THE LIBRARY OF THE UNIVERSITY OF CALIFORNIA LOS ANGELES THE ALGEBRA THE ALGEBRA OF COPLANAR VECTORS AND TRIGONOMETRY BY R. BALDWIN HAYWARD, M.A. F.R S. SENIOR MATHEMATICAL MASTER IN HARROW SCHOOL FORMERLY FELLOW OF ST. JOHN'S COLLEGE, CAMBRIDGE " Tant que 1'algebre et la geometric ont etc separees, leurs progres ont 6te lents et leurs usages bornes : mais lorsque ces deux sciences se sont reunies, elles se sont prfitees des forces inutuelles, et ont marche ensemble d'un pas rapide vers la perfection." LAORANGE. " Die Arithmetik der complexen Zahlen ist der anschaulichsten Versinnlichung fahig . . . eine die Einsicht lebendiger machende und desbalb sehr zu erapfehlende Versinnlichung." GAUSS. * HonUon MACMILLAN AND CO. AND NEW YORK 1892 The Sight of Translation and Reproduction is Reserve KICHAKD CLAY AND SONS, LIMITED, LONDON AND BUNtJAY. Engineering & Mathematical Sciences Library QA TO THE MEMORY OF AUGUSTUS DE MORGAN FORMERLY PROFESSOR OF MATHEMATICS IN UNIVERSITY COLLEGE, LONDON, TO THE SPIRIT OF WHOSE TEACHING AND WRITINGS THE AUTHOR, ONCE HIS PUPIL, WOULD ASCRIBE WHATEVER OF MERIT IS TO BE FOUND IN THE FOLLOWING PAGES PKEFACE THE present work is an attempt to supply the place of the Trigonometry ami Double Alyebra of De Morgan, which (published in 1849) has long been out of print and is now difficult to obtain. It is in no sense however a reproduction of that work. De Morgan in the first part of his book treats of Trigonometry on the usual lines of the geometrical definitions, and in the second part shows the full significance of the results in the light of his Double Alyebra or what is now often referred to as the Ai-gand* interpretation. In the present treatise starting from the conception of a Vector, that is, a magnitude involving the two elements Length or Quantity and Direction, it is shown that, limiting the directions considered to one plane, we arrive at an Algebra identical, in all respects as to its laws, with the ordinary Algebra developed from the notion of number or magnitude involving the single element Quantity ; but with this difference, that the former involves no unexplained symbols or, as they are commonly termed, " impossible " or "imaginary " quantities. Trigonometry follows naturally as the most important corollary or application of this Algebra, and this sequence I have adopted and endeavoured to express by the title " The Algebra of Coplanar * De Morgan mentions Argand's Evsai (1806) as one of several works bearing on the subject, "which I have either not seen or cannot immediately obtain." See Historical Note, p. xx. viii PREFACE Vectors and Trigonometry." De Morgan says in his Preface, " The term Double Algebra has not yet obtained currency," and I think the same may still be said. I have therefore ventured to use the longer term, "the Algebra of Coplanar Vectors," as expressing more explicitly the extent and scope of the Algebra here developed. The complete treatment of vectors, not restricted to lie in one plane, has been shown in the Quaternions of Sir W. R. Hamilton, the Triple Algebra of De Morgan, the Ausdehnungslehre of Grassmann, and some other developments of Algebra, to require an Algebra differing in some of its laws from those of ordinary Algebra. As the treatment of the subject in this work is somewhat novel, at any rate in the order of its development, a short summary of the successive chapters, calling attention to the salient points in each, will probably be found useful to the reader. The Introduction contains a short summary of the principles and laws of ordinary Algebra, as founded on the notion of Number together with that of opposite senses, in which therefore the literal symbols denote scalar, that is, positive or negative numerical quantities. It is shown that number may here be regarded not simply as the discrete number pertaining to quantities commensurable with a supposed unit, but as number in the higher sense of Ratio (or Continuous Number as it may be termed) as defined by Euclid, in which the distinction of com- mensurable and incommensurable ceases to be relevant. The first chapter explains the notions of a Vector and Vector- Aggregation (including in that term both Addition and Sub- traction), and shows that these lead to the same simple laws us hold for the aggregation of numbers. The results are applied to the proof of a number of well-known Geometrical Theorems. As Vector- Aggregation does not require the Vectors to be limited to one plane, these applications include propositions in PREFACE ix Solid as well as in Plane Geometry. In an appendix it is shown that Velocities, Accelerations, Forces, &c. are Vector-Magnitudes to which the laws of Vector- Aggregation are applicable. Ch. ii. treats of the multiplication of Vectors in the same plane and its consequences. The general multiplication of Vectors in space of three dimensions is beyond the scope of this treatise. A definite vector in the supposed plane being assumed as the " prime vector," every other vector in the plane may be derived from it by extending (or contracting) its length in a given ratio (say a : 1) and turning it through a certain angle (say a) in one of the two opposite senses of rotation in the plane, and so may be denoted as (a, a). If from another vector (b, /3), a third vector is derived by the same process, this last is (ab, a+/3), and it is shown that the operation may be properly termed multipli- cation, so that we may write (a, a) x (b, (3) = (ab, a + ft) : and that the commutative, associative and distributive laws of ordinary Algebra are satisfied by the operation thus defined. The consequences of this definition are then developed, and it is shown that (using* the right angle as the angular unit) and denoting by i, the vector (1, 1), or the unit vector whose inclin- ation to the prime vector in the positive sense is a right angle, the vector (a, a) may be expressed as ai a , or the product of two elements, a, termed its tensor, and i a , its versor. The same vector may also be expressed in the' form of a complex number, as m + ni, the component terms of which I call the project and traject respectively. Thus a vector may be regarded either as the product of tensor and versor, or as the sum of project and * The right angle is at once the most natural, and the simplest and most convenient, unit in the development of the subject, and is here denoted by the symbol L . ' ' Circular Measure " is only introduced later at a point where it naturally presents itself as having special advantages, but even then I have frequently used rectangular measure as in many cases presenting formula in a simpler form than that arising out of circular measure. x PREFACE traject, and the relation of these two modes of presentation of a vector is the key to a great part of the farther development of the subject. The tensor is equal to the modulus of the complex number, (a = v m 2 + n z ), and the versor to the quotient of the complex number divided by its modulus. The chapter concludes with some geometrical illustrations and applications. Ch. iii. treats of the Trigonometrical or Circular Functions. The project and traject of a unit vector (i lt ) are denned as the cosine and sine respectively of its inclination (u) to the prime vector which leads to the fundamental equation i u = cos u + i sin u. Hence we obtain at once the scalar relation cos 2 u + sin 2 u = 1, and the expressions for cos u and sin u in terms of versors ; thus 2 cos u = i n + i'- M and 2tsin u i w - i~ u . Then defining the other circular functions with reference to sine and cosine, the usual relations of the functions of a single angle are proved in all their generality by direct algebraical processes. Ch. iv. begins with De Moivre's Theorem, as a direct conse- quence of the fundamental equation i u = cos ii + i sin u. The ordinary formulae of Trigonometry for two or more angles are then deduced ; and the expansions of cos mi, sin nu in powers of cos u or sin u the expressions for cos 71 u, sin" u in terms of sines or cosines of multiples of u, and the fundamental relations of the sides and angles of a triangle established. The application of these last results to the Solution of Triangles, the Geometry of the Triangle, &c., is not pursued : for these the student is referred to other treatises on Trigonometry. Ch. v. treats of Vector (or Complex) Indices and Logarithms. Assuming that A B , where A , Ji are both vectors or complex numbers, represents some vector, we consider first the case where A = k, a positive scalar, and E = i. Supposing then k l = ri u , the problem is to determine r and u. Assuming that there is a value of k, which makes r = 1 and u = 1 , and denoting it by rj, so that r)' = i, we find from the laws of Indices that in which it remains to determine the value of n. After a discussion of the limits of sin z/z (which leads incidentally to the explanation of circular measure) and of (a 2 l)/z, when z vanishes, it turns out that rj = e and 6 denoting the circular measure of W L , the fundamental identity becomes cos u + i sin u = i u = r) iu = e ie , whence the usual exponential expressions for the circular functions. The interpretation of A B or (ri u ) m+ni easily follows : it turns out to be a function having an infinite number of values in vectorial geometrical progression, which are conveni- ently represented in a diagram. The general Theoi-y of Logarithms naturally follows, but is here discussed independently, and illustrated by diagrams and geometrical constructions. Ch. vi. treats of Excircular (or Hyperbolic) Trigonometry. The functions, which are usually called Hyperbolic, I have ventured to term Excircular, because they are in a certain sense, as we see in their geometrical interpretation, the Circular Functions turned inside out. They are not Hyperbolic, as opposed to Elliptic, but Rectangular-Hyperbolic. I hope to carry Mathematicians with me in giving the names of Excircle to the Rectangular llyperbola, and Excircular Functions to the Functions treated of in this chapter. The functions are defined as those obtained by putting iu for u in the fundamental identities, and the formulae corresponding to those established for Circular Functions are deduced. Their geometrical relation to the Excircle is then exhibited, and a number of the properties of the Excircle xii PREFACE deduced. The chapter concludes with graphic constructions representing the vectors sin (u + vi), cos (u + vi), &c. In Ch. vii. the fundamental properties of the roots of unity are discussed both as an instructive exercise in the application of the principles established in the previous chapters and for the sake of the importance of the analytical results. Ch. viii. deals with the general properties of Infinite Series, and the conditions of their Convergency and Divergency, so far as seems, necessary for an Elementary Treatise. It commences with a complete discussion of the Geometric Vector Series showing that it is convergent when r is less than 1 and divergent when r is greater than 1; but that when r = l, although the sum of any number of terms is finite, it does not approach to a definite limit, but fluctuates (within a certain circle whose u\ radius = |- cosec -), except only in the limiting case when u = Q 2/ or the series is scalar, in which case it is divergent. All this is made evident by geometrical representation, which also furnishes a simple and interesting construction for the centre of convergence in the convergent case, and the centre and radius of the circle of fluctuation in the case of the fluctuating series. An elementary discussion of the general properties of infinite series follows, aided by an {Illustration which, I believe, will throw much light for the elementary student on some of the difficulties of the subject. The terms of a series are considered as the links of a chain, the scalar or tensor factor of each term determining the length of the link and the versor factor its inclination ; then the sum of the series is the vector drawn from one end of the chain to the other, and convergency or divergency depends on whether this vector is finite or infinite, fluctuation corresponding to the case where the vector though not infinite is PREFACE xiii indefinite in length or direction. The discussion includes some remarks on the Continuity or Discontinuity of Infinite Series, and the conditions subject to which the fundamental laws of Algebra are applicable to them. The discussion is then confined to the Power Series, and tests for its Convergency, Divergency or Fluctuation established and applied to a number of examples. Oh. ix. treats fully of the Binomial Theorem in its most general form, the Exponential and Logarithmic Series, and the Trigono- metrical Series directly derived from each. Then follows the summation of various series by the help of these known series, and also of certain series by the method of differences, and the chapter concludes with an account of Bernoulli's numbers and certain series involving them. Ch. x. treats of Factor Series, first as to their general properties with a Geometrical Illustration, and then deducing the factor series for sin u and cos u from the resolution of z n - a n into factors. An extended form of Cotes' s and De Moivre's Properties of the Circle is obtained, and applied to the geometrical interpretation of these series. The chapter concludes with sei'ies involving the sums of the inverse integral powers of the natural numbers, and by means of these Bernouilli's numbers are expressed, and the conditions of Convergency or Divergency of the series at the end of the previous chapter determined. Ch. xi. deals in like manner with the expressions of cotan, tan, &c., in series of partial fractions. Ch. xii. concludes the work with such an account of some fundamental properties of Rational and Integral Functions of a Complex Variable as naturally fall within the limits of an elementary Treatise. If z denote a vector in one plane, and w a vector in another plane, and w =/(), where fz is a rational and integral function of z of the n th degree, (z, w being each drawn from a certain fixed origin in its plane,) for every point on the z plane there is one and only one corresponding xiv PREFACE point on the w plane, and it is shown conversely that for every point on the w plane there is a point, and if one, then n points, corresponding to it on the z plane. The n points corresponding to the origin or w = 0, for which fz = Q, are termed the radical points of the function. Cauchy's test for the number of radical points within given limits on the z plane is then established, in a simpler manner (I venture to believe) than that usually given. A short account of Conjugate Functions, as exemplified by Rational and Integral Functions, is then given with their fundamental Geometrical Relations. The chapter concludes with a detailed discussion of a Cubic Function with scalar coefficients, the different cases into which it resolves itself being illustrated by a number of diagrams. In addition to specific acknowledgments in the text of the sources, whence I have derived matter which demanded such acknowledgment as hardly yet included within the limits of the recognised common property of all Mathematicians, I wish to acknowledge generally deep obligation to Prof. Chrystal's great work on Algebra, especially in the chapters relating to Infinite Series, for the fuller development of which beyond what falls properly within the limits of my own work Prof. Chrystal's treatise will form the best and most available guide. From Mr. Hobson's excellent Treatise on Plane Trigonometry also I have derived one or two valuable suggestions. Messrs. Levitt and Davison's recently published Elements of Plane Trigonometry, in which the subject is very ably and accurately treated, has in its later chapters much in common with the present treatise, as was to be expected since like my own it is largely based on De Morgan's -work. Their stand- point and the. order in which they have developed the subject 'however differ essentially from those of the present work. It is perhaps well to joote that in one point of notation in which we have independently hit on the same device, we have unfor- PREFACE xv tunately carried it out in opposite directions. I have distin- guished the pi'ime A-alue of an inverse function by using a capital initial letter (e.g. Tan" 1 *), while the same written with a small initial denotes the general value (tan"" 1 ^) : Messrs. Levitt and Davison indicate the prime value by the small, and the general value by the capital letter. The present work assumes on the part of the student a competent knowledge of Elementary Algebra and Elementary Plane Geometry as usually understood, but not necessarily any previous knowledge of Trigonometry. It will probably, however, be most instructive to those who have already studied some elementary work in which that subject is treated on the usual geometrical basis. R. B. H. HABBOW, September 1892. IN the year 1806 there was printed at Paris an " opuscule" entitled Essai sur une Maniere de rej/resenter les Quantites LtM'jitiaires dans les constructions Geonietriques, without the name of the author, but to be obtained " chez Madame Veuve Blanc, Horloger, rue S. Honore." The work appears to have remained unknown, at any rate unnoticed, by Mathematicians, until there appeared in the Annales de Gergonne, tome iv. 1813-1814, a note by J. F. Francais on the same subject, which elicited from M. Argand* a letter addressed to M. Gergonne, accompanied by a copy of the Essai, with this alteration in manuscript 011 the title-page, "Chez M. Argand, rue de Gentilly, No. 12." The letter, which was published in T. iv. of the Annales, contains a summary of Argand's Essai, and was followed in T v. by an article further developing his ideas. A second edition of the Essai was published in 1874 with a preface by M. J. Hoiiel (whence I have obtained the above information) and an appendix containing, besides Argand's papers in the Annales, notes of Fra^ais, Servois, and Gergonne, who took part in the discussion. Though M. 1'Abbe Bu6e, in a paper read June 20, 1805 and published in the Phil. Trans, for 1806, argued that the symbol v - 1 should be regarded as expressing perpendicularity, he was not so successful in working out the consequences of this inter- * Probably "Jean-Robert Argand, fils de Jacques Argand et de Eve Cauve," born at Geneva, 22 July, 1768. Little more is known of his personal history than that he was residing at Paris in 1813. b xviii HISTORICAL NOTE pretation as Argand, for whom Hankel (quoted by Hoiiel) justly claims the honour of being " the true founder of the theory of complex quantities in one plane," at any rate so far as priority and completeness are concerned. Argand concludes his Essai with the following remarkable passage : "La Methode dont on vient d'exposer 1'essai repose sur deux principes de construction, 1'un pour la multiplication, 1'autre pour 1'addition des lignes dirigees : et il a ete observe que ces principes resultant ^inductions qui ne j)ossedent 2 KIS un degre suffisant ^evidence, ils ne pouvaient, quant a present, etre admis que comme des hypotheses, que leur consequences ou des raisonne- ments plus rigoureux pourront faire admettre ou rejeter." It was reserved for a series of writers during the first half of the present century (several of whom appear to have arrived independently at the same interpretations) to supply, in con- nection with discussions on the foundations of Algebra, the true logical basis which Argand felt to be wanting for his interpretations. The following list of the principal writings of this class is taken from De Morgan's Trigonometry and Double Algebra. List of some writings on the subject of Algebra, in which the peculiar Symbols of Algebra are discusstd. London, 1685, folio. John Wallis. A Treatise of Algebra, both historical and practical. Reprinted in Latin, with additions, in the second volume of Wallis's Works, Lond. : 1693, folio. Naples, 1687, folio. Giles Francis de Gottignies. Logistica Universalis. London, 1758, 4to. Francis Maseres. A Dissertation on the use of the Negative Sign in Algebra. London, 1 796, 8vo. William Frend. The Principles of Algebra.* * An opponent not only of imaginary but of negative (juantities. Perhaps this work suggested M. Buee's memoir. I have a letter in my possession HISTORICAL NOTE xix Cambridge, 1803, 4to. Robert Woodbouse. The Principles of Analytical Calculation. Philosophical Transactions for 180G. M. 1'Abbe Buee. Memoire sur les quantites Imaginaires. (Read June 20, 1805.) See also the review of this in vol. xii. of the Edinburgh Review, April July, 1808 (written by Playfair). London, 1817, 4to. Benjamin Gompertz. The Principles and Application of Imaginary Quantities, Book I., to which are added some observations on porisms London, 1818, 4to. Benjamin Gompertz. The Principles and Application of Imaginary Quantities, Book II., derived from a particular case of functional projections Paris, 1828, 8vo (small). C. V. Mourey. La vraie theorie des Quantites Negatives, et des Quantites Prelendues Imaginaires. Dedie aux amis de I' evidence. Cambridge, 1828, 8vo. John Warren. A Treatise on the Geometrical Representation of the Square Roots of Negative Quantities. Philosophical Transactions for 1829. John Thomas Graves. ' An attempt to rectify the inaccuracy of some logarithmic formulae.' (Read December 18, 1828.) Philosophical Transactions for 1829. John Warren. Con- sideration of the objections raised against the geometrical representation of the square roots of negative quantities. (Read Feburary 19, 1829.) The same volume contains John Warren ' On the geometrical representation of the powers of quantities, whose indices involve the square roots of negative quantities.' (Read June 4, 1829.) Cambridge, 1830, 8vo. George Peacock. A Treatise on Algebra. from M. Buee to Mr. Frend, dated June 21, 1801, by which it appears that the former was desired by a gentleman in whose house he was living (as tutor, perhaps) to write a private reply to Mr Trend's objections. This letter evi- dently contains the germs of the views which he afterwards published. See the Annual Report of the Royal Astronomical Society for 1842. According to Dr. Peacock, M. Buee is the first formal maintainer of the geometrical signifi- cation of J 1. 6 2 xx HISTORICAL NOTE Cambridge, 1837, 8vo. Anonymous [Osborne Reynolds]. Strictures on certain parts of ' Peacock's Algebra,' by a Graduate. Philosophical Transactions for 1831. Davies Gilbert. 'On the nature of negative and of imaginary quantities.' (Read November 18, 1830.) London, 1834, 8vo. Report of the Third Meeting of the British Association for the Advancement of Science. This volume contains George Peacock, ' Report on certain branches of analysis,' a most valuable historical discussion on, among other things, the advance of Algebra. I cite from it the following works, which I have either not seen, or cannot immediately obtain. Paris, 1806, Argand. Essai sur la maniere de representer las Quantites Imaginaires dans les constructions geometriques. Also papers or observations by Frangois, Argand, Servois, Gergonne, in the Annales des Mathemaliques for 1813 (and I suppose the following year). Also a paper on the arithmetic of impossible quantities, by Playfair, in the Philosophical Transactions for 1778 ; with a Reply, by Woodhouse, in the same work for 1802, entitled ' On the necessary truth of certain conclusions obtained by aid of imaginary expressions.' London, 1836, 8vo. Anonymous [George Peacock]. A Syllabus of a Course of Lectures upon Trigonometry and the Application of Algebra to Geometry. London, 1837, 8vo. A. De Morgan. Elements of Algebra. 2nd edition. London, 1837, 8vo. A. De Morgan. Elements of Trigo- nometry and Trigonometrical Analysis, preliminary to the Differential Calculus Edinburgh Philosophical Transactions, Vol. XIV., Part 1. D[uncan] F[orbes] Gregory. ' On the real nature of Symbolical Algebra.' (Read May 7, 1838.) Ladies' Diary. London, 1839, 8vo (small). Thomas White. ' On the algebraical expansion of quantity, and on the symbol HISTORICAL NOTE xxi Y/ - 1, which is usually considered to denote impossible or imaginary quantity ' (at page 59). Cambridge Philosophical Transactions, Vol. VJI., Part 2. A. De Morgan. ' On the Foundation of Algebra. (Head Dec. 9, 1839.) Cambridge Philosophical Transactions, Vol. VJI., Part 3. A. De Morgan. ' On the Foundation of Algebra,' No. II. (Read Nov. 29, 1841.) Paris, 1841, 8vo. M. F. Valles. Etudes P/iilosophiques sur fa science du calcul. Premiere Partie. No more yet published. Cambridge Philosophical Transactions, Vol. VIII., Part 2. A. De Morgan. ' On the Foundation of Algebra,' No. III. (Read Nov. 27, 1843.) Cambridge, 1842 and 1845, 8vo. George Peacock. A Treatise on Algebra. Vol. I. Arithmetical Algebra. Vol. II. Symbolical Algebra and its applications to the geometry of position. London, 1843, 12mo. Martin Ohm [translated by Alexander John Ellis], The Spirit of Matliematical Analysis, and its relation to a logical system. To the foregoing list should be added the great name of Gauss, who in a notice ("Anzeige"), dated 1831, April 23, of his "Theoria Residuorum Biquadraticorum. Commentatio secunda " (Werke, ii. pp. 174-178) speaks of the Arithmetic of Complex Numbers as " der anschaulichsten Versinnlichung fahig," and concisely, but very clearly, lays down the principles of such " Versinnlichung " by geometrical representation. He observes that, although in his memoir he has adopted a purely arithmetical treatment, he has also given sufficient indications, for the reader who] thinks for himself (" selbstdenkende Leser"), of " eine die Einsicht lebendiger machende und deshalb sehr zu empfehlende Versinnlichung." There can be little doubt that Gauss made great use of the geometrical representation of " imaginaries " (" eine wenig schickliche Benennung ") in his researches, although he presented his results for the most part in what he terms a purely arithmetical form. TABLE OF CONTENTS INTRODUCTION ART. 1'AOER 1. Plan of the work 1 2 5. The fundamental laws of Algebra 2 6 6. The laws true for Ratios or continuous Number 6, 7 7. Extension of meanings of + and . Scalar Magnitudes . 8 8. The complex number a + b*J 1 "impossible" in the Algebra of pure number or ratio 9 9. Summary of principles 10 10. Scalar Algebra linear. May not the complex number find an interpretation in planar or two-dimensional space ? . 11 CHAPTER I VECTORS AND VECTOR AGGREGATION 1. Definition 12 2. Vectors in the same line. Scalars 12 3. Equality of Vectors 13 4. Aggregation of Vectors 13 5. Associative and Commutative Laws of Terms 14 6. Application to Geometry 16 7. Vector of any point expressed by given vectors 16 8. Illustrative Examples 1728 Examples on Chap. I 2830 Appendix on Vector Magnitudes 30 35 CHAPTER II MULTIPLICATION OF COPLANAR VECTORS 1. Introductory 36 2. Relation of Vectors to the Prime Vector 36 xxiv TABLE OF CONTEXTS ART. PAGES 3. Vector Multiplication 38 4. Commutative Law of Multiplication 39 5. Associative Law 40 6. Distributive Law 41 7. Reciprocal of a Vector 42 8. Vector Division 43 9. Scalar Powers of a Vector 43 10. Multiplicity of Values of Scalar Powers . . . . 44 11. Illustrations 46 12. Vector as product of Tensor and Versor 48 13. Interpretation of V 1 49 14. Uniqueness of i 49 15. Vector expressed by a Complex Number or the sum of Project and Traject 50 16. General Conclusion 51 17. Illustrative Examples 51 55 Examples on Chapter II 56, 57 CHAPTER III TRIGONOMETRICAL RATIOS 1. Introductory 2. Definitions of the Trigonometrical Ratios. The fundamental equation z' = cos u + i sin u 59 61 3, 4. Fundamental Relations of the six ratios 62, 63 5. The ratios in terms of versors 63 6. Variations of the ratios with the angle, and their graphs . 63 67 7. Ratios for the reversed angle, supplement, &c 67 69 8. Values of the ratios for certain acute angles 69 72 9. Expressions for all angles having the same sine, cosine or tangent 72, 73 10. Inverse Functions 73 76 11. Some Trigonometrical identities proved from the vercor forms 76 78 Examples on Chapter III 78 CHAPTER IV I)E MOIVRE'S THEOREM AND GENERAL TRIGONOMETRICAL FORMt'L* 1. Introductory 79 2. De Moivre's Theorem 7982 3. Formulae for sine, cosine, tangent of u v, 2u, &c 82, 83 TABLE OF CONTENTS xxv ART. PACES 4. Formate for expressing sums and differences of sines' and cosines as products of sines and cosines, and the converse . 84, 85 >. Sub-multiple angle formula; 85 89 6. Functions of the sum of any number of angles 89, 90 7. Functions of nu in powers of cos u, sin 11, tan u 90 93 8. Series for cos nu and sin rait/sin u in descending powers of cos u 9395 9. Series for cos nu and sin nu in ascending powers of sin u or COSM 95 97 10. cos n w, sin n ?( in terms of cosines or sines of multiples of u . . 97 99 11. To express cosu . sin"w in terms of cosines or sines of multiples of u , 99 102 12. Formulae connecting the sides and angles of a triangle . . . 103 Examples on Chapter IV 104, 105 CHAPTER V VKCTOR INDICES AND LOUA1UTHMS 1. Question proposed. To interpret AR, where B is a vector (or complex number) 106 2. If k is a positive scalar, & = i igi)fc, where t\ is a definite numerical constant 107 3. Hence 2 cos u = V + "n~ iu , 2i sin u = rj'" -n~ iu . . . . 108 4. Limit of sin u/u, when u vanishes 109 5. ( 'irculav Measure Radian 110 6. Limit (a' l)/z, when s vanishes Ill 7. Determination of c, where c is such that limit e* l/z = 1 when c = 111114 7T S i, = C 2 = 4-810475 114 9. Exponential Expressions for the Trigonometrical Functions . 115 10. Vector Power of a Vector Geometrical Interpretation . . . 115 117 11. Paiticular Cases discussed 117 119 12. General Theory of Logarithms 119 13. Logometers to base TJ 120 14. Logometers to any numerical base 121 15. Illustrative Diagram 121 124 16. Logometers to a vector base Illustrative Diagrams 124 127 TABLE OF CONTENTS CHAPTER VI EXCIRCULAR OR HYPERBOLIC TRIGONOMETRY ART. PAGES 1, 2. The excircular Functions obtained from the circular Functions by putting iu for u 128 131 3. Range of values of cosh, sinh, tanh Hyperbolic Amplitude (amhtt) 131132 4. Formula for Excircular Functions 132 134 5,6. Geometrical Interpretation of the Excircular Functions . . 134 137 7. Properties of the Excircle deduced from Excircular Functions 137 142 8. Graphic Construction of the Vectors sin (u + vi), cos(u + vi) 142 144 9. cosh (u + vi) and sinh (u + vi) 145 10. tan (u + vi) 145146 11. cot (u + vi) 147 12. tanh (u + vi) and coth (u + vi) 147 Miscellaneous Examples (Chapters I. VI.) 148 151 CHAPTER VII ROOTS OF UNITY 1. Introductory 152 i_ i_ 2. Geometrical representation of l"and ( 1)" 152 155 3. The Roots as powers of the Principal Root 155 4. Primary and Subordinate Roots 156 158 5. Roots of any order determined by roots of orders, which are powers of primes 158 159 6. The sum of the n th roots = 159-160 7. Sum of the 7?i th powers of the n th roots 160161 8. Sum of the products of every r of the n th roots 161 162 9. Applications of the foregoing results : (1) Rationalising Factors (2) Cardan's Solution of a Cubic (3) Gauss's Theorem that, n being a prime of the form 2* + 1 the circumference of a circle can be divided into n equal arcs with the usiial conventions of Geometrical Construction , Examples 167 168 162166 TABLE OF CONTENTS xxvii CHAPTER VIII INFINITE SERIES CONVERGENCY AND DIVERGENCY Geometric Series. ART. PAGES 1. Geometric Vector Series, 1 + ri + rH ' + . . .Character- istic, S Sum of n terms, S n 169171 2. Reduction of S and S n to vector forms, and the resulting Trigonometric Series 171 174 3. Geometrical Interpretation Tensor ratio (r) = 1, the series fluctuating 174 177 4. Tensor ratio < 1, the series convergent. Construction for the Characteristic 178182 5. Tensor Ratio > 1. The series divergent 182 6. Summary of Results. Illustration from a coiled chain . . . 182184 Infinite Series in General. 7. Definitions 184 8. General Properties of Infinite Series. Absolute and Relative Convergence 184188 9. Rate of Convergency 188190 10. Continuity or Discontinuity of an Infinite Series 190194 11. Fundamental Laws of Algebra in relation to Infinite Series . 194 197 12. Power Series 197, 198 13. Continuity of a Power Series 199 14. Convergency or Divergency of Power Series Circle of Con- vergency " 199201 lf>. < 'use where the tensor (r) = radius of Convergency (k) . . . 201 203 1 (j. Examples Application of the Tests to Particular Series . . . 203 206 Examples 207 209 CHAPTER IX EXPANSIONS AND SUMMATIONS 1. Trigonometrical Series corresponding to Algebraical Expansions 210, 211 2. Binomial Theorem 211213 3. Convergency or Divergency of the Binomial Series 213 215 4. Illustrative Diagrams 215 218 5. The Binomial Series equal to the prime value of (1 + z) . . 218 220 6. General Form of the Binomial Theorem : Index a complex number Geometrical Interpretation^ 220 224 7. Trigonometrical Series derived from the Binomial Theorem . 224, 225 8. Exponential Theorem Three Proofs ... 225230 xxviii TABLE OF CONTENTS ART. PAOF.fi 9. Series for sin 6, cos 0, sinh 6, cosh 6 230 232 10. Logarithmic Series 233 236 11. Series for Tan-l x, Tanh-l x 236238 12. Calculation of the value of JT 238 240 13. Series for sin - 1 x, sinh -la;, cos-l x, cosh -l a: 240 242 14. Summation by means of the foregoing series 242 244 15. Sum of selected terms of a known series 245 248 16. Summation of Trigonometrical Series by the Method of Differences 248252 17. Bernoulli's Numbers 253, 254 18. Expansion of x/(e*> - 1) 254,255 19. Series for coth x, cot x, tanh x, tan x, cosech x, coscc a; ... 256 258 Examples 259265 CHAPTER X SERIES OF FACTORS 1. Infinite Products Convergence or Divergence Geometrical Illustration 266270 2. Resolution of z* an into factors 271, 272 3. Geometrical Interpretation, including Cotes's and De Moivre's Properties of the Circle 273275 4. Resolution of z-n - njn cos + 2 " into factors 275, 276 5. Factor series for sine and cosine 276283 6. Geometrical Illustration 283285 7. Factor Series for sinh and cosh 285, 286 8. The Factor Series for sine and cosine are periodic 286, 287 9. Series for log sin u &c 287290 10. Wallis's Theorem. Deduction of approximate value of |, when n is large 290292 11. Deductions from the series for sines &c 292 12. Relations between Bernoulli's Numbers and Sums of inverse powers of the Natural Numbers 293 13. Limits of Convergency of the series dependent on Bernoulli's Numbers 294, 295 Examples 296299 CHAPTER XI SERIES OF PARTIAL FRACTIONS 1. Resolution of l/(s m - 1) into the sum of n fractions 300 302 2. Resolution of !/(* + 1) into the sum of n fractions 302, 303 TABLE OF CONTENTS xxix ART. PAGES 3. Cot u and tan u expressed a.s the mean of n cotangents . . . 303 306 4. Geometrical Interpretation 306 309 ">. I 'ot u and tan u as the sum of an infinite series of fractions . 309 311 6. Series for cot u deduced from the Geometrical Interpretation . 311, 312 7. Cosec u and sec u as the sum of an infinite series of fractious 313 8. Excircular Functions coth, tauh, &c., as the sum of an infinite series of fractions . 314 Examples 315 317 CHAPTER XII HATIONAL AND INTEUUAL FUNCTIONS 1. Definition, Notation, &c 318, 319 3. Expression offz in powers of s - zo ... 319 322 4. Continuity of/:. Every rational and integral equation has a root 322325 5. Expression of/~ in vector factors 325, 326 b'. Tests for the number of roots within given limits Cauchy's Theorem 326329 7. Conjugate Functions 329 334 8. Discussion of a Cubic Function 334 343 9. The Binomial Function, z* 1 343 COPLANAR VECTORS AND TRIGONOMETRY THE ALGEBRA OF COPLANAR VECTORS AND TRIGONOMETRY INTRODUCTION 1. Algebra is a science which originated historically in the notion of number, and the expression by words (for which symbols were gradually introduced), of the simple fundamental opera- tions on numbers ; namely, the direct operations of addition and mtiltiplication with their respective inverses, subtraction and division. The lectures on Algebra, which Newton delivered as Lucasian Professor at Cambridge, were published under the title of Arithmetica Universalis, and it is from this point of view of " Generalized Arithmetic" that the subject is still treated, and rightly so, in elementary works on algebra. The modern developments of algebra, embracing all organized systems of symbols combined according to definite laws, extend to subject-matter of which number or quantity constitutes but one element, or in which this element is entirely absent, and the letters and symbols denote operations or forms only. It is with one of these developments, in which the laws and resulting formulae are those of ordinary algebra, but the meanings of the B 2 INTRODUCTION symbols are extended by the introduction of a new element, so that no uninterpretable forms or so-called " impossible " or " imaginary " quantities remain, that this treatise is concerned. The method it is proposed to adopt is, to start with the notion of a Vector ; from this to deduce the laws of combination of vectors in the same plane, showing that these laws are identical in their symbolical expression with those of pure number ; and then to show, among other results, that Trigonometry emerges as an immediate consequence and thus becomes affiliated to, or rather included in, this branch of algebra. 2. In order to prepare the way for our special subject, a short review of the foundations and scope of ordinary algebra, with its consequent limitations, may fitly here be introduced. Algebra starts with the use of letters (a, b, x, y, &c.) to denote numbers unspecified, and the signs of the four funda- mental arithmetical operations. Thence are derived the following fundamental laws : I. The Associative Law : a. of Terms. That in an aggregate of terms any pair or set of consecutive terms may be grouped into a single term : e.g. a + bc=i(a + b)c=a + (bc). ft. of Factors. That in a product of factors any pair or set of consecutive factors may be grouped into a single factor : e.g. axbx.c = abx.c = axbc. ax b-t-c = (a x 6) -f- c = a x (b -f- c). II. The Commutative Law : a. of Terms. That the sum or aggregate of a number of terms (whether positive or negative) is independent of the order of aggregation : e.g. a+bcac+b= c + b + a, &c. INTRODUCTION 3 ft. of Factors. That the product of factors is independent of the order of multiplication (including under this term division, as multiplica- tion by the recijrrocal), 1 a 1 ab = ba, a . ~ = - = -r . a. e.g. b b b III. The Distributive Law : That the product (or quotient) of an aggregate of terms multiplied (or divided) by any factor is the aggregate of the products (or quotients) of each term multiplied (or divided) by the same factor : ab a b e.g. m(a + b) = ma mb, =--- m m m . IV. T/ie Law of /Signs : a. for the signs + and , + ( + a)=+a, + ( )= a, ( + a)=-a, ( a)=+a. ft. For the signs x and -=- , x ( x a) = x a, x ( -r a) = -f a, -f ( x a) = + a, -f ( -=- a) = x a. V. Tlie Laws of Indices : 1. a m .a n = a m + n . 2. (a m ) H = a mn . 3. (ab) m = a m b m . which are rendered complete and general by the interpretations '*, 1 a. a" = Va m . 8. a~ n = -, y. af=l. a n 3. These laws are the necessary and sufficient basis of ordinary algebra as a formal science, so that from them alone all the results of algebra are deducible. It is an essential condition of such a science that its laws should be universally true, without exceptions, and it is a consequence of this condition that, from whatever primary meanings for the symbols its laws may be derived, any previously B 2 4 INTRODUCTION undefined combination of symbols must be interpreted in accordance with these laws, and not arbitrarily. Hence results the principle, enunciated by Peacock and termed by him the " Principle of the Permanence of Equivalent Forms," which asserts that * "all results of algebra which are general in form, though they have been established as true only for restricted meanings of the symbols, must also be true when the symbols are general in value as well as in form." 4. For a detailed discussion of these laws as based on the notion of number the student is referred to the treatises of Chrystal, C. Smith, Aldis, or Hall and Knight, to which may be added the fresh and original treatment of Oliver and Waite. It will be well however to summarize here the successive stages in a logical sequence,^ by which the laws of algebra may be conceived to be derived from the notion of number. 