in rectangular coordinates. may. be given. in the forms:. y-fix). X = fi{t) y = Fix,y) = o.. (a). (b). ,. (c). 2.201 (a). If. r. tan r. is. foil). .. The parametric. form.. the angle between the tangent to the curve. dx. and the. x-axis:.

—. '. GEOMETRY 2.204. OQ =. y '. — yy. Vi +. = distance. ^ ". _[_. '^^. ••. yy. +. f r> of O: ^. = distance. y. of. y) '-^ -tt^, I + y. y' (^v'. -. y. -. I. +. '. projection of. PQ =. projection of. xy' 77 '. y. normal from origin =. radius vector on tangent. ^. JCoordmates r-. Va. PS =. radius vector on normal.. r . Coordmates. 05 =. from origin =. -. y r-. 2.205. of tangent. 37. ••. re. of S:. X. +. yy'. ^,. !-{ y-. (x -^^. + yy')y' ,./ + y^. I. •.

MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS. 38. The perpendicular from = o at the point x, y is:. 2.212 F{x, y). Concavity and Convexity.. 2.213 lies. as. entirely. y" =. -p;. on one side is. origin. the. upon the tangent. dF. dF. ox. oV. to. the curve. neighborhood of a point P a curve concave or convex upwards according. If in the. of the tangent,. it is. The. positive or negative.. shown. positive direction of the axes are. in figure 2.. 2.220. Convention as to. The. signs.. positive direction of the. normal. to the positive direction of the tangent as the positive ^'-axis. positive v-axis.. The angle r. measured positively. is. is. is. related. related to the. in the counter-clockwise. direction from the positive .r-axis to the positive tangent.. 2.221. where. Radius of curvature = p; curvature = i/p.. 5 is. the arc. drawn from a. I. dT. p. ds. fixed point of the curve in the direction of the. positive tangent.. 2.222. Formulas. for the radius of curvature of curves given in the three. of 2.200.. P =. (^). =. -^. y". dx^. (b). m^m dx d-y. dt If 5 is. ^^. df. dy. d'^x. dt. df. taken as the parameter. _d_x p ~ ds. ^. [. \dfj. '. (d~y\-. (d~s. \dt- J. \dt. y]. t:. ^ _~. d^_d_y ds"-. ds. ds'. /dFV. j. /(PxV. \. [dsy. fdFV. "^. iy^yt f. (c). (f (d'x\r j. p. dx^\dyj. ^. dxdy dx dy. *. fd^yY. \. [dsV. J. ?. ]. "^. 3/ \dj). forms.

GEOMETRY The center PC = p. If p. 2.223. of curvature. that. is. positive. C. is. lies. 39. on the normal at P such on the positive normal (2.213) if negative,. a point. C. (fig.. 2). ;. on the negative normal. 2.224. The. 2.225. The chord. circle of. curvature. of curvature. is. a. is. circle. with. C. as center. and radius =. p.. the chord of the circle of curvature passing. through the origin and the point P. 2.226. The. coordinates of the center of curvature at the point ^. = X - p. sin. T tan r. rj. If. I',. = y. dy. ^ dx. p cos T. m' are the direction cosines ^ = x r] = y. 2.227. +. =. +. I'p. -\-. m'p.. of the positive normal,. x,. y are. ^, rj:.

MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS. 40 2.231. The envelope. to. a family of curves, F{x, y, a) = o,. I.. where a. is. a parameter,. obtained by eliminating a between. is. (i). and. dF. IR = 2.232. If the. curve. is. °-. given in the form,. x=fi{t, a). 1.. y=Mt,a),. 2.. the envelope. is. obtained by eliminating. and a between. /. (i),. (2). and the func-. tional determinant, (see 1.370). Wf)^° 2.233. The. Pedal Curves.. locus of the foot of the perpendicular from a fixed. point upon the tangent to a given curve. the pedal of the given curve with. is. reference to the fixed point.. 2.240. Asymptotes.. The. line. y = ax is. an asymptote to the curve y = f(x) a = b. 2.241. If the. curve. =. +b. if limit. ^^^. rr f. \. f (x). ^„[.m-xf'(xn. is. x=Mi),y=Mi), and. if. for. a value of. for that value of. and the distance 2.242. /. t,. ti,. fi. or/2 becomes. infinite, there will. be an asymptote. if. the direction of the tangent to the curve approaches a limit. of the tangent. An asymptote may. from a fixed point approaches a. limit.. sometimes be determined by expanding the equation. of the curve in a series,. ^-Z/-'+S*-:k. _.. limit. the equation of the asymptote. is. "V^. *=. bic. I. 7 J akX^.