5. First, regarding the signs +, , x, -H as signs of operation only, we may consider the literal symbols as advancing through the following stages. 1. Where the letters denote integers. Here the laws, with such restrictions as the meanings of the letters impose, are very obviously deducible, and are in fact the laws on which all arithmetical operations are based. The expression a b, where b is greater than a, is unintelligible or "impossible," as is indeed the form - or a~ b, unless a is an exact multiple of b. 2. Where the letters denote finite fractions, including integers as a particular case. * This must be understood with some qualification, when the results involve infinite series. See C. viii. t The logical order in any science is generally neither the historical order of development, nor the exact order in which the student will have acquired his knowledge. It is rather an order in which on a review of the subject as a whole after a certain familiarity with it has been obtained, the fundamental principles and the consequences deduced from them may be conveniently arranged so as to show their affiliation and interdependence. INTRODUCTION 5 The laws, as far as regards terms, are deducible exactly as in the previous case for integers, but those involving factors depend on a suitable extension of the meaning of multiplication, when the multiplier is a fraction. Such extension may be made by adopting the following general definition of multiplication. Multiplication consists in doing with the multiplicand what must be done with the unit to form the multiplier. From this it follows that to multiply by a fraction (e.g. ) is to form the same fraction () of the multiplicand. "With this ex- tension of meaning the laws involving factors also are readily proved. The laws of indices also assumed to be true for fractional indices lead to the interpretation of a n as necessarily denoting n i the nth root of a m or v a m . 3. Where the meaning of the letters is still further extended so as to include incommensurable quantities, and thus to justify their use as representing the quantity, relatively to a given unit, of any magnitude whatsoever. Every incommensurable number lies between two finite frac- tions, as and - , whose difference can be made as small n n n as we please, and on the assumption that what is true of two magnitudes which can be made as near to one another as we please is true of any magnitude which always lies between them, the same laws which have been established for finite fractions are seen to hold good for incommensurable magnitudes also. This last extension from the commensurable to the incommen- surable by treating the latter as attainable as a limit from the former is necessitated by the impossibility of completely repre- senting continuous magnitudes by numbers. Number is essentially discrete or discontinuous, proceeding from one value to the next by a finite increment or jump, and so cannot, except in the way of a limit, represent, relatively to a given unit, 6 INTRODUCTION a continuous magnitude for which the passage from one value to another may always be conceived as a growth through every intermediate value. In Euclid's Fifth Book we have the development of the general notion of ratio for magnitudes generally, whether continuous or discrete. Thus treated, ratio, while including the notion of number, is independent of the distinction of commensurable and incommensurable, and is applicable to the complete and exact measurement of any magnitude relatively to a given unit. It is instructive to see how the laws of algebra result from this higher conception of ratio, or as it might be termed, continuous number, and this we proceed summarily to indicate in the following articles, which, as an alternative mode of deduction of those laws, may be omitted without loss to the sequence of the argument. 6. a. Let A, B be two magnitudes of the same kind, and therefore having a ratio, and let the ratio A : B be denoted by a : then if U denote a unit or standard quantity of some particular continuous magnitude (as length or time), a magnitude X of the same kind can always be found such that X : U A : J3, whence a = X : U. ft. Then the associative and commutative laws for terms being obviously true for the magnitudes X, Y, Z..., they are also true for the ratios a, b, c... which measure them with respect to a common unit U, if we define a +.b or (X : U} (Y : U) as equivalent to the single ratio XY: U. y. For factors, the product ab must be regarded as the ratio compounded of the ratios a and b. Then if a = X : U, b = Y : U, e = Z : U, Y' can always be found such that Y' : Z : : Y : U, and X' such that X 1 : Y' :: X : U, and therefore a = X' : Y', b = Y' : Z, c = Z : U. Hence, the ratio compounded of A : B and B : C being, according to Euclid's definition, A : C, we have ab = (X' : Y') . (Y' : Z} = X' : Z, and bc = (Y' : Z) . (Z : U)=Y' : U, whence ab . c = (X' : Z} . (Z : IT) = X' : U and a . bc= (X 1 :').(': Z7) = J? : U, so that ab . c = a . be which proves the associative law for factors. INTRODUCTION 7 5. Also since Y' : Z :: Y : U, alternately Y' : Y :: Z : U, hence bc = (Y\ U} .(Z: U} = (Y' : Z) . (Z : U)=Y' : U and cb = (Z : U) . (Y : U) = (Y' : F) (F : U) = Y- : U, therefore be cb, which proves the commutative law. e. If #, a denote respectively the ratio A* : U and its reciprocal ratio U : A; then aa' = (X : U) . (U : X) = X : X=l or a'= l - a Also since a, b may be written as a : 1, b : I respectively, if - denote the ratio a : b and bt>'=l, j = (a : 1) . (1 : b) = (a : 1) . (// : 1) = a . - or ab'. Hence the quotient alb being the same as the product of a and the reciprocal of b, the laws proved above for multiplication hold also for the inverse, division. f. To prove the distributive law. b= Y : U, and m- Z' : X :: Z : U, and Z" : Y :: Z : U, Let = A" : U, b= F : U, and m = Z : U, and let Z', Z" be so taken that then also Z'Z" : XY:: Z : U. Hence m (ft b} - (Z : U) . (A' F : U) = (Z'Z" : X Y).(XY:TT) = Z'Z" : U = (Z' : U}(Z" : U) = (Z' : X) . (X : U) (Z" : F) . (F : IT) = ma + mb. r). The same reasoning which establishes the laws of signs for numbers is applicable to magnitudes generally and their complete measures by ratio, and the same is true of the laws of indices, so long as the indices are commensurable numbers. The form a m , however, where m is incommensurable, can hardly be interpreted, except as a limiting case deducible from that where m denotes a finite fraction. 8 INTRODUCTION 7. The laws of algebra having thus been shown to result from the widest conception of number or ratio while the signs + and denote merely the operations of addition and subtraction, a farther step in their generalization consists in showing that these signs may be regarded without inconsistency as signs of affection no less than as signs of operation. There are certain classes of magnitudes which may be called purely quantitative magnitudes, of which we can conceive the indefinitely continued diminution by subtraction down to anni- hilation, but for which, the zero being attained, no farther progress of the same kind is possible. If a, b denote the quan- tities relatively to a given unit, of two amounts of a purely quantitative magnitude, a b is "impossible" when b>a, and a by itself is unmeaning. The population of a country or the strength of an army is such a magnitude. It may be diminished to annihilation, but here the process of necessity stops. So the mass or quantify of matter in a given space may be indefinitely reduced, but a zero or vacuum forms a definite limit. We can " draw the well dry," but cannot continue the exhaustion. The same is true of heat or energy. The absolute zero of temperature is a definite ideal limit, beyond which abstraction of heat is impossible. Thus population, mass, heat or energy are instances of purely quantitative magnitudes, and the first has the further limitation that it can be expressed only by integral numbers, so that a fractional answer to a question as to the population of a place is an "impossibility." There are however other, and far more numerous, kinds of magnitudes, which are termed scalar magnitudes, for which the process of diminution need not be regarded as terminating with annihilation. Thus a man may not only expend all his money, but by con- tinuing the process may run into debt. A man may go to a INTRODUCTION 9 certain distance north of a place, but as he diminishes that distance by returning south he need not stop where he started, but may proceed indefinitely southward from his starting point. Instances such as these, which may be indefinitely multiplied, at once suggest an interpretation for a b, where b > a, and thus lead to the general conception of a negative quantity as intel- ligible per se, and denoting the exact opposite of some given quantity, and of the suitability of the signs -f and to express this opposition of affection in the literal symbols to which they are prefixed. It should be observed that this extended meaning of the signs + and is not exclusive of, but may be used in conjunction with, their original meanings as signs of operation : for in fact sub- traction may be regarded as reversing the affection and adding the resulting quantity. So also multiplication by a negative factor becomes multipli- cation by the number with reversal of the affection, and a negative index is found by interpretation, subject to the condition of the laws of indices remaining true, to denote the reciprocal of the same factor with a positive index. 8. The fundamental laws are now completely intelligible without any restriction as to " impossible " cases, but the develop- ment of algebra on this basis leads to new combinations of sym- bols, which present fresh difficulties of interpretation. It is found that every algebraical expression is reducible to the form a + b\/ 1, where a and b are completely intelligible, or, as they are termed, " possible " or "real" quantities, but V 1 is un- explained and is a purely symbolical representation of something, neither a positive nor a negative quantity, whose square is equal to 1, and so is termed an "impossible" or "imaginary" quan- tity. No further extension of the fundamental notion of number or ratio is possible, and thus ordinary algebra remains burthened with the purely symbolical form a + JV 1> termed a complex number, of which no intelligible interpretation can be given. 10 INTRODUCTION 9. It has been our object in the preceding discussion to bring clearly into view the following fundamental facts and principles of algebra as a symbolical calculus. 1. That the results of algebra are not directly the consequences of the meanings assigned to the literal symbols, but only in- directly through the fundamental laws of combination deducible from those meanings. 2. That for a given system * of algebra the fundamental laws of combination must be true without any exceptions, so that any extension of meaning of the literal symbols, or interpretation of previously unexplained combinations, must be made subject to the condition that these laws remain unchanged, save by the removal of some previous restriction on their generality. 3. That the " Principle of the Permanence of Eqxiivalent Forms " is a consequence of the universality of the symbolical laws : and that therefore it follows from the fundamental principles of algebra that (for instance) the Binomial Theorem having been proved for the case where the index is a positive integer, it requires no farther proof to show that it is also true for any index, f positive or negative, integral or fractional. Euler's proof is only an application of this principle to a special case, in which the consequence is, or is supposed to be, more obvious to the elementary student than in the general form above stated. 4. That formulae in which some or all of the letters denote complex numbers, must be regarded as true symbolically, but uninterpretable, or rather waiting for a suitable interpretation. 5. That the "impossible" or "imaginary" quantities of algebra are only "impossible" or "imaginary" relatively to the meanings of the literal symbols which have been assumed, and * There are systems of algebra, in which the fundamental laws are not identical with those of ordinary algebra. For instance, in the algebra of quaternions, the commutative law (ab ba) is not true. t Subject however to certain limitations, when the number of terms of the series is infinite, as shown in C, viii. INTRODUCTION 11 that as by the extension of those meanings several " impos- sibilities " have disappeared, so by some further extension the remaining impossibility, the complex number, may cease to be uninterpretable. 10. The magnitudes considered in ordinary algebra are either purely quantitative, that is, such as can be represented by number or ratio only ; or scalar, that is, such as can be represented by number or ratio with the addition of the signs + and to indicate two opposite senses, in either of which the magnitude may be conceived to be indefinitely extended. The type and representative of all scalar magnitudes is length measured forwards or backwards along an indefinitely extended straight line. Ordinary algebra may thus be said to be linear or one-dimensional, and the appearance of the symbol v 1, which being neither a positive nor negative quantity is as a numerical or scalar magnitude uninterpretable, suggests the possibility that its interpretation may be found in planar or two-dimensional space. That this is the fact together with the consequences resulting therefrom we proceed to show in the chapters which follow. CHAPTER I VECTORS AND VECTOR-AGGREGATION 1. Definition of vector. A. vector is a magnitude which is completely defined by the two elements length or distance, and direction. Thus a finite straight line drawn from one point to another is a vector. 2. Vectors in the same line. Scalars. If A, B are two points, the straight line drawn from A to B, or the step which carries (vehit) a point from A to B along the straight line AB, is the vector AB, which we shall denote thus, AB. The opposite step from B to A, since it exactly neutralizes the step from A to B, may be expressed as AB, so that BA = IB, orZ5 + 2O = 0. A B C A C B If C be a third point in the same straight line, it follows that AB + J3C = AC, or AB + BC + GA = 0. The student should verify this for any relative positions of A, B, C along the line. Thus if vectors are considered which lie along a given line, they differ only in the elements of quantity and sense, and so are completely represented by the positive and negative quantities of ordinary algebra. These positive and negative quantities, the "possible" or "real" quantities of ordinary algebra, are termed " scalar 8." VECTORS AND VECTOR-AGGREGATION 13 3. Equality of vectors. Lines which are parallel to a given straight line, and therefore to one another, are said to have the same direction. Vectors which in this sense have the same direction, may be measured one towards one end, and the other towards the other end of the line. For this opposition of direction along the same line we shall use the word sense, and the opposite senses will be indicated by the signs + and , as with scalars. Two vectors may then be said to be equal, when they have the same quantity (or length) and the same sense in the same direction. Thus if A BCD is a parallelogram, AB and DC are equal vectors, or the step from A to B is equal to that from D to C, but AB and CD, though equal in other respects, differ in sign, so that AB = -'CD = DC, and AB + CD = 0, or AB-DC = 0. We shall see that, when we come to define direction by means of an angle, the sense of a vector may be merged in its direction by properly measuring the angle, but at present it will be con- venient to retain the distinction. 4. Aggregation of vectors. The fundamental law for the aggregation (under which term we include both addition and subtraction) of vectors is the following. C 14 VECTORS AND VECTOR-AGGREGATION If A, B, C be any three points, the aggregate or sum of the vectors A B and BC, or the steps from A to B and from to C is the vector AC or the step from A to C, so that using the sign + to denote this aggregation or addition, From this, since BC = -CB, ABCB = AC, therefore AB-CB+CB=AC + CB, or AB = AC + CB, in accordance with the definition above. Also TB+SC+CA=ZC+CA = (), or the vector-sum of the sides of a triangle taken in order in making a circuit of the perimeter is zero. Hence we infer that the signs + and connecting vectors may be used as in ordinary algebra, and to denote either signs of affection, or sense, or signs of operation. 5. Associative and commutative laws of terms. Let AC be a diagonal of the parallelogram A BCD, Then from the foregoing AC = AB + BC, and also AC = AD + DC, but by the definition of a vector, AD = BC, and DC = AB, therefore AC BC + AB, so that A B + BC = BC + AB. Also since BC = - CB, AB-OB=-C + AB. VECTORS AND VECTOR-AGGREGATION 15 Hence it appears that in the aggregation of vectors the order of the terms is indifferent, or the commutative law is true. 91 J Further, let AB, BC, CD be any three vectors, and let VC = m) = AB, AC' = BC, C 7 D' = CDo& in the figure : then (AB + BC) + CD = AC + CD = AD, and also AB + (BC + CD) = AB + BD = AD. and TB-'CB+'CD=JB-(CB-^D}=TB-DB=AD. Hence in the aggregation of vectors, any two terms may be grouped into a single term with due regard to the law of signs, or the associative law is true. From these laws it follows, denoting the three vectors by a, (3, y respectively, that a + /3 + y = /8 + y + a = y + a + /3 = &c. as may easily be verified from the figure, as thus : It may be left to the student to show that BD' = ft a + y ; and, completing the parallelepiped of which AB, BC, CD are edges, to express the vectors joining any two points of the figure in terms of a, /?, y. 16 VECTORS AND VECTOR- AGGREGATION 6. Application to geometry. From the simple laws of aggregation of vectors, a large body of geometrical truths may be readily deduced. With the view of familiarizing the student with the notion of vectors and their aggregation, the remainder of this chapter will be devoted to the investigation by this method of some well-known geometrical theorems. It will not be necessary here, as it will be found to be in the following chapters, to limit ourselves to vectors in one plane. 7. Vector of any point expressed in terms of given vectors. Let the vector OA be denoted by a, and let P be any point in the line OA indefinitely extended in both senses, and let x denote the ratio OP : OA, x being reckoned positive or negative according as OP has the same sense with OA or the contrary ; then OP = xa. Let P be any point in the plane of two given vectors OA, OB, and let PM, PN be drawn from P parallel respectively to OB, OA to meet OA, OB in M, N, and let OA, OB be denoted respectively by a, ft : then if OM=xa and ON=yf$, OP = OM+ MP = OM+ ON= xa + yft. Hence the position of any point P in a plane may be defined by the vector drawn to it from the fixed point 0, expressed in terms of two given vectors OA, OB (or a, yS) by means of the two VECTORS AND VECTOR-AGGREGATION 17 ratios or scalars x, y reckoned positive or negative according as OM, 0^ have or have not the same senses respectively as OA, OB. It is plain that for a given point P there is one and only one value of x and y, and that for given values of x, y there is one and only one point : hence, if xa + yft = x'a + y'ft, it follows that x = x and y = y . Lastly, let OA, OB, OC be three vectors not in the same plane denoted respectively by a, ft, y, and let P be any point in space : let PN, parallel to OC, meet the plane OAB in N, and NM, parallel to OB, meet OA in M, and let OM=xa, MN=yf$, NP = zy, then OP - OM+ MN+ JVP=xa Thus the position of any point P in space may be denned by the vector drawn to it from the fixed point 0, expressed in terms of three given vectors (not co-planar) OA, OB, OC (or a, ft, y), by means of the three ratios or scalars x, y, z, reckoned positive or negative according as OM, MN, NP have or have not the same senses respectively as OA, OB, OC. For a given point P there is one and only one value of each of the scalars x, y, z, and for given values of x, y, z there is one and only one point P : hence, if xa + yft + zy = x'a + y'ft + z'y (a, ft, y not being co-planar), x = x', y = y', z z'. 8. Illustrative examples. In the examples which follow the vectors of the points A, B, C to an origin or OA, OB, OC are denoted by a, ft, y respectively. c 18 VECTORS AND VECTOR-AGGREGATION If the origin be changed to 0', and 00' be denoted by to, and O'A, O'B, O'G by a', (3', y', it is evident that a = + y'. 1. To find the vector of the middle point (M) of AB. In this case MB = AM= -MA, or MA + MB = ; B C O A but OM+ MA = 6>Tand OM + MB = OB, therefore by addition, 20M+MA + MB = and 20M=OA whence OM= If the parallelogram OACB be completed, OC a -\-(3, therefore OM=\OC, so that the middle point of AB is also the middle point of OC. In other words, the diagonals of a parallelogram bisect one another. 2. To find the vector of any point P in the line AB in terms of the vectors of A and B. Let the point P be defined as that which divides the line AB in the ratio y : x, so that AP : PB = y : x or x . AP = y . PB, whence AP = -^- . ABandPB= . AB. x+y VECTORS AND VECTOR- AGGREGATION 19 Then OP=OA+AP = OA + y - AB = *, O'A = a', O'B = P', O'P = p, then a = a> + a', /?=> + )8', p = a> + p', xa + yP _ x (a> + a') + y (ft) + /?') _ a?a' + yft' x + y x + y xa whence p = P. If P be between A and B, x, y are both positive : if it be in AB produced, y is positive and x negative : and if it be in BA produced, x is positive and y negative. In all cases x + y is positive, as we are at liberty to assume, since only the ratio and not the absolute values of x, y are involved. Also the expression for p depends, not on the absolute values of x and y, but only on their ratio, so that it really involves only one arbitrary element* corresponding to the fact that there is only one arbitrary element in the position of P, its distance (positive or negative) from either A or B. OC I/ y. If we take - = m, then = 1 m. and the expression x + y x + y for p becomes p = ma + (1 - m)p, involving explicitly only the single arbitrary element m. By reason of the symmetry, however, the form involving both x and y will be found generally the more elegant and convenient. 8. If a, P, y be the vectors of three points A, B, C connected by the relation xa + yp + zy = 0, the condition that they are collinear is x + y + z = 0. xa + yP xa + yP For then z (x + ?/), and y = = z x + y c 3 20 VECTORS AND VECTOR-AGGREGATION e. If PM, PN be drawn to OA, OS, parallel to OB, OA respec- tively, it follows, from the expression for OP, that OM = a, x + y B, whence ON : OA = x : x + y = BP : BA, and ON : OB = y : x + y = AP : AB. This is a vector proof of Euclid, vi. 2. 3. The mean point of any number of points. Let the vectors to the origin of the n points A, B, C ... be a, 8, y . . ., then - 2 ', or the mean of these vectors, is the vector of a point, which depends only on the relative positions of the points themselves, and not on the origin. It is termed the mean point of the n points. To prove this, we have only to observe that if another origin 0' be chosen so that 00' = w, and a, (3', y' ... be the vectors to the new origin, a = w + a, (3 = o> + /3' . . ., and therefore n n so that the same point is determined by a + B + y + . . , , a' - ; - ' - and by = w + 4. The mean point of three points, A, B, C. Let /*, be the vector of the mean point M : then /u, = , o which may be expressed in either of the forms 1 + 2 ' " ~TT2 ' 1+3 VECTORS AND VECTOR-AGGREGATION 21 A' Now - --- , , are the vectors of the middle points 2i A 2 A', B', C' of BC, CA, AB respectively, and the forms above show, by the form for p in 2, that M is a point in each of the lines A A', BB', CO', dividing them in the ratio 2:1. Hence the "medians" (AA', BB', CC'} of a triangle (ABC} meet in a point, the "centroid " (or mean point of the points A, B, C}. i ( + y y +a a+ P I Again, since p. = ' j ' + * -- \- ~- \ , M is also the ( 2 i 2 j mean point of A', B', C' : or, the centroid of ABC is also that of A'B'C' . Also the vector of the middle point of A A' is ii a + M, which is equal to M jp- + - -r-^jj the vector of ~ \ 2 / \ ^ 2 / the middle point of B'C' . Hence AA', B'C' bisect one another, and AB' AC' is a parallelogram. It may be left to the student to prove that AM + M+CM=Q, and A'M+'M+C'M=0, also that WC' = \CB, C 7 !' = \AC, Til' = \BA. 5. The mean point of four points, A, B, C, D. Reasoning as in the foregoing case, , 3 P+y+* a f /3 + y + 8 3 since . - - 4 1+3 -if is a point on the line joining A to the mean point (A') of B, C, D, and dividing it in the ratio of 3:1. Hence the lines joining 22 VECTORS AND VECTOR-AGGREGATION the vertices of a tetrahedron to the centroids of the opposite faces all pass through the mean point of the vertices. Also y + 2 and therefore M is the middle point of the line joining the middle points of the opposite edges AB, CD. Hence the lines joining the middle points of each pair of opposite edges of a tetrahedron all pass through the mean point of its vertices and are bisected in that point. It may also easily be proved that, if A', ', C', D' be the centroids of the faces opposite to A, B, C, D respectively, (1) M is also the mean point of A', B', C', D'. (2) The middle point of AA' is the mean point of A, B', C', D'. (3) The mean point of B', G', D' divides A A in the ratio of 2 : 1. (4) B'c 1 = ICB, VECTORS AND VECTOR-AGGREGATION 23 (5) The middle points of A'B', C'D' divide the line joining the middle points of AB, CD in the ratios of 2 : 1 and 1:2. 6. Of the n points A, B, C, D . . . suppose that x coincide with A, y with B, z with C . . . . : then x + y + z+ . . . =n, and the vector of the mean point becomes x+y+z+. . . In such a case it will be convenient to call x, y, z . . . the weights of the points A, B, C . . . respectively. Thus if there are two points A, B having weights x, y respec- l'(L ~\~ 7//J tively, the vector of their mean point is --- , which, as we have seen, denotes a point P in the line AB, dividing it so that AP : PB = y : x. Thus any point P in the line through A and B may be regarded as the mean point of A, B by assigning suitable weights (positive or negative) to these points. 7. Let the three points A, B, C have weights x, y, z respec- tively : then p being the vector of their mean point P, zy p = x + y + z a. The expression for p may be written in the form yfi + zy z) y + z Hence, if L be the point in BC which divides BC so that BL : LG = z : y, P is a point in the plane A BC and in the line AL, which divides AL so that AP : PL = y + z : x. In like manner it may be shown that, if CM : MA = x : z, P is a point in BM which divides it so that BP : PM = z + x : y, and that if AN : NB = y : x, P is a point in CN which divides it so that CP :~PN=x + y : z. 2\ VECTORS AND VECTOR-AGGREGATION It is plain that by assigning suitable values (positive or negative) to x, y, z, P may be any point in the plane ABC, so that the vector of any point in the plane ABC may be expressed xa + yB + zy in the form - x + y + z Hence, if xa + yB + zy + u8 = be the relation connecting the vectors of four points A, B, C, D or determining either in terms of the other three, the condition that they should be co-planar is It should be observed * that, assuming x + y + z to be positive, if x, y, z are all positive, P lies within the triangle ABC : and that * On a spherical surface three great circles divide the surface into ciyld regions, which may be distinguished by signs as above for a plane with - - VECTORS AND VECTOR-AGGREGATION 25 the other possible permutations of signs correspond to the six regions outside the triangle, as shown in the annexed diagram, in which the signs are written in the order of those of x, y, z respectively. /?. The ratio compounded of the ratios EL : LC, CM : MA, AN : NB is xyz : xyz or 1. Hence the theorem, known as Ceva's Theorem, that if lines drawn from A, B, C through a point P meet the opposite sides (or these produced) of the triangle ABC in L, M, ^respectively, BL CM AN = . - . = 1 : or,* as it is usually expressed in geometrical LC MA NB treatises, BL . CM . AN = LC . MA . NB. y. Let MN meet BG in L'. Then, since the vectors of M and zy + xa. , Xa + yB N are and respectively, the vector or any point in z+x x+y p zy + xa q xa + yfi J/A may be expressed as . - h - , and there p + q z + x p + q must be some value of p/q for which this point is L'. Now a will p q disappear from this expression if + = 0, z+x x+y so that _ - z + x x + y y z in which case the expression reduces to 1 1 vB zy - - - which is the vector of a point in BC and therefore the vector of L'. Hence L' is such that L'B : L'C = z : y and L'M : L'N = q : p= --(x + y) : (z + x). Similarly if NL, LM meet CA, AB for the eighth region. If the surface becomes a plane by the radius becoming infinite, one of these regions goes oft' to infinity. * The multiplication of vectors not having been yet considered, we are not entitled to write BL . CM . AN = ~LC . ~MA . ~NB. 2t> VECTORS AND VECTOR-AGGREGATION respectively in M', N', the vector of M' is and that of N' Xa ~yP and cjf . M ~ A = - x:Zf AN' : NB= -y : x. x-y Hence if A.', p!, v denote the vectors of L', M', N' , zyXa. , xa- y($ z x x y and therefore (y z) A.' + (s x) p + (x - y) v 0, whence (by 2 8) L', M', N' are collinear, since (y z) + (zx) + (x - y) = 0. 8. The line L', M', N' has been derived from the point P, for which the ratios x : y : z have definite values. It is plain that the steps might be reversed, so that starting from any line L', M', N' definite values of the ratios x : y : z would be obtained, and thence definite points L, M, N and a definite point P, whose vector is - -. Hence between the point P and the line x + y + z L', M 1 , N' there is this correspondence that, given either, the other is determined with reference to the triangle ABC. The ratio compounded of the ratios BL':L'C, CM': M'A, AN': N'B ^BL' CM' AN' is zxy : yzx or 1, so that -= . = = 1. Hence the L'C M'A ' N'B theorem, known as the Theorem of Menelaus, that if a transversal meet the sides BC, CA, AB of the triangle ABC in the points L', M'. N' respectively, - . =- . = = 1 and the converse. L'C M'A N'B 8. Let P, Q be two points in the line AB, such that, OA, OB being denoted by a, ft, x+y x +y Then E . -"and ^ . , PB x QB x VECTORS AND VECTOR-AGGREGATION 27 and therefore the (an/tar monic or) cross ratio of the range A, P, Q, B, . APiAQ . x'y which is ^= . ~=, is BPl BQ Let another transversal meet the pencil (APQB) in A', P', Q', B', then if OA r =m.OA, OB' = n. OB, OF=p. OP, x y x y a x y xa + yB m n m n' m n JT= OP =p. = p . - . : --=. . OP , x + y x +y x y x + y m n since the last factor is the vector of some point in A'B' and there- fore of P', whence 0P' or w p Similarly QQ' x+y , . xa + yfi or q = and 00 = -. OQ x ^l * 2/ Hence so that AP y in , AQ y m =-. and -= . B'P' n x ^ BPl 1BQ 28 EXAMPLES or tJie cross ratio of the range A'PQ'E is the same as that of the range APQB. Hence the same ratio is said to be also the cross ratio of the pencil (APQB}. CC 3C If = - - or x'y + xy = 0, the cross ratio = 1 , and P, Q y y divide AE externally and internally in the same ratio, or harmonically. Hence if P, Q be points which divide AE harmonically, _ \jj. - x+y x y The development of the geometrical consequences of these results is beyond the scope of the present work. EXAMPLES. 1. Verify geometrically the formulae a + fi a-_l_ a + P a~P _ 2 2 = "' ~~2 "2 " ** 2. The line passing through the middle points of AE, AC bisects every line drawn from A to EG, and therefore is parallel to EC. Also verify Euc. VI. 2 generally by vectors. 3. If the vectors OA, OB, OG are denoted by a, /?, y respec- tively, and / is the centre of the inscribed circle of the triangle ABC, 01 = - -. where a, b, c are the lengths of EC, CA, a + o + c AE respectively. 4. If I v I 2 , I 3 are the centres of the escribed circles, express OI V 0/ 2 , 0/ 3 in terms of a, /3, y, and prove that s.OI=(s-a).Of l + (s-b)Ol. 2 + (s-c) 0/ 3 , where s - . EXAMPLES 29 5. If the inscribed circle touches EC, CA, AB in D, E, F respectively, find the vectors OD, OE, OF and prove that Also that AD, BE, GF meet in a point K, such that where r v r.,, r% are the respective radii of the escribed circles. 6. Find the points where the line through / and M (the mean point of A, B, C) meets BC, CA, AB respectively, and shew that if b + c 2a, IM is parallel to BC. 1. If a straight line meets BC, CA, AB in D, E, ^respectively, the middle points of AD, BE, CF axe collinear. 8. If N is a point in the median AM of the triangle ABC such that MN : NA : : I : m, and if BN meets CA in P, then CP : PA : : 21 : m. If also ON meets AB in Q, PQ is parallel to BC. OCCL -l~ ?vo *J si'v 9. If - - - is the vector of a point in the plane ABC, x + y + z and that of its isogonal conjugate with respect to the triangle yy z * A w ,- ,1 ABC is , then -^ - -. x+y+z a*- b L c 2 10. If A, B, C, D be four points having equal weights, and A', B, C', D' are the mean points of the groups BCD, CDA, DAB, ABC, prove that (a) The mean point of A', B', C', D' coincides with that of A, B, C, D. ((3) The mean point of B 1 , C', D' divides AA' in the ratio of 2 : 1. 30 APPENDIX ON VECTOR MAGNITUDES (y) B'C' is parallel to BC, C'A' to CA, &c. (8) The mean point of A, B', C', D' is the middle point of A A'. 11. Investigate corresponding properties, when A, B, C, D have unequal weights. 12. If a, ft, y are three edges of a parallelepiped meeting in one of its corners, express the diagonals in terms of a, /?, y : and thence shew that they meet and are bisected in one point. Also that the lines joining the middle points of opposite edges are bisected in the same point. 13. If A', B', C' are points in OA, OB, OC respectively, and BC, CA, AB meet B'C', C'A, A'B' respectively in D, E, F, prove that D, E, F are collinear. Also if BC', BC intersect in D', CA, C'A in E', AB', A'B in F, the planes ABC, A' B'C', D'E'F' all intersect in the line DEF. Also AD', BE', CF' meet in a point ((?), and aD', bE', cF' in a point (g), and 0, G, g are collinear. 14. If a system of points, of equal or unequal (but fixed) weights, is displaced in any manner, the displacement of its mean point is the vector mean of the displacements of its several points. APPENDIX ON VECTOR MAGNITUDES. 1. IT has been observed in the Introduction (p. 8) that Magnitudes, which may be made the subject of mathematical treatment, are of various kinds. Some, such as Population, Mass, Energy, Heat, &c., of which the negative is inconceivable, are purely quantitative. Others, such as length measured on a given line in one sense (forwards) of whicli the negative is length measured in the opposite sense (backwards), time after a given epoch of which the negative is time before the same, debit of which the negative is credit, gain of which the negative is loss, and so on, are scalar, or can be represented by the positive and negative quantities of Ordinary Algebra. There are others, however, which, like the vectors we have considered in this chapter, require for their com- plete definition the two elements, quantity and direction. Such magnitudes can be completely represented by vector line?, which have the same directions and senses as the magnitudes themselves, and lengths (or APPENDIX ON VECTOR MAGNITUDES 31 tensors) proportional to their quantities. These may be properly termed Vector Magnitudes, if it can be shown that they can be combined accord- ing to the laws of Vector Aggregation. Again there are other Magnitudes, which, besides the elements of a Vector Magnitude, require for their complete specification additional elements as, for instance, forces acting on an extended body or system, (in which case besides quantity and direction, the particular line of action must be given), (Clifford's rotors) : or screws, wrenches, &c., which have pitch as well as quantity, direction, and position (Clifford's motors). Corresponding to these several kinds of magnitudes are the funda- mental subdivisions of Algebra in its most general sense. The treatment of purely quantitative Magnitudes is contained in Arithmetical Algebra, in which a b, where b is greater than a, is an "impossible" quantity. Scalar Magnitudes are the subject of ordinary or Scalar Algebra, in which J 1, or such expressions as a -j- b V- 1, are " impossible " quantities. To Vector Magnitudes, if they are limited to one plane, corresponds Ordinary Algebra without any " impossible " quantities (the " Double Algebra " of De Morgan), as it is the purpose of the present work to show. For the treatment of Vector Magnitudes in all their generality, and other magnitudes transcending these in complexity, special Algebras are appropriate, as the Quaternions of Hamilton, the Triple Algebra of De Morgan, Biquaternions of Clifford, &c. We proceed to show that the Elementary Magnitudes of Dynamics, Velocity, Acceleration, Force (acting at a given point) are truly Vector Mai/mtudes. For this purpose it is only necessary to prove the following proposition : 2. The rate of change of a Vector Magnitude is itself a Vector Magnitude. Let OA represent at a given instant a Vector Magnitude, which after t units of time becomes changed to OB, then by the law of Vector Aggregation AB represents the total change of the Vector Magnitude in the interval of I units of time. If the change takes place uniformly, that is, by equal steps in the same constant direction for any equal parts of the interval, the extremity of the vector always lies on the line AB, and moves along that line with uniform speed, so that the change that would 32 APPENDIX ON VECTOR MAGNITUDES be effected in the \\nit of time is completely represented by the vector AC, if AC '= j x AB. In other words the vector ~AC, or -j- (where t is, it will be observed, a pure number), represents the uniform rate of change of the vector magnitude at any instant during the interval. If the change takes place not uniformly, but so that either the direc- tion of the steps into which the total change can be subdivided or the length of the steps for a given fraction of the interval or both are variable, the vector represents the mean rate of change during the t interval. The extremity of the vector then moves with variable speed along a curve line from A to B, and as the time t is taken smaller and smaller, so that the point B gets nearer and nearer to A, the mean speed ^ approaches continually a certain limit v, which we may call the t speed at A of the extremity of the vector, and the direction of AB approaches a limiting direction, that of the tangent to the curve at A. Hence if AC be taken along the tangent AT at A, equal to v, the limit of A n T- when t is diminished without limit, AC will represent the rate of v change of the vector at the instant when it coincides with OA. Besides thus estimating the complete rate of change of a vector as a single vector having a quantitative, and also a directive element, it may be estimated as the sum of two vectors having given directions and there- fore each specified by a single (quantitative) element. Confining our- selves to vectors in one plane, let Ox, Oy be two given directions in that plane ; then, if AH, BK are drawn parallel to Oy to meet Ox in H, K, and AL parallel to Ox to meet BK in L, the vectors AL, LB, represent the changes of the vector magnitude in the directions Ox, Oy respectively, which, combined by the vector law of addition, AB = AL + LB, are equivalent to the total change. Then, taking the case of uniform rate of change, the complete rate of change 1 1 1 . r . AB is made up of the component rates r.AL, 7.Z5 in the directions t it Ox, Oy, respectively by the vector law of addition ~ ,AR = \-~A~L + - t .LB. APPENDIX ON VECTOR MAGNITUDES 33 1 1 . Hence if AC = j.A, and AD = j-AL, since AC = AD + DC, it follows that DC = ^-LB, and it may be stated that the rates of change H K represented by AD, DC, combine by the vector law of addition into the rate represented by AC. In the case of variable rate of change, the same conclusion would be arrived at by taking the limits of r.AB, &c., in- t stead of the actual rates or mean rates, as explained above. Hence we conclude that Rates of change of Vector Magnitudes are themselves also Vector Magnitudes. 3. Velocity, a Vector Magnitude. The velocity of a point is the rate of change of a vector defining its position at any instant. Hence, if OA, OB, in the preceding section are vectors defining the position of a moving point, which passes from A to E in t units of time, AC represents the velocity of the point when at A, and of its two elements the length AC represents the tensor element, termed the speed of the point, while its direction is the direction of the velocity. Also, since AC is the vector sum of AD and DC, the lengths AD, DC, represent two speeds along given direction?, which may be re- garded as component velocities, which, combined according to the vector law of addition, give the complete or resultant velocity. This is the first elementary proposition of Kinematics, known as the " Triangle (or Parallelogram) of Velocities." 4. Acceleration, a Vector Magnitude. The Acceleration of a point is the rate of change of its velocity. Hence, if OA, OB are taken to represent the velocities of a point at a given instant and after an interval of t units of time, respectively, by similar reasoning to that regarding velocity, the vector ^/6'represents the acceleration of the point at the first instant, and of its two elements the length A C represents the tensor element, while its direction is the direction of the acceleration. Also it is plain in like manner, that the total acceleration, represented by AC, may be regarded as made up 34 APPENDIX ON VECTOR MAGNITUDES by vector addition of component accelerations, represented by AD, DC, which are the quickening s, or rates of change of speed, in two given directions. Hence the proposition of Kinematics known as "The Triangle (or Parallelogram) of Accelerations." 5. Force, a Vector Magnitude. It follows from the second of Newton's " Laws (or Axioms) of Motion," that forces have the directions of the accelerations which they produce in a particle, and are measured as to quantity by the acceleration multiplied by a certain purely quantitative constant belonging to the particle on which the force acts, which is regarded as the measure of the quantity of matter the particle contains, and is termed its Mass. Hence, the product of a vector by a pure number being a vector having the same direction, what has been seen above to be true of velocities and accelerations must be true also of Forces. Hence the fundamental proposition of (non-linear) Dynamics, known as the " Triangle (or Parallelogram) of Forces." 6. Illustrations. To develop the consequences of these fundamental propositions is the proper object of special treatises on Kinematics and Dynamics. We may, however, by way of illustration, call attention to the dynamical reading of one or two of the geometrical consequences established in the present Chapter. Thus, if OA, OB, 00, represent three forces acting on a particle, and is the mean point of A, , C, the resultant of these forces is re- presented by 3.0G, and in like manner if G were the mean point of n such points, the resultant would be represented by n.OG. Hence, supposing to coincide with G, the forces represented by GA, GB, . . . form a system in equilibrium, when applied to the same particle. Again, if moves off to an infinite distance, the vectors all become parallel, and it follows that the resultant of a system of equal parallel forces acting through the points A, B, C . . . acts in a parallel direction through G, their mean point, which is then known as the centre of the system of equal parallel forces. This may easily be generalized to meet the case of unequal parallel forces, and so to that of the weights of the several particles of a system, when the mean point becomes their centre of gravity. Again, if the points of a system are regarded as material points having masses proportional to the (geometrical) weights of the points ( 8-6), it may be inferred (as in Ex. 14 above) that the velocity of the centre of mass of a material system is the vector-mean of the velocities of the several particles or bodies of the system, so that this becomes the proper measure of the motion of translation of the system. The same is easily seen to be true of the accelerations. Then the equal and opposite forces APPENDIX ON VECTOR MAGNITUDES 35 of a stress between two particles producing equal ami opposite mass- accelerations (or rates of change of momentum), thes% ]&st may be left out of account in determining the acceleration of tle> centre of mass and that acceleration is the vector-mean of the accelerations of the' several particles due to the external forces only. It may therefore be stated that the motion of translation of a material system, or the motion, of its centre of mass, is the same as that of the whole mass supposed! collected at its centre of mass and acted on by forces equal to and in the same direction as the actual external forces. CHAPTER II MULTIPLICATION OF COPLANAR VECTORS 1. IT has been shown in the preceding chapter that the laws of aggregation of vectors are in accordance with the fundamental laws of ordinary algebra, and this without any limitation as to the vectors themselves. In the present chapter we proceed to show that for vectors limited to lie in the same plane, or coplanar vectors, a natural extension of the term " multiplication " leads to laws also identical with those of ordinary algebra. The consideration of the pro- ducts and quotients of vectors generally in space of three dimensions leads to the higher algebra of Sir W. Hamilton's Qua- ternions or De Morgan's Triple Algebra, the Ausdehnungs Lehre of Grassmann, &c., with laws divergent from those of ordinary algebra, with which alone the present treatise is concerned. 2. Relation of vectors in a given plane to the prime vector. Let a definite vector (01) be taken, whose length is the unit of length, and let this be called the prime vector and denoted by 1. Then any other vector in the plane may be defined by the ratio of its length to the length of the prime vector and its inclination to the same. This ratio, a purely quantitative (non-scalar or signless) quantity, we shall call (after Hamilton) the tensor of the vector. The inclination is measured by the angle between the prime vector 01 and a vector drawn from equal (c. i, 3) to the given vector : this angle being the angle turned through by MULTIPLICATION OF COPLANAR VECTORS 37 a line revolving about from coincidence with 01, till it coincides with the vector so drawn, and reckoned positive or negative according as the rotation is one or other of the two senses of rotation in the plane, known as clockwise, or the sense of rotation of the hands of a clock looked at from the front, and counter- clockwise, the contrary sense. In this treatise the latter (the counter-clockwise) sense is regarded as positive, and the former as negative, and all the figures are drawn accordingly. For the present too the right angle, which we shall denote by the symbol L , will be taken as the unit angle, so that if u is the measure of an angle, 90 u is its measure in degrees : or U L = 90w. Hence a vector, whose tensor is a and inclination a L , may be formed from the prime vector 01 by taking a length along 01 which shall have to 01 the ratio of a to 1 and turning it about through the angle a L . This vector we shall represent by OA and denote it for the present by (a, a). Thus if CD be a line equal and parallel to OA and in the same sense, CD = OA = (a, a) : and, if OA' be equal to OA in AO pro- _ . __ |_ _ _ duced, the inclination of OA' being a + 2 , DC = OA' = (a, a + 2) so that, since (a, a) + (a, a + 2) = 0, or (a, a + 2) = (a, a) : or, the addition of two right angles to the inclination of a vector is equivalent to the reversal of its algebraical sign. Also, since the rotation of OA through four right angles or any multiple 38 MULTIPLICATION OF COPLANAR VECTORS (positive or negative) of four right angles brings it back to its original position ; if n be any positive or negative integer, (a, a + 4w) (a, a) and (a, a + in + 2) = (a, a). It should be observed that, having regard to the algebraical sign of an angle as denoting the sense in which it is measured, if AOB denote the angle which OB makes with OA, BOA will denote the angle in the contrary sense which OA makes with OB, and consequently that , BOA=-AOB, or AO + OA = 0. Also that, whatever be the relative positions of OA, OB, OC, = AOB + BOC, or 3. Vector multiplication. Multiplication has been defined (Introd., 5) as the operation of " doing with the multiplicand what must be don,e with the imit to form the multiplier." Now to form the vector (a, a) the prime vector (1,0) has its tensor altered in the ratio a to 1, and its direction changed by rotation through the angle a : hence if the vector (b, ft) is multiplied by (a, a), its tensor b must be changed to ab, and its direction changed by rotating through the angle a L , so that its new inclination is (a + (3). We may therefore write, (a, a) x (b, ft) = (ab, a + ft), MULTIPLICATION OF COPLANAR VECTORS 39 regarding the factor before the sign of multiplication as the multiplier. To give geometrical expression to the foregoing, let OA, OB be the vectors (a, a) and (b, ft) respectively, then if on OB a triangle OBC be described equiangular to 01 A, the side OB corresponding to 01 and the angle BOG having t/te same sense as the angle 10 A, OC will be the vector (ab, a + ft), or the product of OB multiplied by OA. For, since the triangles are similar, OC : OB = OA : 01, or OC : b = a : 1, or OC^ab, and the angle IOC ' = 10 > + BOO We have therefore OAx~OB = ~OC. 4. Commutative laiv of multiplication, We have seen that, regarding (a, a) as the multiplier, (a, )x(6, ) = (a6, In like manner, regarding (6, ft) as the multiplier and (a, a) as the multiplicand, (b, j8)x(a, a) = (ba, ft + a), but since the commutative law holds both for sums and products of numerical and scalar magnitudes, so that ba = ab and ft + a = a + ft, (ba, ft + a) = (ab, a + ft), and therefore (a, a) x (b, ft) = (b, ft) x (a, a). Hence the commutative law is true for the multiplication of vectors. The same may readily be proved geometrically, For IB and AC being joined, since OC \ OB :: OA : 01, therefore alternately OC : OA :: OB : 01, and the angle A OC = A OB + BOC = AOB + 10 A = 10 B, 40 MULTIPLICATION OF COPLANAR VECTORS whence the triangle AOC is similar to the triangle 10 B : so that 00 will be obtained either by describing on OB a triangle similar to 10 A, or by describing on OA a triangle similar to IOB. Hence OAxOB=0*OA. 5. Associative law of multiplication. Since (a, a) x (b, ft) = (ab, a + ft) {(a, a) x (b, ft)} x (c, y) = (a&, a + ) x (c, y) = (a&.c, a + ft + y). Also since (b, ft) x (c, y) = (be, ft + y), (a, a) x {(6, 0) x (c, y)' = (a, a) x (be, ft + y) = (a.&C, a + + y). But, since the associative law holds both for sums and products of numerical and scalar magnitudes so that a b x c = a x be and (ab.c, a + ft + y) = (a.6e, a + /3 + y), and therefore {(a, a) x (b, ft)} x (c, y) - (a, a) x {(b, ft) x (c, y)}. Hence the associative law is true for the multiplication of vectors. The geometrical verification of the foregoing, the details of which we leave as an exercise for the student, may be obtained from the annexed figure, in which OA, OB, OC being the three MULTIPLICATION OF COPLANAR VECTORS 41 vectors, OD is the product of 02, 'OB, and 6~D' that of OB, 00, and OE is the product at once of OD and OC, that is, of (02. 01?) . 0(7 and of 02 and 077, that is, of 04. (0. The student is also recommended to construct his own figure for three vectors drawn at random ; or to scale for particular vectors, as (1, | L ), Q, 3 L ), (2, - 1 L ). 6. The distributive law. Let the vectors (a, a), (b, ft) be represented by BC, GA respec- tively, then their vector-sum (c, y) will be represented by BA. Also let any other vector (in, p.) be represented by OM. Produce OA, OB, OC to A', B', C' respectively so that OA' = m.OA, OB' = m.OB, OC' = m.OC; then it is known from elementary geometry that B'C', C A', A'B are respectively parallel to BC, CA, AB and in the same ratio, each to each, as OA' to OA, and hence WC' = mBC, CTA' = mCA, B 7 !' = mBA. Now B'A' here fore m.BA =* m. BC + mCA , 42 MULTIPLICATION OF COPLANAR VECTORS or, since BA = BC + CA, m (BC + CA) = mBC + mCA, that is, m |(a, a) + (b, ft)\ m. (a, a) + m, (b, ft). Now let the figure OA' B'C' be turned about 0, so that OA' turns through the angle //, or IOM, let its new position be OA"JJ"C", then all the other lines of the figure have turned through the same angle /m. Hence (m,fji).C, G"A" = (m, p.).CA, and J3"A" but jrA?> = jpC 7 ' + (rJ7> t therefore (m, p.). BA = (m, p).BC + (m, p.).UA, or (m, /*) (BC + CA) = (m, /t) j5(7 + (m, jt).(Li, that is, (m, /*) {(a, a) + (b, ft)} = (m, /*) . (a, a) + (m, p) . (b, ft), which proves the truth of the distributive law for vectors, with the interpretation of multiplication given above. 7. Reciprocal of a vector. Two vectors are reciprocal, when their product = 1 , so that (a, a) and (b, ft) are reciprocal, if (a, a) x (6, ft) = 1 or (1, 0). But (a, a) x (6, ft) = (ab, a + ft), 1 and therefore ab = 1, a + ft = 0, or b = -, ft = a, 1 or the reciprocal of the vector (a, a) is the vector (-, a). ct Geometrically, on 01 describe the triangle 70-4' equiangular to MULTIPLICATION OF COFLANAR VECTORS 43 .407, the side 01 of the former corresponding to OA of the latter, and the angle 10 A' being equal to, and having the same sense as, AOI or the opposite sense to 70.4, then OA' is the reciprocal of OA. For if OA = (a, a), since OA' : 01 : : 01 : OA or OA' : 1 : : 1 : a, OA' = - and 10 A' = -a, whence OA' = (-, -a). a a 8. Vector-division. Division is the inverse of multiplication, so that if (a, a) x (&,) = (aft, a (ab, a + 0) - (a, a) = (6, 0) or ( y Let a6 = c, and a + ft = y, then 6 = - and /3 = y a, so that (a, a) v a' / Hence the quotient of one vector divided by another is the vector, whose tensor is the quotient of the tensors, and inclination the difference of the inclinations or, more definitely, the sum of the inclination of the dividend and the inclination, reversed in sense, of the divisor. Also, since ( -, y - a) = (c, y) x (-, a) \a \a / / \ ~ Vr> It * 7 \ (a, a) (a, a) or the quotient of one vector divided by another is the product of the former and the reciprocal of the latter. 9. Scalar powers of a vector. The interpretation of (a, a) m , where m is a scalar, follows im- mediately from the foregoing articles. 1. If m is positive and integral, (a, a) m = (a, a) . (a, a) . (a, a) . . . . to m factors = (a m , ma) by the laws of multiplication. 44 MULTIPLICATION OF COPLANAR VECTORS fry 2. If m is positive and fractional ( = - suppose), / - P \ q then, since ( ai, -a J = (a?, pa) = (a, a)P, IP ,|a) = (a,a). 3. If m is negative, since the rules of indices are to hold good universally, (a, a)~ m = ^- = = ( - , - ma) = (a~ m , - via). (a, a) m (a m , ma) x a m Hence universally for any scalar value of m, (a, a) m = (a m , ma), assuming by the law of continuity or by limits that the result is true, when m is incommensurable with unity, no less than when it is commensurable. 10. Multiplicity of values of scalar powers of a vector. Since (a, a) = (a, a + 4/&), where n is any positive or negative integer, (a, a) m = (a m , ma + 4mn). If i is an integer, 4ww is a multiple of 4, and (o w , ma + mri) = (a m , ma), so that (a, a) m has only one value. 7) If m is fractional and = -, p being prime to q, the form 9 nf\ ^-TtT) ma + kmn or - a H will define q different directions, as n <1 3 receives any q consecutive values in the series ... -3, -2, -1, 0,1,2,3..., since no two values of n differing by an integer less than q can make the two values of differ by an integer. Also it will MULTIPLICATIOX OF COPLAXAR VECTORS 45 define no more than q different directions, since the q + 1 , q + 2 th , -) of the equiangular spiral, whose pole is 0, passing through the points / and A v and they may be said to be in vectorial geometrical progression, with the common ratio OA l or (a, a). If a = 1, the tensors are all equal to 1, the equiangular spiral becomes the unit circle, and the powers are the radii or unit vectors"^/, ~01 V 0/ 2 . . . and OI- v ~OI- 2 . . . 2. fractional powers of the vector (1, 0) or unity. 4n Since (1, 0) = (1, 4n), (1, 0)J = ^1, \ which has the three MULTIPLICATION OF COPLANAR VECTORS 47 distinct values (1, 0), (1, - ), (l, - J for n = 0, 1, 2, repeated in the same order for the successive triads (3, 4, 5), (6, 7, 8), &c. These values are represented in the figure by the three radii of the unit circle drawn to the points /, I v 7 2 which divide the circumference into three equal arcs. Hence unity is said to have three cube roots, of which 01 or unity itself is alone scalar, the other two 0/j, OI 2 are unit vectors or vectors, whose tensors are unity. It should be observed that the successive integral powers of 0/j are 01,,, 01, 0/j repeated in the same order continually, while those of OI 2 are OI V 01, 01. 2 repeated in the same order continually. All the integral powers of 01 however are 01 itself. Since (a, a) = ox(l, a), (a, 0) = Va . (1, 0)4 = Va (l, ) > o / denoting by IJa the arithmetical cube root of the number a. Hence any positive scalar has three cube roots, which are the products of the three cube roots of 1 multiplied by \/a, or the vectors drawn from to the points where 01, OI V OI 2 cut the circle, centre 0, and radius = V- 3. Let us examine (1, 1 L )^ = (1, 4n + l)*=(l, * In the unit circle OJ, perpendicular to 01, is the vector (1, 1), and 61', opposite to OI, is the vector (1, 2) or the scalar, 1. 48 MULTIPLICATION OF COPLANAR VECTORS Then if, starting from 01' the circumference is divided into five equal parts by the points 1', 5, 3, 1, 4, the vectors 01, 02 or 01', 03, 04:, 05 are the five values of (1, to the values 0, 1, 2, 3, 4. or ( - l)i, corresponding 12. Vector as the product of tensor and versor. Let the unit vector (1, 1), that is, the vector OJ whose tensor is 1 and inclination to 01, reckoned in the positive sense, is a right angle, be denoted by *. Then t a = (1, l) a = (1, a) and ai a a,. (1, a) = (a, a). Hence the vector (a, a), whose tensor is a and inclination a L can now be expressed as the product ai a , a mode of expression which may henceforth naturally supersede the arbitrary one which we have hitherto adopted. Of its two factors the first a, as an operator by multiplication on another vector, alters the length or tensor of that vector in the ratio of a to MULTIPLICATION OF COPLAKAR VECTORS 49 1, while the other, i a , turns the direction of the vector through the angle a, and hence may be called (after Sir "W. Hamilton) its versor. Thus every vector is the product of the two elements, its tensor and its versor, the former being a purely quantitative element and the latter a power of the vector i. It should be observed that, although a is the measure of the inclination in right angles, the versor i* is a factor or operator depending on the angle only and not on its particular measure with reference to any assumed unit. In fact if Jc denote another l vector whose inclination is m<- so that k = i m , then i = km and a i- = km, so that the same versor may be expressed as i a or ZP, if R = and R is the measure of the inclination, when the unit m contains m right angles. For example, if the unit be 1 and & 90 = i, i a and & 90a are different expressions for the same versor. 13. Interpretation of V 1 . Since i = (l, 1), therefore i 2 = (l, 2)= 1, whence *=/!, so that V 1, the " impossible " quantity of ordinary algebra, now appears as the unit vector J making with the prime vector 01 a right angle in the positive sense, and in multiplication \/ 1 or i is the versor which turns the vector multiplied through a right angle in the positive sense. Also 01', OJ' being the unit vectors opposite in sense to 01, OJ respectively, i z = - 1 - 07', t 8 = - V^l = OJ' t 4 = + 1 = 07. Hence n being any positive or negative integer, 14. Uniqueness of i. It is also important to observe that, although the vector OJ = (1, 4n + 1) for all integral values of n, i must be regarded as denoting the unique vector (1,1) and no other of the series (1, 5), (1, 9) . . . which coincide with it. Otherwise i a would in R 50 MULTIPLICATION OF COPLANAR VECTORS general be a function having many distinct values, whereas re- garding it as the versor of a vector having a given value of a, it must be unique : in fact it must be the prime value (that is, the value when n = 0) of i( 4n + 1 ) a , the generalized a th power of i, which determines different vectors for different values of n, except in the special case where no. is an integer. 15. Vector expressed by a complex number. If m, n are scalars, m + n\f 1 or m + ni is termed a complex number. Let OM=m. 01, then OM = mi = m, and if MA makes with the prime vector 01 a right angle in the positive sense and is equal to n.OI, MA = ni. But OA = OM+MA, therefore OA = or the complex number m + ni represents the vector OA. Hence, M being the foot of the perpendicular let fall from A on the direction of the prime vector, if we call OM the project and MA the traject of the vector. OA, we have the vector expressed as the sum of two elements, its project and traject, corresponding to the so-called possible and impossible parts of the complex number. Thus the same vector may be expressed either as the product of its tensor and versor, or as the sum of its project and traject, and it is desirable to determine the relation between them. Take MA' MA, then OA' = m ni. Also since OA' = OA, and MOA'=- MO A, if OA = ai a , OA ' = ai ~ a , so that ai - = m + ni and ai~ a = m ni, MULTIPLICATION OF COPLANAR VECTORS 51 but ai a ai ~ a = a z and (m + ni) (m - ni) = m 2 n 2 ^ 2 m 2 + ?i 2 , therefore a 2 = m 2 + w 2 or a = Vfl that is, the square of the tensor = the sum of the squares of the project and traject, and the tensor is the modulus of the complex number, which represents the vector. Hence also the versor i a = , a complex number whose V m 2 + n z modulus is 1. (Observe that we have here incidentally proved Euc. i. 47, since we have shown that OA 2 = OM Z + MA 2 , OMA being a right angle.) 16. It has now been shown that, with the meanings of al- gebraical symbols and their combinations which have been found appropriate for co-planar vectors, all the fundamental laws of ordinary algebra hold good : hence all the results of ordinary algebra may be read with these new meanings, and, reserving the question of the meaning to be assigned to vector-indices for a later chapter, all such results are definitely interpretable so that the "impossibles," or " imaginaries " of ordinary algebra are now fully explained. In the succeeding chapters we shall apply our results to the establishment of Trigonometry on a purely algebraical basis. It will be instructive to conclude the present chapter with the geometrical verification of a few formulae as illustrations of their extended meanings. It will be necessary to select only the simplest cases, because even for simple formulae the diagrams become complicated, but the student is recommended to test others for himself by actual geometrical drawing. 17. Illustrative examples. (1) (l+i)(l-t) = l-t* = l + l=2. In the figure 07= 1, 7j= i, IJ'= - i, therefore OJ = 1 + i, ~OJ' = l-i, E 2 52 MULTIPLICATION OF COPLAKA.R VECTORS and if JK be drawn pei-pendicular to OJ to meet Of produced in K, the triangle OJK is similar to OIJ', so that OK OJ. OJ', but OK is also obviously 2 01, therefore OJ. OJ' = 2, in accordance with the formula. (2) Let (a + U} (a - bi) = a 2 - VP = a 2 07=1, ~OA=a, ~AB = U, AJ?= -bi, then OB = Make the angle OBC equal to OH?, then BOA being equal to B'OA, the triangle BOC is similar to 10 B' and therefore OC=OB.~OB' = (a + M) (a-bi). Draw CH perpendicular to OB and HK to (L4. Then OK : OA : : OH : OB : : OA : 01, therefore OK .01=0 A* or OK=a?. MULTIPLICATION OF COPLANAR VECTORS Also KC:KH::KH:OK::AB:OA::b:a. KC : OK : : KH 2 : OK 2 : : b* : a 2 , and OK=a z , .-. KC = b*. Hence (3) 53 Let OM=a MA = bi and therefore OA=a + bi, then, if OAB be described similar to 01 A, = OA z = (a + bi)' 2 . Draw BHC perpendicular to OA and meeting 01 in C, and draw BN HK perpendicular to 01. K M C I Then since OH, perpendicular to JJC, bisects the angle BOG, BH= HC and therefore NK= KC and BN= 2HK. Also KH : MA : : VII : OA : : OM : 01 (since 01 A, OAB are similar). KH.OI=OM.MA or KH=ab, whence Further but and and ON=OK-KN=OK-KC, OK : OM : : OH : OA : : OM : 01. OK.OI=OM* or OK=a\ CK:KH::KH:KO::MA:OM::b:a, KH=ab, /. CK = P. 54 MULTIPLICATION OF COPLANAR VECTORS Hence so that OB Hence we have verified geometrically the formula _ Let OA, OB, 00 be three vectors denoted by a, ft, y respectively, then BC = y-(3 CA = a-y AB = ft-a, and OA . BC = ya-ap, OB. CA = ap-/3y, OC . AJB = Py-ya whence OA .~BC+OB .~CA+~OC .~AB = 0, or ~OB.^KC=OC .TB+~OA.~EC. Hence in any quadrilateral the vector-product of the two diagonals is equal to the sum of the vector-products of the two pairs of opposite sides, due regard being paid to the senses in which the vectors are taken. This result reduces, when 0, A, B,C are concyclic, to Ptolemy's Theorem (Simson's Euc. vi. D). For the inclination to the prime vector of a vector product is the sum of the inclinations of the factors, and therefore double the inclination of the bisector of the angle between them. If then the bisectors of the angles between the diagonals OB, AC and those of the angles between the two pairs of opposite sides OA, BC and OC, AB have the same inclinations or are parallel, MULTIPLICATION OF COPLANAR VECTORS 55 the versors of the three vectors are equal, and the equation holds between the tensors OB . AC = 00 . AB + OA . EG. But these bisectors are parallel, only when 0, A, B, C are con- cyclic, as the student may readily prove from elementary geometry, therefore Ptolemy's Theorem is proved true. In the general case, on OC describe OGD similar to ACS, Tjn ~Trj then _ or OC . 'AB = OD . AC, OD AB and since OB . AC = 00 .^LB+OA .BC, therefore C>B.1C = OD .1C + OA . BC. Hence (OB - OD) AC = OA.BC, or DB.AC = OA.BC, from which it follows that the triangle DBC is similar to OAC. But when two vectors are equal, their tensors and versors are separately equal, so that, taking the tensors, OC.AB = AC. OD and OA.BC-^AC . BD, whence OC , AB + OA.BC = AC (OD + BD) and .-. > AC . OB, so that the sum of the rectangles contained by the two pairs of opposite sides is greater than the rectangle contained by the diagonals, unless 0, D, B are collinear, which is the case if 0, A, B, C are concyclic a well-known extension of Ptolemy's Theorem. Also from the equality of the versors, if OA, CB meet in E and AB, OC in F, and if OD, BD meet AC in H, K respectively, the bisectors of the angles at H, F are parallel, and likewise the bisectors of those at K, E. When 0, A, B,C are concyclic, H, K merge in G the intersection of the diagonals, so that the bisectors of the angles E, F, G are parallel, as assumed above. 56 EXAMPLES EXAMPLES 1. Draw the following vectors: a. (l, 1) ft. (2, 1) y . (2,-i) 8. 3t* 77. 4 + 3* 0. V3-t, find the projects and trajects of a, /3, y, 8, e and the tensors and versors of , 77, 0. 2. Find the reciprocals of the vectors in the previous question, and find the products and quotients of a and /3, of a and y, of 8 and e, of and 77 and of 8 and : and verify the results geome- trically. 3. If OP, Op are reciprocal vectors, P, /, />, /' are concyclic, and OP, Op are equally inclined on opposite sides of 01. 4. The product OR of the vectors OP, OQ may be constructed thus : Let OP, OQ meet the unit-circle in P , Q respectively and on OP, OQ take Oq, Op respectively equal to OQ, OP ; take the arc IR equal to the sum of IP , IQ : then R will be the second point of intersection of OR with either of the circles R Qp or R.qP. 5. If Q P' parallel to QP meets OP in P' and arc IR arc Q^PQ, and if OR meets the circle, centre 0, and rad. OP', in R, then OR = OP/OQ. 6. In the figure of Euc. iv. 10, taking the base as the prime vector, express all the other vectors in the figure both by tensor and versor, and by project and traject. 7. Verify the following geometrically (a, /?, being any two vectors). (1) (-) = . (2) (l+)* = 2i. (3) \^ = i. . If (4) (l+a) 2 =l+2a + a 2 . (5) EXAMPLES 57 8. Exhibit in a figure all the values of 1^, $, ( 1)*, and express them as the sum of project and traject. 9. Prove by vectors that, if on AB, BC, AC similar triangles AFB, BDC, AEC are similarly described, BDEF is a parallel- ogram. 10. In the figure of Ex. 4, 17, express the vectors AF, CE in terms of a, /?, y, using (if required) the relation la. + m(3 + ny = as the condition for expressing any one in terms of the other two. 11. Prove that in the same figure OB . BC . CO + OA . AB . B0 = OA . AC . CO + AB . BC . CA. 12. Prove that the vector-product of the two diagonals of a parallelogram is equal to the difference of the squares of two adjacent sides. Hence shew that if on the sides AB, AD of the parallelogram A BCD triangles ABE, ADF are described outside the parallelogram and similar respectively to ACB, ACD, then EF is equal and parallel to BD, and verify by geometry. 13. If the locus of a point P is a circle, and Op is the re- ciprocal of the vector OP, the locus of p is also a circle. Prove this, and thence shew that, if M = - - ,, where z is the vector cz + d from to a point which describes a circle, u is the vector of a point Q which describes another circle. CHAPTER III TRIGONOMETRICAL RATIOS 1. TRIGONOMETRY, a science deriving its name from that special branch of it in which it originated, namely, the relation between the sides and angles of a triangle, is in a more general sense the science of periodic magnitude : or rather of certain periodic magni- tudes, that is, magnitudes whose values recur in regular sequence over and over again, as the quantity on which they depend con- tinually increases in a given sense. This quantity is generally the angle, through which a line, revolving (in a plane) about a fixed point in it, turns in passing from a given fixed position to any other, and the magnitudes are the trigonometrical functions or ratios of the angle. The properties of these ratios are usually deduced, and (we believe) rightly so for the beginner, from geometrical construc- tions ; but they admit also of a complete discussion on the basis of algebra, regarded, as has been done in the preceding chapters, as the Algebra of Coplanar Vectors. This discussion will occupy the present and several succeeding chapters, in which though nothing will be assumed from the geometrical treatment (with which the reader will probably be more or less familiar) and every relation will be established ab initio on an algebraical basis, the more elementary relations will be succinctly treated and the geometrical developments less fully discussed than in the ordinary treatises. TRIGONOMETRICAL RATIOS 59 2. .Definitions of tJie trigonometrical ratios. Let XOX', YOT' be two rectangular axes, that is, indefinitely extended lines at right angles to one another, and on OX let 01 be taken as the prime vector or 1, and on OF let OJ be taken a unit vector (that is, one whose tensor is 1) and denoted by i. Let OP be any other vector, whose tensor OP = r . 01 or r, and whose inclination is u*, u being measured positively (counter- clockwise) or negatively (clockwise) from 01 to OP as explained in the last chapter, so that OP = ri H . From P let PM be drawn perpendicular to XOX' (as shown in the figure for four different positions of OP in the four quadrants into which XOX', YOY' divide the plane, and which are called first, second, third and fourth in the order of positive revolution, that between OX and OY being the first), and let the project OM=x . 01 or x and the traject MP = y . OJ or yi, x, y being scalars, positive or negative according as they have the same or contrary senses to Of, OJ respectively : then, since the vector is * In this work we shall usually denote the measure of an angle in right angles by the letters u, v, w, &c. , reserving the Greek letters a, ... 6, <(>... for circular measure. We shall not introduce this latter measure until we find the need for it. 60 TRIGONOMETRICAL RATIOS the sum of its project and traject or OP = OM+ MP, ri u = x + yi, x it . or *" = - + -i. r r ), and the reverse by an arrowhead to the left (< ), it will readily be seen that the limiting values of the several ratios for the angles 0, 1 L , 2 L , 3 L and their signs, and the sense of their changes between these, are those expressed in the following table : 64 TRIGONOMETRICAL RATIOS a> .11 + & rj I s . . i 5 o rt i +t t +t TRIGONOMETRICAL RATIOS 65 The same results may be more clearly exhibited by means of curves or graphs, in which the abscissa ON, taken along X'OX or the axis of abscissae, is proportional to the angle, and the ordinate J\'Q perpendicular to the same is proportional to the ratio O N FIG. 1. O J- FIG. to be represented and the graph is the locus of Q. The graphs for one complete period from to 4 L are shown in the diagrams. The graphs for sine and cosine are the continuous and dotted curves respectively in fig. 1 , in which ON = u, NQ = sin u XQ' = cos u. TRIGONOMETRICAL RATIOS Those for tangent and cotangent are the continuous and dotted curves respectively in fig. 2, in which ON= n, NQ = tan, 1?Q' = FIG. 3. FIG. 4. Those for secant and cosecant are the continuous and dotted lines respectively in fig. 3, in which ONu, ^^ = sec u, 2\ r $' = cosec u. TRIGONOMETRICAL RATIOS 67 Another mode of exhibiting the results is by means of the polar graphs, that is, by laying off lengths on the revolving vector OP equal to the values of the several ratios in each particular position of OP, and finding the loci of the points thus determined, the lengths being laid off from in the sense of PO produced back- wards when the values are negative (fig. 4). Then the graphs for the cosine and sine respectively are the circles on 01, OJ as diameters, these being traversed twice for one complete period. The graphs for the secant and cosecant are the tangents to the unit circle at I and J respectively, also traversed twice for a complete period. The graph for the tangent is a curve of four branches, all touching 10 1' at 0, to which the tangents to the unit-circle at /, /' are asymptotes : that for the cotangent a like curve touching JOJ' at and having the tangent at J, J' for its asymptotes. (These curves are of the fourth order in rectan- gular coordinates.) The arrowheads indicate the sense in which the describing point is moving along the curve, as the vector revolves in the positive sense, and the numbers the quadrant to which the branch of the curve corresponds. 7. Ratios for the reversed angle, supplement, complement, &c. a. The addition of any multiple (positive or negative) of four light angles leaves the ratios unchanged, since the position of vector, project, and traject are thereby unchanged, so that the ratios are periodic functions of the angle, the period, or interval of recurrence of the same values, being an angle of four right angles. Algebraically i* n =l, and /. i 4w + = i, whence cos (4w + u) + i sin (4w + u) cos u + i sin u, so that cos (4n + u) = cos u, sin (4w + 11) = sin u, tan (4n + u) = tan u. (3. The reversed angle or u. ?'-" = cos ( u) + isin( u), and also i~* = cos u i si F 2 68 TRIGONOMETRICAL RATIOS equating projects .and trajects separately, cos ( u) = cos u, sin (-*)= sin w, and therefore tan ( - u) = tan t. y. The angles 2 + u, and 2 tt or the supplement of *. Since t 8 = -1, i^+u _ cos w 4 s i n W) f2-M _ _ cos u + { s i n M> that is cos (2 + u) + i sin (2 + w) = cos w - i sin u, cos (2 w) + i sin (2 u) cos w + i sin it ; whence cos (2 + u) = cos u cos (2 u) = cos w sin (2 + u) = sin u sin (2 u) = sin w. tan (2 + w) = tan M tan (2 u) = tan u. 8. The angles 1 +u, and 1 u or the complement of . Since a 1 + lt = i.*, t 1 - = *.*-, cos (1 + w) + i sin (1 + w) = i (cos u + i sin w) = sin u + icosu, cos (! *) + i sin (1 tt) = i (cos M i sin it) = sin u + i cos ; whence cos (1 +u) = sin u cos (I ?/) = sin u sin (!+*)= cos u sin (1 w) = cos u tan (!+*)= cot u tan (! *)= cot u. e. The foregoing formulae enable us to find the values of the trigonometrical ratios of any angle in terms of those of an angle in the first quadrant or an acute angle. 1QKQL 2 L x 1 \ L Thus since 1050 = -^- = 11 - = (12- J) , the ratios for yu o \ o ] L 1050 are the same as for or 30, so that, by /3, o TRIGONOMETRICAL RATIOS 69 sin (1050) - -sin 30, cos (1050) = cos 30, tan (1050)= -tan 30. 555L 1 L 1 L Soako -555== ._ 6f -.(-S+i-g) , 1 L therefore, by a and y, sin ( 555) = sin = sin 15, 1 L cos (555)= cos = cos 15. 8. Values of the trigonometrical ratios for certain angles. There are a few angles, for which the values of the sine, cosine, etc., can be easily and directly obtained from geometry. It will be instructive to see how the same can be obtained algebraically. a. 45 or }, . Let x + yi be a unit- vector (so that x^ + y"!), such that (x -f- yi) 2 = i, or generally (a; + yi) 2 = t" 4n +', where n is an integer, then x + yi = i 2n +* = cos (2n + ) + i sin (2n + ), whence x has two values, cos i and cos (2 + ^) or cos i, and y also two values, sin \ and sin (2 + i) or sin , corresponding to the two values of 2n + ^ within the first complete period. These values may be found thus : (x + yi) 2 = i, and.', x- y- + 2xyi = i, whence x 2 y z = and 2xy = 1 . The scalar (or real) solutions of these equations are 1 1 11 so that cos - = sin - = = 2 2 V2 l u 9 L ft. 30 and 60 3 or - and ^ . o o ) 3 = 1, or generalizing (x + yi) 3 = i 4n , . ^ ^ 6 = cos + % sin ' o o 70 TRIGONOMETRICAL RATIOS .-. x = cos or cos i or cos f y = sin or sin .'. or sin ;: = 1 or sin ^ or cos f , =0 or cos ^ or .sin 4|, 4-T& for the three values of within the first period. 8 Now since (x + yi) 3 = 1, expanding x 3 + Sx^yi Sxy 2 - y s i = 1 , or x (cc 2 3y 2 ) + iy (3x 2 y 2 ) = 1 , whence a? ( 2 3?/ 2 ) = 1 and y (3o3 2 y 2 ) = 0. The value y = from the second equation gives + I from the first. Combining 3x z y z with x- + y z = 1 , 2 _ i ,.2 ;i x 4' y v whence by the first equation /I 9\ 1 V3 aj - = ~ y= b ' 1 L 2 L 1 1 L 2 L V3" Hence sin = cos = - and cos = sin = 332 o o L or A. similar proceeding gives, from (x + yz) 4 = i, 1 L 3 L 1 sm - = cos -= 22-^2, .3 1 - 1 L 1 and sin = cos = - V2 + y'2 1 U OL QU AL 8. g-,^, , g,orl8, 36, 54, 72. , ' Let (a; + yi) J = 1 = z 4 ", . '. a; + yi = cos --- + i sm -- , whence period, whence, talcing the five values of in the first complete TRIGONOMETRICAL RATIOS 71 4 8 12 16 x = cos U or cos - or cos - or cos or cos 555 5 4224 1 or cos - or cos - or cos - or cos - o 5 55 ,4,8 12 16 y = sm or sm - or sin - or sm or sin 5555 .4 2 2 4 = or sin - or + sin - or sin- or sin -. 5 o 5 5 so that there are three different (scalar) values of x and five of y. To determine these, we have expanding and arranging as a com- plex number, x : > - 1 Oz 3 / + So;/ + i(5x*y - 1 2 ?/ 3 + y 5 ) = 1 , or x(x* - 1 Oafy 2 + 5y 4 ) = 1 and y(5x* - 1 Oa; 2 ?/ 2 + y 4 ) = 0. .From the second equation (omitting the factor y, which gives which, by the condition y~ = 1 a; 2 , reduces to 16a; 4 - 12*8+ 1=0, 3 \/5 5 + \/5~ whence x z = - and .-. y*=- -- Hence it would follow that x , giving four values of x, but by reason of the first of the two equations, these are reduced' 1V5 V5-1 V5+1 to two, namely x = -- - or x = - or -- , -- .For y however there are four values, and y = _ ' _ r-J^L . 