1. GEOMETRY degree in. If of the first. 4. represents a rectilinear asymptote;. x, this. if. of a higher. =. o, singular. degree, a curvilinear asymptote.. 2.250. Singular Points.. equation of the curve. If the. is. F. {x, y). points are those for which. dF_dF_ dx Put,. A If. A<o A>o. the singular point the singular point. through. A. = o. point. —. dF dF d'F is. one. '. ^^. _. dx'. \dj. dy'^. a double point with two distinct tangents.. an isolated point with no. real. branch of the curve. it.. the singular point. —'. ^ ^ fJ~^ ^^,-,^J. =. is is. is. branches, with a ^, If. dy. d'F. d^'F. T-5'. -T-j'. an osculating point, or a cusp. The curve has two tangent, which meet at the singular point.. common. .. .. ^ simultaneously vanish at a point. .u the singular .. ,. ^. ,. ^. •. ,. •. 1. of higher order.. PLANE CURVES, POLAR COORDINATES 2.270. The equation. of the curve. is. given in the form,. r=f{d). In figure 2.271. OXy and. 2,. is s,. OP. =. r,. measured the arc,. is. angle. XOP. angle. B,. =. r,. angle pPt =. in the counter-clockwise direction. Then,. tan. .. <A. ^. sin <p. +. =. ^. =. rdd ——. dr. ds cos. from the. The. (j>.. initial line,. measured from the positive radius vector to the positive. r = e 2.272. XTP. so chosen as to increase with 6.. in the counter-clockwise direction. tangent.. =. A. (p. =. dr. ^. as. ^.. angle. is.

MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS. 42 2.273. ^''. -73. +. r cos. dr cos " Jo. ~. f". •. z) C7. Sin. =. tan T. ,. fl tf. •. drV. sin. ]. i. rf^. ^^M'^-'G^) PR. 2.274. =. r^i + (^j'=. polar tangent. = V/r- + (— = polar PFK=v/^=.(|J=po, ;;. 1. normal. j/3. 2.275. OQ =. —. Oi?. = r-—. = polar subtangent. OV. =. =. -Tn. =. ,. polar subnormal.. = p = distance. of tangent. from. origin.. V/'=^(IJ dr. OS =. 2.276. If. M =. —. = = distance of normal from origin,. ,. -, the. curve. r. = f{d). is. concave or convex to the origin according as d-u. is. positive or negative.. At a point. of inflexion this quantity vanishes. sign.. 2.280. The. radius of curvature. is,. drV '^. +. [Td. dh. yy. 2.281. If. w = - the radius. of curvature. is. r. iuV. ^ 1. and changes.

GEOMETRY 2.282. where. If the. 5 is. equation of the curve. is. 43. given in the form,. the arc measured from a fixed point of the curve,. (dr\ IdrX-. d'-r -V. ^7? ? + \Js}2.283. If. p. is. the perpendicular from the origin. P = 2.284. If. upon the tangent. dr. r-. 2.. P = P. to the curve,. d~p. + ^,. w = r. 2-285. 2.286. JS Polar coordinates of the center of curvature,. r-+. drV. „. +. dr. ''. rf^. ,dd) If 2C is the. 'dd^-i. x,. ^dd. 2,287. Bi:. 2. \dd). +. r-i,. dd''. chord of curvature (2.225): dr. p 2C=2Pj^=2p-, ^. 2.290. Rectilinear Asymptotes.. and. r{a. if. -. B). approaches a. If r. r sin is. an asymptote to the curve. approaches. limit, b,. r. =. {a -. f{B).. 00. as. approaches an angle a,. then the straight line B). =. b.

:. MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS. 44. 2.295. curve. If. Equation of a plane curve. An intrinsic equation of a plane one giving the radius of curvature, p, as a function of the arc, s,. Intrinsic is. r. is. the angle between the a:-axis. and the. (Is. d-T. m. y. 2.300. The. +. cos T-ds. '•*£. sin T-ds.. X = Xo. pds. positive tangent (2.271). general equation of the second degree:. anX^. +. 2avixy. +. a^^.f. +. 2ai3.T. an. ayi. flis. (h\. 0-22. ^23. Q31. O32. Oss. Ahk = Minor. of Ohk-. Criterion giving the nature of the curve:. +. 2any. dhk. =. +. dkh. 033. = O.

GEOMETRY. 2.400. Parabola (Fig.. 2.401. 0, Vertex;. 45. 3).. F, Focus;. ordinate through D, Directrix.. Equation. parabola,. of. origin at O,. 'f=. 4<7X. X =. OM,. MP,. y =. OF = 0D = a FL = 2a = semi. latus. rectum.. FP = D'P. FP = FT = = X + a.. 2.402. MD Fig. 3. NP. = 2Va(a. ON' = \l-. (x. +. x),. +. 2a),. TM. =. OQ =. MN =. 2X,. xJ-^,. 2a,. OS =. FB perpendicular to tangent TP. FB = Va{a + x), TP = 2TB = 2Vx(a + Tb' = The on the. FT X FO FP X TP and UP' =. tangents. directrix at. U. at. Radius. P. +. XTP.. = N/f. bisects the angles. FPD' and FUD'.. 2(x. +. a)'. _. I. WF. Coordinates of center of curvature:. =. a. -\-. X. x).. of curvature:. ^. 2a.. 2a)J-. at right angles.. P =. +. FO.. tan ^. The tangent. (x. = x. at the extremities of a focal chord. T = angle. 2.403. ON. 3a:. +. 2a,. rj. Equation of Evolute:. 270/ = 4{x - 2a)^. = -. 2x\/'^'. PFP' meet.

MATHEMATICAL FORMULAE AND ELLIPTIC FUNCTIONS. 46 2.404. Length. of arc of parabola. 5. Area. OPMO. +. = Vx{x. measured from vertex,. +a. a). log. (y. I. +~ + y - j. •. = - xy. 3. 2.405. Polar equation of parabola: r. = FP,. e. = angle XFP, 20 I. 2.406. Equation of Parabola. tangent, and. r,. in. —. cos d. terms of p, the perpendicular from. the radius vector. FP:. p"' I. 2.410. Ellipse (Fig. 4). ^. = semi. r. latus rectum.. F upon. the.

GEOMETRY 2.412. 47. Parametric Equations of Ellipse, X = a cos. y =. (f),. b sin. cf).. = angle XOP', where P' is the point where the ordinate at as center and radius a. eccentric circle, drawn with </). OF = 0F' =. 2.413. e. =. meets the. ea. eccentricity. FL = — =. P. -. a(i. =. j. e^). = semi. latus rectum.. a-. FP + F'P =. a. F'P = a-\- ex, FP = T = angle XTT'.. ex,. 2a.. hx tan T. aVc. NM = ^", a^ PT. .. OiV =. ^"^ -. "'^'''. e^xV a^. —. Va2 _. DD'. 2.415. MT "^. - <T^V^^T-;3, PS = '. Va-. 6. e"X'. :v. DD' and PP'. T'T;. 0Z)2. =. a2. =. ab.. +. +. are conjugate diameters:. 62.. of Ellipse referred to conjugate diameters as axes:. = OD'. OP. Radius. a''. a =. A|2 y. 6'2. =. oiv'. , '. 2 X'-. b'. -,. y. '""'. PSxOD. or^. a'. X. g2^.2. parallel to. Equation. -. = -, Or' =. X. „„. 2.414. OT. e2.r,. ~. ^ ,8. '''^'. =. ,. ,. or sin-. a+. a^ sin-. jS. tan. r-o;. 0- cos-. 6'2. +. 6''. cos-. jS. of curvature of Ellipse:. '^. ~ ab. a^b^. angle. FPA^ = angle F'PN = tan. oj. P cos. CO. =. eay -7|-,. FP ^ F'P. co,. angle. = angle. a. tan. XOP XOD i3. = -. ^ or.

48. MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS Coordinates of center of curvature:.

GEOMETRY 2.423. OF = OF' = e. =. 49. ea.. Va- +. eccentricity. b"". Fig. 5. FL = - =. aie^. -. i). = semi. latus rectum.. a. F'P =. ex. +. a,. T = angle tan r. FP = ex-. a,. F'P - FP =. XTP.. bx. =. a\/x~. —. a'^. NM = ^, ON = a/-. X. e% OT =. ~, X. X. OT =. -, y. 2a..

MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS. 50 2.425. Radius. of curvature of hyperbola,. P = angle. F'PT =. angle. FPN. angle. ab. FFT.. angle. =. = - - FFT.. co. F'FN =. co'=. - + F'FT.. aey. tan. = -^.. CO. p cos. CO. -\/g^.V". —. FP. F'P. a^. Coordinates of center of curvature,. Equation. of. Evolute of hyperbola,. (f-(f 2.426. In a rectangular hyperbola b = a;. each other.. Equation. the asymptotes are perpendicular to. asymptotes as axes and. of rectangular hyperbola with. origin at 0:. -.. xy = 2.427. Length. of arc of hyperbola, 6". L. /. 2.428. ^^^. •. I leyo. VI —. ^ "^ ^=^, v^. _ =. kA. ^^ sin^. ^ _. an'. e. Polar Equation of hyperbola: r. = F'P,. e =. XF'P,. r. ^'. ^a e. r. = OP,. (9. = ZOP,. Equation of right-hand branch. dicular from. F upon. the tangent at. T2 p^ I. of. P. I. =. cos. r2 e-. 2.429. ^„„ tan. I. 1,. 6-1. hyperbola in terms of p, the perpen-. and. 2. I. -. + -a. r. COS". ^. (7-1. = semi. r,. the radius vector. •. latus rectum.. FP,.

GEOIVIETRY. Cycloids and Trochoids.. 2.450 If. SI. a circle of radius a. on a straight. rolls. The. radius, a, describes a cycloid.. X = a(4> y = a(i where the x-axis. is. line as. base the extremity of any. rectangular equation of a cycloid. — sin — cos. is:. 4>),. 0),. the base with the origin at the initial point of contact.. the angle turned through. by the moving. 4> is. (Fig. 6.). circle.. Fig. 6. A = C = The tangent. vertex of cycloid.. to the cycloid at. P. radius of curvature at. P. is. AP. Area. 2. X chord AQ.. parallel to the chord. PQ Length. =. =. circular arc. QD. and equal. to. 2. X. chord QD.. AQ.. = 8a; a = CA.. of cycloid: s of cycloid:. at A.. chord AQ.. parallel to the. is. Arc. The. drawn tangent. center of generating circle,. 5 =. 3x0-.. A point on the radius, b>a, describes a prolate trochoid. A point, b<a, describes a curtate trochoid. The general equation of trochoids and. 2.451. cycloids. is. X =. acf). y =. (a. —. +. (a. +. d) (i. sin. d). -. cos. (f>,. (j)),. d = o Cycloid,. d>o d<o Radius. Prolate trochoid,. Curtate trochoid.. of curvature:. (2av. P = ay. +. +. ad. d-y. +. d-.

MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS. 52 2.452. An epicycloid is described by a point on a on the convex side o a fixed circle of radius h. An described by a point on a circle of radius a that rolls on the con-. Epi- and Hypocycloids.. circle of radius. hypocycloid. is. a that. cave side of a fixed. Equations of. rolls. circle of radius b.. epi-. and hypocycloids.. Upper. sign:. Epicycloid,. Lower. sign:. Hypocycloid.. X = y =. The. origin. is. ±. {b. \. /I.. 4>^. cos. a). -. A.. •. sin <p. (o =h a). cos. a. •. sm. ^. The. at the center of the fixed circle.. ±. ^. J.. cp.. .x-axis is. and. centers of the two circles in the initial position. by the moving. 0,. is. the line joining the. the angle turned through. circle.. Radius of curvature: 2a{b. p = —T. ±. a). —a. .. Sin. b zL 2a. 2.453. In the epicycloid put 6 =. 2.454. Catenary.. y =. 1.. 2.. The curve becomes a. o.. The equation may be -. a{e''. +. y = a cosh. -. ,. (p.. 2b. a log. Cardioid:. written:. c ").. -•. y. Vy-. =b. a. The. radius of curvature, which. is. equal to the length of the normal,. —. p = a cosh^ 2.455. Spiral of Archimedes.. A. is:. a. point moving uniformly along a line which. rotates uniformly about a fixed point describes a spiral of Archimedes.. equation. is:. r. =. ad,. or. Vx^ + The. polar subtangent. Radius. =. = a. polar subnormal. tan~^ -•. X. =. a.. of curvature:. _ ^ 2.456. y-. Hyperbolic. r(i (9(2. + d'-y+ d~). _. r2. spiral:. rd =. (;-^. a.. + a^)J + 2a2. '. The.

:. GEOMETRY 2.457. Parabolic spiral: r-. 2.458. Logarithmic or equiangular r. =. a=. =. a-d.. spiral:. ae"^,. n = cot. q;. =. const.,. makes with the radius. angle tangent to curve. 2.459. Lituus:. 2.460. Neoid:. 2.461. Cissoid:. rVd = r. (x^. +. /).T r. 2.462. (.i;2. Lemniscate. (6. = a. —. a-{-bd.. = 2 ay-, = 2a tan 6. sin 6.. +/+. +. Conchoid:. 2.465. Witch. =. = ^. 2a^ix'^. x^y"^. =. (i. x^y. =. 4^2(20. 3;2)2 y-2.. 2.464. ^2)2. 2ah^ cos 20 =. ^^2^2 _^. —. b^. -. f),. 2a^ cos 2^.. +. v)-(a-. -. of Agnesi:. Tractrix:. J4^. a^.. in Cassinoid). 0^2. 2.466. =. a.. Cassinoid: r^. 2.463. 53. —. >')'. 'f).. vector..

MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS. 54 2.602. Cz. +. The perpendicular from D = o is: -. a = 2.603. is. A^x cos 6. (xs,. ys,. Xi,. >'i,. Si. upon the plane. Ax + By +. + Bv, + Czi + D vOP + B' + C^. Axi. the angle between the two planes:. Aix. 2.604. the point. Equation S3):. yi 2i I. + Biy + Ciz + A = o, + B^y + C2Z + A = o, AxAi + B,B2 + C1C2. of the plane passing. through the three points. (xi, yi, Zi) {x2, yi, Z2).

GEOMETRY The. 2.625. 55. perpendicular distance from the point. -. X. y. -. 2i)2}i. -. .vi. -. s. >'i. Xo,. to the Hne:. y^, 22. zi. is:. d =. {xi-XiY. {. 2.626. and. .V2,. The y2,. +. -. ()'2. ji)-. +. -. (z2. -. {/i(x2. xi). + mi(y2 - yd + «i(z. - Zi)}. direction cosines of the hne passing through the 22. two points. Xi, yi, Zj. are: (V2. -. -Vi),. -. (>'2. yi),. -. (22. Zi). |(.V2-Xi)^+(>'2->'l)^'+(22-Zl)2}^* 2.627. The two. Hnes:. X =. niiz. +. X = nhz. pi,. +. p2,. and y =. .. intersect at a point {nil. The. iiiz. +. nh). (qi. =. noZ. (pi. -. y. qi,. +. q^,. if,. -. -. -. 92). -. (wi. jpz). =. pi). o.. coordinates of the point of intersection are:. miP2-moPi ^ ^ nil. The equation. -. WZ2. ^. niq2-tu_qi. -. Wi. '. of the plane containing the {ill. -. Ui) {x. -. miz. -. pi). two. =. ^. ^. lines is. -. {nil. po. -. ^ q^-qi. pi. mi - nh. '. ih. W.2). ni. -. ^. n^. then. {y-. niz. -. qi).. SURFACES 2.640. A. single equation in x, y, z represents a surface:. F{x, 2.641. The. ,. m, n. o.. dF. dF. dx. dv. dz. m-m-m. The perpendicular from. The two. dF. dFV. the origin. p = 2.643. =. direction cosines of the normal to the surface are:. I,. 2.642. y, 2). Ix. 1. i. upon the tangent plane at. + my +. principal radii of curvature of the surface. given by the two roots of:. x, y, z is:. nz.. F. {x, y, z). = o. are.

56. where:. MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS k. dJF. p. dx'.







i.

—. —. GEOMETRY 2.649. The. ,. I,. —. d{u,. m, n. If the. d{u,. '. v). equation of the surface. The. ,. I,. 1. (rt. 52)p2. -. {. (1+ q'y -. '. -(IJ-(I)T. The two principal radii given by the two roots of:. -. form 2.650 are:. ^. of curvature of the surface in the. 2.652 are. to the surface in the. -(i)-(g^ ^^^. m, n =. ]. yi, Zi is:. normal. direction cosines of the. ^\d{u, v)). y),. the equation of the tangent plane at Xi,. 2.651. v)). diu, v) ^^_/. '. is:. =fix,. 2. v). ^^. ^U(«,. [diu, v)l 2.650. 57. normal to the surface in the form 2.648 aje:. direction cosines to the. 2pqs. +. + p-')i}Vi + f +. (i. q'^. p. +. (i. +. p2. form 2.650. +. ^2)2. ^. ^^. where. 2.653 is. If Pi. and. ax'. ^dj. ^dj_. df. ^. ^. p^ are the. dy. "^. ay. dH. ax2'. ^~dxdy. ^~a/'. two principal radii of curvature of a surface, and p making an angle with the plane of pi,. the radius of curvature in a plane. cos^. I. P~. sin^. </). Pi. (/). P2. '. curvature in any two mutually perpendicular planes, and pi and p2 the two principal radii of curvature: 2.654. If. p and p' are the. radii of. I. I. I. ~. "^. P 2.655. P'. I '. "^. Pi. P2. Gauss's measure of the curvature of a surface. _. I. is:. I. P~P^2*. SPACE CURVES 2.670. The equations. of a space curve. =. may. (a). ^i(^, y, 2). (b). ^=m,. y=Mt),. (c). y = 0(x),. z. o,. =. be given in the forms:. FoXx, y, z). xf/ix).. =. z=Mt).. o..

MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS. 58 2.671. The. direction cosines of the tangent to a space curve in the. dFi dF2 J 1=. dy. dFi dF2. dz. dz. f SFy dz. dy '. form. ,. (a) are:.

GEOMETRY. 59. -. y'(y'z" -z'y") -x'jz'x". x'z"). where. W+. L = 2.Gn. The. y"". + z"'\^\{y'z" -. z'y"y. +. -. {z'x". x'z'. J+. {x'y". direction cosines of the binormal to the curve in the. - y'x"YY. form. (b) are:. S. where. 5= 2.678. {{y'z". - z'y"f +. If s, the distance. the parameter,. is. d-x. d-y. ,. (x'y". ,. the principal radius of curvature;. ,,. The. +. x'z")'. -. measured along the curve from a. I". 2.679. -. y'x"r-]K fixed point. on. it is. t: ,,. where p. {z'x". = p. _. dh. and. ^dv. dh. dz d"V. ds. ds'^. ds ds-. \ds ds-. ds ds'y. Idx d^y. dy d^x\. \ds ds^. ds ds^j'. radius of torsion, or radius of second curvature of a space curve f.v'2. (dl"\-. I. + + (dm"Y y">-. z'')\. /(9;?/'V]4. is:.