4 4 L 1 L \/5-l 2 L 3 L \/5 + l Hence cos sm = - , cos = sin = - . , 554 554 72 TRIGONOMETRICAL RATIOS The student may with advantage compare the foregoing with the geometrical investigations (Johnson's Trigonometry, pp. 60 67). It is plain that by a similar process equations for determining the values of the ratios for other fractions of a right angle might be obtained, but the solution of these would generally depend on the solution of an equation of a degree higher than the second. The equation having definite numerical coefiicients, any particular root might be obtained by Horner's process to any degree of approximation required, and thus with sufficient labour the value of the ratios for the given fraction of a right angle. [It will be a good exercise for the student to show that 1 L 1 cos = - V^p where z l is the greatest root of the equation / 2i s 3 -7;s 2 + 14s- 7 = 0, which, by Horner's method, will be found to be 3'801937737 so that cosy =cos 12 51' 25f = '9749279124 ] 1 L 5 L Ex. Find sine and cosine of and 6 6 9. Expressions for all the angles ivhich have the same sine, cosine, or taiigent. Let W O L denote a particular angle, and U L any other angle such that : a. cos u = cos UQ. Then, from the vector expressions for the cosine, jtt + I -u jii,, _f_ i~ _ J4)i + o 4. {- (4it+ K,,^ since i in 1, n being an integer. This is satisfied by iu i*n+v or ^u._ j-(4n + ,,) and by these only. TRIGONOMETRICAL RATIOS 73 Hence u = kn + U Q or 4w u , which, since n may be either positive or negative, are both included in the formula U = ft. sin u = sin w . Then i" i- " = i^-ru,, _ {-(4+,,) the two solutions of which are i = i** + o and i u = - i~ 4 " -. o = p - 4 - ()) whence w = 4?& + w or (4w 2) - u , or u = 2>iu () , + or according as n is even or odd, or, as it may be conveniently expressed, n being any integer u = 2n + ( I) n w . y. tan u = tan w fl . '_{- io_i-n Then = . which reduces to i-" =i-"o, or i-("-"o)= 1 =i 4 " therefore w w = 2w, or u = 2n + M O . 1 0. Inverse functions. Letyic denote a function of x, that is, a quantity depending on x, and therefore changing when x changes, and connected with x by seme definite law, which may be either algebraical, wheny.r denotes some ordinary algebraical expression, or transcendental, when its dependence on x cannot be expressed algebraically in finite terms (of scalars) as i x , cos x and the other trigonometrical ratios, log x and other higher transcendents. If fx be substituted for x infx, we have f(fx) which is naturally denoted byf^x, and then f(f~ 2 x) is denoted by f z x, and so on. In accordance with this notation fx denotes x itself, for then f(fx)=fa' and so f~ l x denotes a quantity such that f(f~ l x) =fx = x, or f~ l x is that function of x, of which the function./ produces x. Any operation, which reverses a given operation on a given operand (a;) so as to reproduce the operand, is termed the 74 TRIGONOMETRICAL RATIOS inverse of the operation (e.g. division is the inverse of multipli- cation, the square root that of the square, fec., &c.) : hence f~ l x is the function of x inverse iofx. It is obvious that in the same w&yf~ z x,f~ z x. . . are inverse iof-x,f*x ... so that/ 2 (/- 2 a;) = a;, f 3 (f~ 3 x) = x ---- : also that y =fx, if x f~ l y. The general law is (the same as that of indices)/ TO (/ n ) =f m + n (x) for positive and negative integer values of m and n. It is to be observed that, though the analogy of this notation with indices of any algebraical quantity is patent, it is only an analogy, since./ by itself does not denote any quantity and has no existence independent of the subject of operation x. [Ex. 1** . JL-, then/% - jiy/* - ji-. ./, - -- /-IT- X f- l r- ~' J ~ - ~1+W To apply this to the trigonometrical functions. To be consistent cos 2 ;e, cos 3 # .... ought to denote cos (cos x), cos (cos (cos a;)) .... respectively, but in practice, as these functions do not often occur, they are used, where no confusion is likely to arise, to denote the powers of cos a;, which are more correctly expressed as (cos a;) 2 , (cos a;) 3 . . . Negative powers of the trigonometrical ratios are always expressed correctly as (cos a;)" 1 , (cosx)~ 2 , tfcc., &c., and the notation cos" 1 ;*;, tan -1 a;, ... is reserved to denote the functions inverse to cos x, tan x, *kc., such that cos (cos -1 x) = x, tan (tan ^ic) = #, &c., and therefore cos" 1 a; denotes, and is read as, the angle whose cosine is x, tan -1 x as the angle whose tangent is x, &c. This notation is very convenient, when the angle is denned by the value of one of its trigonometrical functions instead of its measure in right angles, or degrees, and it is therefore desirable to recognize the relations established in the foregoing sections in the shape which they assume under this notation. Thus if cos u = x, = cos~ 1 x, and since sin u = Jl cos 2 ** = \/i x*, u = sin- * Vl Vl-cos'w VI x" . ,._t aT1 -i also tanw= = , M tan COSM TRIGONOMETRICAL RATIOS 11 T 1 also, since sec u= = - , u sec- 1 - , cos u x a: and similarly == coBec"" 1 -7== --, and u = VI a: 2 so that, _ x cos- 1 ^ sin- 1 A/1 x' J = tan" 1 -- = cot -1 /^ -=sec x VI ar 1 = cosec Vl a; 2 It is in like manner easily proved that 1 VI +x 2 x The student should write down for himself the corresponding equivalents for sin." 1 x, cot" 1 x, sec -1 x, cosec" 1 x. It is most important to observe that, even if fx is a single- valued function or one which has only one value for .a given value of x, the in verse./- 1 a; may be, and generally is, a multiple-valued function. Hence though f(f ~ l x) = x, we can only say that some one of the values oif~ l (fx) = x and cannot assert without limita- tion that y- 1 (fx)=f(f~ l x) , or generally that, while /"* (f n x) =/'"+" (*)>/"(/"'*) =/'"(/"*) Thus, as we have seen in the last article, if cos u x, w (or cos" 1 a;) is one or other of the infinite number of angles included in the formula 4?i u , where is a definite one of those angles. It will be convenient to take the angle u as that one, which when defined by sin, tan, cot, or cosec, lies between 1 L and + 1 L , or that which when defined by cos or sec lies between and 2 L , and term it the prime angle of the series, and to dis- tinguish it when written with the inverse form by making the initial letter of the function a capital. The formulae of the last article then become, 76 TRIGONOMETRICAL RATIOS cos" 1 x = 4w Cos" 1 x or cos" 1 (COSM O ) = 4n sin~ l x = 2n + ( l) n Sin- 1 aj or sin" 1 (sinw ) = tan~ a a; = 2rt + Tan~ 1 a; or tan ~ l (tan w ) = the angle being measured in right angles and n being zero or any integer. If an angle is denned by the value of one only of its trigono- metrical ratios, even if limited to lie within an interval compris- ing a whole period, it will always have two distinct values, and so will generally other functions of that angle, and thus the ambiguity of sign necessarily attaching to the square roots in the formulae above is accounted for. Thus sin (cos~ l x) = sin (4w Cos" l x) = sin ( Cos ~ l x) = \/l x 2 cos (sin- 1 ;*;) = cos |2>i + ( l)"Sin- 1 a;} = ( - l) n cos {( !)" Sin" 1 ;*;} = ( 1)" cos (Sin -!#) = l-a;2 tan (sin- 1 x) = tan (2n + ( I)' 1 Sin- 1 a;) = tan (Sin- 1 x) x ~ Vl-a 2 ' 11. Examples. By the formulae in this chapter expressions involving the trigonometrical ratios of one angle may be reduced, identities connecting them verified, and trigonometrical equations involving one angle only in certain cases solved. The desired results may usually be most simply obtained by the scalar formulae of 2 and 3, of which abundant examples may be found in works on Elementary Trigonometry ; but they may also be obtained by the versor expressions in 4 and algebraical reduction, and for the sake of familiarising the student with the nature of the work involved, a few examples will be now worked out by this method. (1) Prove that cosec u sin u cos u . cot u, cosec u sin u = ^ - -rr % ~ t 1* TRIGONOMETRICAL RATIOS 77 cos u . cot w. sin w 1 cos u (2) Prove that sin u ^"' ^ 1 + cos u sin u (u u\ / u 42 i~z) \i2 + i-z) t"2 i 2 I+COSM (2 + i tt + i-)i (u _\2 iz-i 2) i" + {-*-% 2 cos u 2 COSM 1 1 COSM sn u smu (3) cos 4 u + sin 4 M = 1 2 sin 2 u cos 2 w c) // '9u '_9\9 i CM 6 /=23\(* -* 2tt ) 2 + 8 } = ! + !({ + *~ tt ) 2 (*"- *~") 2 = 1 + ^ ( 2 cos u ) 2 ( 2 * sin M ) = 12 sin 2 M cos 2 u. (4) Solve the equation tan u + sec w = 2. Putting the vector expressions for tan u and sec u, ^ + V-V B = 2, (1 - 2i) P' + 2t . t* - (1 + 2t) = 0, 2?) -iV^~iTr whence , u = -2t -*2 . so that t"= i or t"= z 1 2t 1 ( 4) 78 EXAMPLES FOR PRACTICE Therefore either cos u and sin u= 1 , from which u = (4ra + 3) L or n x 360 + 270, or cos u=i, sin u : ^, corresponding to a vector in the first quad- rant, so that u = n + Cos - l * s = in + Sin- * ;.) . EXAMPLES FOR PRACTICK. 1. Verify the following, by using the vector forms for the ratios : (1) sec 2 u + cosec 2 u = sec 2 u . cosec 2 u. (2) (sin u + cos w) 2 = 1 + sin 2w. 1 + cos u 1 cos u (3) v .. = 4 cot u cosec u. 1 COSM 1 +COS U sec 2 it (4) sec2w=~ s 2 sec 2 u /K\ * 2 l-cos2w (5) tan j w = = jr- 1 + cos 2w . 3w w (6) sin M + sin 2w = 2 sm cos - ,. sin 3u - sin u (7) - -^- = cot 2w. cos w cos o*t 2. Solve the equations : (1) sinw + cosM = l. (2) cot u + cosec u = 3. (3) sin2w = tanw. (4) cos2w + 3sin = 2. (5) tan w + cot u = 4. (6) 1 + cos u + cos 2u + cos 3it = 0. 3. Prove, and verify geometrically : u (1) l+i = 2cos|. P 1 1 *'' (2) r^i = 2" cosec 2 * , .. *" - I* U - V (4) . - = i tan v ' u v CHAPTER IV DE MOIVRE'S THEOREM, AND GENERAL TRIGONO- METRICAL FORMULA 1. IN the last chapter from the two forms of a unit-vector or the versor of a vector, i* and cos u + i sin u, all the relations between the trigonometrical functions of an angle have been deduced as immediate consequences, and other results, involving one angle only, obtained. The identical equality of these two forms leads at once to the theorem, known as De Moivre's Theorem, whence we shall in this chapter deduce the relations between the functions of two or more angles, angles and their multiples, and certain trigonometrical expansions. 2. De Moivrc's Theorem. Since *'" = cos u + i sin u and i v = cos v + i sin v, and by the law of indices i u . i v i u+v , therefore (cos u + i sin u) (cos v + i sin v) cos (u + v) + i sin (u + v). Also since in i-* = cos v ^ sin v and = *"*"" = *""", ^ v therefore cos u + i sin u . . (cos u + i sin u) (cos v ^ sin v) cos v + 1 sin v cos (u v) + i sin (u v). 80 DE MOIVRE'S THEOREM These formulae directly express the fact that the product or quotient of two unit-vectors (or versors) is the unit-vector (or versor), whose inclination is the sum or difference respectively of their inclinations. It should be observed that, like all the formula} of Algebra, they are universally true, whatever be the magnitude or sign of the angles u, v involved. Together with the develop- ments which follow they constitute the famous theorem of De Moivre,* which, in the words of De Morgan (English Cyc. Eiog. Die.) "has had the effect of completely changing the whole character of trigonometrical science in its higher departments." More generally, if u v u. 7 , u., . . . u n denote the measure of n angles in right angles, since i u \ . i un - . . . in = ii+2+ . .. +, (cos MJ + i sin Uj) (cos ?*., + i sin u 2 ) . . . (cos u n 4- i sin u n ) Lastly, since (i") n = i nu by the laws of indices, (cos u + i sin u) n = cos nu + i sin nu, a formula which, like that from which it is derived, is quite general, and true, whatever be the value of n. If n be integral, both sides of this equation have but one value, and the equation is unambiguous. P If n = -, a fraction in its lowest terms, we have seen (c. ii. 10) ? ?_ _P thati?, and therefore (cos u + i sin ) os ?/ so that tan- = - - = / i 2 1+cosw smw v 1+cosw The following formulae for cos 3u, sin 3w may be deduced by the student : cos 3w = 4 cos 3 w 3 cos w, .sin 3ti = 3 sin u 4 sin 3 w. G 2 84 DE MOIVRE'S THEOREM 4. Formula: for converting Sums or Differences of Sines and Cosines into Products of Sines and Cosines, and conversely. These formulae are readily deducible from those obtained in the last section, but it will be instructive to obtain them independently. We have cos u + i sin u = i u , cos v + i sin v = i", therefore (cos u + cos v) + i (sin u + sin v) Also U+V/ U V _ u ~ v \ ~( .T , T" I = ^ \i + ^ / (u + v . u + v\ u v cos - +i Bin 51 . 2 cos - 2 2 ' '2 u+v uv Whence cos u + cos v = 2 cos -- . cos - > . u+v uv sin u + sin v = 2 sin . cos - 2 2 cosw cos v + i (sin u sin v) = i i = i 2 yi 2 i~ 2 ('u + v . u + v\ .u v cos + 1 sin ( }.1i sin a ' Jj . U + V . U V U + V . U V 2 sin sin - H t . 2 cos - sin - - - -- therefore . u+v . uv . u+v . vu cos u cos v = 2 sin - sin -- = 2 sin sin - 22 22 u+v . uv sin u sin v = 2 cos . sin - 1 2 Again, since = i u + t~" and 2i sinw = i u i~ v AND GENERAL TRIGONOMETRICAL FORMULAS 85 4 cos u cos - ({ + i-) (i + i-) - i M +" + i-< + ") + i- + i - (- r ) = 2 cos (u + v) + 2 cos (uv), or 2 cos u cos v = cos (u + v) + cos ( v). Similarly it may be proved that 2 sin u sin v = cos ( v) cos ( + v) 2 sin ?* cos v sin (?t + v) + sin (M v) 2 cos w sin v = sin (?t + v) sin ( v). 5. Sub-multiple at\gh formuke. The formulae just proved (with the exception .of those for the functions of w/2) involve only single-valued functions, and are there- fore without ambiguity arising from the presence of radical signs. The formulae for the functions of sub-multiples of an angle in terms of some function of the angle, on the contrary, necessarily involve ambiguity to a degree, which it is easy to ascertain a priori, and thus to infer the general nature of the formula to be expected or the possibility of obtaining such a formula. a. Functions of /2 in terms of cos u. From co8 = cosM , we infer that w = 4w + w (C. iii. 9), therefore cos - = cos ( 2n + - ) = ( l) n cos + - = + cos 2 \~2' v ~2 2 Thus in this case there is simple ambiguity of sign correspond- ing to the expressions found above containing a single radical sign. For a particular angle the particular sign of the radical is determined by the known sign for sine, cosine or tangent of half that angle. Sfi DE MOIYRE'S THEOREM ft. Functions of ii./2 in terms of sin u. From sin u = sin ?/- , we infer that it, = In + ( 1)"< , therefore . U ( ,\'Xl I W ( , v"W ft 1 sm-=sm| + (-l) - j>, cos - = cos { + (!) - j , and taking n successively eojual to 0, 1, 2, 3, these expressions have four values, namely, two different numerical values, each with either sign, thus : . U . U } U Q . U It. sin - =sm or cos or sin -~ or cos - u u n . u u (} . 11 , cos - = cos -j or sin or cos or sin , while tan - has two values only, namely, tan or cot To this result correspond the formula;, which we will now investigate. / u . u\ 2 2 u , 2 u . n u Since \^cos - + sin - J = cos - + sin - + 2 sin - cos - = 1 + sin it, u , u I -. - cos - + sin - = / 1 + sin u. a a U . U I- - ; Similarly cos - sin - = \'l sin u, therefore 2 cos -- = {v 1 + sin u 2 sin - = |\/1 + sin u + Vl sin u}, each expression presenting a fourfold ambiguity corresponding to that which we were led to expect above. The particular signs of the radicals corresponding to any particular value of the angle u may be obtained by determining the signs of cos w/2 + sin u/2 and cos uj2 sin u/2 for such value by considering AND GENERAL TRIGONOMETRICAL FORMULAE 87 which of the two, cos u/2 or sin u/2, is numerically the greater, and whether it is positive or negative, A general rule however may be obtained thus : u . u , \ u u . u ,- \ u u . u , f\ u\ u . u ,- - + sm- =V2sin(^- +-Jand cos- -sin - = V2 ' Vl + sin u must be taken with the same sign as sin \- + -\ and \/l sin u must be taken with the same sign as cos f - + -J so that, if KOK', LOL' are the bisectors of the angles J'OI, IOJ respectively, the two radicals must be taken with the signs of the sine and cosine respectively corresponding to that one of the quadrants KL, LK', K'L' or L'K in which the terminal line of the angle u/2 measured from 01, lies. [Thus if u = or 210, so that sin u = sin (2 + -")= sin - = -, L \ oj o 2 u 7 - = - which measured from OK lies in the second quadrant, therefore whence cos - + sm -= + V 1 ~9 = )S 6" n r cos - = - , 6 2V2-, 1 ^o = ~ 7_l_-M/_3 '6~"Vf" 88 DE MOIVRE'S THEOREM u 1 cos u 1 + J 1 sin 2 UP i i i r. But tan - = a formula which gives for 2 sin sin u any particular case two different numerical values, both having the same sign as sinw, in accordance with what we were led to expect above. y. As another instance, given sin u to find the functions of u/3. From sin u = sin u , we have u = 2n + ( 1 )" w , and sin| = sin( 2 -%(- l)"-*)^ | or sin(|~) (4 u n \ I u \ /8 u \ /10 ^n\ - + I or sin f 2 -^ J or sin I - + ) or sin ( - ^ ), six values corresponding to the values 0, 1, 2 ... 5 of n, which . u n . /2 w ft \ /2 M O \ reduce to three, sin or sm 1 - } or - sm I - + 77 ), since 6 \o 61 \o 61 / M o\ w n /" u a\ in. 2 * n \ /2 u a \ *(*- i)= sin i' sm (3 + i) =sm ( 3 -r -^) "(5-5) /8 u. } \ . /2 w \ /10 \ . /2 m ( + 3 ) " - S1 " ( 3 - + 3 )' Sm ( 3 - 1) ' - Sm ( + 3 Hence it may be inferred that the three values of sin w/3 would be obtained by the solution of a cubic equation, as is seen to bo the case, since sin u 3 sin - 4 sin 8 - o o This result is illustrated by the diagram, in which /, I r /, I y 7 4 , /. 2 are the extremities of unit-vectors corresponding to the angles 0, -, -, 2, -, respectively, and IP being the arc corresponding to -A 333 3 the arcs I I P r 7 /* 2 , / 3 P 3 , / 4 P 4 , / Pr t are taken alternately backwards and forwards on the unit-circle each = IP, so that OP, OP^ . . . OPr, are the six vectors corresponding to the inclinations AND GENERAL TRIGONOMETRICAL FORMULAE contained in the formula " + (!). It is then obvious that o o for OP and OP y OP l and OP,, OP 4 and OP r> , the trajects or sines of the inclinations are equal : also that for the same three pairs, the projects or cosines are equal but of contrary signs, so that to determine cos u/3 from sin u, a cubic in cos 2 tt/3 might be expected to present itself for solution. 6. Functions of the sum of any number of angles. Since (cos MJ + i sin a ) (cos w 2 + i sin 2 ) . . . (cos u n + i sin n ) = cos(u 1 + u 2 + . . . + w n ) + i sin (u l + u 2 + . . . + u n ), if A',i-r, r denote the sum of all the products of n r cosines and r sines of the n angles u v u. 2 , . . . u n , in the multiplication of the factors on the left-hand side of the equation the term S n -r, r will be multiplied by i r : whence, remembering that i 4n = 1, t' 4n + 2 = 1, and i**+ l = i, i 4n + 3 = i, and grouping together scalar and non- scalar terms, cos (wj +u. 2 + . . + u n ) + i sin (u l + u 2 + . . + u n ) 90 DE MOIVRE'S THEOREM .and therefore COS (Mj + U., + . . .+?/) = #,(> >S' n _2 i 2 + # )l -4 > 4 -#rt-C,G + ... + (-!)' ,s;_o r . 2r +... Sin (M! + Wo + . . . + U n ) = S n -i,i- Sn-?,, 3 + $-&, 5 S n -7. 7 + ... + ( -l) r >S' n _ 2r _ lj2 r+l + ... If /S^ denote the sum of all the products of r tangents of the angles, it is plain that > n - r , r = cos MJ cos u. 2 . . cos M M . S r , and the above formulae take the form cos (u^ + M 2 + . . + w n ) = cos Wj . cos M O . . . cos ,{! >S' + /S" 4 3 C> + . . + ( l) r /So r -f . . .J sin (MJ + M 2 + .. + ) = COS Mj . COS U. 2 . . . COS U n \>\ A.V., + **?. ^ 7 + . . +' ( 1 ) r S r +\ + } and hence . S l -S^ + S. -... + (-l tan M + + ..+ M = ---- " 7. Functions of nu in powers of cos u, sin u, Ian u. Since cos nu + i sin nu = (cos u + i sin )", expanding the second side by the Binomial Theorem, and grouping together * scalar and non-scalar terms, n.nl n _ 2 . , * . _ cos mi + 1 sin nu = cos" u -- - - . cos u sua* u n.n 1 . n 2 . n 3 _ 4 4 - - cos -z* sin 4 u , 1 J!i o 4 ./ n 1 W.W l.W 2 _ o 1 {w cos u sm M - - cos u sin 3 u + . . . 1 . 2i . o * Observe generally that if fx =a = (ar. - a-jX 2 i*"=l, . 4 "+ 2 = -1, AXD GENERAL TRIGONOMETRICAL FORMULAE 91 n . n 1 n . n 1 . n 2 . n 3 = (COS" U) I - - -_ tail" U + n'~a~~A~ * an U v T ~7> " 1 n.n L.n 2 + i (n tan u - tan 3 M + ...)} 1 ._. 3 J Whence, if cos n n is scalar, as it is if n is integral, B ( n.n I n.n 1 .n 2. i 1.2 an-w+ ~TT2T3~.'4" tyt n-i 1 = cos" u . cos w ~ 2 u . sin 2 u + . . . i . - n.n l.n 2^ , ) sin nu cos" n tan u - tan 3 u + . . . n .n 1 . n 2 . , = n . cos" u sin u - - cos n ~ s u , sin 3 u + . . . 1.2*3 and therefore n.n l.Ti-2 n tan u --- - i - - tan d u + ... i . 2i . o tan mi n n.nl t n.n l.n 2 .n 3 1>2 .3.4 If n is not integral, cos" u has a scalar value if cos u is positive, and also if cos u is negative and n is a fraction which in its lowest terms has an odd number for its denominator : in other cases, that is, where n is a fraction which in its lowest terms has an even number for its denomin- ator or when n is incommensurable, cos" u has no scalar value and the above series are not true. The following investigation gives the true result in all cases. Let the series above given for cos nu and sin nu be denoted by S, S' respectively, when cos u is positive, and by ( I)"*?, ( l) n S', when cos u is negative (so that S, S' are scalar) ; and let u be the prime angle of the series for which COSM = COSZ/ O and sinM = sin?/ , so that = 4r+ , r being any integer : then cos nu -\- i sin nu = | cos 4r -{- u -}- i sin (4r -J- )} n =(cos 4r + * sin 4r) M (cos + i sin ) = (cos 4rn -f- * sin 4r/r) (+ iS'~) if cos is positive. = (cos (4r + 2}n + * sin (4r + 2)7?) (S + ? '"S") if cos a o ' 8 negative. 92 DE MOIVRE'S THEOREM Hence - /?' sin 4/-# or S*cos(4r-f- 2)n >S" sin (4/- -{- 2), as COB O is positive or negative. sin nu = Sain 4rn -\- S' cos 4rn or tfsin (4r + 2)n -f- >$" cos (4r + 2), as cos w () is positive or negative. If n is integral, these expressions reduce to cos nu = S, sin nu = *S" if ji be even ; and to cos nit = 8 or - *S f , sin?zw = $' or - S", if n be odd, according as cos M is positive or negative. P if n = -, a fraction in its lowest terms, 1 f l \ when r = or a multiple of y, accord - 2p , , 2p I i n g as cos u j s positive or negative, 9 and the complete expressions have q distinct values corresponding to consecutive values of r in the series . . - 3, 2, 1, 0, 1, 2, 3 ... 5 L 3 [Ex. Let u = or 150, and = 2 .2 V , 21 1 so that cosw = sin - = , smw = cos - = 5, tanw = - ; ^-, & & o & V3 and nu = 4r + = 3r + and (4r + 2) = (2r then cos f 3r + - J = S cos f 3r + '-} - S' sin ( 3r + ^ sin ^3r + |) = ^ sin \3r + g ) + ^' cos (r + g), , /x^sxi/ |.-i i a.-^.-j.-ft i where ,9 = ( ) ( 1 - * ~ - . - + yT^Vr 32 " ) */?( j. 3X1 T _ 3 x ! 5 9 ! , \ ~ V 64 \ "" 4.8 ' 3 4.8.12. 16 3 3 "*" ' V 3 ^ x -^-^ , 2 1.2.3 3 \ 1.2.3.4.5 T"<3 3x1.5 1 . 3x1.5.9.13 1_ 192 \l ~ 478.12 3 + 4 . 8 . 12 . 16 . 20 3 2 ~ ' ' '/ AND GENERAL TRIGONOMETRICAL FORMULAE 93 o that, putting r successively equal to 0, 1, 2, 3, we find 5 l S + S ' ( R , 5 r = sin - = - .- cos l6-J-7 V \ 4 If ?i is incommensurable, the expressions for cos nu, sin nu reduce respectively to S, S', only when cos it is positive and r : all the rest of the infinite number of values involve both S and S'. It is to be observed that, unless n is a positive integer, these formula; are applicable only if tunu is between -1 and + 1, since beyond these limits the infinite series obtained from the Binomial Theorem is diver- gent. If tantt is numerically greater than 1, cotw is less than 1, and convergent series may similarly be obtained from the formula cos nu-J-t sin?Mt = (cos(4/ i -f-l) w + f sin (4r+l)n)(sintt ) w (l icotM ) or (cos (4r 1) u -f- z'sin (ir 1) M ) (sin ) n (l i cotM ), according as sin u is positive or negative. The development is left to the student. [8. Series for cos nu ami in descending powers of cos u. 81YI U By putting 1 - cos 2 u for sin 2 tt in the expanded form of (COSM + i sin w) M , it will be readily seen that we may assume cos nu + i sin nu (A Q cos" u + A z cos*- 2 u -\- An cos' 1 - 4 M + . . .) + i sin u (AI cos"- 1 u + ^ 3 co8- 3 u +. . .) where A w A v A.,. . . are functions of n to be determined. If we collect the coefficients of COS M W from the several terms in the expanded form in 7, . .n . . , = 1.2 1.2.3.4 but by the Binomial Theorem n.n l , n . n 1 . n 2 (l + 1). = 1 + n + - 1:Y - + 1T2 - ; 3" + - - and n . n 1 n . n 1 . n - 2 therefore, adding 2 = 2^ or A = 2-i. 94 DE MOIVRE'S THEOREM The remaining coefficients may be found in terms of J as follows. Differentiating * both sides of the equation above with respect to u, - n sin nu + i n cos nu (nA cos"- 1 u + n - 2 A% cos ~ 3 u -\- n - 4 ^ 4 cos " - -f n - 3 A 3 cos"- 4 w + . . .)} whence, multiplying by - i, and putting 1 - cos* u for sin 2 , n (cos nu + i sin nit) = jn^i cos" + (ft - 2 // 3 - n - 1 ^) cos" - - + (n - 4 -// 5 - ;i - 3 ^3) cos" - * + ...] + i sin w hiA cos"- 1 u + n - 2 J 2 cos"- 3 M + - 4 ^ 4 cos"- 5 M + ) Hence, equating the coefficients of like powers of cos u in the two expressions for n (cos nu + i sin M), it 4 A^ n 3^/3 = /iJ r = (n - G)J , &.( whence A l = A, (n z - n - 2 2 )A 3 = - n 1 . ft - (ft 2 - n - 42) J 5 = --.- 4^ 3 , . . . and generally (n 2 - JT^ 3 ) ^ar+i = - (ft - 2r+ 1) (ft - 2/-) -/-.v-i or 2r. 2 (H - r) A-2r+\ = - (ft - 2r + 1) ( - 2;-) ./ L v_ i. We have then 2 s . 1 . (n - 1) J 3 = - (ft - 1) (ft - 2) AI 2 2 . 2 . ( -- 2) A- a = - (n - 3) ( - 4) J 3 2 2 . 3 . (ft - 3) A-i = - (ft - 5) (n - 6) A~ &c. 2 2 . r(n - r) Azr+i = - (ft - 2/- + 1) (ft - 2r) Jo r _ * In the articles in small print I have not scrupled to use the processes of the Differential Calculus when it seems the simpler and more natural course to do so. The student, who desires a proof independent of the Differential Calculus, will find an ingenious, though somewhat artificial one, in Johnson's Trigonometry, pp. 271-273. Serret (Cours d'Alg. Super ieur, Let;on 14) deduces the series from the expansion of xn+ in powers of x + -, which he investigates, and which conversely we might deduce from the scries obtained above. AND GENERAL TRIGONOMETRICAL FORMULA 95 whence ~-. 1 and since - 2r so that . r - ( -l)r- ("-'-!) (*-r-2) . . . (*-8r+ 1) _ T 1.2.3...)- - TC ~ 2cosMW = (2cosw) - -.(2 cos )- 3 _|.- ~L_ (2 cos M)-* + 1 1.2 - r - 2) . . > - 2r 1 . 2 . 3 . . . r (2co8M) , t _ 2r+> &c 7 = (2 cos )-! - -Z . (2 sinw 1 1.2 (" " r - 2) . . . (n- .( 2cosu)-5-., 1 . 2 . 3 . . . r ( _ t These series can only be used, when n is a positive integer ; for if u is not a positive integer, they become infinite series which are essentially divergent, the limiting ratio of one term to the preceding being sec 2 w, which is always greater than 1.] [9. Series for cos nu, sin nu is ascending powers of sin u or cos u. In the expanded form of (cos u + i sin ), if + Vl - sin 2 u be substituted for cosw (supposed positive) and its powers expanded by the Binomial Theorem, the series may be arranged according to ascending powers uf sin win such a manner that we may assume cos nu -\- i sin nu A-\- A^ sin 2 u + A* sin 4 u + . . . + i (A! sin u + J a sin 3 u + where J , A lt A z . . . are functions of n to be determined, and it is easily seen that A$ 1, A-^ = n. Differentiate twice, then since * * Strictly it is only true for circular measure that (sin x) = cos z, &c. dx For any other measure (sina;^=^cosa;, &c., where A- is a constant depend- ent on the unit of angular measure. In the investigation above this constant would appeal- on both sides of the equation, and so may be omitted. DE MOIVEE'S THEOREM d 2 / d 2 (cos H.Z-) = - n- cos HA- - - (sin nx] = - ?* 2 sin nx, a.? 2 (a- 2 and - (sin r #) = - (? sin r ~ 1 x cos .r) = r . r 1 sin r - '- x . cos 2 x r sin r &* we find (^ \ /7i \ 2 + 1 / n.n-l . . . (<> + 2 / 2" sin" u = 2 cos 2u + . . . ( I) 2 ** . 2 cos ?i 2 M + ( I) 2 . 2 cos nu, if w is even : and n + 3 n + 5 n.n 1 . . . n.n\ . . . 2" sin nu = ^ . 2 sin u - . 2 sin 3w + ... ( 1) 2 w . 2 sin w 2w + ( 1) 2 .2 sinnu, if w is odd. Thus also sin u is purely periodic, if n is odd ; but has a mean value if n is even, the same as that of cos u, as might have been anticipated a priori. It will be a good exercise for the student to deduce the series for sin u independently from the expression 2i sin u = i u i - " by a course of reasoning similar to that employed for cos u above. 11. Expression of cos m u . sin n u in terms of sines or cosines of multiples of u. It is frequently required, in the expansions employed in astronomical and other similar investigations, to reduce a term containing the factor cos u sin u into a series of sines or cosines of multiples of u of the same character as those found for cos u, sin"w in the last article. No general formula of a simple character can be given for this purpose, but an easy process applicable to any particular case may be readily obtained as follows. Denoting the coefficients in the expansion of (1 +x) m by the Binomial Theorem by 1 TO , 2 OT , 3 m . . . , we have H 2 100 DE MOIVRE'S THEOREM 2 cos m u = (JU + i - u)m = fin +\ m , i< - 2)n + + . . . and the upper sign corresponding to n even, the lower to n odd. Hence it is plain that the product 2 m + n i n cos TO Msm 1l tt will be equal to a series of the general form + J 2 t'( TO + n ~ 2 >" i ~ ("' + "), reducing, if n is even, to a series of cosines of multiples of u commencing with (m + n)u and diminishing by 2u from term to term : but, if n is odd, reducing to a like series of sines. Further, if m + n is even, there will be a single middle term of the form -4m+n* or A m+n ; but if m + n is odd, there will be two middle terms grouping into one of the form A m+n _i (i u i~ M ). Hence if m, n are both even or both odd, the series would appear to consist of a mean term and periodic terms of cosines or sines respectively of 2, 4 . . . (m + n)u. It will be seen however that, when m and n are both odd, the middle term A m+Jl , being the sum of pairs of equal terms with contrary signs, will vanish, so that the mean term is or the series is periodic. If one is even and the other odd, the series will be purely periodic and involve cosines or sines of u, 3u, 5u . . . (m + n)u, according as n is even or odd. To obtain the coefficients of the several terms, we observe that 2m+n ^ cos m u cos n u = (jit + j-u^n . ^tt_ t '-) = (i 2u -i~ 2u ) M (i + i~ ) - ", if m > n : then the result may be obtained by expanding (i 2u i 2 ") and multiplying the expansion (n m) times by i u i~ u , if mn. In the expansion by the Binomial Theorem of (i 2u - i ' Zu } m . the indices in the successive terms being written 2mu, (2m - %)u, (2m - 4)w ... - (2m - 4)w, - (2m - 2)u, - 2mu, the corresponding coefficients are 1,0, - m . . . m, 0, 1 : and to form the coefficients of the product of such a series bj t" + i" M or i u i' u , it is only necessary to add each coefficient to the next following without or with change of sign of the first of the two. The process is best shown by an example, for which let us take cos 8 u sin 4 u. Then 2 ' - i 4 cos 8 u . sin 4 u = (i u + i - M ) 8 (i u i - tt ) 4 = (i u + i - ) 4 . (t 2 i - 2 ")4. The coefficients in the expansion of (i~ u t'- 2 ") 4 of the terms involving jSu J6u ^4u P 2(?/i 2 + jj^s? cos { - Tan -i - + - } i k.i* k , then, since (&)* = (A;*)", so that (&) = ($/) and iKA") = nty Now, since log (&) = n log k and ^ (A,-") = n log (A;) log A;' so that, since, n having any value, L n has any value, the value of T//-A; , r is a constant independent of k ; and this may obviously be taken * As far as a priori considerations arc concerned, this assumption might have turned out to be impossible, in which case &* would have been, relatively to our subject-matter, coplanar vectors, uniuterpretable or an "impossible" quantity. 108 VECTOR INDICES AND LOGARITHMS us 1 by properly taking the base of the system of logarithms, hitherto left arbitrary : hence, denoting the base by rj, fyk log,, k. Also, since ^>(ld n ) = (/)", log ft (*")}- log $(*), whence, by similar reasoning (log taking the place of ^) log ( by means of which either of these measures may be converted into another. We have also limit A - } - 1. a =\ a / 6. Limit VECTOR INDICES AND LOGARITHMS 111 a z 1 . a z 1 Another limit of fundamental importance is that of when m z vanishes. It is plain that, if finite, it must be a function of a, so that we may write limit _ = (a") = limit - = n x limit - = n (a n ) a = Clog a, The base of the logarithm above has been left indefinite, and the value of C will obviously depend on the base chosen. Let e denote that base for which (7=1, then limit - = d> = logea, 2=0 % where e is a numerical constant to be determined from the con- e z _ I dition that limit -- = 1. z=o z 7. Determination of e. * 1 Let e be a quantity such that - =1 and therefore 112 VECTOR INDICES AND LOGARITHMS I / 1 \ TO then e limit ,. e = limit A (l + z) z = limit I 1 + - I , z zz^v\ / m co \ 7>i/ if z = , where m may be taken to be a positive integer. Expanding by the Binomial Theorem and reducing, ,, 1.2 1.2.3 \~ \ 1 . 2 . . . r Now a, b, c . . being positive quantities all less than 1, (1 -)(!-&) (I -c)...> l-(a + b + c+ ...) For (1 -)(! b) = l -a- b + ab >l a + b (l-o)(l-6)(l-c)>(l-o+7 and so on. r . (r - 1 ) and this product is positive and <1, since r is not greater than m, so that, u r being some positive fraction less than 1 , we may write r . r 1 2m Therefore IN ill ."2. 3 1.2.3.4 1.2...* ___ 1 2? 1 . 2 2w ' ' 1.2... m- 2 2/n VECTOR INDICES AND LOGARITHMS 113 / 1\ so that ( 1 + ) differs from the sum of the m + 1 terms \ in/ 1 1 1 + 1 + , + + 1.2 1 . 2 . . . m by less than Now the series above has its terms less than those of the series 2 2 2 2 3 2'"- 1 ' of which the limit, when m is increased without limit, is 1+ L-oi-3: therefore this series has a finite limit less than 3. Hence / l\ m 1 1 (1 + -) differs from 1 + 1 + -! ... 1 m) 1.2 1 . 2 . . . m 3 by less than , 2m and when m is increased without limit, this difference vanishes, so that / IV 1 I e. = limit 1 + - ) -1 + 1 + - -- + . . . + r + . . . (ad inf.) "= \ in/ 1.2 \r and e is thus proved to be a number greater than 2 and less than 3. The series for e converges with tolerable rapidity, and it is easy to calculate by taking a sufficient number of terms that to 9 fractional places 6 = 2-718281828. . . a result which the student should verify. 114 VECTOR INDICES AND LOGARITHMS It is easy to prove that e is an incommensurable number. For if not, let e = where m and n are integers, then e An m An 1= integral terms -| + , -^-. ^ + . . . n+l (n+l)(n + 2) so that we have two integers differing by 1 1 which is less than 1 1 I TT7 "7* n+l (n+iy (n + iy (n+l)(l - L) that is, the difference of two integers is a fraction, which is absurd. ir 8. Value, of 77 e 2 Resuming the equation ( 3), yjiu _ ~- iu r .2iu I since ^-. = -n~ lu r 2iu '2iu and, when u = 0, limit r> - iu = 1 , and by 6 limit - - = log e r], 2iiu . sinw TT while by 4, limit = x > TT - we have -^ = logi; or t] = e' 2 , Jj so that VECTOR INDICES AND LOGARITHMS 115 Taking TT = 3-1415926 . . . and Iog 10 e = -43429448 . . . log, 77 = -6821881 ... and 77-4-810475... Hence the remarkable result that (4-810475... )' = t. 9. Exponential Expressions for the Trigonovietrical Functions. To sum up the results of the preceding investigations, if an angle contain u right angles or radians, so that = ^u, we have cos u + i-sin u = i u = T"* = e ie - - 2t 2i 2i 1 P* - 1 1 T/ 2iu - 1 1 e 2ia - 1 and .\ tanu=tan0 -. - =-', ., = -. -ss i i 2u +l i r-^+l i e**+ I 10. Vector Power of a Vector. We are now in a position to interpret generally the expression A B , where A and B are both vectors. Let r, n be the tensor and versor respectively of the vector A , so that A = ri", and let m, n be the project and the traject of the vector B, so that B = m + ni, then, since ^* = *> A B = (r-t*)"-J- = i rr jiu\m+ni _ r m r in ~imu -t'^nu _ r m y-nu ^i(nlog,jr+iitu) I 2 116 VECTOR INDICES AND LOGARITHMS or, with complete generality, putting u + 4A. for u (X an integer), Hence A B is a vector, whose tensor is r m ij~ n *ij~***, which for different values of A positive and negative gives the terms of an infinite geometric series continued upwards and downwards from the fundamental term r m -r)~ nu with the common ratio iy~ 4n : and whose versor angle, expressed in right angles, is n log,r + mu + 4A.m, which in like manner gives the terms of an infinite arithmetic series continued upwards and downwards from the fundamental term n log^r + nu with the common difference 4m. The same may be more concisely expressed thus : The values of A B form infinite series of vectors in vectorial geometric progression formed from the fundamental vector r'"i7~""t" lo V'+ n " t forwards and backwards by the common vec- torial ratio rj-* n i* m . This leads to the following geometrical interpretation. \ VECTOR INDICES AND LOGARITHMS 117 Taking OI as the prime vector, and 00 as the fundamental vector, so that the tensor OC = r m r)~ nu and the versor angle IOC = (n log,,?- + mw) L : also taking OH as the vectorial common ratio, so that the tensor OH = r/~ 4n and the versor angle IOH = 4i L ; on OC describe the triangle OCC r similar to OIH, OC being homologous to 01 and the angle COC r measured in the same sense as IOH, and on OC V similarly a second triangle OG^G. 2 , and the like on OC., and so on ; and farther on OC describe the triangle OCC -\ similar to OIII, OC being homologous to OH the angle COC~\ measured in the sense opposite to that of 10 II, and on OC-i similarly a second triangle OG-\ C-z and the like oil OC-z and so on : then the infinite series of vectors . . . UC-s OC- 2 OC-i OC ~OC l OC, OC's . . . , in which 'OG^ 'OG. OH\ constitute the several values of A B . It is easily seen that these vectors are radii of the equiangular n ~ n spiral, whose equation is r = OC . t] ' for the values, u~ 4A?. 1 1 . Particular Cases. (1) If n = 0, the index is scalar and the case has been dis- cussed in Chap. ii. Regarded as a particular case of the general interpretation above, the vectors OC V OC. . . . have all the same tensor r m ; which are coincident if m is an integer, but, if m= p/q (p being prime to , if r = {(log^r) 2 + (u + 4&) 2 j* and tan u' = Hence the logometer is any one of a series of vectors, all having the same jsroject log^r, but whose trajects are the terms of an arithmetic progression continued upwards and downwards VECTOR INDICES AND LOGARITHMS 121 from u (or as many units of length as there are right angles in the inclination of the vector) with the common diffei ence 4. The same result may be stated thus : The logometer of a vector (ri u ) to the base 17 has an infinite number of values, forming a vectorial arithmetic progression with the common difference 4t, arid extending upwards and downwards from a fundamental vectorial value defined by the project, log,,?- or the logarithm of its tensor and the traject, (u) equal in numerical value to the number of right angles in its inclination. It is plain that the logometer can only be scalar, if u- or a multiple of 4 right angles, that is, if the original vector is a scalar. 14. Loyometers to any numerical base. Let denote any numerical base, A, (ri u ) log,r .u + 4k . u + 4A: then X^H" = - -+,-_ = log^r + t - k*i lg, lo g>, *g^ In the particular case where =, since -n e 2 so that 1 = - log,, e or = 2 log* 2 if the inclination in radians = 0, so that = - u, X e (ri tt ) = X e (re 16 ) = log e r + i(H The change from the base t] to the base , being effected by the multiplication by the scalar factor 1/log^, alters the tensors of the logarithms only in a given ratio without changing their inclination. 15. Illustrative Diagram. the foregoing results are illustrated in the annexed diagram, in which OA representing a given vector, OA' is another vector 122 VECTOR INDICES AND LOGARITHMS having the same inclination, and OA" another having the same tensor. Then 01, OE, OH being 1, e, rj respectively, if OM=log 1l OA or logflA", Om = - . OM=log e OA or log e OA", a OM' = loff OA' VECTOR INDICES AND LOGARITHMS 123 and MO = M'C' = i x measure of 10 A in right angles, MO" = i x /CM" me = i x /0.4 in radians, then 00, 00', 00" are the fundamental logometers to the base -q of OA, OA', OA" respectively, and Oc, Oc", those of OA, OA" to the base e. Farther if the values of X^OA are the infinite series of vectors . . . OO- 2 , 00 -i, 00, 00 v 00 2 . . . If the line MCC l is shifted so that M comes to M', to C', C l to C"i, &c., the values of A, OA' are the infinite series of vectors . . . 00". 2 , 0C"_i, OC", 00"! 0<7 2 . . . If all the points in MCC l are shifted along the line through the same distance (namely, i x measure of AOA" in right angles)* so that C comes to 0", C l to C" v &c., the values of X^OA" are the infinite series of vectors . . . 0(7" _ 2 , 00" -i, 00", OG\, OC" 2 . . If the vectors of the points in MCO l are extended to the perpendicular through in, so that . . . (7_i, C, C\ . . . come to . . . c_i c c x . . . , the values of X e OA are the infinite series of vectors . . . 0c_i, Oc, Oc l . . . and tho.se of \ K OA" are in like manner . . . 0c"_ lp Oc", 06'j" . . . 124 VECTOR INDICES AND LOGARITHMS Observe that if the inclination of OA is given, according as OA is greater or less than OH, OM is greater or less than 01 : if OA = OE, m coincides with / : if OA = 01, both M and m co- incide with 0, and the logometers to every base coincide in direction with i : if OA is less than 01, OM is negative. In all these changes the C"s move on lines parallel to 01. Ex. If OD, OD', OD" . . . are the logometers of OC, the funda- mental value of A B , and through D, D', D" . . . lines are drawn perpendicular to the vector (or m + ni) intersecting lines parallel to DD'D" drawn through M, M v M z . . . points on the 01 axis separated by the common interval 4n, OM being equal to log OC : then the vectors drawn from to the inter- sections of these two systems of lines are all values of \(A B ). 16. Logometers to a vector base. The conception of a logarithm may be still further generalized by taking for the base a vector, instead of (as hitherto) a purely numerical quantity. Let us examine the logometer of R ( = ri") to the base A ( = ai''). Then since X, j? = log, r -\- i (u -f- 4) and X, A = log, n + i _ log, a +1(1-4. 4*0 or, if a, 6 are the circular measures of ^ L , L respectively, lo&r + where /., &* are unconnected integers. The logometers therefore form an infinito-infinite series of vectors, each being the quotient of some one of the logometers of R to the base r) by some one of those of A to the same base. The meaning of this result will be made clearer by its diagrammatic representation in Figs. 1 & 2. Let .... OA- V 7L4, OA l ... be the vectors of the series X^ //, and. . . . 7)E_ 1 ,~OIi i Ofi^ . . . those of the series X^, the points . . . . A_^ A, A . . . lying on the perpendicular to 01 through M, where OM=log^a, while . . . . R, R l . . . lie on that through N, where 0^"=log r. To form the quotients of the vectors of the latter series divided by OA, we must divide each tensor by the tensor OA and VECTOR INDICES AND LOGARITHMS 125 turn it negatively through the angle equal to AOL The first operation is effected for all the vectors by shifting the line of Ks, parallel to itself till N comes to A', such that UN'=ONj<>A or OA"/ON=0//OJ ; then, P-, FIG. 1. if OA meet the line of Jt's in this new position in A\ the second opera- tion is effected by turning the triangle OA'N 1 about O until OA' coincides with 01. If then A' conies to K and N' to JV", the line of R's finally coincides with the infinitely extended line through A' and N". 126 VECTOR INDICES AND LOGARITHMS FIG. 2. VECTOR INDICES AND LOGARITHMS 127 Now 6A"= 0,1'= OA . Oi\'/OM= 01. ONJOM, since ON'. OA = ()N. 01 from above, therefore and since ON"K is a right angle, A"' is on the circumference of the circle, diameter OK. Hence one series of values of \ A R is found thus. On 01 take OK=\o a r, and on OK as diameter describe a circle, draw ON", to meet the circumference in N", making the angle KON"=AON, then if . . . P_!, P, Pi ... are taken on A"A' so that . . . A"P_,, A"P, N"P l . . . are equal respectively to ... K'R' _ v N'R', N'R\, . . . the series of vectors OP-^ OP, OP^ . . . form one series of values of \ A R. In like manner, if A'"!, N"- v A" 2 , N" - 2 , . . . are points on the cir- cumference, such that the angles KON'\, KON"-^ . . . are equal to A V ON, A-^ON . . . respectively, there will be like series of vectors drawn to the R's on each of the lines KN'\, KN"-^ A~A'" 2 , KN" _ 2 , . . . forming for each a series of values of X.,7?. Hence while logometers to a numerical base are vectors drawn from to points distributed at equal intervals along one line perpendicular to Ol, logometers to a vector base are vectors drawn from to points similarly distributed along each of a pencil of lines all passing through one point A' on 01, such that 6A"=the logarithm of the tensor of the given vector to the tensor of the given vector-base as base. Farther it is to be observed that (as shown in Fig. 2) all the points P, correspond- ing to the logometers of Oil to the several bases . . . OA- V OA, OA l . . . lie on a circle through 0, A divided by OK into segments containing angles equal to the supplementary angles at P. Similarly the points PI, P 2 . . . lie respectively on circles divided by OK into segments con- taining angles equal to the supplementary angles at R. The centres of these circles are the intersections of the line bisecting CIA" at right angles with the lines . . . OR-^ OR, OR l . . . Hence the complete series of values of \ A R is the series of vectors drawn from to the intersections of the pencil of lines through K corresponding to . . . A-\, A, AI, . . . with the circles through O, K corresponding to ... R . 1? R, RI . . . EXCIRCULAR OR HYPERBOLIC TRIGONOMETRY 1. THE measure of an angle is essentially scalar, but the as- sumption of a vector form for such measure leads to a series of functions which have a close analogy to the trigonometrical functions and which have relations to the rectangular * hyperbola corresponding very closely with those of the trigonometrical functions to the circle. A discussion of these functions is neces- sary to give completeness to our subject, and will be found useful in its applications to higher Analysis, particularly in the integra- tion of certain algebraical forms. 2. Returning to the formulae (Chap. V. 9), u, 6 being the measures of an angle in right angles and radians respectively, fit + I - It It + y - ttt '+-- i0 Put iu for u, or iO for 0, then r7 i ' 2 + T?- f3M n- u + rf l cos in - = - or cos iQ = * Observe that these functions have relation not to the hyperbola generally, but to the rectangular hyperbola, which is among hyperbolas what the circle is among ellipses. It would be impossible to use the octosyllabic, though correct, term " rectangular hyperbolic " functions ; I propose, therefore, "to call the "rectangular hyperbola" the "excircle," and its functions "ex- circular." EXCIRCULAR OR HYPERBOLIC TRIGONOMETRY 129 and or sin = i r The quantities 222 are termed respectively the hyperbolic cosine and hyperbolic sine of the angle, whose measure in right angles is u and in radians 6, and they are usually denoted as cosh u, sinh u. The rectangular- hyperbola, from the point of view in which we shall have to re- gard it in the sequel, would be not inappropriately termed an excircle, and these functions the excosine and exsine respectively, and as these terms have the advantage of brevity in speech, we shall generally make use of them. The notation however, cosh, sinh, tanh, is so well established, that I have not ventured to substitute for these excos, exsin, extan, or ecos, esin, ectan, as it otherwise would have been natural to do.* Hence we have = cosh u = cos iu and = cosh iu = cos u, if* + r)~ iu rft _ - sinh u i sin iu and = sinh iu = i sin u, and therefore cosh u + sinh u = if 1 , as cos u + i sin u = rj iu cosh u sinh u = tj~ u , as cos u - i sin u = rj~ iu * It will be found however convenient to read at any rate the unpro- nounceable syllables sink, tanh, as ccsin, extan. 130 EXCIKCULAK OK HYPERBOLIC TRIGONOMETRY so that cosh 2 u sinh 2 u = 1 , as cos 2 u -I- sin- t = 1 , and (cosh u sinh u) n = ij nu cosh nu sinh nu, as (cos u i sin u) n = cos nu i sin nu. Farther, if we define tank, cot/t, sech, coseck similarly to Ian, cot, sec, cosec, TjU y-U r j iu -n~ iu tanh u = i tan iu = i tan u = tanh iu = - r ) u + r ) -" if + 7?-' coth u i cot I'M cot w = coth iu sech w = sec iu sec w = sech iu cosech u = - i cosec iu cosec u = i cosech iu. We have also the inverse forms of the above formulae, as cosh~ ] cc= -icos~ 1 a? cos~ 1 a; and the like, which the student should write down for each function. r? " + r? -u Also from x = cosh u = - ~ . a 1 r) u = x v/cc 2 - 1 = x + v/ic 2 - 1 or - 73= x + Va: 2 - 1 whence u = \^(x \ 2 - 1) = log^(a; + v 2 - 1) + 4^', so that, if u = log^(a; + va; 2 1), and writing Cosh with a capital C to denote the particular angle u , ~ l x= Hence cosh" 1 ^; has the same multiplicity of values that cos^ 1 * has, but with the difference that, while in the case of cos" 1 * all the values (if any) are scalar, in the case of cosh ~ l x two only (if any) are scalar. A like remark applies to the other inverse functions. The comparative formulae (which the student should verify) are EXCIRCULAR OR HYPERBOLIC TRIGONOMETRY 131 Cosh - ] a; = log, (ce+ \/* 2 1) and cosh" 1 * = Cosh" 1 * + 4H, as cos" 1 * = Cos" 1 * + 4&. Sinh" 1 a; = log 1 (a;+ V* 2 + 1) and sinh" 1 a; = (-l) fc Sinh- 1 a; + 2H, as sm- 1 *^ ( 1 -1- y Tanh- 1 * = 1 log - 1 x and tanli ~ l x Tanh " l x + '2ki, as tan ~ l x = Tan ~ l x + 2k. If we use circular measure, the first becomes Cosh" 1 * = log e (* + V* 2 - 1), cosh" 1 * = Cosh" 1 * + Qkiri as cos " l x = Cos " l x + Skir, and similarly for the others. 3. Range of the values of cosh, sinh, tank. Hyperbolic Ampli- tude. f,H ^-ij ll. Since cosh= ' ^ , so long as u is scalar, cosh u has a 2i positive scalar value, not less than 1, or it ranges from + 1 to + GO. 17* -77"" Again sinh u = ^ , and therefore, u being scalar, sinh u A has any scalar value whatever, positive or negative, or it ranges from QO to + oo . _- 2u -l Also tanh u = -..-, and therefore, u being scalar, 17" + 77"" 77 2u + l tanh u has any scalar value between the limits 1 and + 1. Let T denote an angle between the limits 1 L and + 1 L ; then consistently with the above we might assume sec T = cosh u or tan T = sinh u or sin r = tanh u, K 2 132 EXCIRCULAR OR HYPERBOLIC TRIGONOMETRY but, by the relations cosh 2 u sinh 2 u = 1 compared with sec 2 T tan 2 T = 1 sinh u tan T and tanh u = - with sin T = , cosh u sec T the last two follow from the first, so that for the same angle T we have sec T = cosh u tan T = sinh u sin T = tanh u and cos T sech u cot T = cosech u cosec T = coth w. The angle T has been termed the Hyperbolic Amplitude of u, and denoted as amh u. Since ^2_i 1 + sinr ^2 + 1 ' l _ s i n T therefore ~2w _ ]_ 1 _j_ s | n T amh M = T = Sm~ 1 ^K -, u or, using radians, ,- - 1 -t-sinr /TT T amh = bin i -^ 1 - , = log,,, ^ = lo tan( / + - It will be observed that, while for scalar values of the angle, the circular functions sin, cos, tan are periodic, the excircular functions sinh, cosh, tanh are non-periodic. This fact corresponds to the limitation of the hyperbolic amplitude to be a positive or negative acute angle. 4. Formulae for excircular functions. The formulae established in previous chapter for circular func- tions may, by means of the relations in 2, and putting iu for u, be transformed into formulae for excircular functions. The following list contains the principal formulae which it is desirable to remember. EXCIRCULAR OR HYPERBOLIC TRIGONOMETRY 133 The formulae below For Circular Functions become for Excircular Functions, cos z u + sin 2 u = 1, cosh 2 u sinh 2 w = 1, 1 + tan 2 u = sec 2 u, 1 tanh 2 u = sech 2 u, cos (u v) = cos cos v + sin u sin v, cosh (u v) = cosh M cosh v sinh w sinh v, sin (?/. v) = sin cos v cos w sin v, sinh (M + w) = sinh u cosh v cosh u sinh , sin 2w = 2 sin M cos M, sinh 2w = 2 sinh u cosh M, cos 2 M = cos 2 w sin 2 w = 2 cos 2 - 1 = 1 2 sin 2 w, cosh 2w = cosh 2 u + sinh 2 u = 2 cosh 2 1 = 1 + 2 sinh 2 w, tan u tan v tanh w tanh v r . 1 + tan u . tan v 1 tanh u tanh i? \ "i ^Q ~r ^K ^ -- , A> , + O q + O E + tanh (1^ + ^ + ^ + . . .) = i 5^ - I + o 2 + o 4 + . . . W+V W V COS II, + COS V " COS - . COS , J J , u+v uv cosh 7^ + cosh v = 2 cosh . cosh , 2 2 . U + V U V sin u + sin v = 2 sin - -- . cos , 2 L . u+v uv sinh u + smh v = 2 sinh - . cosh - , . u + v v u cos u cos v = 2 sin . sin , '2i 2i i, u + v \,~ cosh u cosh v 2 sinh - . Sinn - , A u+v u-v sin u sin v = 2 cos - .sin 2 2i , . , sinh u sinh v = 2 cosh . smh 134 EXCIRCULAR OR IIYPERKOLK' TRIGONOMETRY It will be instructive for the student to obtain these formulae independently from the definitions of the excircular functions. 5. Geometrical Interpretation of Excircular Functions. On the unit circle IJl' take the arc Ip = u and the arc It = r or amh u : draw pm, tl perpendicular to 01, and t M, the tangent to the unit circle at t. Then OM= sec T = cosh u and Mt = tan T = sinh u hence, if PMP' is drawn perpendicular to 01, making MP = MP'= Mt, E.\( Ii;i ri.AIl OR HYPEUnOLK' TRIGONOMETRY 135 P will correspond to p, so that, as Om, mp are cos u, sin u re- spectively, so OM, MP, are cosh u, sinh u respectively, and OM, MP' are cosh ( u), sinh (u) respectively. And, since cosh 2 u sinh 2 u= 1, OM 2 MP 2 =I, so that, while the locus of p is a circle, the locus of P, which may appropriately be termed the " excircle," is the rectangular hyperbola, whose semi- axis is Of ; and, as the vector equation to the circle is Op cos u + i sin u, that to the hyperbola is OP = cosh u + i sinh u. We proceed to show that, if P, Q are points on the excircle corresponding to p, q on the circle, the areas of the excircular or hyperbolic sector POQ and the circular sector pOq are equal. Let the angle I0q --- V L and let Qff be the ordinate of Q, then the A POQ= &OQN- &OPX- trapezium PMNQ, = i (ON . NQ - OM . MP- (MP + WQ) (ON- OM}}, = * {OM .NQ-ON. MP}, = J, (cosh u sinh o cosh v sinh t), = \ sinh (v - u), from 4, and the triangle pOq = J- sin (v u) ; triangle POQ sinh (v - u) triangle pOq sin (v - 11) Exit, since sinhw r] u -ij sinh u w 2u - 1 TT , sin M ir limit u=0 __ =1. It. u=0 = -, and It. =0 = - therefore sinh u sinh w u ~ w _ , " =0 sinw " sin u It. tt=0 136 EXCIRCULAR OR HYPERBOLIC TRIGONOMETRY and litnit sinh ( v - M > _ 1 limit = j . i , sin (-w - M) so that the triangles POQ, pOq are ultimately equal, when Q, q come to coincide with P, p. Hence since two finite sectors POQ, pOq can be divided into the same number of elementary sectors, each corresponding pair of which are ultimately equal, the limits of the sums of the elements, or the sectors themselves, are equal. Instead of regarding cos u, sin u, &c., as functions of the angle lOp we might regard them as functions of the corresponding sector of the unit-circle, and then regarding cosh u, sinh u, &c., in like manner as functions of the excircular or hyperbolic sector IOP, the analogy between the circular and excircular functions is complete. The sector 10 P sector lOp = u quadrants of the unit circle = - u or i Q square units. If the angle IOP = < radians, e 2 * 1 tan < = tanh 6 -^ so that therefore the area of the sector IOP in terms of is 4 log c tan f - + < j square units. 6. If IOP V PiOP & P Z OP 3 . . . are the hyperbolic sectors cor- responding to the successive quadrants of the circle, and M^ M z . . . the feet of the ordinates to P v P 2 . . ., OM l - cosh 1 L = 2 . 5, OM 2 = cosh 2 L = 11.5, OM 3 = cosh 3 L = 55 . 5, 03f 4 = cosh 4 L = 245 . 5, approximately. These values show the rapidity with which the arcs IP V P 1 P 2 , P 2 Pa , corresponding to successive quadrantal arcs of the circle, increase. The successive sectors, corresponding to successive complete revolutions of a radius of the circle, form a series of more and more elongated sectors with smaller and smaller angles, of which no finite number will fill up EXCIRCULAR OR HYPERBOLIC TRIGONOMETRY 137 the space between the curve and its asymptotes, so that this area is infinite. Riemann, in his researches on the complex variable, found it con- venient to conceive a revolving radius after completing each revolution to continue its path, not on the same plane, but on a plane superposed upon the previous one, thus moving on a continuous surface without passing through the same position a second time. In the present case, i f we suppose an infinite number of circles superposed on the unit circle above and below it and all of them cut along the radius 01', and then each terminal radius of the one (the revolution being supposed in the positive direction) united with the coincident initial radius of the next above it, we shall have a Riemann's surface (a screw surface with a zero pitch), whose folds, infinite in number, correspond to the successive sectors, also infinite in number and each equal to the area of the unit circle, which fill up the area between the excircle and its asymptotes. It will be observed that it is only one branch of the hyperbola that is thus determined as corresponding to the complete Riemann's surface, as appears from the fact that cosh u is for scalar values of u essentially positive. To obtain the other branch, we must take vector values u -f- i or + 2/. The former gives cos (u + i) -\- i sinh (u -f- i) cosh u -f- i sinh w, which gives the second branch as the image of the first with respect to the OJ axis. The latter gives cosh (u -j- 2i) -j- i sinh ( -j- 2z) = (cosh u + i sinh w) which gives the second branch by turning the first through two right angles. 7. Properties of tJie Excircle deduced from Excircular Func- tions. It is instructive to see the manner in which many of the well- known properties of the hyperbola are at once deducible from the formulae of excircular functions. The following will serve as illustrations and suggestions for the mode of treatment of similar geometrical investigations. The vector equation to the excircle, OP = cosh u + i sinh u, becomes by substituting the exponential expressions for cosh u, sinh u, 138 EXCIRC!ULAR OR HYPERBOLIC TRIGONOMETRY _r .- 2 2 * 1 , 1 -I * ^ = cos i z sin .V -- Take OK = ^ $, that is, a vector whose tensor is -rfj J2 and whose inclination to 07 is half a right angle, and OK' = 5 i~ ' , v ~ that is, a vector whose tensor is f]~ u / ^2 and inclination half a right angle negatively : then OP = OK + OK' and OP is a diagonal of the rectangle OKPK', whose area Observe also that, if u is positive and very large and therefore OP very large, -q~ u is very small and PK (or OK") is very small and may be made as small as we please by taking u sufficiently large : so also if u is very large and negative, PK' is very small and diminishes without limit as u increases numerically. Hence the excircle approaches without limit to OK, or OK' as the point P is taken farther and farther from 0. We have proved therefore that "the excircle has two asymptotes at right angles to one another (hence called the Rectangular Hyperbola), equally inclined to its axis on opposite sides of it, and that if perpendiculars are let fall from any point on the curve to the asymptotes, the area of the rectangle between these and the asymptotes is constant and equal to half the square on the semi-axis 01." Let Q, Q' be two points on the excircle corresponding to the values u + v, u i?of the angles lOq, I0q' on the circle, or shortly the points (u + v), (uv) respectively, then EXf'IRC'ULAR OR HYPERBOLIC TRIGONOMETRY 139 .jjtt + U OQ cosh (u + v) + i sinh (u + v) or OQ = 2- $ + l - j-i, V 2 x/2 __ r) - jj- OQ' = cosh (M v) + i sinh (w v) or #$' = -'-- - i* + -- i~ J. v 2 V 2 ff r is the middle point of the chord QQ', 07=$(OQ + OQ'), .'. 07= % (cosh u + v + cosh u - v) + i . -J- (sinh u + v + sinh u v) = cosh v (cosh ?t + i sinh ) = #/* cosh v, so that the middle point of the chord QQ' lies on OP. In another form 0V = cosh v (-!"- 1* + ^J *-*\ \ , ^ are the inclinations of OP or 0V, QQ' respectively, OP = cosh u + i sinh u = (cosh 2 w + sinh 2 w)i (cos ^> + i sin <) = (cosh 2w)i e^ 07 OP cosh v = (cosh 2w)i cosh v . e 1 '*, where tan ^> = tanh w . F^ = sinh v (sinh M + i cosh w) = sinh v . (cosh 2w)* e'* : where tan \j/ coth u : so that tan < . tan \j/ = 1 or ^ and ^ are complementary, and OP = (cosh 2w)*, OT= (cosh 2w)* cosh v, F^ = (cosh 2)* sinh w. Hence, if POQ, POQ' are equal sectors on opposite sides of OP or (M), ^^' has a constant inclination to the axis equal to the 140 EXCIRCULAR OR HYPERBOLIC TRIGONOMETRY complement of 10 P, and in all positions it is bisected by OP, and OP is therefore termed the diameter of this system of parallel chords. Also 0V 2 - VQ Z = cosh 2w |cosh 2 v - sinh 2 *} = cosh 2u = OP' 1 , whence VQ* = OV 2 -OP 2 = (0V- OP} (0 F+ OP) = PV .. P' \ r , if P' be a point in PO produced such that OP' = OP, which are well-known properties of the rectangular hyperbola. Farther, if R, R' are points on the asymptotes such that OR = -if cosh v 72 . i*, OR' = >;-" cosh v J2 . i~*, and * ^ _ a* v ( JL-ii- 17 , i- V J2 v/2 = cosh v (sinh u-\-i cosh u). ' = OR'-OV= -cosh v - **-' i- cosh v (sinh w + i cosh M). Comparing this with the expression for VQ, it follows that RR' coincides with QQ', and V is the middle point of both RR' and QQ', so that QR= Q^R' = (cosh v - sinh ) JL 1 <- A \ /31 /2 / KQ= -RQ' = (cosh v + sinh v) ( ^" t* - ^75 ' A \ /2 /2 / When V= 0, ^, ^' come to coincide with P, and RR' coincides with LPL', the tangent at P, and since cosh 0=1 and sinh ;= 0, EXCIRCULAR OR HYPERBOLIC TRIGONOMETRY 141 LF = PL = I-- t* -T" -l, OL = rf'J-2 . $ = 2 Off, v - v - 1 017"= >r tt V 2 -* = 2 OJT. Hence T= F# = PL cosh v,OK = OL cosh v, Oft' = OZ 7 cosh v, VQ = PL. sinh v. so that the triangles OKK', OLL', ORR' are similar, with the sides KK'y LL', RR' parallel and in the ratio of 1 : 2 I 2 coshv, and their areas respectively , 1, cosh 2 v of the unit square on 01. Taking the tensors of 0V, VR, VH,', we have OV=VR= VR' = cosh v . (cosh 2)*, and so OP = PL = PL' = (cosh 2w)*. Again OL . OL' = r) u V- 2 # . 17-" s /2 . *-* = 2, hence, if 0^= J2,~OL . OL' = OS' 2 , so that the triangles OZtf, O^S'Z/' are similar, )l_ L \sin ( - v) sin (u - io)l 4 g[Bm.(g-*OBin(s-u>) _ cos \ = cos Isin (M - v) sin (it - w) sin j sin 5. If i u -\- i v = m + ni, prove that sin ( + r) = OTra -j- n 2 m 2 + 2 cos (M - ) = - J - 1, cos ?t cos v = w2 Aj2 and cos 2w -f cos 8c - - 9 (//< 2 + n 2 - 2). 2 2 MISCELLANEOUS EXAMPLES 149 6. If cosh u + cosh v m and sinh u -\- sinh v = ?/, i / \ ~nt it , . . in n 6 sinh (u + v) = ----- , cosh (u-v) = - 1, m 2 n*- 2 /// - |r // - and cosh 2w + cosh 2v = (w 2 - n 2 - 2). m 2 - 2 ., sin 2w + i sinh 2w / . tan ftt + vi) = cos 2w + cosh 2v and tanh ( + ) = ^*i2 M + isin2. cosh 2 + cos2v 8. Prove that (1 + i sinh )* + (1 - sinh v) * = 2 cosh |x/ -4- ( - 1) A | J, where X is any integer : and that the expression has the four values 2 cosh -, 2 sinh -. 2' 2 Verify this by putting iv for . 9. Prove that, if u and u + A are positive acute angles, sin ( -f- /<)/sin u is less, and tan (u -j- /t)/tan M greater than 1 H : and that, as M increases u from to 1 L , sin / decreases continuously from - to 1 or from 1 to 2 -, according as u is measured in right angles or in radians, while tan uju TT increases continuously from - to oo or from 1 to co . 10. If sin v = m sin u, where m is a constant less than 1, as u increases from to 1 L , tanv/tanw decreases continuously from m to and sin (u t>)/sin (u -\- v} increases continuously from to 1. 1 -f- m 11 Prove that ( cos - ) lies between 1 and 1 y, m being positive \ n/ 2ra 2 ' and 6 the measure of an angle in radians : and thence that the limit ($\ n cos-j , when n becomes infinite, is 1. Also the limit of cos- ) = e 150 MISCELLANEOUS EXAMPLES 12. The complete value of i* is a series of scalars in geometrical pro- gression. Does it follow from a 1 = b l that a I? 13. The prime value of (1 + i) 1+f . (1 -i} l ~ i = '410. . . and that of 14. Prove that * + i " * = 2 cosh (1) and i* - i ~ * = - 2 sinh (1). 15. Prove that i m + ni + r m "'"' = \/2 (cos 3m + cosh 2n) i", where tan v = tan m . tanh w. 16. Prove that A- (m + ro) =-- A log, (TO* + n 2 ) + i tan-i - and X,"* "*""' = 2itan-i -. m m m in In these formulae rectangular measure is implied, adapt them to the case of circular measure. Show that the two sides of the equations have the same multiplicity of values. 17. Prove that {cos ( + vi) + i sin ( + vi)} m+ni = ,-(""+){"-". )} m+ni = ^ 1 V+"' . i lo V - where = A/ - (cos 2w -}- cosh 2w) and tan w = tan u . tanh v. m 19. cosh (M + vi) + i sinh (M + W = ^cosh 2w sin 2w . i w where tan w = tanh u tan (w + ) 20. If u, v are the values of the sectors TOP, 10Q of the excircle, and the chord PQ meets the asymptotes in R, R, prove that (!) US = t+ . a , off (2) ra B_^ -'_";-,. (4) PR . PR QR . QR' = cosh (w -f- r) = const, for all chords parallel to PQ. (5) OR . OR' = 2 cosh 2 t -= const, if the area of the sector POQ is constant. MISCELLANEOUS EXAMPLES 151 21. Show that, if a line cuts off lengths h, k from the asymptotes, the vector radii of its intersections with the excircle are \ (h Examine the cases where h, k are of contrary signs, and deduce the properties of conjugate diameters. 22. Prove that (see fig. p. 134) Q,Q' being the points u -\- v , u- respectively OQ . OQ' = cosh 2e + i sinh 2w = OP 2 + 2 sinh 2 f, and OQ . OQ' = {cosh 2 4- r . cosh 2w - }*. 23. Prove that Tt'Q . QR = PL* and R'Q . QR = OP 2 . 24. Prove that OH .OR' =- OR . OR' = cosh^. 25. The tangent at P passes through T. 26. If OP = cos (a -}- ri), cosh c - cos M = IP, cosh t? + cos ** = cosh 2w + cos 2w = 2 OP 2 , cosh 2w - cos 2 = 2./P . IP. 27. If OP = x + yi, 1'P IP I'P 4- /P (1) sin- 1 (z + yi) = sin-i - -f-icosh" 1 - ^- * I'P -ip r p 4- IP /f\\ i / i *\ -..tj-~~ji . . i . j. j. ^^ J.J. (2) cos- 1 (a; -4- yO cos- 1 -- -j-tcosh- 1 - - -- 2 2 (3) tan - 1 (* + y ) = i (PJJ' + PJ'J) + i log, ~~- 28. Prove the identities : y{sin 4a cot (w ii). cot (w v)} = 2 (cos 2w -f cos 2v + cos 2>) (sin 2u + sin 2v + sin 2). iSj cos 4w cot (K? - u) cot ( - r) = 2{cos (2 + i) + cos (2w + i) + cos (2w + i)} {sin (2 4- i) CHAPTER VII ROOTS OF UNITY 1. We have seen (c. ii. 10 and 11) that the n ih root or l/n th power of any (scalar or) vector has n and only n distinct values ; the simplest of which, or that which is first reached in revolving in the positive sense from the prime vector, we have termed its prime value, (this being in the case of a positive scalar its arith- metical value) ; and that the other values are obtained by multiplying the prime value by the n values of the n ih root of the prime vector or 1 . These quantities, the n th roots of unity, have many remarkable properties, the simplest and most important of which we proceed to discuss in the present chapter. i 2 2. Geometrical rejrresentation of 1 * and ( - l) n . Since, when A. is any integer, 01 or 1 = i 4A = cos 4A. + i sin 4A, r '- * ? 4X . 4A Ol n or l n == ^ n = cos - + i sin , n n i so that any value of l n is a unit vector inclined to the prime vector at the angle 4A./, that is, some multiple of one *"' part of 4 right angles. Hence, if the circumference of the unit circle is divided into n equal arcs at points denoted by /, 1, 2, 3 .... (n 1), the radii drawn to these points are the several values of the n th roots of unity, and it is thus plain that there are n, and ROOTS OF UNITY 153 only n, distinct vectors representing the values of 1". Of these values one only, (01) is scalar if n 'is an odd number, and two only, (01 and 01') if n is even. The diagrams, drawn for the J or 2' c;i>f.s, n = 5 and n = 6, make this evident. It should be noted that, when n is even, the extremities of the vectors form a system of points symmetrical with respect to both 10 1' and JOJ', while, when n is odd, the system is symmetrical with respect to 10 1' and not with respect to JOJ' : in the former case too the 154 ROOTS OF UNITY system is symmetrical with respect to the centre 0, the points being the opposite extremities of a set of diameters, but riot so in the latter. i The values of ( 1 )" might in like manner be deduced from the consideration that = cos ( 4X 4 2 ) + i sin (4\ + 2) ; but we may also deduce them as follows. 1 2 The prime value of ( 1) K = i n , and the general value is 2 4A 1_ i*. i n ,so that the system of vectors corresponding to ( - 1) " is i simply that corresponding to 1" advanced through the angle 2/?& L , so that each vector of the former bisects the angle between two consecutive vectors of the latter. These vectors are represented in the diagram above by the broken lines drawn to the points 1', 2' ... n', and it is obvious that, if n is even, none of the values are scalar, while, if n is odd, one only (01') is scalar. The n lh roots of 1 are the roots of the binomial equation z n 1 = 0, so that if n is odd, omitting the factor z 1 corre- sponding to the arithmetical root 1, the non-scalar roots are the roots of the equation z n-l + 371-2 + 2 -3 + . . . + Z Z + Z + 1 = : while if n is even, omitting the factor x z 1 corresponding to the scalar roots 1, the non-scalar roots are those of 3n-2 + ;~n-4 + . . . +. 2* + 2 _|_ 1 = 0. So also the w th roots of 1 are the roots of the binomial equation 2 + 1 = 0, so that, if n is odd, omitting the factor 2 + 1 corresponding to the scalar root 1, the non-scalar roots are those of the equation ~H-I _ jf-a + . . . + 2 -' - s + i = o, ROOTS OF UNITY 155 while, if a is even, there are no scalar roots. Ex. What are the roots of ^ n-i zn-z -|_ . . . -?'* -|- z - 1 = 0, /* being an even number '? 3. The Roots as Powers of the Principal Root. 4A The general expression for the ?i th roots of unity is i n , an ex- pression which has (as we have seen) n distinct values for any n consecutive values of A, repeated in cycles of the same n values in the same sequence, as the values of A are continued upwards or downwards beyond the limits of the first set. These values 4 are the consecutive powers of i n , which may therefore be con- veniently called the principal n th root. Denoting the principal 4 - 4 4 root by a, we have a = i n = cos |- i sin -, n n represented by the vector 01 : and the n roots are 1, a, a 2 . . . , a -i corresponding to the vectors 01, 01, 02 ... (n 1). Since a-* = a".a~i' = a~P, the roots may often be conveniently grouped in pairs, corresponding to points symmetrically situated on opposite sides of the prime vector 01, thus : n-l 2 . , when n is odd 1 when n is even a~S a~- . . . , a It is to be observed for any pair of, a~P, that a p + a~ p = i n + i H = 2 cos f9 156 ROOTS OF UNITY 4 1 _ 4 J? 4;> anda p a~P = i " i n = 2i sin , n as is evident from the diagram. 4. Primary and Subordinate Roots. If n is a prime number, the successive powers of any one of the roots (except 1 ) will determine all the roots in a special sequence. For, taking the root a p , (a p )3 and (a?) ' can only be equal if q = q or pq and pq' differ by a multiple of n, in which case, p being prime to n, q and q' must differ by n or a multiple of n, and this for values of q, q' less than n is impossible. Hence, giving to q the successive values 0, 1, 2 . . , (w 1), we shall have all the roots in the sequence 1, a p , a Zp , . . . , a( n ~ ] ^'; a sequence determined by the successive remainders of p, 2p, 3p . . . (n l)p when divided by n. Thus in the case of = 5, the sequence for a being 0,. 1, 2, 3, 4, that for a 2 is 0, 2, 4, 1, 3 : that for a 3 is 0, 3, 1, 4, 2, or that for a 2 reversed in order : that for a 4 is 0, 4, 3, 2, 1, or that for a reversed in order. If the successive points on the unit circle are joined in accordance with these sequences as in Fig. 1, a determines the regular convex pentagon 71234, and a 2 the regular crossed pentagon J2 4 1 3, while a 4 (or a- 1 ) and a 3 or (a~ 2 ) determine the same respectively traversed in the opposite sense. If n is a composite number, the same reasoning holds good if p is prime to n. But if p is not prime to n, and pfn reduced to its lowest terms becomes p'/ri, (a p )? and (a p )i r will be equal if pq and pq differ by a multiple of ri, that is, if q and q differ by n' or a multiple of n'. In this case the successive powers of the root a? determine only n' out of the n roots. The complete period determined by the first n powers (0, 1, 2 . . n 1) splitting up into subordinate periods of n' roots, repeated as many times as n contains n'. Those roots, whose successive powers determine all the n roots, may be termed primary, while the other roots, which determine only a certain set out of the n roots, (in number a submultiple of n) may be termed subordinate. Then if n is a prime number, all the roots (except 1) are ROOTS OF UNITY 157 primary roots ; but if n is a composite number, there are only as many primary roots as there are numbers less than n and prime to it, and this number is, if n v n. 2 , n z . . are the prime factors of n, (by a well-known formula)* n(l - ) (1 )(! )... "l n -2 n s or, T fc i - 1 n 2 k s~ l . . . (n l 1) (n 2 I) . . . . if n = n^. 2 *2. . . . If p is a divisor of n so that n pm, any p ih root is also an w th root : for if ft p = 1, (3* = ((3 p ) m = 1 = 1. Hence among the secondary n ih roots of 1 will be found all the p ih roots. No primary root of one order can be also a primary root of another order. For any primary root of the w th order is of the 4p form i n , where p is prime to n, and p/n cannot be equal to p'/ri, where p' is prime to n', unless p = p' and n = n. _ ~-\ I or o 9 or * See C. Smith s Alg. , Art. 380, Ch. xxviii., or Hall and Knight, Art. 431, Ch. xxx. 158 ROOTS OF UNITY As an instance, let n= 12 = 2 2 x 3. The principal root is a = z' 4 - = fi = cos - -4- i sin - = ~~ 4- * i 3 3 28 There are 2 1 X 3 (2 - 1) . (3 - 1), or 4, primary 12 th roots, and these are Joining the extremities of the successive powers of each vector in order, we have the regular dodecagon and the regular crossed dodecagon, each taken in the two opposite senses, as shown in the diagram. The secondary roots are a 2 , a 3 , a 4 , a c , a 8 , a 9 , a 10 ; of which a 2 , a 4 , a c , a 8 , - , _ _ _ a 3 , a 6 , a 9 are the fourth roots ', - 1, - i 1 VJF. 1 \/3~. *, o 8 , are the cube roots --- 1 ----- t, - i ; a 6 is the square root - 1. Joining the points determined by these sequences, the first gives (Fig. 2, 2) the regular hexagon, the second the square J//'/', the third the equilateral triangle / 24, and the last the diameter //'. 5. Roots of any order determined by roots of orders, which arc, powers of primes. Let p be prime to q, and let a,/3 be the principal jt> th and q ih roots of unity respectively, then the pq 01 roots of unity are the several terms in the product (1 + a + a 2 + . . . + a?- 1 ) (1 + ft + /? 2 + . . + 09 -i). To prove this, observe that the terms in each factor are all differ- ent and, except 1, there is no term the same in both. Next every term in the product, as a r /3 e , is a p root of 1, these terms are all the pg il> roots. ROOTS OF \JNITY 159 Let r be prime to both p, q and therefore prime to pq, and y the principal r th root. Then the terms in the product of the sum of the pq ih roots by that of the r th roots are, for a like reason, the pqr th roots : so that the pqr terms in the product (1 + a + a 2 + . . + aP- 1 ) (1 + (3 + /3 2 + . + pl~ 1 } (1 + y + y 2 + . . + y- 1 ) are the pqr th roots of unity. It is obvious that this reasoning may be extended to any number of factors p, q, r, s . . . , each of which is prime to the rest. If n is a composite number, whose prime factors are n v w. 7 , n 3 . . . and = // t *i. nj 1 !. n.fa . . . , then n^i, n^, . . . are prime to one another, and therefore, by what we have just proved, the n th roots of unity are all the products that can be formed from the n^'iih, n.^ih . . .roots. Thus since 12 = 2 2 x 3, the products of the 3 cube roots and the 4 fourth roots give the 12 twelfth roots. Again since 360 = 2 3 x 3 3 x 5, all the 360 th roots may be obtained as the products of one of the eighth roots, one of the ninth roots and one of the fifth roots. The multiplication may be performed mechanically thus. Take three equal circles divided the first into 5, the second into 8, and the third into 9 equal arcs. Place the second on the first so that the circles coincide, the zero of the second being made successively to coincide with 0, 1 , 2, 3, 4 of the first, and in each position mark on the first the points coincident with the eight points ot the second ; then the points on the first (none of which will be coincident) will determine the 40 fortieth roots of 1. Do the same with the third, placing its zero successively on these 40 points, and there will then be 360 points on the first corre- sponding to the 360 th roots of unity. This process is obviously quite general, and might be applied to demonstrate the general case discussed in this article. 7. The sum of the n tb roots = 0. The points on the unit circle corresponding to the n roots are symmetrically placed with respect to 01, and also with respect to 160 ROOTS OF UNITY 01, 02, . . : hence their mean point must lie on each of these lines and must therefore be itself, so that 01 + 01 + 02 + . . . + (n - 1) - or 1 + a + a 2 + . . . + a"- 1 = 0. This may also be proved algebraically thus. Let S = 1 + a + a 2 + . . . + a"- 1 ; then 08 = a + a 2 + . . . -f a"- 1 + a n = S, since 'a n = 1 whence, a not being 1, S = 0. 4 4 Substituting for a the trigonometrical form cos \- i sin - . n n we have both 1 -f cos - -f cos - + ...+ cos = n n n 48 4(n - 1) and sin - + sin - + . . + sin ' = 0. n n n The truth of this latter series is obvious, since the terms cancel in pairs, the first with the last, the second with the last but one, and so on. The same is true of the first series, if n is even, the terms in the first half being equal and of contrary signs to the terms in the second half : but, if n is odd, the terms do not cancel in pairs. All this is evident from the geometrical representation. 2 \/5~-4- 1 Ex. Bv the first series prove that cos - = 4 . 5 4 8. Tlie sum of the m ih powers of the n th roots = or n, as m is not, or is, a multiple of n. Let S m = 1 + a + (a 2 )"* + . . +(a n ~ l ) m then a m S m = a. m + (a 2 )" 1 + (a 3 )'" + . . + (a"- 1 ) + (a*)" 1 = S m , since a" = 1 therefore S m = 0, unless a m = 1 , or m = n or a multiple of n. If m = n or a multiple of n, a m , a 2m , . . . each = 1 , so that S n , S zn , >S sn , . . . each = n. ROOTS OF UNITY lei Otherwise, 8 M = 1 + a"' + (a) 2 + (a) 3 + . . . + (a" 1 )"- 1 , whence, if a is a primary root, the n terms of S m give the n ?t th roots, whose sum (by 7) is 0. If a m is a secondary root, m not being prime to n, let m = km' and n = ku', then a k is an ' th root, and therefore 1 + a fc + a 2fc + . . + a< w ' - 1 > fc = and S m splits into k subordinate periods of n' terms, each separately = 0, and therefore S m 0. As in the last article, the following trigonometrical series result : 4m 8m 4( l)ra 1 + cos 1- cos -- + . . . + cos - = generally, n n n but = n, if m = kn ; 4? Sm 4(?t l)z sin + sin - - + . . . + sin = always. n n n 9. The sum of the products of every r of the /t th roots = 0, r beinfj less than n. If ft is any it th root of unit, x n 1 = x n - /?", so that x' 1 1 is divisible by a; ft : hence, a denoting the principal n ih root, a* _ 1= (a; - 1) ( - a)(x - a 2 ) (a - a"" 1 ). Fuller development of the consequences of this resolution into factors is reserved for a later chapter (Ch. X.). Here it is only necessary to observe that, since the coefficients of x n ~ l , x' 1 '' 2 , . . . x 2 , x all vanish in the product of the factors above, and these coefficients are respectively the sums of all the products of the roots taken 1, 2, 3 ... n 1 together, these sums must all vanish. If the product of all the n roots is taken, that product, which is equal to a 2 , is + 1 or 1, as n is odd or even. x' 1 - 1 Further, since (a; a) (u; a 2 ) . . . (x a n - *) = - 1 = B ~~ J. x n-\ + -?i-2 + _ _ _ + x + 1, the sum of the products of the roots, excluding 1, are + 1 or 1, according as they are products of an even number or of an odd number of roots. M 162 ROOTS OF UNITY These results may be seen to be true from the following general considerations. Every product of a number of roots is itself a root. For the product of any two or more vectors of the series Of, 01, . . . 