6o. MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS dx ds.

X. TRIGONOMETRY. III.. 3.00. tan x sec-. =. X =. versin x. 3.01. sin. sm x. I. =. sec x. >. X =. =. I. +. =. i. -. -. ,. 2. sm. Vi +. =. csc^x. tan^a;,. sin. =. esc x. >. =. -. i. = (_,).y'i^2^. : (x) = V. 2. Vi +. tan X. _. tan'^x. (,ot. « «« 3.02. cos X. -. cot -• (i. ,. +. =. sin y cos. (x. =. cos V sin. (x. =. cos (- x). =. cos^. N. /. ,. X -. -. =. -. +. I. -. +. -. y),. y). +. y),. /l. +. cos 2X. 2. cos^. —. 2. +. ,. .. sin- ->. X. I. I. Vi +. 2. tan2 X. I. I. I. +. tan x tan -. tan x cot. tan. cot X. :^. Vn- cotlx. sin 2X 2. sin. x. +. y),. 2. =. cos y cos (x -f cos. cos (x. }»). +. — y)-. sin y sin sin. 3^. 61. sin. —. ^^. 2. 2. 2. =. 3;. 1-2. =. X tan -. 2. cot ^. (x. -. 2. X. cos x),. sin j cos. .. tan^ -. cot -. +. -• (i. cos y sin (x. 2 I. cot x. -. 2. tan-. +. +. „ X sm^ - =. 2. 2. y). y 4. tan^^. tan -. (x (x. -. y),. x =. cos" ->. 2 ,. X. i,. 2. = tan. cos x). =. >. tan x. X tan -. 2. =. coslx-. =. =. ^Q^ ^. cof-^x. =. sin x, haversin. 2. —. cot x. ^y y.cos^ - -. >. X X - cos - =. +. cot-x, sin^x. cos x, coversin x. 2. >. x. sin. +. i. 1,1. -.. cos x. cos X. i. sin^.

MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS. 62 3.03. tan x =. -. sin 2X. tan (- x). +. I. I. —. ^+ I. COS. 2.V. COS (x. -. y). sin. +. >'). (.V. _. cos 2X. -. I. cos 2X. sin. (x. COS. (.v. COS 2X. Sin 2X. + y) + sin + + COS. (.v. {x. }'). cos jx. +. y). sin. -. y). {x. -. cot. -. a;. — y) — y)' 2. cot 2X,. X. tan. tan. 2. tan 2. I. -. tan -. I. +. -. X tan 2. .. 3.04. The. I. -. i. -. tan^ -. 2. 2. I. tan -. I. +. 2. X tan 2. e^'^. values of five trigonometric functions in terms of the sixth are given. in the following table.. (For signs, see 3.05.).

TRIGONOMETRY functions table.. by the root. of. some quantity, the proper. 63 sign. may. be taken from this.

,. 64. MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS _ ~. X. sin. +. I. _. COS. —. I. ±Vi +. COS X. sin. a;. tan"^. —. .Y. '. ic. I. tan X. 3.11. 3.111. Functions of the sin. {x. ±. Sum and y). cos (x. ±. y). -. =. cos X cos y (tan x. =. tan X. tan. tan. tan y. ,. ,. tan (x. ±. cot (x. ±. 3^. .. +. tan y),. ^ T. ^. sin. cos (x. ± (x. .. 3;), -^^'. ^. y). +. cos {x. y^ sin. -. cos x cos. cos X cos y (i. =. cot X =F tan y -^ n / ^ cos ix y) tan y cot X. ^. ic. =^^. =F i. cos X sin. (cot y. tan x. T. ±. ;y. ^. tan y. tan X tan y. /. .. sin. cot y tan x. .. a;. \. tan y),. T. cot y =F tan x. =. y). sin y,. tan. ±. 3'). y). cos x sin y,. =. I. 3.114. \. ± — X T. Angles.. =. = 3.113. ±. sin x cos y. -. 3.112. Two. Difference of. =. T. (x. _ T. s. y),. tan x).. (tan x. ±. tan. y)..