0(nl) gives another vector of the same series. For every term of the product a m , there will be another term a~ m . For instance take the term a? aa r , then the term a n 'P a n -i a n ' r = a~P. a~i. a~ r and these terms will be different and symmetrical with respect to 01, except in the case of p + q + r = n or a multiple of n when the product will be unique and equal to 1 ; or, in the case of n even, when p + q + r = -oran odd multiple e of -, when the product will also be unique and equal to 1. Hence the series of roots obtained from the several products can be arranged in pairs symmetrical with respect to 01 or 1. In like manner they can be arranged in pairs symmetrical with respect to 01, or 02, or O3 . . , and therefore their sum must be 0. The single exception to this is the unique product of all the roots, which = 1 or 1, as n is odd or even. 10. Application of the foregoing results. Having now established the principal elementary properties of the roots of unity, which find useful applications in various branches of analysis, especially in the Theory of Equations, we will conclude the chapter with a few simple instances of such use. (1) Let a be the principal m th root, and /3 the principal n ih root^ of unity, m being prime to n ; then the product of all possible factors of the form aa r + b(3 s , where r may have any value from to m 1 inclusive, and s any value from to TO 1 inclusive, will be a scalar function of a and b. For, the values of a r being the ra th roots of unity, the sums of their products taken 1, 2, 3 . . . n 1 together all vanish, and ROOTS OF UNITY 163 therefore in the product (a + b) (aa + b) . . (aa" 1 ~ ] 4- b) the only terms which have finite coefficients are those containing a m and 6 m , so that, since the product a.a 2 . . a m ~ l = +1, when m is odd and 1, when m is even, (a + b) (aa + b) . . . (aa'"- 1 + b) =- ( - l)"*-^* + b m . For b, write successively 6/3, bfi 2 . . . bft"- 1 in this identity, and multiply the results ; then the product of the mn binomial factors of the form aa r + bfi 8 is equal to Then since n is prime to m, 1, (3 m , /3 2m , . . . /Jfr- 1 )" 1 are all n ih roots and are all different, so that they are the n n ih roots of unity. Hence, reasoning as before, the product last written = (_ l)(i-l) a mn + (- l)(n-l) Jron. Since m is prime to n, m and n cannot both be even. If they are both odd mn is odd, and this result takes the form a mn + b mn ; if m is odd and n even, the form is a mn b mn , while if m is even and n odd, it is b mn a mn . For instance, let m = 2, n = 3 and put vie for. a and v/2/ f r ^> then o> denoting the principal cube root or - H . i, .2 J ( s/a; -f- vy) ( Va; + v y) ( \lx + 3 = 1 and 1 + w + w 2 = : then (x + i/ta + zw?) (x + /to 2 + 2)) = x 2 + y z + z 2 yz zx xy and (x + y + z) (,c + y 2 ) (x + yu? + ZH>) = x 3 + y* + 3 3 3xyz M 2 164 ROOTS OF UNITY Hence X A + y* + Z A 3xys = I) or, arranging according to powers of x, if 03= y 2 or -y<& so> 2 or --yoj 2 zw. Comparing this equation with the cubic equation in the form x 3 - qx + r = y 3 + s 3 = r and Sys = q, whence y 3 z s = \/ r 2 4\*- j 3 - , r and - y =,y -- , whence the three roots of the cubic are expressed in accordance with the solution, known as Cardan's. (3) Gauss's famous discovery of the possibility of dividing the circum- ference of a circle into n equal parts without assuming any postulate of construction beyond the usual conventions of Geometry, when n is a prime number of the form 2 A +1, depends on the theory of the roots of unity. The following outline of the argument will perhaps be of interest to the student. For full details see Peacock's Algebra, eh. xlv. It is shown in the Theory of Numbers that, if n is a prime number, there are always values of m less than u, such that the remainders after division by n of #;, ? 2 , m 3 . . ni n -^ are all different and so form a sequence in a definite order, without repetition, of the numbers 1, 2, 3 . . , n - 1 ; and that the number of such values is the same as that of the numbers less than n and prime to it. Hence, if any one of these values of >n is taken, the series a m ,a" 3 , a"' 3 . . . ,a mH ~ l , where a is the principal ;i th root, determines all the th roots of unity, except unity itself, in a definite sequence ; which has this property, that the substitution of any one of the series for a does not alter the sequence, but merely the point at which it begins. Thus if n = 5, the series a 2 , a?, a'-* 3 , a* 1 * gives the roots in the order a 2 , a 4 , a 3 , a ; and in this putting a 2 for a, the sequence becomes a 4 , a 8 , a r ', a 2 , or, since a 6 = 1, a 4 , a 3 , a, a 2 ; putting a 3 for a, it becomes a, a 2 , a 4 , a 3 and so on. The crossed quadrilateral in the figure exhibits these sequences, in passing round it in the sense indicated by the arrowheads. ROOTS OF UNITY 16". p 2 , . . . p n -i denote the th roots of 1, (excluding 1 itself) in a sequence of this kind, and let o> be the principal iT^T th root of 1 so that w, 2 , . . w re - 2 , and -i (= 1) are the several n-i th roots. Then, taking = pn-2 - + .. . + pn-40) n ~ 3 +pn-3<> n ~ -f- . . . -|_p n _ la) n-3 But f.-i = (W)"- 1 = (w 2 ^)"- 1 = = (co*- 2 /)"- 1 , since w"- 1 = 1, therefore the expression for t n ~\ wliich may be reduced to the form tfj + r/ 2 co + ('30* + + (fn-io> n 2 , is unchanged when the coefficients p r p 2 , . . are changed from term to term, the cyclical order remaining unchanged. Hence a^ (t. 2 , . . e/n-i must be numerical quantities, not involving p 1? p 2 . . . explicitly. In the case of n = 5, t* = - 1 -f 4 is known, it can be expressed as one of the iT^i th roots of an expression of the form a + bi. If - 1 is of the form 2'', = 1) t = p l -f- PZ + . 4 Pn-i = - 1 by the known property of the sum of the th roots. Let T l be the value of tnis root for the root o>, and T 2 7' ;i . . Tn-z, its values when o> 2 , co :! . . to"- 2 arc; substituted for o>, thon we liave the >( i equations Pi + P2 4- Pa 4" + l>n - 1 = - 1 Pi + P 2 W 4- Ps" 2 4- = 4" p/i-lw"-2 = 2*1 Pl 4- P2 2 + P3<" 4 + + Pn-Ko 2 = 7*2 whence by addition, remembering that the sums of the powers of , (a 2 . . . ., as shown on the right hand side of the following, we have (n-l) p 2 = - 1 + 7>-2 + TV 1 ' 3 4- + Tn-w ( _ 1) p s = - 1 + 7>2n-4 + r 2 o,2-C + . . . + T n _vJ &C. Thus the ^I~i roots, different from 1, of the equation ar B - 1 = 0, where n is a prime number, are completely determined,* if the roots of a?*- 1 1=0 are known, and this is the case, as we have seen, when n is of the form 2P + 1. The simplest case is that of p = 2 and therefore n = 5, and the detailed working out of this case will give a good indication of the laborious character of the investigation for other particular cases and of the great care that is necessary to obtain correctly the values of T]T y . . to be employed. The next case is that of^> 4 and therefore n = 17 ; this as well as the case of n = 5, is worked out in detail in Peacock's Algebra with some simplifications in the process, which, (the object of this summary being merely to show the general character of Gauss's investi- gation) it is needless here to discuss. The next value of p, which gives 2P + 1 as a prime number, is 8 and therefore n = 257. * By the algebraical solution of an equation with numerical coefficients is understood the expression of its roots exactly by means of definite arithmeti- cal operations on finite numbers. This excludes the practical solution of the 4\ binomial equation x*- 1 = in the form x i n or cos +isin , from n n which, by the aid of a table of sines, approximate values of the roots can at once be obtained. The difficulty or impossibility of algebraical solutions in different cases is analogous to the difficulty or impossibility of geometrical constructions under the limitations imposed by the conventional postulates of construction in Geometry. EXAMPLES 167 EXAMPLES. 1. The odd powers, from 1 to 2nl inclusive, of the principal nth root of 1, determine all the nth roots, and their sum is zero. Also the sum of the even powers, from to 2n 2 inclu- sive, is zero. 2. The sum of the mth powers of the wth roots of - 1 is zero, unless m is a multiple of n, when it is equal to + n or - n, according as it is an even or an odd multiple. 3. Prove that x 2 + y z + z z - yz - zx - xy is unchanged, if x, y, z are each increased by the same quantity t. 4. If U=x 3 + y^ + z 3 - 8033/2;, and U becomes U', when x + t, y + t, z + t are substituted for x, y, z respectively, then U' = (3m + l)ff, if t = m (x + y + z). [In Exs. 5 to 8 w denotes a, primary cube root 0/1.] 5. Express the product (x + y 2 ) (y + zw + xuP) (z + x) 3 = 3 (x 3 + y a ) and (x + yY + (ajw + yw 2 ) 4 + (o;w 2 + yw) 4 = 1 8a: 2 # 2 , 7. If r n denote the number of combinations of r things out of n, prove that (x + y) n + (tx + u?y) n + (u?x + y + uPz, x + u?y + is 3 (a? 2 + Zyz), and that of the cubes is 3 (x 3 -\-y 3 + z s 9. Prove that (&+^+&-3xps)(af'+y"+'*-3x'y'z) = X s +Y* 4 - if X = xx' + yz' + y'z, Y = yy' + zx' -f z'x, Z = zz' -}- xy + as'y. 10. If w is a primary fifth root of 1, the sum of the squares of + a> + wa> 2 , and aw 3 + yi 4- 2o> 2 + ww 4 is 10 (xz -f yw). 11. If a chain of n links is coiled so that each link is inclined to the preceding at the angle 4/n L , the relative position of the two ends of the chain will be unaltered if the lengths of the links are all increased or decreased lay the same amount, CHAPTER VIII INFINITE SERIES -CONVERGENCY AND DIVERGENCY BEFORE proceeding to those developments of our subject, in which the free use of infinite series is required, it is necessary to consider in some detail the general character of Algebraical Series, whether having a finite, or an infinite number of terms, in the light of Vector Algebra, and especially with reference to the question of their Convergency or Divergency. It is beyond the scope of the present work to give even a summary account * of the modern theory of convergency of series, as established by the investigations of Cauchy, De Morgan, Bertrand, and later writers, and this is the less necessary as, except in extreme cases (which may generally be dealt with by some special consideration applicable to the particular case), the ordinary simple tests of convergency or divergency are sufficient for the series which occur in Elementary Mathematics. These simple tests are derived from an examination of the character of a Geometric Series. We shall begin therefore by a complete discussion of such series. 1. Geometric Series. A Geometrical Progression or Series is one in which each term is formed from the preceding by multiplication by the same * For such an account see ChrystaFs Algebra, Part II. Chap. xxvi. and the references in the historical note in the same chapter. 170 INFINITE SERIES factor. We may, without loss of generality, take the first term to be 1, then if z is the constant factor, the series is 1 + z + z* + . . . + z n + . . . Let S n denote the sum of the first n terms of this series, so that then S n = 1 + z(l + z + . . . + z n ~ 2 ) = 1 + z(S n - z n -*), 1 - z n 1 whence S n = -^ = S- z n S, if S= I -z 1 -z The term, which we have here denoted by *S', is independent of n and is a function, from which by the ordinary algebraical operation of division the series may be obtained to any number of terms. Being then essentially connected with the form of the series continued without limit as to the number of its terms, and without regard to the question of its value (whether finite or infinite), as depending on the value of z, S may properly be termed the "characteristic function" or "characteristic" of the series. Let z be a vector whose tensor is r and versor i u or (cos u + i sin u), then S = S-Sr n i. Now if r < 1, by taking n sufficiently great r' 1 may be made as small as we please, and therefore ultimately when n is in- creased without limit, the term Sr n i nw (since S is finite) becomes evanescent, and the limit of S n is S. The infinite series is then said to be convergent and the finite quantity S, the limit of the sum of the terms as a greater and greater number are taken without limit, is called its sum : so that we may state that, when r < 1 and the series is therefore convergent, the sum of the series is equal to its characteristic. If however r > 1, the term /Sr n i nu increases without limit with CONVERGENCE AND DIVERGENCY 171 n, and as the series has no finite limit, it is said to be divergent and the characteristic is no longer identical with the sum. The case of r = 1 will require more detailed examination, but before proceeding with this, it will be well to reduce the forms of S and S n to the standard forms of a vector or complex number. 2. Reduction of S and S n to vector forms. Since s- 1 - ri u 1 -r cos u - ir sin u therefore (1 - r cos u) + ir sin u 1rcosu . r si (1 - r cos u) 2 + r z sin *u 1 whence, if r sn u cos v sn v 1 r cos u r sin u (I - 2r cos u + r 2 )* cos v + i sin v 1 V - _____ _ - _ nV (1 -2rcosw + r 2 )i (1 - 2r cos w + r 2 )* ' We have thus expressed the characteristic of the series both as the sum of a project and traject, and as the product of a tensor and vestor. The series, being 1 + ri u + rH- u + r' J i 3 " + ...... may be put into the form 1 + r cos u + r 2 cos 2w + r 3 cos 3w + . . . + i(r sin u + r z sin 2u + r 3 sin 3u + . . .) which, if r < 1, is equal to S. Hence, equating projects and trajects, if r < 1, 1 + r cos u + r z cos 2w + r 3 cos 3w + . . . (ac r sin w + r' 2 sin 2w + r 3 sin 3^t + . . . (a C//2 Since OI^I^', OC = CI^ and the angle OCI^OI^I^u, therefore COI 1 = CI 1 = 1 - -, and CI l bisects the angle O/^. 2 Farther CI V / x / 2 being respectively equal to CO V OI V and the included angles equal, (7/ 2 = CI^ and it bisects the angle I^I. 2 I Z . CONVERGENCY AND DIVERGENCY 175 Similarly CI S , <7/ 4 . . . bisect the angles at Y 3 , / 4 , . . . respectively, and therefore C is the centre of a circle passing through all the points 0, J v I. 2 , I 3 . . . Hence the characteristic of the series is the vector OG drawn from to the centre of the circle on which lie all the extremities of the vectors OI V /L/., . . . and of which those vectors are equal chords subtending the angle u at the centre, and forming a regular polygon inscribed in the circle, closed or unclosed, according as u is or is not commensurable with a right angle. 176 INFINITE SERIES u i-- It is plain from the figure that S= OC = J cosec - i -, as we found analytically in the preceding article. Farther S n =OI n = OC + CT n = OC + CO i, since CI n is obtained by turning CO about C through the angle nu : hence H w \ cosec . i. i~* i * - iz nu nu nu i~t ^7i 2 - 1 sn- as before proved. It appears then that $ n is the sum of two unit vectors, one (OC) fixed and the other adding u to its inclination for each in- crease of the number of the terms of the series by one, so that the sum of the series fluctuates about the central or mean value OC, its tensor always lying between the limits and 206' or cosec - . The series cannot therefore be said to be either conver- 2 gent or divergent, since it neither approaches continually to a finite limit nor increases without limit. It may appropriately be termed a, fluctuating series. If u is a small angle, the radius OC is very large, and the limits of the fluctuation are widely apart, while, if u = 0, the circle becomes in the limit the straight line 01 infinitely extended through 7, and the series being then 1 + 1 + 1 + 1+.. .is obviously divergent. As u increases the radius of fluctuation diminishes, until when u = 2 L , C coincides with the middle point of 01, and &n 2 + I *'"" = I + ( ~ 1)" = or 1, as w is odd or even, as is CONYERGEXC'Y AXD DIVERGENCY 177 obvious from the series itself, which becomes in this case the oscillating series 1-1 + 1-1 + .. . As u increases beyond 2 L , C passes to the other side of 01, and the radius of fluctuation increases till at u 4 L , it again becomes infinite, and then, passing to infinity on the other side of 01, repeats the same cycle of variation, if u is supposed to go on increasing. Hence a geometric series whose common ratio is a given unit vectoi-, that is, a series of unit vectors, each inclined to the pre- ceding at the same angle, is neither convergent nor divergent, but fluctuating. The characteristic of the series is the radius of the cii*cle of fluctuation, and is the mean value of the sum of its terms, differing from the sum of any number of terms by one of the vector radii of the circle. The tensor (or modulus) of each sum is the length of some chord of this circle and therefore ranges between and its diameter. In the limiting case, when the angle between the vectors vanishes, the series becomes scalar and all its terms positive, the radius of the circle of fluctuation infinite and the series divergent. The series also becomes scalar if the angle is two right angles, but its terms are then alternately positive and negative, the diameter of the circle of fluctuation is the least possible and equal to unity, and the series is oscillating. Between these two extremes the circle of fluctuation may be of any magnitude, being larger or smaller as the angle between the vectors is nearer to zero or two right angles. The student should construct diagrams for a few cases, where u is a simple fraction, e.g. i, |, |, &c. The diagram below corresponds to the ease of u = ^ L or 150, and it will be observed that the sum of the series has twelve distinct values as the number of terms is increased, the same values being repeated over and over again continually in the same order. It is easy to show that if M = -, where p is prime to q, the sum has 9 either ly or 'lq or q distinct values, repeated over and over in periods. N 178 INFINITE SERIES 4. Tensor ratio < 1. Series convergent. Next consider the series 1 + ri u + r 2 i 2 " + . . ., when r < 1 . Take r. //a - ri, PjjPg = r 2 . 7 2 7 ;i = r 2 P p _~3 / r _ r 3 ,'37t * 3-* 4~~ ' 3*4~ r * ' and so on ; then the polygon OP^P^P^ . . . will have its successive sides parallel respectively to those of OI-^I^I^ . . . , but of lengths diminishing continually in geometrical progression, and therefore lying within the latter, and so inside the circle of fluctuation, as represented in the diagram. Then and the characteristic of the series 1. Series Divergent. Making a construction similar to that in the preceding article, the polygon Op l p 2 . . will have its successive sides parallel to those of O/j/g . . ., but of lengths continually increasing and therefore lying outside the latter, and so outside of the circle of fluctuation, as represented in the diagram. The position of P will be determined as before by the inter- section of the it-circle and the r-circle, but now, r being > 1, .V and N' will lie on the side of M, and the r-circle will intersect the w-circle on the side of the perpendicular to 01 through M. Also since PP n = PO . r n i nu , PP n increases without limit as n increases, and P n continually recedes farther and farther from P. Hence the sum of the series has no limit as the number of terms is increased, and the series is therefore divergent. In this case, although the characteristic of the series ( = OP} is still definite, it cannot be said to be the sum of the infinite series, as it was in the case of convergency, when r < 1. 6. Summary of Results. The foregoing results may now be conveniently summed up as follows : Regarding the terms of a geometric series as a series of an infinite number of links of a chain, if the chain is stretched out to its full extent so that the links are all in the same straight line, it extends to infinity, or the series is divergent, if the ratio (r) of the length of each link to the preceding is greater than or CONVERGENCY AND DIVERGENCY 183 equal to unity, but extends only to a finite distance, or the series is convergent, if r is less than 1. If now the chain is coiled round so that each link is inclined to the preceding at the same fixed angle (u), it still extends in wider and wider circuits to infinity, if r > 1. But if = 1, or the links are all equal, the coils all lie within a certain circle, the circle of fluctuation, the extremities of all the links being on the circumference of the circle. This circle is very large, if the in- clination of each link to the preceding is small, but diminishes as this angle increases, until finally when it becomes 2 right angles or a straight angle, the links are all superposed on the first, which is then the diameter of the circle of fluctuation at its minimum. The series is then an oscillating scalar series, its terms being alternately positive and negative, whose sum oscillates between the two values and 2. Hence if r=l, the series is fluctuating, and its sum has no single limit but fluctuates between a finite or infinite number of finite values within the limits and the diameter of the circle of fluctuation, except in the single extreme case, when the angle is zero or the chain is fully extended, in which case the sum is infinite or the series divergent. If however r is less than 1 or the successive links continually diminish from the first, the fully extended chain extends only to a finite distance, and when coiled as before, its extremity con- tinually approaches without limit to a definite point on that half of the plane which lies on the / side of the line which bisects the prime vector (or first link) at right angles. The sum of the series then has always a definite limit, or the series is always convergent. When the chain is most extended, its length is 1/1 r, and when most compactly packed by the successive links being superposed in opposite senses on the first, its extremity is distant from the origin by 1/1 +r ; in other positions this distance is intermediate between these extremes. Whatever be the value of r, the characteristic of the series has a definite value (OP) corresponding to a definite position of P on the plane, which is on the O side or the / side of the perpendicular 184 INFINITE SERIES to 01 through its middle point according as r is greater or less than 1. In the latter case only however is the sum finite, and therefore quantitatively equal to the characteristic. Infinite Series in general. 7. Definitions. The general form of an infinite series of complex terms ov vectors may be written thus : rj"! + r^-i + . . . + r n i tl " + . . . where r v r 2 . . . being moduli or tensors are positive numbers, and r n , u n are functions of the integral number n, and, it may be, of one or more other variables. If the sum of the first n terms of such a series has a definite finite limit, when n is increased without limit, the series is said to be convergent. If the modulus (or tensor) of the sum of n terms increases without limit with n, the series is said to be divergent. It may be however, as we have seen in the foregoing discussion of a geometric series, that the sum of n terms is always finite but yet has no definite limit, when n is increased without limit, either because the modulus (or tensor) of the sum becomes in- definite, when n is increased without limit (e.g. sin nu), or because its versor is indefinite through its inclination becoming infinite. In such a case we term the series fluctuating, and when its terms reduce to scalars, oscillating. 8. General properties of Infinite Series. A few of the simpler general properties of infinite series may now be stated. Many of these become almost self-evident, if we regard each vector-term as a link of definite length (the tensor of the term) and the series as a chain of an infinite number of such links. If the links have all the same direction ; that is, if u v u. 2 . . . u n . . . are all equal, the chain is extended to CONVERGENCE 7 AND DIVERGENCY 185 its full length, and if in this state the length of the chain is finite, the distance between its extremities must in all states remain finite, however the chain be folded or coiled, or whatever be the values of u r ^t. 2 . . . It is thus obvious that if the series r l + r., + . . .+r n + . . . is convergent, still more is the general series r^i u ^ + r. 2 i u ^ + . . . + ri u n+ . . . convergent. The series is in this case said to be absolutely convergent. A chain of an infinite number of links can obviously be finite when fully extended, only if from and after a certain point on the chain (if not from the beginning) the successive links con- tinually diminish so that the n th link may be made as small as we please by taking n sufficiently great. Thus the series cannot be absolutely convergent unless the limit of r n , when n is infinite, is zero. The converse however is not true, for the series may be diver- gent although lt,, =co (r n ) = or the length of the chain may be infinite, although the links become ultimately infinitely small. A well-known instance of this is the series which, by grouping the terms thus, i+i+a+i)+a+Hi+i)+-.- is obviously greater than 1 + g + f + | + T \ + . . . or 1 + i r + + . . . , of which the sum is evidently infinite. When such a chain is folded or coiled, it will obviously depend on the law of folding, whether it extends to a finite or to an infinite distance. In other words, when lt. n=0 o(r n ) = 0, but 2i (r m ) is infinite, it will depend on the law determining the 00 values of t^, ii. 2 , u s . . . whether the series 2i (?*n* u ") is con- vergent or divergent. 186 INFINITE SERIES A chain may be most compactly coiled by folding each link over the preceding one so that the links of odd order have one direction and sense and those of even order exactly the contrary. It is obvious that in this case, if the links are ultimately infinitely small, the distance between its extremities is finite. In other words, if u^ u 2 = u z = . . . and u. 2 = ii 4 = u G = . . . , but u. 2 = 2 + u v so that the series reduces to *" a ( r i ~ r 2 + r 3~ r 4 + - .) or, if w = 0, to r l - r z + r a - r 4 + . . . , the series has a definite finite limit. Hence an infinite alternating scalar series (that is, one whose terms are alternately positive and negative scalars) is convergent, ij the successive terms diminish without limit. Between the extreme cases of 2i (r n ) divergent and Si {( - !)"*} convergent, in the absence of any general criterion, the convergency or divergency of 2i (*"") must be investigated for each particular case. [As an instance, take the series 1 + \i u + ^i 2u + \i +... in which r n = -, so that r n = ultimately, when n is infinite. 3 When u = 0, the series becomes l+^ + i + j + ... which, as we have seen, is divergent. This is represented in the diagram by the links 01, 12, 23, 34 ... all lying in the same straight line and the chain extending to infinity. When u is a small angle, (in the diagram nearly 10), the CONVERGENCY AND DIVERGENCY 187 chain is seen extending in a regularly bent line, each link inclined to the preceding at the same angle and for the few earlier links getting further away from to a great distance, but then bending round and obviously xiltimately tending to a point at a finite though large distance from 0. When u = L or 60, the chain is coiled as in the figure, and its ultimate extremity is Z, such that (as we shall afterwards be able to prove) OZ= . i*: that is, OZ=^tr . 01 and the angle 10 Z= i L or 30. When u = 2, the series becomes 1 - ^ + J 4 + y ~ ^ + > an d as shown in the diagram, the chain is coiled in its most compact form, all the links lying on 01, the odd links in the sense 01, and the even links in the opposite sense, and the end therefore ultimately approaching the limit Z, such that (as it will be easy to show later on) OZ=log e 2 = -693. . . . Hence the series above is convergent for all values of u, except only for the extreme case, where u = or 4A. L , A. being a positive integer. J 3 - 6| < , A A As a second instance, take the series i u + ^i~ u + ^i u + ^i~ w + ... This like the former reduces to 1+^ + ^ + ^+.. .when u = ; and when u 1, to i(l - ^ + ^ - \ + . . .). In the former case the series is divergent, in the latter convergent. They are repre- sented in the diagram by the chain stretched out to its full length along 01 and coiled most compactly along OJ. For intermediate 188 INFINITE SERIES values of u, it is obvious that the projects of the links form the series cos u(l +% + % + %+), which is divergent, and the trajects the series sin u(l-^ + ^-^+...) which is convergent, and there- fore the series as a whole is divergent. Hence the series is in all cases divergent, except only when u = l L or 2A+1 L , for which values it is convergent, the divergence however being towards a point at an infinite distance along a line parallel to 01 at the distance log e 2 . sin u from it. In these two examples if we regard the infinite series as functions of u, the first is at once seen to be a periodic, function which, commencing with an infinite value when w = 0, is finite and has a continually diminishing modulus or tensor from u = to u = 2 ; a definite finite value, when u = 2, (to be afterwards shown to be log e 2) ; and then an increasing modulus from u = 2 to u = 4, when it again becomes infinite : the same series of changes is then repeated over and over again in successive periods. We may expect therefore that, if an equivalent finite expression for the infinite series can be found, it will be a periodic function of -, which is continuous, except (like cot-) for the values = a \ 2if or 4A.. This will later be shown to be the case. The other series is a function of w, which is finite only when t = 2\+l, and is therefore essentially discontinuous, although its separate terms are continuous, that is, change gradually by infinitely small increments for infinitely small increments of u.] If r n does not vanish but has a finite limit, when n is infinite, although the series of moduli (or tensors) r l + r 2 + ...r n + ... is obviously divergent, the vector series r^i + rfl u - + ... may, as we have seen in the case of the geometric series, when the common ratio is a unit vector or i u , have a sum which is finite, but indeterminate, or, in other words, the series though not con- vergent may be fluctuating. 9. Rate of Convergency of a Convergent Series. Let S n , represented by the vector OP n , denote the sum of n terms of a series, and S, represented by OP, the finite limit of CONVERGENCY AND DIVERGENCY 189 S' n , when n is increased without limit, then R n , the remainder of the series after n terms, is represented by P n P and is equal to S S n . Let R n = t..S.i v where e is a positive number > the modulus of the ratio R n //S or the tensor of P n PIOP: then after a certain value of n (if not from the beginning of the series) e must con- tinually diminish as n increases and ultimately vanish, when n is infinite. According as e is greater or less for a given value of n, or according as n is greater or less for a given value of e, the convergency of the series may be said to be less or more rapid, or more or less slow. Hence the magnitude of e when n is given, or that of n when e is given, may be taken as a measure of the slowness of convergency of the series. As an illustration, con- sider the geometric series discussed in 1 4. In this case R n = S - S n = Sr n i nu , whence c = r n and v = nu. Then if r is a small fraction, for a given value of n, e is very small, and the convergency is rapid : as r increases, for the same value of n e becomes larger and the convergency slower : until when r is nearly = 1, c becomes large, or if e be a given small fraction, n is very large and the convergency excessively slow ; and ultimately it may be said to be infinitely slow, when r=l and the series ceases to be convergent by becoming fluctuating. Since e depends on r only and not on u, all the geometric series which have the same value of r have the same rate of con- vergency, whatever be the value of u. In the diagram of 4. 190 INFINITE SERIES while P moves along the equitensal circle NPN' the rate of con- vergency remains constant, while if P be the centre of a small circle whose radius varies as OP, in order that P n , P n +i, . . . may be within that circle, n must be greater and greater, (or more and more links of the chain must be taken) as P moves on the equiclinal circle I^PC from / x to C, becoming infinite as P comes up to C , or along the circle between /j and C the slowness of convergency varies from to oo.* It should be observed that, since r nu, r becomes infinite with n (unless u 0) and therefore the versor i r indeterminate, corre- sponding to what is in itself obvious that P n P has no ultimate determinate direction. Also that when r = 1 and the convergency ceases, P n P is ultimately finite instead of evanescent, but has no ultimate determinate direction, or, as we have seen, the series is fluctuating. 10. Continuity or Discontinuity of Infinite Series. We have seen, in the instances discussed in 8, that the sum of an infinite series of terms, which are continuous functions of a variable, may be discontinuous for certain values or beyond certain limiting values of the variable. It is important there- fore to examine more generally under what conditions such discontinuity may occur, or rather under what conditions con- tinuity may be certainly predicated. Let z v z 2 , z. A . . . z n be a series of complex numbers or vectors. and let Z n = z l + z. 2 + . . . + z n , and let Z be the limit of Z n , \vlicn n = oo, the series being convergent. Also let x , . ^ . . . be a like series of continuous functions of the scalar t x, which * Continuing this discussion beyond the value r=l, it would appear that the value of e being greater, for a given value of n, the greater T is, this value would properly measure the rate of divergency of the sum of the series from the value of its characteristic function. t For the sake of simplicity we limit the discussion in the text to functions of a scalar, though properly understood the result is equally true of functions of a complex number or vector. The discussion however of the nature of functions generally of a complex number and their continuity is beyond the scope of this work. On this point the student may consult Hobson's Trigonometry, pp. 247 251. rOXYEROENCY AND DIVERGENCY 191 reduce to z v z 2 , . . . z n for the particular value x = a, then, if <"n = i + 2 + - + " an d %' is its limit, when n is increased without limit, the series being in general convergent, the question is : whether we may assert that Z = Z, when x = a1 Stated geometrically, let OP = Z, OP n = Z n , OQ = Z', OQ n = Z' n , then, when x = a. by hypothesis Q n coincides with P n ; can we then say that Q coincides with PI In other words, does PQ diminish without limit, as x approaches without limit to a and therefore Q n to P n ? Observe that we proceed from Z ' n to Z by first taking x = a which reduces Z' n to Z n (brings Q n to P n ) and then making n infinite (bringing P n to P). On the other hand we proceed from Z' n to Z' (from Q n to Q) by first making n infinite : shall we then reduce Z' to Z (bring Q to T 3 ), by taking An instance will show that this is not necessarily the case. Take the - ,. ax* + bxy -f- e// 2 + mx + ny -,- .. function ' - a -?- ----- -Zj regarding x, y as quite m- (t'x* -\- o .ry -\- ejf + m x ~T n y 192 INFINITE SERIES . . . . , ax 2 -4- mx ax + m dependent, rutting 7/ = 0, this reduces to - - or , and a'^ + m 1 * a'x + m' then putting x = 0, it farther reduces to . On the other hand first m putting x = 0, and then y = 0, we obtain successively f and Again, if we suppose y and x connected by the equation y = l:x, the n' , (a + bk -4- dPix + (m + nk) ., . fraction reduces to ,\ ^ , T ^ ' ^^ / ', so that on this sup- (a 1 + "K + ck?).r + (m + n'k) position, when .r = and therefore // = Q, its value is . ' + n'k Thus it appears that, as long as .r, y are quite independent, the value of the function for ,r = and y=-0 is quite indeterminate, and can only be rendered determinate by assuming a particular relation between x and y. For a given value of n, let Z - Z n or P n P = e . i v Z or ei v OP, then, since the series z 1 + z. 2 + . . . is convergent, OP is finite and e may be made as small as we please by taking n sufficiently great. For the same value of n, let Z 1 '- Z ' or Q n Q = ei v 'Z' or e'i v 'OQ, and suppose the series t + 2 + . . . to remain convergent as x approaches the limit a, and also at the limit, when x = a, then e' also may be made as small as we please by taking n sufficiently great. Now when x = a, Z' n becomes equal to Z n or Q n coincides with P n : suppose that Q then comes to Q' so that OQ' = \i. x=a Z' , then PQ' cannot be greater than P n P + P n Q' or e . OP + e . OQ' : but. both c, e' may be made as small as we please by taking n sufficiently great, and therefore PQ' cannot have any finite value, so that Q' must coincide with P, and lt.-B =a Z' = Z. If however as x approaches to the value a, the rate of con- vergency of the series Z' becomes slower, and at last, when x = a, infinitely slow, we cannot assert that ' can be made as small as we please for a sufficiently great value of n and therefore PQ' does not necessarily vanish, and H. x - a Z' may be different from Z. Hence If an infinite series, whose terms arc, continuous vector functions of a scalar variable x, is conceryent for values of x continually CONVERGENCY AND DIVERGENCY 193 approaching to the particular value x = a, for which its terms reduce to those of a definite convergent series, the limit of its sum, wJien x a, cannot differ from the sum of t/te latter series, unless the first becomes infinitely slowly convergent wJien x = a. The foregoing discussion is essentially that given by Prof. Sir G. Stokes (Math, atid Phys. Papers, i. p. 280) in a paper "On the critical values of the sums of Periodic Series," read before the Cambridge Philosophical Society, Dec. 6, 1847. This paper (I believe) contained the first published statement of the dependence of the discontinuity of an infinite series on the infinitely slow convergence of the series for a particular value of the variable. The same result was published in a note by Seidel in 1848 in the Transactions of the Bavarian Academy. The following example adapted from one given by Stokes will illustrate the argument of this article. Consider the two series 1 1 1 "*" 1.2* 23 3 . 4 + 3*)-r ^ x x 4- r (1 + n-U) (!+*> They may be put into the forms V ' 1 + -0 * U -h* " 1 +"2^/ whence it is seen that their sums to n terms are respectively 1 7 j and 1 -r- -, and their sums ad infinitum are each 1 as long as x \-\-nx* differs by however small a quantity from 0. 194 INFINITE SERIES If we add the two series, term by term, we have the infinite series, 2T37T+ x) 1 + ( n i + 3 - 1) a + . - 1 x* n . ( + 1) (1 + ^T"i *) (i 4. jwr) " which is convergent and of which the sum must therefore be 2, what- ever the value of x. If however in this last series we put .r = 0, the series is reduced to the series 2 23 (+!) ' ' whose sum is 1. Hence there is discontinuity when x Q. In fact, using P n , Qn as in the proof above, while H + 1 + 1 ' 1 + nx so that P n P, QnQ differ by the finite quantity p^p , which is nearly 1, when x is small. Hence for a given small value for x (in order to get within a given distance e from the limit), the number of terms required to be taken in the series (3) would be much greater than in the series (1), or the former converges much more slowly than the latter. As x diminishes, the difference in these numbers of terms is greater without limit, or the series (3) converges infinitely slowly close to the critical value of x. 11. The Fundamental Laws of Algebra in relation to Infinite Series. The Associative and Commutative Laws of Algebra are no* necessarily true for the terms of an infinite series, as they are for those of a finite expression. The order of succession of the terms is, in fact, of the essence of an infinite series, so that if that order is changed, the series thus obtained has not necessarily the same value, or does not represent the same function, as the original series. To show this, an instance will suffice. Consider the series z l + z 2 + z. A + . . . + + ..., and form a second series by displacing the terms of even order so that two of odd orders are followed by CONVERGENCY AND DIVERGENCY 195 one of an even order, and let the series be v l + v 2 + v 3 . . ., where v^ = z l + z.^ + z. 2 , v. 2 z 5 + z 7 + z . . ., and generally Then and therefore we cannot assert that 2 1 (m) = S^m), unless lt. ra =o> (2n+2 + 2 2n+4 + . . . + 2-in) = 0. Now if the z series is absolutely convergent, (that is, if 2i'n is CO CO M finite,) S^n or ^(^nt n ) is in all cases finite, the remainder after n terms vanishing ultimately when n becomes infinite, so that in this case the limit above is always 0, and the v series has the same sum as the z series, and the commutative and associative laws do hold. If however the series is not absolutely convergent, so that 2^,1 is divergent, and the convergency of S^n or 2 1 (r n " n ) depends on the law of the values of u v u 2 , u s . . ., it may be that the limit of 2n+2 + #2n+4 + z in does not vanish when n is in- finite, and then the original series and the altered series have different sums. Recurring to the illustration of a chain, suppose the chain to consist of alternately white and black links : then if the order of the links is altered so as to have two white links followed by one black link, it will be obvious that at the end of n such triple links there will be n black links displaced which have not been used, namely, those following the (2w) th link as far as the (4w) th . Now if the chain (though the number of its links is infinite) is of finite length when fully stretched (the case of an absolutely convergent series), both the links themselves and the sum of any number of them after the n th continually diminish as n is increased and ultimately vanish when n is infinite, so that the unused black links may ultimately be neglected. But if the chain, when fully extended, is infinite in length, although the successive links diminish and ultimately vanish, it may be that the n unused black links form a chain of finite length, or even extending to infinity unless suitably folded or coiled. o 2 196 INFINITE SERIES As an instance take the series 1 - A + -J : - 4 + . . . or 2,( ), V > which ( 8, p. 186) is convergent and has in fact the sum Iog e 2, while 2J- ) is (as we have seen) infinite. Then 1 1 1 8n~3 Vn ' + = 4?^ 3 4.n 1 2n (^n and the new series is t~ 5 13 21 ' l3x2 5. 7x4 9.11 x 6 ' ' '' a convergent series, of which the first n terms exceed 3n terms of the original series by ~ - + r -. + ...+ ;. Now this quantity is obviously less than - and greater than , and therefore ultimately, when n ~ co , it has a finite value between ^ and |, so that the sum of the second series differs by a finite quantity from that of the first, although it was derived from it merely by an alteration of the order and grouping of the terms. Again consider the series i-^ + -f + ..., which is shewn below [ 15, (6)] to be an oscillating series, and, without altering the order, group the terms in pairs : we then have the series _L J_ JL _ L_ 1.2 3.4 5.6 ' (2n+l)(2n + 2) ' which is convergent. In fact, since 1 = (l-J^\-(l l \ 2 \ 2n+lJ \ 2n + 2/ -i-i-r-i- \ "+l 2n + 2/' \2n+ the first series is the sum of the second (convergent) series and the oscillating series CONVERGENCY AND DIVERGENCY 197 Thus the law of Association of Terms must not be used abso- lutely in the case of all infinite series. The foregoing instances are sufficient to shew that, unless an infinite series is absolutely convergent, the laws of Association and Commutation of Terms can only be applied upon careful examination of the particular case. Of. Chrystal's Algebra, xxvi. 12 to 14. 12. Power Series. It is beyond the scope of the present work to consider further the subject of infinite series in general ; and we shall therefore confine our attention in what follows to series of the general form as this form includes almost all the series which naturally arise in connexion with our subject. In the above series we shall regard z as representing any complex number or vector, which may be replaced by x + yi, where x,y are scalars, or by ri u , where r is the modulus of z or its tensor and therefore equal to the positive root \'ic 2 + y z , and i u its versor or u its argument measured in right angles. Coefficients , a v a. 2 , . . . we shall regard generally as positive scalars, negative terms in the series being rendered positive by adding 2 (or 4A. + 2) to the index u. In this way an alternating series, for instance, is the same function of u + 2 that the same series with all its terms positive is of u. In a few instances it may be desirable to regard the coefficients also as complex numbers or vectors : in such cases we shall write them explicitly as t a o, a^" 1 , a i a 2, . . . The representative chain of links, when the coefficients are scalar, is (like the geometric series) one that is coiled so that each link makes the same angle with that preceding it, the lengths of the several links being those of a geometrical series altered in given fixed ratios according to the values of the coefficients 198 INFINITE SERIES , j, a. 2 . . . If the coefficients are complex, the coil is further altered by bending at each angular point through the angles a (J , a i a 2 Writing the series in the form + a^i" + 2 r 2 * 2u + . . . + a,,ri nu + . . . it is obvious that, if the series obtained by putting u = 0, (so that it becomes scalar) + 1 r + 2 r 2 + . . . +a n r n + . . . is convergent, still more is the general series convergent. If the chain stretched out to its full length, (that is, so that the links are all in the same line,) has a finite length, any coiling or folding must bring its ends nearer together. Hence, if the scalar series, which we may call the r-series, is convergent for all values of r, so is also the general series, which we may call the (r, u) series. More commonly however the r-series is convergent only when r is less (or not greater) than some fixed quantity k ; then the (r,u) series is certainly convergent when r is less (or not greater) than k, but it may be that when r = k, the (r, u) series is convergent, though the r-series is not so. We shall also show (what is not in itself obvious) that it is divergent, when r is greater than k. It would seem that this last assertion is not necessarily true, when the coefficients are complex. Though the regular coiling (due to the versors, i u , i 2 ", etc.) may not bring the ends of the infinite chain to a finite distance from one another, it is con- ceivable that some farther folding (due to the versors i a i, i ?...) might do so. A circle, whose centre is the origin and radius k, includes the extremities of all vectors from the origin whose tensors are less than k : this circle is termed the circle of convergence for the given series, and its radius k the radius of convergence. CONVERGENCE AND DIVERGENCY 199 13. Continuity of the Power Series. For all points within the circle of convergence, that is, for all values of z, for which Mod() or r is less than k, the series is a continuous function. For, take a value z near to z and also in the circle of conver- gence, then the rates of convergence of the series for both 2 and z are finite, (since, if convergent for any value of r, the series must be still more convergent for a less value of r, the terms in the latter case having all less moduli than in the former), and there- fore, when z ultimately coincides with z, the sums of the two series become equal, by the reasoning in 10. If, however, r = k, so that the extremity of z is on the circumference of the circle of convergence, the rate of convergence of the series for the value 2' may become infinitely slow, as 2' approaches without limit to 2, and then, as we have seen, the limit of the sum of the series for the value z' is not necessarily the same as the sum for the value z. 14. Convergency or Divergency of the Power Series. Taking the series in the (r,u) form, and denoting the remainder of the series after the first n terms by /?, we have and since the sum of a finite number of terms is finite, the sum of the first n terms is finite, and it remains to consider under what conditions /? is finite or infinite, or whether, though finite, it becomes indeterminate. Let = k n , and let the limit, when n is infinite, of - - , which we shall denote as |_ * , be k. Then if n be taken sufficiently great, though still finite, the series of fractions , fL~ . _ ft... i , . . . will be successively nearer and nearer l/k and 200 INFINITE SERIES by continuing the series sufficiently far a fraction will be obtained which differs from l/k by as small a quantity as we please. Hence when n is sufficiently great but finite, .,1 1 <*)+) a n + * n+4 either ->-->->-> . . . > 7 fc n a n+l a n+2 a n+3 k 1 a n +2 n+3 fln+4 1 or T" *c. ~~ *c. - *c. *c. . <. ~ K n Q>n+i O>n+ 2 O>n+3 K T -j.1 **n+2 *+> a n-)-i f *n+o . , j- , In either case, since - . - , is intermediate a n a between =i and I , also ^- 3 = ^ . sl . ^1, and there- V & a n a n+2 a n+l a n fore -^ - is intermediate between 3 and , and so on, a n ; Ar Hence the series will be comprised between the two geometric series and 1 + ~ i u + f- 9 t 2 * + . . . k n k n z But, by taking n large enough, k n can be made as near as we please to k, and therefore the latter series as near as we please to the former ; hence the character of the series-factor of R n as to convergency or divergency is determined by that of T We have seen that this series is convergent, if - < 1 and K r divergent, if - > 1, and therefore, the factor o n r"i n " being finite k and determinate, R n is finite if r < k, and infinite if r > k, CONVERGENCY AND DIVERGENCY 201 Hence the infinite series a + a^i" f a. 2 rH- u -f . . . is convergent, if r < L , and divergent if r > L In other words, the series + a^z + a 2 z 2 + . . . + a n z n + . . . or a + a^i" + . . . + a n r n i nu + . . . is convergent or divergent, according as r< or > L ' ; that is, according as the extremity of the vector z drawn from the origin lies within or ivithout the circle, whose centre is the origin and radius = k or L ' , termed the circle of convergence. Of course, if k = 0, the series is divergent for all values of z ; while if k = x> , it is convergent for all values of z. The remaining case, where r Jc, that is, mod. (z) = radius of convergence or the extremity of the vector z lies on the circum- ference of the circle of convergence, requires further examination. 15. Case where the tensor = radius of convergence, or r k. If r = k or L * , the limiting geometric series of the series- +! factor of R n become 1 + i u + i- u + i Su + ... k . F . A; 3 . and 1+ * + ?*+--*.+ ... n K n the former of which is, as we have seen, a fluctuating series the 1 u radius of whose circle of fluctuation is - cosec , and which in A 2i the limiting case of u = becomes divergent ; while the latter is convergent or divergent as k n is greater or less than k, that is, as the successive terms of , - - , . . . continually approach n+! + 2 without limit to k by decreasing or by increasing. 202 INFINITE SERIES Also we have R n = a n k n i \ 1 +-" +1 . H + - n+2 Fi 2w + . ( a,,. (hi First, suppose L(a n &") to be infinite, then R n is also infinite and the series is necessarily divergent. Next, let [_(a n k n ) be finite, then R n will be finite or infinite as its series-factor is finite or infinite. Now if k n > k, the series-factor of R n is comprised between a fluctuating and a convergent series, and therefore cannot be divergent, but it may be fluctuating. On the other hand if k n < k, the series-factor of R n is comprised between a fluctuating and a divergent series, and therefore cannot be convergent, though it may be fluctuating. Whether R n is actually fluctuating or not requires examination in each particular case. If it is fluctuating, 1 u the ultimate radius of fluctuation = \_(a n k n ) ~ cosec -, 2i 2i and the magnitude of this radius may be taken as a measure of the deviation of the series from convergency. Lastly, let L(a n & n ) = 0, then if k n > k, so that the series-factor of R n is either convergent or fluctuating, R n ultimately vanishes, when x co , and the series is convergent, the radius of fluctuation diminishing as n increases and ultimately vanishing. If however k n < k, the expression for R n , in the case where its series-factor is divergent, takes the form x GO and no general conclusion can be drawn. The foregoing reasoning supposes u to have a finite value different from 0. In the extreme case, where u = 0, the radius of fluctuation becomes infinite, the circle of fluctuation a straight line and the fluctuating series divergent, and the series itself the scalar series of ordinary algebra with all its terms positive. In this case, unless \_aje n = 0, the series must be divergent, as is obvious, since the series becomes then + a ik + 2 & 2 + . . . + a n k n + . . . Whether, when [_a n k n = Q, the series is actually convergent or CONVERGENCY AND DIVERGENCY 203 divergent must be determined by .such tests of a simple character as may be found applicable to the particular case, which are given in the usual text-books of algebra, or by the more refined tests of Cauchy, De Morgan, Berfcrand, 1, divergent. If r=l, the series is 1 +2i u + 3i Zu + . . ., which is obviously diver gent, as also appears from the fact that L (#&") = L (n + 1) = oo . Here k = L = 1 , and the series is convergent or divergent, as r is less or greater than 1. 204 INFINITE SERIES If r = 1, the series i u + | i 2 " + \ i 3u + . . . , and since 1 n the series is convergent : except only, when u = 0, in which case the series is 1 + \ + \ + . . ., which as we have shown above (8) is divergent. (5) 1 + -| z + 1 . z 2 + . . . or 2^ . . Here # = l_ = L -^-7 - ^ = 1, so that the series is n / n + 1 ?i (n + 2) convergent, if r < 1, divergent if r > 1. If r= 1, the series is . = therefore the series cannot be divergent. It is in fact fluctuating, for it may be put into the form (1 + i u + i 2u + i 3u + ...) + (i* + \ i 2u + & i 3u + ...), that is, a fluctuating series + a convergent series or finite quan- ni tity : and the radius of fluctuation = | cosec - 2 If u = 0, the radius of fluctuation is infinite, and the series, which then becomes l + 2 + f-f| + . . ., is divergent, as is evident. If u = 2, the series is 1 2 + f *- + . . ., which being put in the form (1 1 + 1 1 + ...) (! ! + ! + ...) is obviously an oscillating series, the limits of the oscillation being log e 2 and 1 log e 2, since (as will be proved) log e 2 = 1 3 + 3" i + - n / n ( 6) i+j.+t*+. :.+, SFT -V. . . r a, ( j Here, as in the last example, k=l and the series is convergent if r < 1, divergent if r > l r CONVERGENCY AND DIVERGENCY 205 n(n + 2) n It r = 1, k = -j-* .r ' < i or K, and l_ ((t n k ) = L_ , = 1, therefore the series cannot be convergent. It is in fact fluctuat- ing, for it may be expressed as of which the first member is fluctuating and the second conver- gent. If u = 0, the series is l + 2 + f + f + . . ., which, being greater than 1 + i + 1 + ^ + . . . , is obviously divergent. If u = 2, the series is 1 + f + .. . and its sum oscillates between 1 log c 2 and 2 - log e 2. /7\ * z " JL z * * 3 o a n + b r l+'b + a + b a n+l + f t Here k = L T - =a, if a > 1, but =l,ifa< 1 or =1. a. If a > 1, the radius of convergence being =a, the series is convergent if r < a, divergent if r > a. Oi n CL n ~^ 1 + b If r - a, \_ a n k n = \_ -, = 1, and k n = > 1 so that a + o a n + o k < 1 : hence the series cannot be divergent. The series being K,i 1 ai u a 2 i zu in this case - - -\ --- H + . . . may be put into the 1 + 6 a + o a* + o form (I +i + i 2u + . ..)-(>(- -+-i t ' + _ --t2 + .. \ \l+b a + b a* + b ) of which the first member is fluctuating (radius of fluctuation u\ = cosec - j and the second is convergent, since it is the series itself when ? = 1 , which is less than the radius of convergence, a ; 206 INFINITE SERIES so that when r = a, the series is fluctuating. In the limiting case of u = 0, the series is 1 a a? i i - 1 O . T I ' l+b a + b and is divergent. ft. If a=l, the series reduces to a geometric series, and is therefore convergent, fluctuating or divergent, as r <, = , or > 1. a n+l _j_ y. If a < 1, &= L - j- 1, and the series is convergent if Ot "T" r < 1, divergent if r > 1. 1 1 a' l+1 + 6 If r-1, Lo^-L^j^p and *. = -^; 6 & > 1 : hence the series cannot be convergent. In this case the tc n series is i~TT i l+o a + b a in which latter form the second member is convergent, since a < 1 and the first fluctuating. Hence the series is fluctuating, and the ultimate radius of fluctuation is cosec -, which becomes '<> 2i infinite when u = 0. EXAMPLES 207 EXAMPLES. 1. Using the figure and notation of 2 and 4, (1) Prove that ~RP=!L- iu+2t> 1 r 2 , and - 1~OP 2 -1 /l + 2r = o = ~o \f 2(AP' 2 ^ l-2r 2rcosw + r r sin u 2r sin zt where tanw= - and tanw;= - . 1 rcosu l-r j (2) If S=pi v , prove and illustrate by diagrams that a. riw + p-ii- = l. 0. p-2 = l-2rcosM + 2 and ?- 2 =l -2p- y. ~ cos M = 1 + - cos v + . cos 2; + . . . r p p^ - sin = - sin v + - sin 2r + . . . r p p j ?* provided r < 1 and cos u > - . 2 8. - cos w sec u-\ - cos 2 sec 2 u -\ cos 3 sec 3 -} cos 2. - sin u sec M + sin 2 sec 2 u -\ -- sin 3u sec 3 u + . . =sin 2w. 2 2 2 2 3 (3) Shew that 0, Pj, P a . . . are points on an equiangular spiral, whose focus is P. (4) Determine P by a construction from the expressions for the project and traject of OP. (5) Given the position of P determine the series by geometrical construction. 2. Prove that, z denoting the complex number or vector ri u , the radii of convergence of the several series 2f > (na2 n ), 2!(-^ ), S^w 1 \ 71 / are equal to that () of the series 2^ (on^ 71 ). 208 EXAMPLES Also that, if 2f(a n ,?n) is convergent when r=k, so also is 2( I 2 re ), \ n > while 2 (nctnZ n ) may be fluctuating or divergent : and that, if 2(an2 n ) is divergent, so also is 2 (mtn2 re ), while 2"> \ z n ) may be fluctuating * \ n * or convergent. 3. For the following series find k (the radius of convergence): and determine, when r = k, if the series is convergent, fluctuating or diver- gent. In the case of fluctuation, determine the radius of fluctuation. Examine specially the limiting case, when u Q or the series becomes scalar. (1) 1 + 1.2^ + 2.3^+.. . + W (7i + l)*'* + . .. \ / T c. * i n ~ i ~T~ _ ~, r ~r 2.4 2.4.6 For L 2V4T6T8 I v O EXAMPLES 209 1 . 1 . 3 z' 2 . 1 . 3 . 5 z 3 . (19)2-|j-,~}. (See Hall and Knight's Higher Algebra, p. 246, ed. 1887.) (20) 1 + " : ^ z + q - + 1 -/ 3 -g+. 1 02 1-7 1.2.yy+l , a . a + 1 . a +2 j./B + 1 . ,,. CHAPTER IX EXPANSIONS AND SUMMATIONS IN the ordinary treatises on Elementary Algebra, the develop- ments of a few finite expressions into infinite series of terms are established for scalar values of the symbols : in the present chapter the extension of these to vector values or complex numbers, and some important results thence deducible will be considered. It will be convenient to premise the following general method of obtaining Trigonometrical Series from any given Algebraical Expansion. 1. Trigonometrical Series deducible from Algebraical Expansions. Let z denote any function, which can be developed by algebraical processes into the series (finite or infinite in the number of its terms) . . . + a n z n + . . . Then z, having been proved (it may be) to be the generating function of this series with some restriction on the meanings of some or all of the general symbols involved (as, that they are scalar, or positive, or integral), must by the principle of the " Permanence of Equivalent Forms " (see Introd. p. 4) be also the generating function of the same series universally ; and understanding the sign of equality in this sense, we may assert for all values of z, that z = a + a^z + a.j.z 2 ' + . . . + a n z n EXPANSIONS AND SUMMATIONS 211 It must always be borne in mind however that, in the case of the series consisting of an infinite number of terms, quantitative equality can only be asserted when the infinite series has been shown to be convergent. Let z be a vector or complex number, denoted by x + iy or ri u or rt i9 , where x, y are scalars, r, a positive scalar, u, 0, the measures in right angles and circular measure respectively of the inclination of z to the prime vector, then z = (ri u ) = + api* + a. 2 rH- u + .. .+ a n r n i nu + .. . = ct + 1 r(cos u + i sin ?t) + ... + ar n (cos nu + i sin nu) + ... = a + a^r cos u + a. 2 r z cos 2u + . . . + a n r n cos nu-\- . . . + 1(0,^ sin u + 2 r 2 sin 2u+ ... + a n r n sin nu + . . .) Let (frz or (x-\- iy) be reduced to the form U + iV, where U, V are scalars, then, equating projects and trajects of the vector z, (7 = a + a^r cos u + 2 r 2 cos 2u + . . . -f a n r n cos nu+ ... V = a-p sin u + 2 r 2 sin 2u+ ... + a n r" sin nu + ... The quantities U, V are termed conjugate functions, and the two equivalent series conjugate series. This result may be used, as will be shown in the sequel, either to obtain new trigonometrical series from a known algebraical expansion, or to sum certain given trigonometrical series by means of a known algebraical summation. 2. Newton's Binomial Theorem. The theorem is that m.m-\ m.m-\..m-n+\ j-g -z z +---+ j 9 a"+... The right-hand side of this equation has a finite number of terms if m is a positive integer, but in all other cases it is an infinite series. It is proved, in the case of m a positive integer, by considering the product of the m factors, l+z v 1 + 2 2 , . . . , 1 + z m , in which P 2 212 EXPANSIONS AND SUMMATIONS it is clear, from the process of multiplication, that the terms consisting of the product of n z's will be as many in number as there are different combinations of n out of the ra letters z v z 2 . . . , z m . These terms will all reduce to z n , if each of the quantities z v z 2 , . . , z m is equal to z, and their sum will then be m .m- 1 ...m-n+i m . ., . ,. ,, Nm . , ,, z n , so that the series tor (1 +z) m will be that 1 . 2...n written above, terminating with the term z m . In the foregoing result, deduced directly from the fundamental laws of combination of algebraical symbols, there is no restric- tion whatever on the meaning of z ; and the restriction that m is a positive integer may be removed by the consideration that, the form of the equation being perfectly general, and the meanings of indices in the successive cases of fractional, negative, or complex indices having been obtained in accordance with the same fundamental laws as were shown to hold for positive integers, the result must, by the Principle of the Permanence of Equivalent Forms,* be true without any restriction as to the value of m. It is however to be specially observed that in passing from the case of m a positive integer to any other case the series changes from one of a finite number of terms to an infinite series. Hence, though the formal equality of the finite expression and the in- finite series, regarding the latter as derived from the former as its generating function, will still hold, it cannot be asserted of the series regarded as the quantitative sum of its terms that the finite expression and the series are quantitatively equal, unless it is proved that the limit of the sum of the terms of the series is finite and definite, or, in other words, that the series is convergent. In fact (as has been shown in the preceding * It is usual to give a distinct proof of the theorem for fractional and negative indices, and the one most commonly adopted is that known as Euler's proof. An examination of this proof however will show that it does in fact assume the principle we have just used, though perhaps in a case in which it is easier for the beginner in Algebra to see its truth than when asserted as above in its complete generality. EXPANSIONS AND SUMMATIONS 213 chapter 11) it cannot be asserted of series, which are not absolutely convergent, that the fundamental Laws of Algebra hold without limitation. 3. Convergency or Divergency of t/te Binomial Series. Confining ourselves to the case, where in is a scalar, the general term being m. (m- l)...(m n+ 1) * the ratio of the n + 1 th term to the n ih is m-n+l f m+l\ ( m+l\ . or 1 1 ri u or I 1 ri u + z . n \ n J \ n / In this last form the coefficient (1 ) is positive for all \ n I values of n greater than m + 1, and its limit, when n = o>, is 1 : hence, by the test established in the last chapter, the series is convergent if r < 1, and divergent if r > 1. If r= 1, the remainder after n terms is equal to the product of (l - ?!+l) ( 1 - "II- 1 ) . . . (l - "^1} *> V l/\ n / \ n / and a series, which ultimately, as n is increased, approximates without limit to the series which, except in the case where u + 2 = 4X, (X being an integer) is a fluctuating series, whose radius of fluctuation , u + 2 u = xcosec - =sec-- 214 EXPANSIONS AND SUMMATIONS Hence the remainder is ultimately a fluctuating series, whose radius of fluctuation is the limit of Now, since the series (m +!)(- + - + . ..H --- h . . .) is divergent, \1 2 n (cf. 10, Ex. 4 in the last Chapter), the infinite product of factors * in the expression for the radius of fluctuation is 0, if m+l is positive, and oo if m + I is negative. Hence in the case of r=l, the Binomial Series is convergent, if m+l is positive, and divergent if m+l is negative ; while, if m + 1 = 0, it is truly fluctuating, since in this case the series reduces to the case of a Geometric Series, of which a unit-vector is the common ratio, discussed in the last Chapter. In the exceptional case of r 1 and u = 4A - 2 (X integral), 3_U-2_ _ i } so that (1 +z) m = (l - l) m = or oo, according as m is positive or negative. The series in this case becomes m . m - 1 m.mlm - 2 1.2.3 and we may expect to find that it converges to 0, if m is positive, and is divergent, if m is negative. Taking the sum of the first n + 1 terms, it may readily be proved that 1 . m.m-l...m * See C. Smith's Algebra, Chap. xxvi. or Chrystal's Algebra, Chap, xxvi 2227. EXPANSIONS AND SUMMATIONS 215 For if the product on the right-hand side of this identity is denoted by u n , 1 - m 2 - m n-l-m -in 12 n-l = (-!)" but u n = u l + (u. 2 -u l ) + (u. A -u 2 ) m .m 1 m.m .m . m l...(m n+ 1) Now the product u n , when n is increased without limit, is (as we have seen above) ultimately or GO, according as in is positive or negative ; therefore the infinite series above converges to 0, if in is positive, and is divergent if m is negative. 4. Illustrative Diagrams. The preceding section may be well illustrated by a few diagrams drawn for particular cases, when r = 1 or z is a unit-vector. 1. Let in = or 4, then the series are 1 IN 5 - 3 -2 a 5 . 3 . 1 M[f 5.3.1x1 . 4it 2* 2. 4* 2.4. 6* ~ 2.4.6.8* 5.3. 1x1.3.. -1 i ou &c. 2.4.6.8.10 !_? 5 _'T-' M _ 5 _:Il?-3 5.7.9.11. 4M - 2*" + J . i* jM " 2TO* " + 2.4.6 . 8 l "~ l T ' 216 EXPANSIONS AND SUMMATIONS The diagrams representing the first four terms of these series in the case where u = |, are figs. 1 and 2 respectively, the succes- sive terms being OA V A^A y A 2 A 3 , A 3 A, in each, and OA^ there- fore their sum. The remainder after the first three terms of the first series will be a vector drawn from A s to some point within the circle of fluctuation passing through A 3 , A 4 and divided by 2 L 4 L A S A into segments containing the angles , respectively. o o The limit of the series is, by the Binomial Theorem, (1+^)5, which reduces to 3i$ and is represented by OZ, such that 6^=3-93 and ZOA 1 = '75 . It appears that Z lies within the circle through A 3 , A v the radius of which (being equal to ^^'- Q % sec - = ^3 = '18 nearly) EXPANSIONS AND SUMMATIONS 217 affords a sort of measure of the degree of approximation obtained by taking the first 3 or 4 terms. The convergency of the series is thus obvious. FIG. 2. The generating function of the latter series becomes (1 or 3~U~*, and its divergency is evident. (2) The cases represented in figs. 3, 4 are those of 1.1. 3. . 1.1.3.5. n _ 4 -u _ I _ i A _ I.'.. 1 2 * O _ . _ __ 2.4.6 2.4.6.8 4 _ taken for the value M = . The values of OZ are i~$, $ respec- tively. Observe the slowness of the convergence in fig. 4 as FIG. 3. 218 EXPANSIONS AND SUMMATIONS compared with fig. 3. As u diminishes, this slowness increases, until finally, when u 0, the second series becomes divergent. FIG. 4. 5. The Binomial Series equal to the prime value of (1 +z) m . The expression (1 +z) m is, as we have seen (p. 45), a function which has in general an infinite number of values, which how- ever are all coincident, if m is an integer, and reduce to a finite number of distinct values, when the index is scalar, and a finite fraction. The Binomial Series however is a single-valued func- tion. It is necessary therefore to determine to which of the values of (1 +z) m the series must be taken to be equal. It will be sufficient here to confine our investigation to the case, where w is a scalar. Let 1 + z = pi, then pi v 1 + ri u 1 + r cos u + ir sin u, so that p cos v = 1 +r cos u and p sin v = r sin u, whence r sin u p (l -f 2r cos u -f r 2 }* and tan v = 1+rcosu It follows that sin u sin v sin u cos v cos u sin v sin (u - v) p r p cos v - r cos u 1 and if r be not greater than 1 (as must be the case for conver- gency), since sin (u v) lies between 1 and + 1, sin v is limited EXPANSIONS AND SUMMATIONS 219 to lie between r and r. Also p and 1 + r cos u being both positive, cos v is positive, so that the value of v (within the limits 2 L and + 2 L ) is a positive or negative acute angle, whose sine is numerically not greater than r. If this angle is denoted by v a , the general value of v is v + 4X, (X integral), hence the general value of (1 +z) m is p m i''o+4A i if m is integral, this has only one distinct value ; but if m is a finite P fraction = -, where p, q are integers prime to each other, it has q distinct values corresponding to the values 0, 1, 2...g 1 of X : while, if m is incommensurable, it has an infinite number of distinct values for each different value of X. Now if u = or a multiple of 4 L , v = 0, and both z and the binomial series are scalar ; but p m i^ m , when m is not integral, is scalar only if X = 0, if m is incommensurable, or if X = or some /ft multiple of q, if m -. Hence, if u = 0, X must be taken as : and ? as v changes continuously with u, X must also be taken as 0, when u has any value. The annexed diagram will throw light on the foregoing dis- cussion. In this 01 is the prime vector, IP = z = ri u , so that OP = 1 + z, and to represent the case of a convergent series for which r < 1, IP < 01. Then OP = p and the angle 70P = v L , and since P is some point on the circle, radius r and centre /, or A PA', the limits of p are OA, OA' or 1 +r and lr, while if OB, OB' are tangents to the circle A PA' drawn from 0, OP lies between OB and OB, and the limits of v are the positive and negative angles BOI, B'OI, whose sine (BI} is equal to r. Hence 1 + z m or OP m will be one of the vectors OQ , OQ V OQ 2 . . . whose tensors are each p m or OP m and inclinations Q OI=m . POI, Qfll = m . POI+ 4m L , Q 2 OI= m . POI+ 8m L . . . respectively. (The diagram is drawn for m = -.) But it is clear from the figure that the limit of the sum of the vectors in the Binomial Series, 220 OI + m.IP~+ EXPANSIONS AND SUMMATIONS IP 2 + . . . is the vector 0# and not OQ V 1.2! or OQ% . . . , so that the series is equal to p m i mv o, v being the acute angle POL FIG. 5. Ex. If m < 1, the locus of Q corresponding to the circular locus of P, when r is given, is an oval lying within the circle Examine the nature of the locus for values of m > 1. 6. General Form of the Binomial Theorem. The foregoing results may be thus summarized. If z = ri u , and 1 4- z = pt"o, where p 11 + 2r cos u + r 2 ]i, and v is a positive or negative acute angle, such that tan v n = \ +r cos u the Binomial Theorem that m .m 1 .. .m-n - 1 . 2 . . . n EXPANSIONS AND SUMMATIONS 221 1 . -i m.m 1 . . . w TO+ 1 . + - - r**"" + ... 1 . 2 . . . n is quantitatively true (the infinite series being convergent) if r < 1 : but not quantitatively true (the infinite series being diver- gent) if r > 1, unless m be a positive integer, in which case the number of terms of the series is finite. If r = l, the theorem is quantitatively true, provided m > 1 ; but not so, if m = 1 or m < 1, for then in the former case the series is fluctuating and in the latter divergent : with the single exception that, if u = 4A. + 2 (A. integral), it is quantitatively true, if m > 0, but not so, the series being divergent, if m is negative. If the index m is a (vector or) complex number, it may be shown' just as for scalar values of m, that the series is convergent or divergent? and therefore quantitatively equal to the prime value of (1 -f- z) m or not, according as r is less or greater than 1. The case, where r=l, requires farther examination. Denoting the successive coefficients of the powers of z by l m , 2 m , 3 OT ..., we have to inquire under what conditions the series is quantitatively equal to (1 + &")"*, when m=p. -f n/, where /i and v are scalars. _ m.m 1 . . ,m n-\-\ The coefficient of i nu or n m = may be con- 1 . z ... n veniently written in the shape (m + 1 1 --T- Let the successive factors of this product be fc^i, & 2 z' A 2, . . kni^n, so that whence 222 EXPANSIONS AND SUMMATIONS and -/*-! then u + 1 Now, when p is large, \ p is small and kP = l- - very nearly neglecting the terms in its expansion containing powers of - higher than the first, so that after some large but finite value p, we may write ( P ~t~ ^\( P' ~^~ ^-\ But the series 11 I 1 n increases without limit with w, hence the product k p kp+i . . . k n is infinite, when n is increased without limit, if jt + 1 is negative, and 0, if /* + 1 is positive : therefore the series is convergent if p. > 1 and divergent if /* fact, as will be shown hereafter (see Chap. x. 7), - ; therefore the limit of the product Te^ k 2 . . . k n is finite and the series is ultimately truly fluctuating, the radius of fluctuation being ultimately _ _ _ _ / sinh 2 v u / n 2 " - n - 2 " u / - . i sec 3 or / 5 -- X | sec -, V TV V "TTJ/ which is finite except only in the case of u = 4p -j- 2 or = - 1. EXPANSIONS AND SUMMATIONS 223 The preceding investigation will be elucidated by the following geometrical interpretation : OI = 1, IM^ = p, M l ATi = vi, /A, = JJL -f- vi and OKi = 1 + /* -\- vi. Then if OK y OK 3 , 6>A' 4 ... are respectively , i, J . . . of OA^ or 1 + m, 1+TO - and so on, . OM nearly, when /j is large, and TO = . Now however large OM or p. + 1 may be, only a finite number of A''s fall without the unit circle, centre /, and for those within the circle 7Ap, 7Ap+i. . . are all less than 1, tending to 1 as their limit, but the 224 EXPANSIONS AND SUMMATIONS limit of their product is 0, since they differ from 1 by a quantity of the order -. Hence the remainder of the series after n terms tending always to the product of a fluctuating series and a zero radius of fluctuation is convergent, as long as p, + 1 is positive. If /i + 1=0, M coincides with 0, and K, K v A 2 ... are all on the tangent to the unit circle at and so are all without the circle, but IKp tends to the limit 1 as p increases continually, differing from 1, when p is large, by a quantity of the order - 2 , and consequently the limit of the product IKi . IK 2 . . . IK n is finite, so that the series is truly fluctuating, the limit of this product being the ultimate radius of fluctuation. If p. + 1 is negative, A', K^ K 2 . . . are all without the circle on a secant, and IKp is greater than 1 by a quantity of the order -, in which case the limit of the product 7A'i . IK 2 . . . I K n is infinite and the series is therefore divergent. 7. Trigonometrical Series derived from the Binomial Theorem. If we replace i u , &c., by De Moivre's form cos u + i sin u, &c., we find after rearrangement, and using the notation * l m , 2 TO , 3m ... for the coefficients of the series, p" 1 (cos m v + i sin mv ) 1 + 1 TO r cos u + 2 TO r 2 cos 2u + 3 TO r 3 cos 3u + . . . + i |l m r sin u + 2 TO r 2 sin 2u + 3 m r 3 sin 3u + . . . j, whence TO (1 + 2r cos u + r 2 ) 2 cos mv = 1 + \ m r cos u + 2 m r 2 cos 2u + 3 m r' A cos 3 u + . . . and m (1 + 2r cos u + r 2 ) 2 sin mv = l m r sin u + 2 m r 2 sin 2u + 3 m r 3 sin 3u+ ..., true for all values of m, provided r < 1. * This notation was suggested by De Morgan, to be read 1 out of m, 2 out of m, &c., being, when m is a positive integer, the number of com- binations of 1, 2, ... things, out of a set of m things. EXPANSIONS AND SUMMATIONS 225 If r = 1 , p = |1 + 2r cos u + r 2 H = (2 . 1 + cos tU = 2 cos - > sin u u tan v ft = - = tan -> 1 + cos u 2 and these series become (v and therefore - lying between 1 and 2 + 1), (u\ "* / u\ cos - ) cos ( m - } = 1 + l m cos u + 2 m cos 2w -j- 3 m cos 3w -f . . -i/ \ 2/ (A*\ 7/1 / rtf^ cos - ) . sin ( m - I = 1 m sin w+ 2 m sin 2ti 4- 3 ;n sin 3?t + . . . ^/ \ a I true, provided m > 1 . Ex. 1. Deduce from these, when in is a positive integer, the formula; of Chap. IV. 10. Ex. 2. If m be the measure of an angle in right angles, prove from the above that 2 cos m - -l. 2m-2 - + &c., 1.2.3 and verify the result independently. 8. Exponential Theorem. The Exponential Theorem asserts that for all values of z (scalar or complex), a z is quantitatively equal to the infinite series ) 2 z n log e a) n +...+ -- 1 f- Q 226 EXPANSIONS AND SUMMATIONS It may be deduced from the Binomial Theorem as follows : First Proof. If x is a positive scalar < 1 (or a vector, whose tensor < 1), it is quantitatively true, as we have shown above, that 1.2 -(- -!)...(- -n+1 x \x . 2 ...n z(z-x) z.(z-x) ... (z-n-lx) - -' + ... +- -' 1.2 1 . 2 ... 74 whence, if denote the n ih term of the series, z n lor u n +i = . n and for any finite value of n, % It. X O u n+} =~ -It. x =0 u n> so that z z 7 .a;=0n- - 7- K 1 TO ri in 2 z z z ...- It. a; O^D but tj = 1 , therefore 1.2.3...^ Also since w n +i = \ x ~ ~ ) M ' ^ n ^ e taken so large that is very small compared with x, the sum of m terms after the EXPANSIONS AND SUMMATIONS 227 w th term approaches without limit to u n (l x + x z ...( x) m ~^) or 1 _ / , _ /-\ m ftfi u n - :p , which ultimately = , when a; = 0, and therefore is 1 + X Yflt ultimately evanescent, when x = and n is infinite. - z z n Hence \t. x =o(l + x) x = 1+2 + n 1.2 Putting 2=1, a series whose limit ( = 2-718281828...) we have already denoted by e : whence, since ( Ll* l J =lt. a!= o(l+a;) a! - 2 Z Z" i72 + '" + |w" and, since a z = (e lo g a )* = e* l Se, Second Proof. The theorem may also be proved, assuming the Binomial Theorem for a positive integral index only, in the same manner as it was used in Chap. VI. to obtain the value of e. a x l We have seen that the limit x= o = log e a, so that if x e zx -\ a e z , limit 3=0 - = x **-! ~ Let = z and therefore e z = (1 +zx) x , then e is the limit of x c and a that of e 2 , when x vanishes. Q 2 228 EXPANSIONS AND SUMMATIONS Let x = , where m is a positive integer, then <* = a finite series of m + 1 terms. Now (see ch. v. 7) A l\/ t 2\ / t r-l\ /I 2 r-l\ (1-- (1-- )...(1 -)< 1 and >!-(- + -+...+_ ) V mj \ in/ \ m / \m m m / r.r-l ~&T' T . 1' 1 it may therefore be taken to be equal to \6 r - , where O r is between and 1. Hence <-!+.+ + + + - + + ... But the infinite series l+z+ + + ... is convergent for all I I values of z, and therefore 3 , 4 ... being all positive and less than 1, the series 1 + 6 3 z + 4 ^ ^ + ... is also convergent, hence as m i.s 1.2 increased without limit, "the product -- ( 1 + 0z + 6, ^ ^ + - 2m\ * 1 . 2 ultimately vanishes, and as e then becomes e, we have, for all values of z, EXPANSIONS AND SUMMATIONS 229 whence, as before, e = 1 + 1 + - + - - + ...= 2'71 8281828. . . 1 . 2i 1 . i , O z z (log e a) 2 and a' = I+zloa+--z L +... Third Proof. The following proof, independent of the Binomial Theorem, though following the lines of Euler's Proof of that Theorem, is modified from Cauchy. 2 2 Z n Consider the series I +z + ~~~ + ... -\ --- f- . . . , which we have 1.2 in shown (ch. viii. 16 (2)) to be convergent for all values of z, and let it be denoted by./ (2). Then also f(y) = 1 + y + ^ + . . . + p + . . , and if the product L . I /^ fy . fz be obtained by actual multiplication, it is easily seen that the term of the n, th degree in y, z, is n.n-l \n ... ... .2 \ n ~P\P or Let e be the value olfy, when y= 1, and therefore then 230 EXPANSIONS AND SUMMATIONS and f(z + m) = e .f(z + m - 1 ) = e z f(z + m - 2) = . . . e m fz, whence, putting 2 = and observing that /(())= 1, fm = e m , or, if m is a positive integer, m 2 m n e m = 1 + m + + . . . + + . . . 1.2 | Now this equation, being perfectly general in its form, must by the fundamental law of the permanence of equivalent forms be true for any value of m no less than for m restricted to be a positive integer, hence for any value of z whatsoever z z z n e* = l+ z + - + ...+ - + ... 