TRIGONOMETRY 3.12. 3.121. Sums and. Differences of Trigonometric Functions. sin. X. ±sm. y =. i. sin ^(.v. 2. =. (cos. =. (cos y. .V-. +. cos. >>. =. -. cos. ±. y). tan X. -. ±. cos y. tan y. ±. cot I (x. +. >'). tan ^ (x. -. 3'). cos X sin (x .. ;. sm(x. •. ±. ±. cot y. sm X cos. =. ± sm. T. x. .. y).. -. j)?. (cos y. —. cos x).. sin | (j. -. x). T. tan |(x. 3'). j).. y). cos y. ±y) ^'' =F. ,. (tan X. _^. T. .. ,. tan. y),. 3'). (x. ±. cot (x (i =F. ±. T sm. ±. }')(cot y =F tan x),. tan X tan y. I =F. ~. -^. >>)'. = tan y tan. =. X. y) cos K-^. tan ^(x. sin (x. .. }>. = 2 sin | (>' + x) = —(sin X ± sin. =. cot X. /. (sin '. X. =F y),. 3'). sm X ± sm. cos X. cot ^{x. .v). +. cos ^{x. 2. T y), ± y),. cos y) tan ^{x. w T ^. cos X. ^{x. y) cos. +. tan ^ (a= 7 tan ^ (x3.122. 65. '. }'). tan X tan y) tan (x. sin (x. ±. y). sm X sm y. 1. .. .. = tan l(x±y).. -. cotU--^-"^ y)-. tan ^ (x. +. y). ±. 3;)..

MATHEMATICAL FORMULA AND , ELLIPTIC FUNCTIONS. 66 3.140. -. 1.. sin-. X. +. sin-. y =. 2.. sin-. X. —. sin-. y = cos- y. 3.. cos- X. -. sin-. = y =. i. —. (.v. sin {x. )'). j) cos {x. = + y) + sin- (^ + - sin^ (x — y) = (x + y) + cos^ (x - y) =. 4.. sin-. 5.'. sin^ (x. (.V. cos-. 7.. cos^ (x -h y). —. -. i. >»). y).. +. i. = —. cos- (x -}'). - y). — y).. cos ix cos. a)*.. sin 2x sin 2j.. )'). 6.. -. y) cos (x. cos- x. + +. sin (x. cos. +. cos {x. cos 2x cos 2y.. sin 2X sin ay.. 3.150. wx = mx = nx sin mx =. 1.. cos nx cos. | cos («. 2.. sin. »x. \ cos («. 3.. cos. sin. - m)x + - w)x + m)x -. \ sin («. + +. m)x.. ^ cos {n | sin {n. —. m)x.. ^ cos (w. w)x,. 3.160 gi+iy = gx. I.. ^(.Qg. J. ^'. _|-. sin. }')•. 2. ±. (cos X. 3. =. sin x) ". f. cos nx. ±. wx. sin. i. [De Moivre's Theorem], 4. sin (x. 5. cos (x. ± iy) = ± iy) =. sin. 7. 8. and Cosines. 3.170. Sines. 3.171. n an even. sin. \. «x =. w^ I. j. .. x sinh. y.. e"'^). (c'^. -. *^).. g. cos X. +. « sin x.. ^. _. I sin. (,Qg. cos x sinh y.. i sin. -j;^. of Multiple Angles.. -. (»-. .. sin. x. 2-). .. „. sni-*. ;. ,. x. (/?-. -. 2-) (;?-. -\. ,. sin-. ,. X H. n^{n". ;. -. ;. 3!. I. cos. ±i. integer:. [. nx = « cos x. +. X =. = ^. e^'. g-ix. y. cos X cosh y =F. = |(e'^. cos X. 6. sin x cosh. 4-). .. .. sin^. x. —. .. 5! 2-). .. n^i^n^. ,. sin*. x. -. 2^) (;?2 ^-^. - ^^ -^. sin^. x. +.