1.2 \n It must be observed that in its most general sense e z has an infinite number of distinct values for the same value of z, while the series above has but one value, but since the series is scalar, when z js so, that value of e z must be taken in the equation above which becomes scalar with z. Now (Chap. V. 10) the general value of e z is e x . e { y . tj-^v . i^ x , A. being any integer : and when y = 0, so that z = x (a scalar), it becomes e x i 4 * x , which i? scalar, only if 2\x is integral ; a con- dition which can only be satisfied in all cases, if X = 0. Hence the fundamental vector e x . e*v or e x (cos y + i sin y) is that value of e z , which is equal to the infinite series. This is the vector 00 in the diagrams of Ch. V. 10, 11, in the case where OC = e*, IOC = y radians, OH = ^~ 4 y, IOH=kx right angles. 9. Series (deduced from the Exponential Series} for sin 0, cos 6, sink 0, cosh 0. Let z = x + iy = r(cos + i sin 0), (using circular measure), then . e l > si e = e rnls# |cos (r sin 6) + i sin (r sin #)} EXPANSIONS AND SUMMATIONS 231 r 2 and e*=l+r(cos6 + isinO)+ -. (cos 20 + it sin 26 + . . .) 1 . a = (1 -f-rcos0 + -cos20 K. - .) + i(rsin6-\- ^ sin 20+. . .) whence, equating projects and trajects separately in the two expressions for e z , r 2 r u e rcose cos (r sin 0) = 1 + r cos + 7 - cos 26 + . . . -\ cos nO + . .. 1.2 \n and = r s i n + s i n 2^ + . . . + - sin n6 + . . . , 1.2 ]ri in which, as in the exponential series, the series are convergent for all values of r. From the above equations, putting = (4A+1)-, so that 2i cos = 0, sin 0=1, cos nO cos 4A. + 1 . w- = 0, if n is odd, or (-l) 2 ,if n is even, n-l , x sin nO sin ( 4/\ + 1 . n- J = 0, if n is even, or( - 1) 2 , if n is odd, V 2/ we have the following important series for the cosine and sine of an angle, _ r 3 r 5 , mr = r - + -"-- where ? is the circular measure of the angle. Again, since a 2 Z 3 3 4 ^ 1+ , + _ + . + _ + ... 232 EXPANSIONS AND SUMMATIONS 234 and therefore cosh z = - = 1+.-+-+- + . - -T" 0+ g + g*i7" + - , From these formulae for cosh z, sinh 2, putting iz for 2, we obtain, since cosh iz = cos z and sinh iz = i sin 2, COS =1 -- ;; + -7 --- (- |2 |4 |6 "ilTil" 1 " as we found otherwise above. It is worth observing that, since sin z, cos z are periodic functions, the equivalent series are also periodic. It would be difficult to see d, priori from the series themselves that they are periodic, but that they may be so is plain from the alternating character of the series ; for if each term represent the length of a link of a chain, however much the successive links may increase in length at first, after reaching a certain link of maximum length they begin to diminish and ultimately become infinitesim- ally small, and it is plain that if such a chain is folded over into its most compact form, which is that represented by the series for sine and cosine, there may be different values of z, the quantity determining the lengths of the links, which will bring the second extremity of the chain ultimately to the same point at a finite distance from the first. The series for cosh z, sinh z are clearly non-periodic ; in fact, they represent the chain stretched out to its full length. EXPANSIONS AND SUMMATIONS 233 10. Logarithmic Series. a x - 1 We have seen (Ch. V. 6) that It^o = log e a : hence, if for (l +zY - 1 a we write 1 + z, log e (1 + 2) = \t x= o - Now by the Binomial Theorem, if r, the modulus (or tensor) of z, < 1, ,. * x 1 x.x 1 x .x - I . . .x-n+l 1.2 1 . 2 . . . n ~ therefore (!+)*-! 1.1-*., l.l-aj.2-! 1.2 1.2.3 .1.1 ...(n 1 ) -'" - Zn + "' Now if in this series we put x = 0, it reduces to 9 - 3 * z- - but to show that, subject to the condition that the modulus of z is less than 1, this latter series is equal to log,, (1 +z), the limit- (1 + Z Y -I ing value of - , when x = 0, it must be proved that X the former changes continuously, as x diminishes withoiit limit. (See Ch. VIII. 10.) 234 EXPANSIONS AND SUMMATIONS Now if P n , Q n denote the remainders of the two series after n terms, n/ n + I \ V n + l X n+lj\ n + 2Jn n+l n+l ^ z + n+l\ n + 2 n and the series factors of both P n , Q n ultimately approach without limit, as n is increased, to 1 x + z 2 z 3 + . . . or, since Mod. (z) < 1, to the finite quantity . Hence if we take n I + z to denote such a number of terms that Mod. Q n = e, a numerical quantity which we may make as small as we please, Mod. P n = f (I -i very nearly, so that Mod. P n and Mod. Q n differ by a very small quantity for values of x near to 0. Hence the rates of con- vergency of the two series, measured by the number of terms necessary to be taken to get within a given distance e from the limits of their sums, are very nearly equal, and therefore the second series is the limit of the first, when x vanishes, and we have lo ge (l+z)=z-? +?-?+... quantitatively true, when Mod. (z) < 1. It is evident also that in the extreme case of z=l, the formula is true, the series being still convergent, so that but in the other extreme case of z= 1, it is not quantitatively true, the series factor of P n , Q n being in that case infinite, so that EXPANSIONS AND SUMMATIONS 235 the reasoning above would not hold. The complete logarithm or logometer of a quantity is, as we have seen, a function having an infinite number of values, one only of which is scalar when z is scalar, and all are (vector or) complex, when z is so. Hence the series is equal to the arithmetical logarithm in the case of z scalar, and the fundamental or prime value of the logometer, which we may conveniently distinguish as the logarithm, and which is such that its traject lies between TT and IT, in the general case. The series for log e (l +z), being divergent for values of z > 1, can only be used directly for the calculation of logarithms (to the base e) of numbers not greater than 2, and this not conveniently on account of the slowness of its convergence. Other more con- vergent series however may be derived from it, as we will now concisely indicate. Changing the sign of z, we have * - z log, (!-.)=-*------ --- and since log. If 1 + ', m 1 z n' whence m fm n 1 /mn\ 3 1 /m w\ 3 \ n \m n 3 m n) 5 m n) 3 \m + n 5 \m + n a rapidly converging series, when m and n are considerable numbers differing by a much smaller number. If m n + 1 , we have 23(5 EXPANSIONS AND SUMMATIONS a rapidly converging series for the difference of the Napierian logarithms of two consecutive numbers. If n = 1 , we have l 1 1 11 11 which gives log,, 2 = -6931472... This the student should verify for himself, as also the following. Taking m = 10, n = 8, - = -, and since log, 8 = 3 log, 2, m + n 9 which gives log e 10 = 2-3025851... and therefore This last number, the value of - - , is the factor by which Napierian logarithms must be multiplied in order to obtain the common logarithms to the base 10, and is termed the modulus of that system. To adapt the formulae above to the calculation of common logarithms, it is only necessary to prefix the factor /* ( = -4342945) to the series. 11. Series (deduced from the Logarithmic Series) for tan '.<. tanh~ l x. Let z = re l6 = r (cos 6 + i sin 0} and 1 + 2 = pe 1 * = p (cos + i sin ) so that p cos Q + 2kir), EXPANSIONS AND SUMMATIONS 237 where < lies between TT and TT, and \ (1 + z) = r 1 and have any value except TT or an odd multiple of TT. Hence, equating projects and trajects in the two expressions for X e (1+z) and taking & = for the reasons given above, log e p = log e {1 + 2r cos 6 + r 2 }* = r cos - r 2 cos 20 1 and if r= 1, has any value in the latter series, and any value except an odd multiple of TT in the former. If Q = -, p cos = 1 , and p sin< = + r, so that < lies between - and - ; also cos 2kO = cos kir = ( l) fc , cos (2 +1)0 = 0, sin 2k6 = sin for = 0, sin (2& + 1) = (- l) fc , and the above formulae reduce to - -... A3 a*'} rt*7 , = ,--+---+... true for values of r ^ 1, whether positive or negative, so that Tan -I r, denoting an angle between - and -, must lie between -i 2i TT IT -jand-- The first affords only a veriiication of the general series above being obviously true from tae series for log e (1 + z), but the other 238 EXPANSIONS AND SUMMATIONS is an important result, known as Gregory's series for the cir- cular measure of an angle in terms of its tangent. The series may also be deduced thus through the series for tanh" 1 x. If _e*-e- z _ e zz -l ^ l+x CC DELUll & o ~ j Q ' 2 e* + e - z e^+l lx so that 1 +x z or tanh" 1 x = ^ log,, = -- 1- km, 1 X where k is an integer, whence 05 s C 5 tanh" 1 x = x + ++...+ km 3 5 or Mod. x not > 1. Now tan" 1 ^^ - therefore / fy fw) \ tan- 1 a; = -i. \ix-i + i -= - . -I- V 3 5 = *-- + - . . . + for o O or x 3 a? 5 Tan- 1 x = x- - + --..., 3 5 Mod. x not > 1. 12. Calculation of tlie Value of ir. Gregory's series furnishes the readiest methods of calculating tlie value of ir. If x = 1, since Tan -1 1 = -, the series gives EXPANSIONS AND SUMMATIONS 239 111 222 _ = l__-l u = ---- 1 ------ H ------ u . 4 35 7 1.35.7 9.11 a series which converges too slowly to be convenient for calcu- lation. Since 1 1 Tan-^ + Tan-'^Tan- 1 - - f-Tan-11-j 23 4 6 - /I 1 1 1 I \ , f l _ l i , l 1 \ 4 = \2 ~ 3 ' 2 3 5 ' 2 5 ' 7 \3 3 ' 3 3 5 ' 3 5 which is known as Euler's series, and is more rapidly convergent. (Eight terms of the first part and five of the second give the result correct to 6 figures. The student should verify this.) A still more convergent series, known as Machin's, is obtained from the fact that -. differs little from 4 Tan- 1 - 4 O For ** iJ- and therefore J-4 Tan- \- Tan- ^ The. use of this formula is facilitated by observing that so that -l-Tau-.i. 240 EXPANSIONS AND SUMMATIONS In 1853, Mr. Shanks published the value of TT calculated to 607 decimal places, with which the independent calculation of Dr. Rutherford to 441 places agrees as far as it goes. The figures may be found in De Morgan's article in the English Cyclopaedia, " Quadrature of the Circle." To 20 places the result is TT = 3-14159265358979323846 13. Series for sm-^x, cos-^x, sinh~ l x, cosh-^x. These series can most readily be obtained by the use of the differ- ential calculus, though they may be deduced from the series* in C. IV. 9. Since (sin - 1 ;?.) = and (einh-i x) = dx VI -#2 dx expanding by the Binomial Theorem, we have, d .. .\ ,1 , , 1.3 4 . 1.3.5 , . ,1.3...(2-1) , (sm- 1 x) = 1 H x* H -- x* 4- -- ar> -|- . . . 4- x 1 dx 2 2.4 2.4.6 2.4...2 -. -. -. dx 2 2.4 2.4.6 ' 2. 4... ft. whence, integrating and observing that the value of sin-Jar between - and + - and the scalar value of sinh - 1 x vanish with x, the circular 2 2 measure of .1 ar 8 . 1 . 3 .r 6 , 1 . 3 . . . (2 - 1) a?2+i bl n-i. = .+- . _+__ .-+,.. + -27^^- ' a^Ti +' and i r s i Q r s 10 (%*, _ T\ ? -2n+i 1 .i.O ^ .. ,i.t>... \Zl L .U . Sec Johnson's Trigonometry, Arts. 529 and 637. EXPANSIONS AND SUMMATIONS 241 Now these series are obviously convergent within the same limits as those whence they are derived, since the coefficients of the corresponding terms are less in the former than in the latter, so that from this it appears that the series for sinh- 1 x is convergent both for .? 2 < 1 and x z = 1, and that for sin- 1 .? for .r 2 < 1. But it may be shown that this latter is also convergent for # 2 = 1, though the series whence it was derived is not so. For, when x = I, the general term of the series is . , and from Wallis's Theorem (proved Ch. X. 10) the limit of -- ' \/2# + 1, when n = oo, is finite and = \f - 2 . 4 . . . 2 * TT therefore the general term approximates, as n increases, to /? 1 X/ 2 ' (2 + l)i* and the remainder of the series after n terms approximates to a finite quantity X I 1 H h . . . \ But this \(2* + 1)1 T (2 + 3)3 T (2fl + 5)1 ^ I series is the sum of certain of the terms of the series + -J- 1- . . lm ' <2 m 3 m 3 for the case of m = -, in which case, m being > 1, the series is con- m vergent (Smith's Alg., Art. 270), so that the remainder of the series after n terms is convergent, and therefore the series itself convergent. Hence the formulae above are both valid, if x z < 1 or x 2 = 1, and therefore for sin- 1 :? between the limits - and - both inclusive, and 2 2 for sinh- 1 .? between the limits sinh- 1 (-l) or log e (\/2-l) and sinh -1 1 or log e ( v 2 + 1), both inclusive. Also, since Cos- 1 .? = Sin- 1 .r, and cos- 1 ^ changes from to TT, m while sin- 1 x changes from ^ to ^, 2t 2t TT lar 5 1.3 a.- 5 (Jos- 1 .r= x . ... 2 23 2.4 5 where x may range from 1 to -{- 1 inclusive. Since Cosh- 1 .? = * Cos- 1 *-, the above would give . / 7T . . 1 X s . 1 . 3 X 5 . \ Cosh- 1 .r = (-- + #+ \- \- ... I V 2 232.45 / true from x = - 1 to x + 1 incl., corresponding to the fact that for values of x numerically less than 1, cosh- 1 .? is non-scalar. When cosh l x becomes scalar, the series becomes divergent. 242 EXPANSIONS AND SUMMATIONS To obtain a scalar series for cosh- 1 .?, it may be expanded in a series of negative powers of .?, thus : .. x \ 2 .r 2 2 . 4 .v* whence by integration, if x 2 > 1, and k some constant, 1 1 1.3 1 , = * + log.*--. _. ir4 -... l". 3...2-l _ _ 2 . 4 . . . 2 ' 20, .r2 If .r = oo, = It. {cosh- 1 .r log e .r} = It. {log^, (.r -|- v r 2 1) log, .rj therefore cosh- 1 * = log e 2# - - . s - . ... 2 2.r 2 2 . 4 4.7* 1 . 3 . . . 2 - 1 11 . . OEC. 2 . 4 . . . 2 2 .r as long as cosh - ' x is scalar. This formula is valid not only for x > 1, but also for x = 1, since in that case cosh ' x 0, and it reduces to 1 1.1.3 1.1.3.51. 10g ' 2 = 2 2 + 2-TT 4+2T4T6 ' 6 + &C> and this series is convergent, as may be proved by considerations similar to those given above in that for sin- 1 1. 14. Summation by means of the preceding series. We have hitherto used the method explained in 1 for the development of series from finite expressions : we shall now give some examples of its use for the summation of certain trigono- metrical series. . , cos 20 cos 40 cos 60 (1) To sum the series 1 H I I r^ h . . . 2 |4 |o EXPANSIONS AND SUMMATIONS 243 Denote the sum of this series by U, then if V denote the sum of the conjugate series, sin 26 sin 40 sin 60 ~~~ ~~~ ~~~ whence r+ iV= 1 + - (cos 20 + 1 sin 20) + - (cos 40 + i sin 40) + ... g2t0 g 4t'0 = 1 + |2- + - j + = cosh (e) by 9. gcosfl ^ gi sin _j_ g - cos >e {sine Now cosh (e' fl ) = cosh (cos + i sin 0) = a gcos 9 i g - cos gcos _ g - cos cos (sin 6) + i sin (sin 0) 2i a = cosh (cos 0) cos (sin 0) + i sinh (cos 0) . sin (sin 0). Hence U= cosh (cos 0) cos sin or | (e cose + e~ cose ) . cos (sin 0) V= sinh (cos 0) sin (sin 0) or | (e cos0 - e - cos ) sin (sin 0), so that we have siimmed the conjugate series as well as the given series. (2) To sum the series x cos + a? 3 cos 30 + -5 a: 5 cos 50 + . . . , where x ^> 1. Taking U= x cos + ^x? cos 30 + ice 5 cos 50 + ... and V= x sin + ^ sin 30 + }y? sin 50 + . . . .- = tanh- J (a:e ie ) by 11. Now tanh- 1 (a"e fe ) = i log .^ = | log 5 l-a-e' S l-a:cos0-7a;sm0 or, making the denominator scalar, = i log -- j- l -2x cos + x" R 2 244 EXPANSIONS AND SUMMATIONS so that i&uh- l (xe) ie = \ log (pe 1 '*) = -} log p + i . ^, '2i I - x 2 2x sin it p COS = -5- > 1 -a? 2 so that . , 1 + 2x cos 6 + x z 2xcosO U = i> log p = 1 log -- - - s^Atanh- 1 r-> 6 1 - 2 COS ^ + X Z 1 + C 2 2o; sin (3) To sum sin + sin 30 + sin 50 + ... to n terms. Taking V n = sin + sin 30 + . . . + sin (2n - 1)0, U n = cos + cos 30 + . . . + cos (2n - 1)0, piO _ ->i(2nf 1)9 1 _-,2w.70 YT * ff -*/ OJA . j/._ i \/i ^ ** K. _ (1 - cos 2nO) - i sin 2nO _ sin 2nO . 1 - cos 2w$ - 2i sin ^ ~ 2sin^ + "~2"rinT~ sin 2w0 1 - cos 2w^ sin 2 nO Hence c/ TO = ^r-^ -, V n = =-. = . 2 siri ^ 2 sin sin Ex. 1. Deduce from the above that Ex. 2. Illustrate the above by diagrams drawn for particular cases, e.g. = -, ^g* c EXPANSIONS AND SUMMATIONS 245 15. Sum of selected terms of a known series. If the sum of the series a + i + a^K 2 + ... is a known function of x, the sum of every r th term of the series beginning with any term, or the sum of a p xP + a p+r &v +r + a p +2 t xP+ 2r + , may be found by a genei'al method dependent on the properties of the roots of unity (Ch. VII.), which will be understood, without a formal statement in general terms, by a few examples. (1) To sum the series 1 + 3 n x 3 + 6 n a; 6 + 9 n oj 9 -f . . . , where !, 2 n , 3 ra . . . (read 1 out of n, 2 out of n, &c.) are used to denote the coefficients of x, x 2 , x 3 . . . in the expansion of (1 +x) n . The three cube roots of 1 are i$, i', 1, and if we put o> = t*, then and generally 1 + a> r + or r = 0, unless r is a multiple of 3, in which case it = 3. We may apply these well-known results thus (1+ *) = 1 + \ n x + 2 n x* + 3 n x 3 + . . . ) n = 1 + l n coa; + 2 n o Adding these,_the coefficients of x, a: 2 , and any power of x not a multiple of 3 vanish, by the properties of o> stated above, so that (1 + x) n + (1 + was)" + (1 + oAu) = 3(1 + 3 n a* + 6 n x 6 + 9 n a + . . .) 27T . 2- 1 \/3 but w = i* = cos + 1 sin = - - + - t, o o 2> 2i so that 1 + wx = 1 - + i ._ 2 2 x J3 = (1 - x + a 2 )i (cos + i sin <), where tan -^- 3C So also, since w 2 = - -- i, 2i 2i 1 + o) 2 a; = (1 - x + a; 2 ): (cos - i sin <), 246 EXPANSIONS AND SUMMATIONS n Hence ( 1 + tax) n + (1 + wV) re = (1 - x + x r f . 2 cos n, and therefore 2" 2 / 7T\ If aj=l, l + 3 ( , + 6 n + 9 n + ... = T + -co8(w - ) o o \ o/ a result which it would be well for the student to verify by particular values of n. (2) Tosn*-l-jg+-j+... Here the sum of every fourth term of the series for e x , beginning with the 2nd, is required. The four 4th roots of 1 are i, 1, i, 1, ic 4 a/' X G x 7 x s and writing successively ix, - x, -ix for *, we have, since i 2 = - 1 , $ = - i, &c. x* or 5 x 6 .x 7 x 8 - + |T + '|2-|3- 4 |4 + |5 + "- a; 4 x 5 a; 6 a; 7 as 8 --- + -_- + - + ... a; 4 .a; 5 a; 6 jx 7 x 8 __.___ + ._+_ + ... whence by addition 3,5 3.9 IK + FS + } ID iy EXPANSIONS AND SUMMATIONS 247 ar* a; 9 or 2 jsinh x - i 2 sin x} = 4{a; + + + . . ./ I I or - (sinh x + sin a;) = x + + + . . . 2 ] .' a result easily verified from the expansions in 9. x 7 x ls x 19 (3) To sum x+ - + _+_+... we have x a; 2 x^ x* or* x 6 x~ 1 . log (^- ~^~x) = I -+ 7 ~t" e 7r"l"i; ... 234567 2 L : 2 L 1 J3 Now, if CD = t'i = i* = cos =* s i n ^ = + * n > the sixth-roots of o 322 1 are 1 , CD, ft) 2 , 1 , 2 #, and adding, {log (1 + a;) + to- 1 log (1 + wa;) + w" 2 log (1 + w 2 a;) - log (1 - a;) 5 - ft)- 1 log (1 - x - . or (since w~ l = - - i~-, to" 2 = - - - 1^- j, to 1+a; 1 (1 + us) (1 - ft) 2 a;) . v ^3 (1 - ma;) (1 - a> 2 a;) g 1 - a; + 2 g 1~- oxc 1 + 2 a; - w 3 a; 2 = 1 + a; + x 2 248 EXPANSIONS AND SUMMATIONS and (1 - wx) (1 - uPx) = 1 - (to + (a z )x + w 3 a; 2 = 1 - x z - ix J3 and (1 + toa?) (1 + w 2 *) = 1 - x z + ix x /3, therefore the expression farther reduces to 1, the series is divergent and cannot be summed. Ex. Shew that ITJ + H-B+-- BT- 1 "^*^ 16. Summation of Trigonometrical Series. Certain Trigonometrical Series can be summed by the general method of differences, the application of which to Algebraical Series the student will have seen in ordinary Algebra (see C. Smith's Algebra, Art. 313). If u n is the w th term of a series, w : + u z + . . . + u n , and u n can be expressed as the difference of two consecutive values of another function (v) for the values nl and n of the variable, so that these values being denoted by v n -i, v n respec- tively, u n = v n v n - 1, then EXPANSIONS AND SUMMATIONS 249 and therefore Wj + U. 2 + . . . + U n = V n V , . so that a finite expression is obtained for the sum of n terms. The full development of this subject is to be found in treatises on the " Calculus of Finite Differences." We will shew by a few examples how the method may be applied to obtain series from certain finite expressions, and inversely to sum certain series. 1. Take the formula A + Q sin# = 2 cos sin - 2 2 Then if -6 = 2a sin (6 + 2a) sin = 2 cos (0 + a) sin a, and so .sin (d + 4a) sin (6 + 2a) = 2 cos (6 + 3a) sin a, sin (0 + 2wa) - sin (0 + 2w la) = 2 cos (0 + 2n la) sin a, whence .sin (9 + 2na) sin 9 = 2 sin a (cos + a + cos + 3a + . . . . cos (0 + no) = cos (0 + a) + cos (0 + 3a) + . . . Mil Ot la). Hence, if 6 = 0, cos a + cos 3a + . . . + cos 2n la = sin (2a) 2 sin a j -c /i "" sin 2 (a) and if 9 = -, sma + sm 3a . . . +sm '2n la = '. 2 sin a Again, if no. = TT, cos ( 9 + ^ ) + cos (e +) + ...+ cos ( + 2n 1 - ) = 0, \ n/ \ n) \ n) 250 EXPANSIONS AND SUMMATIONS sn n Ex. Verify these last results by a diagram, and the previous results by taking a = 0. 2. From the formula sin(<4 <9) tan A- tan = COS I {/ _ \ / _ Q\ I cos (0+ 2n -3 -) -cos (0 + 2-l -)[ \ A/ \ A/) EXPANSIONS AND SUMMATIONS 251 /a, / a\ = cos ( - ) cos ( + 2n 1 - ) V 2/ \ 2/ . na L a) = 2 sin . sin \0 + n 1 -}, 2 ( 2) na sin + sin (0 + a) + . . . + sin (0 + n la) = . sin (6 + 74-1 ^ ) . a \ 2/ sin - Hence, putting - + for 0, we obtain the conjugate series, na sin la)= .cos . a sin- These results could also have been easily obtained by the method of 13, and the results of Ex. 3 in 13 are easily deducible from those above. Ex. Let OA j be a unit vector inclined to the prime vector at the angle 6 : describe the circle with the chord OA l subtending the angle a at its centre : take successive chords "1 "2' 23 ' ' ' "!" each equal to OA l : then from the value of OA n obtained from the figure the above results may be at once deduced. 4. To sum e e cosec v + cosec - + ...+ cosec . 1 + cos since cosec + cot = = cot -> sm0 2 fi ft ft fi cosec $ = cot - cot 6, and cosec = cot - cot 252 EXPANSIONS AND SUMMATIONS whence by addition 6 99 cosec + cosec - + . . . + cosec = cot - cot 9. 5. To sum tanh x + - tanh - + tanh + . . . to n terms. 2 22 2 Since sinh 2 x + cosh 2 x cosh 2a; tanh x + coth x = . - = t . , - = 2 coth 2a; sinh x . cosh x ^ sinh 2x Of* IT tanh x = 2 coth 2a; coth x, - tanh- = coth x - coth -. . . , 22 22 1 x la; 1 x - tanh --- = H~rro co ' ;n 0^2 ~~ q~ _ c th _ , therefore tanh x + - tanh 5 + ^5, tanh + . . . la; la; + -- tanh . L , = 2 coth 2a; - --. -, coth Since and when 1 . . " J ^ * /, * coth - - = - . - - it 6 = -- n = oo, = and It. 0=0 - / <=1| tanh 1 x I x 1 tanh x + - tanh - + tanh + ... ad inf. = 2 coth 2a? . 2 2 2 2 2 2 x Ex. 1 . Verify this result by taking x = 0. Ex. 2. What does this become, if ix is put for a; ? EXPANSIONS AND SUMMATIONS 253 17. Bernoulli's Numbers. X The coefficients of the expansion of form a series of e x 1 numbers, known as Bernoulli's numbers (having been first used by James Bernoulli in his work Ars Conjectandi, posthumously published at Bale, 1713), which are of great utility in the expansion of certain functions and the summation of certain series. x It should first be proved that it is possible to expand 6 1 in a convergent series of positive powers of x within certain limits of the value of x. Since xx 1 e x 1 x 2 a? x x z and the infinite series in the denominator is convergent for all values of x ; if limits of x can be found for which 05 numerically, - - can be expanded in the form ( x x- \ 2 / x x 2 \ 3 ( + 3 + ' ' '> ~~ + + 1 ' which is then absolutely convergent, since it remains so, when all its terms are taken as positive. Now -z + TTi + ,T + or - - < 1, if e x < I + 2x: but 2 Jo |4 x taking aj = l, we know that e < 3, and taking x = 2, e 2 > 5, sa that for some value between 1 and 2, which will be found by trial 254 EXPANSIONS AND SUMMATIONS to be very nearly 1.25, e x = 1 + 2x : hence, if x < 1.25, the above series is certainly convergent, and then if the terms were expanded and combined so as to form a series arranged according to powers of x, a convergent series of the required form would be obtained. This proof only shews that a convergent series can certainly be found, if x < 1.25, but it may be that the limit of the value of x for convergency is greater than this, and we shall later on prove it to be so.* / 18. Expansion of e x Let then M /Vi2 /-ViO /y4 snrv**- 1 -* j2 + B * [3 + ^ ii + " " where j5 , B v B. 2 . . . are numbers to be determined. x xe x Changing the sign of x, we have -^ - = =, so that * The beginner might be tempted to proceed thus. Since e~ x < 1, x xe - x . _ x . _ 2a . , _ 3x , ,. e x-\~\- e -x~' a convergent series, and the series for e- x , e~ 2a; . . . are all convergent, and therefore that for e~ x + e~ 2x -\- e~ 3x + ... is so. But on forming this series we find it takes the shape a series of positive and negative infinite terms. This example shews the need of great caution in dealing with doubly infinite series. On this point the student is referred to Clm-stal's Algebra, chap, xxvi., 3234, and chap, xxviii., G. EXPANSIONS AND SUMMATIONS 255 and by subtraction, TP^* - *Y or t *orx = -2B l x-2& i --2B, -=-... e*-l 3 J3 5 |5 whence B l ^, and B s , Br > ...B-2 r - v are all zero. We may therefore write, B$ being obviously 1, x 1 or, more conveniently, anticipating the result that the coefficients are alternately positive and negative, X 1 _ X 2 _ X 4 03 2r Also 1 2 ^-w3 *w4 BB ^ "^ JS J ."o i ."7' it* * ' * f * J2 ' |3 |4 |5 |2r+l Hence the coefficients of all the powers of x in the product of these two series must vanish, so that, selecting the coefficient of x Zr |2r+l 2 j2r |2r -1 |2 |2r-3 ]4 and therefore 2r.2r 1 2r . 2r 1 . 2r-2 . 2r-3 T5 -D2r-4 ^ 11 II 2 |2_ 2 7 2r+l' a formula whence, by putting r successively equal to 1, 2, 3, 4..., the values of B. 2 , B t , -5 . . may be found. 250 EXPANSIONS AND SUMMATIONS The result as far as _6 14 (which the student should verify) is B * = 6' ^ 4 = 30' B = 42' B * = 30' ^ 10 = 66' ^ 12 = 2730' ^ = 6' Ex. Shew also that 2r.2r-l 2r . 2r 1 . 2r-2 . 2r-3 3.4 3.4.5.6 /_iy ( IV " 2r+l ' (r 19. Series for coth x, cot x, tanh x, tan x, cosech x, cosec x. We have proved that " |-^ 4 .* 3 + ^ 5 -.. e * + e -x e ix + i 2 Now coth x = - = - = 1 + - e x e~ x e ix -\ e ix therefore x + \2 ' ]4 '" ^ |6 ' Hence, putting ix in the place of x, since i coth ix = cot x, \ B B B iX = x'~]2 X ~]^ ~]6 cosh 2 x + sinh 2 x cosh 2aj Also coth x + tanh a = -r r-r - = , . , = 2 coth 2x, cosh x . sinh x sinh 2 - 1. 2~ 2 ~sin^-ii= 1 + 1 -2 -3 re +4+5-... n+l , , 2 2 cos _Jt_L = 1-1 - 2 re + 3 + 4n - 5 - 6 n + ... Jl (2) Expand e# cos 5 and e sin in ascending powers of 0, and show that the general terms of the expansions are 2 2 cos- . and L, tl 2 -. n0 2 in 9 r- respectively. (3) From the expansions of the previous question deduce those for cos . cosh 0, sin . cosh 0, cos . sinh 0, sin . sinh 0. (4) From the series for sin 0, sinh 0, &c., verify the formula; A. 2 sin 6 cos = sin 20. B. cosh 2 - sinh 2 6 = 1. C. cos 2 (9 - sin 2 - cos 20. (5) Expand sin - . cos in powers of 0, and prove that, when vanishes, the limit of (sin 6 - 0cos 0)/0 3 = 1. Deduce that of (sin 2 6-6"- cos 2 0)/0*. (6) Prove that X ' l - = log cot- -f- i- + 2km, and thence that, if >4X and <4X+2, cos + - cos 3?< + - cos 5w + = - lg( c t- ) 35 2 \ 2/ 11 TT sintt + -sin3M + -sin5 + ... = -. 35 4 What, if w>4X + 2 and <4X + 4 ? Illustrate the above by diagrams for particular values of u (e.g. I 1 , ^ L , HS f| L ), and explain how discontinuity arises, when u passes through 2, 4, ... right angles. 260 EXAMPLES (7) Prove that, with certain limitations, ( which determine ) r cos 6 + - r 3 cos 30 + lr> cos 50 + ... = 1 Tanh- l2rcos f, 5 D ^ 1 -J ?'" o/i i 1 i *d i IT ,2rsin0 n 30 + -r'sm50 + ... = - Ian- 1 - *) ^ I ?' (8) Prove that, if 6 lie between and STT, 1 sin 20 + 1 sin 40+ I sin 00+ ... = Zt 4 O T: i also cos 0-1 sin 20-1 cos 30 + 1 sin 40+1 cos 50 - ... = (9) If .r is not greater than cosec a, Tan -1 (.r + cot a) = ^ a + ,r sin a sin a - sin 2 n sin 2a 2t 2i o + -- sin 3 a sin ?>n + ... r 3 3 5 (10) Prove that * = 1 4- 1 _ L_ + X 3 1 4. 3 2 ' 3 X 2 2 2 . 4 5 X 2 4 "*" 2 . 4 . 6 ' 7 X 2 (11) Prove that n + 1 cos ?/0 n cos n + 1 1 cos + 2 cos 20 + ... + n cos = - - 4 sin 2 -: ft and >? + 1 sin nQ n sin y/ + 1 sin + 2 sin 20 + ... + n sin ??0 = - ^ 4 sin _ From the last show that I 2 + 2 2 + ... + ? 2 = gn (w + 1) (2 + 1). (12) Prove that CC'-T * f 1 I 1 1 I 1 1 1_ \ Binl ' i -^| 1 + 3*i- H S'P + - r (13) If sin = n sin (0 + ), prove that, 2 being less than ()~, EXAMPLES 261 and 6 = - a - - sin a - - - -sin2a - ..., (>1). n 2 n- Exauiine the case, where n = 1 and a is not ; and illustrate the conclusion by geometry, regarding 0, 6 + a, as two angles of a triangle and n the ratio of the sides opposite to them. (14) Prove the following summations to n terms: +_ i . + ' .+ .. cos a -f- cos 3a cos a -j- cos 5a cos a -\- cos 7 a sin / ( 2n sin 2x ~ sin 2n K. Tan- 1 2 + Tan' 1 + Tan 1+3x4' 1+7 L. Tank- . + Tnh- . 1 _i^ ! + Tanh- j-f^, + ... = Tanh" 1 ??.^. M. 2 cos - + 2 2 cos- . cos + 2 3 cos -cos - cos 4+- 2 2 2 2 2 2 2 2 3 2" (15) Express cot H$ in terras of cot# (n a positive integer), and thence prove that eot<9 + cot(0 + -) + cot ( Q + 2-) + ... to terms = ncotnd. \ n f \ n/ (16) 1 - I n 2cos0 . cos^ + 2 n . 2 2 cos 2 0eos20 - ... = (- l)cos (17) 1 +7icos<9 . cos<9 + ... -|- 1 cos nd "- 1 sin d " or ~ ! 2 ' accordin g as is eveu or I cos 3^ . cos fid , " " | . |>t-f 1 |"- 1 JH + 2 | - 2 | + 3 ~pn EXAMPLES 263 (19) l + 3 + 6,,+ ... =1 (2- + 2 cos"?). j \ 3 J (20) tan + I tan - + ~ tan ^ + ... (to n terms) " L cot - 2 cot 20 and hence and - tan - + 1 tan ? + 1 tan Z + ... = 1. 4 4^8 8 ^ 16 18 T TT J3 J5 = cos (a - j8) cosh (sin ^) sin (cos /3) - sin (a - ) sinh (sin /3) cos (cos^)- (22) sin^ . - sin 20 . + sin 3d ... 1 23 = Cof^cot + cosec 2 6}. (23) (1 !) - (2 n 3) 3 + (4n 5) 3 3 - ... C24.1 1 4- cos ( a "I" ^) 4. cos a cos ( a ~^~ ^) -U cos2 a ' cos ( a 4- 3jS) . """ cos (a - 8) "* cos 2 (a - ft) ~~cos 3 (a - /3) ~ = 0, if cos a < cos (a - /3). (25) 2 sin ^ + sin 3^ + cos 6 sin 40 -f cos 2 sin 50 4- ... = cosec 0. (26) Find the coefficient of x n in the expansion of {log e (1 + .r)} 2 , and show that 0.1og e cos| ^sin20- (27) sin 2 - sin 2 2u + -sin' 2 3 ... = - log (sec x) & 3 2i and 1 1 _ cosh 2 u - cosh 2 2u 4- o cosh 2 3w ... = log (2 vsechar). 264 EXAMPLES (28) cos u . sin (u -f- + In cos (tt -{- 1) sin (u -\- n 1) 4- 2,i cos (w -f 2) sin (u + n - 2) + ... 71 1 = (- l)22' l ~i sin 2w or (- 1) 2 2"- 1 cos2w, as is even or odd. (29) From the formulae, Ch. IV., 8 (p. 95), deduce the following : A. cos a + cos(a-f- | + 08 ( a + I + ... \ n J \ n J + cos ( a + 1_^) = 0. \ n J B. sin a -}- sin ( a + *- ) + sin ( a + - - ) + -|- sin fa + H - 1 } = 0. \ n ' r~< ( i 2?r\ t i 4'/r\ ( i 2ff\ C. COS a . cos I a 4- I . cos a H ... cos I a -4- n 1 - \ n / \ n J \ n / 1 n l f 51 cos na or ( 1)1 { 1 ( 1) 2ccs ua }, as is odd or even. 1). sin a . sin fa + 2n \ . sin fa 4- *?) ... sin fa 4- - 1 ^^ \ uj \ n/ \ nj H ~i , n 1 = (~ 1) 2 - 7j sin a or (- 1)^ a^i (1 cosa),asisoddoreven. E. tan a + tan fa + \ + ... + tan fa + - 1 ^\ \ n J \ n J = n tan a or n cot( ~ua J or ;* tan ( -na J, i .-is n is of the form 2X + 1, ^, or ^X + 2. F. sec a + sec( a + ^ ) + sec ( + / - 1 n } \ n / \ n / n-l = ( - 1) 2 sec a or 0, as /* is odd or even. (30) I n .i- + 4, l .r 4 + 7 n .r 7 +...' = 1(1+ *)~ - | (1 - x + a*? cos (116 + where d - Tan"' '' ^ und ./>!. EXAMPLES (3D . -f + f - ... = 1 - = -LjSinlT'l + Tair 1 ! j. l = \ ~ | < 2 * - J) ' r + jT (24 - 1} ^ - and deduce the series for tan x and tanh x. (34) If tan. f = C^' + C> 3 + C^ + ..., / ., . TJ 2 \so that Cs = Ban prove, from the formula sin .r = tan x cos .c, that ^ ,, n , t C 2 ~ T I and thence calculate the values of C. a C 4 ... to C iy . (35) Prove that the roots of the equation ''" + 1 n cos a . x n ~ l + 2,i cos 2a . .r" ~ 2 + "I" cos ua = 2X I 1 are the n values of the expression sin a cot - cos a for w c n secutive integral values of X. Also that the roots of . .r+ l n sin a . *'-! + 2 n siu 2a . x 11 -' 1 + ... + sin no. = 0, 2X are the values of sin a cot - cos a. CHAPTER X SERIES OF FACTORS THE circular and ex-circular functions, besides being capable of expression by infinite series of terms proceeding by powers of the measure of the angle, as shown in the last chapter, can also be expressed in terms of such measure by products of an infinite number of factors, and by the sums of infinite series of simple fractions. It is the object of this and the following chapter to investigate these series and deduce from them some important analytical results. It is necessary to premise certain proposi- tions as to the convergency of Infinite Products. 1. Infinite Products. Convergence or Divergence. Let z r 2 , . . . z n be a series of vectors, whose tensors are each less than 1, then the product (1 +z 1 ) (1 + z 2 ] . . . (1 + z n ), which we shall denote as n"(l +z p ), may, when n is increased without limit, increase without limit, in which case it is said to be divergent ; or it may have a definite finite limit, in which case it is said to be convergent ; or it may be finite, but have no definite limit : or it may diminish without limit and ultimately vanish. Now, if z = ri u , 1+2=1 +ri u and mod. (1 +z) lies between the limits 1 + r and 1 r, or as it may be expressed, , 1 + 1\ + r 2 + . . . + r n , or (as it may be denoted) 1 + 2 1 (r p ). 268 SERIES OF FACTORS Heiice if the series 2 (r p ) is divergent, II (l+r /) ) = co and CO 05 therefore II (1 r p ) = 0. In this case mod. II (l+z p ) may be infinite, finite, or zero, according to the law by which z p depends 011 p, and in the absence of any general test, the convergence or divergence of the infinite product must be investigated in each particular case. Again, (! ~ O I 1 ~*Wi) = l -r m ~ VH + VWi > ! - - r + i' a -O (i - ' + i) a - *w 2 ) > (! -'. - > wi) (i - '+,) ^ 1 __ / _ rt _ /J m w+1 m + -2> and generally Now if 2 (v } ) is convergent or has a definite finite limit, a finite value of in can be found such that ^, m (r ) differs from that i \ P' cc limit by less than 1 , so that 2 (r ) is a finite quantity less than 1, which may be made as small as we please by taking i/i suffi- ciently great: hence H m+l (l r ), though a finite quantity less than 1, may be made as near as we please to 1 by taking in sufficiently great, and therefore, Ii m (l r ) being a finite quantity GO less than 1, EL (1 r) is finite and has a definite limit less than 1 and greater than 0. Hence also IIj (1 +r ) is finite and has a definite limit greater than 1. It follows therefore that mod. II (1 +z ) is finite and between CO W O3 the finite limits of II, (1-7- ) and II, (1 +r ), if the series 2, (r) I \ p' pr' I \ p' is convergent ; but may be infinite, finite, or zero, if 2 1 (/' ) is divergent. In order to prove that when mod. 11^(1 +z ) is a definite finite SERIES OF FACTORS 269 quantity, IL (1 + z ) converges to a determinate limit, it is necessaiy further to show that, v denoting the inclination of the r sin.u , v, + v-, + V Q + . . . or vector l+n u so that #=laii 1 + r cos u 2. (v ( ) converges to a definite finite limit : since n = mod. n Ex. Take as an example the product (1 +z) (1 +z 2 ) (1 +z s ) . . . fid inf. or 11^(1 +??#"'), and in the figure (drawn for the case 270 SERIES OF FACTORS where r = f, u = -J- L or 30) let OP V OP 2 , OP S . . . represent the vectors z, z z , s*, . . ., all within the unit circle, centre 0, then n"(l + *#") is the product of the vectors r~P v FP 2 , T~P ..... Then since r + r z + r* + . . . has the finite limit - . or the 1 r sum of the tensors of OP V ~OP 2 ... is finite, the modulus of the infinite product n"(l +**!) or the tensor of the product /'/*! . I'P 2 . l'P., . . . has a definite finite limit. Also the angles v v v 2 , . . . or OI'P V 01' P y . . . are respectively r sin u r z sin 2w r 3 sin 3u Tan-i , Tan- 1 - , Tan- 1 1 + r cos u ' 1 + r 2 cos 2u ' 1 + r 3 cos 3u and since the points P n +\, P n +2, ... all lie within the circle, centre and radius OP n , the angles v n +i, ^+2> ^?j+m all lie between sm~ l OP n and sm~ l OP n so that their sum lies between m.sin~ l (r n ) and -wsin- (r n ), which may be made as small as we please by taking n sufficiently great, so that v l + v 2 + v., + . . . has a definite finite limit. Hence the infinite product II" (1 + r^i?") is a definite finite vector. In the figure IQ V Qfyy Q 2 Q S are arcs subtending angles at equal to v v v 2 , v., . . ., and it is obvious that their sum is finite and definite. The limiting vector might be approximated to, by geometrical construction, by describing on I'P, a triangle I'P^-2, s i ml l ar to I'OPy and then on /'II 2 a triangle /'II.,II 3 similar to I'OP. A and so on continually, as in the figure, where l'P v /Ho, /'H 3 are the successive products of 1, 2, 3 ... factors. If r = 1 , the vectors OP l OP 2 . . . are radii of the unit circle and 1'Pv I'P^ cnor ds of the same, and their product, if u is commensurable with a right angle, resolves itself into periods of products repeated over and over again ad inf., and if for any one period the product is finite, which is the case if no P coincides with /', the infinite product will have a tensor or oo , as the product of the tensors of the period is less or greater than 1. If one of the P's coincided with /', there would be a zero factor in each period, and the infinite product would be 0. SERIES OF FACTORS 271 2. Resolution of z n a n into factors. If n is a positive integer, by algebraical division z a and the quotient becomes na n ~ l , when z = a. 4A But a" = t' 4A a n = (i n a)", X being any positive or negative 4A 4A integer, hence z n a n is divisible by x i n a. Now i n has n, and only n, different values corresponding to any n consecutive inte- gral values of X, which may be most conveniently taken in the symmetrical sequence H ~ l n ~ b -3 2-1012 H ~ 3 n ~ l ~1~' " 2 '' 22 when n is odd, or when n is even, and z n a n cannot have more than n binomial factors, therefore 4 8 2n-2 (z-i"a).(z i n a)...(zi n a) _4 _??-' 2 i n a).(z-i~ n a) . . .(z-i " ) if n is odd, or 4 4 2n-4 (z-i n a)(zi n a)...(z i n a) ^ _i 2"-4 ffl ) l(z-i 2 a)..A. 2 (z-i n a)(z-i n a) . ..(z-i n j C = (-o) -J ( if n is even. 4A But s i" 4X = z z 2az cos -- h a 2 , SERIES OF FACTORS hence the series above may be written in the shape 4 8 z n -a n = (z- a) (z z - 2az cos - + a 2 ) (z* - 2az cos - + a 2 ) n n . . . (zt-Zaz cos +2) . . n m > i if n is odd, or 4 * - a n = (z - n) (z + a) (z 2 - 2as cos - + a 2 ) 7 n if rt is even ; in which, if c, a are scalar, all the factors are scalar. 2 In these formulae, if we write i n a in the place of a and therefore a n for a n , 2 6 2n-4 !(-t"o)(*-^*)... (*") ) 2 _ 6 2n-4 I (a-t~o)(-*" n a). . . (z-i n a)f = (s + ) (s 2 - 2as cos - + a 2 ) (z z - 2az cos - + a 2 ) _2 _2_ 2 _2-_2 and, since 2 - i z . i n a = z - i* . i n a = zi n a, 2 6 2n-2 ( (z-i n a)(z-i n a) . . . (z-i * a) J 2 + rt w = J- _2 _o 2n-2 L, even . ((*- n a)(z-i a}...(z-i n a) \ 2 6 = (z z - 2az cos - + a 2 ) (z z - 2az cos - + a 2 ) n n n- - 2? cos - + 2 ) . . . C.-,. SERIES OF FACTORS 273 3. Geometrical Interpretation. Cotes's and De Moivre's Pro- perties of the Circle. Let the circumference of the unit circle, centre 0, be divided into n equal parts (in the figure n = 5) at the points /, A v A. 2 , . . . A n -\ and let OP = z. We will take the case, where a = OI or 1. Then if OP n make with 01 the angle IOP n = n.IOP and s - 1 = (d~py-(oi) = op n - 01= ip n . Also 4 S z - 1 = OP - Of=IP, z-i^ = OP-OA l = A l P, z-i = A,P, and so on, hence, by the formulae A l or ^1., of the last Article, so that both the tensor lP n of OP n OI n is equal to the product of the tensors of IP, A^P . . . A n -iP and the inclination of IP n to 01 is the sum of the inclinations of the same vectors to 01. If P, and therefore P n , are on the line 01, OP and OP n or OJ\ are scalar, and in that case the sum of the inclinations of IP, T 274 SERIES OF FACTORS A^P . . . A n -\P is zero, as is obvious from symmetry, and we have the scalar relation IP n = OP n -OI n IP .A^. A. 2 P . . . An^P, a property, which is known as Cotes's Theorem. H 3=1, P coincides with /, and IP, IP n both vanish, but when z becomes 1, ip ~n _ I n n -i, * * and we have A^I . A 2 I . . . A n .\I=n, ^A 4 A 8 i 12 i 4w " 4 or chd. - . chd. - . chd. ... chd. -- = n, n n n n 2.4.6 , 2^-2 or 2 n ~ 1 sin - . sin - . sin - ... sin - n. n n n n If L \, B.,, li. A . . . B n are the middle points of the arcs JA V A-iAf, A^Ay . . . A n ~if, we obtain in like manner from the formula C l or (7 2 , p n =Opn+O n =BJ> . BP . . . B n P. When OP is scalar, P and P n ai-e on 01, and we have the scalar relation known as Cotes's Second Theorem. SERIES OF FACTORS 275 If P coincides with /, I'P = 2, and this theorem gives BJ.B.J . . . ,J=2, , 2 '6 , 4w- chd. - chd. - ... chd. n n n 1.3 , or 2" -1 sin. - . sin- . . . sin n n n Since (tensor 77> n ) 2 = // V = OP,? -201. OP n cos 10 P n + OI 2 = OP 2 ' 1 - 20 P n cos nu + 1 , for Ul\ = OP' 1 and IOP n = n . IOP and (tensor /7 7 n ) 2 = /7V - 0/V + 2 0/ . 0/' H cos 707 J re + 0/* we have OP'' l -2OP" cos MW + 1 - IP 1 . ^ 2 . A. 2 I* 2 . . . A H -il* OP- 11 + 20P' 1 cos nu + 1 = Ay J2 . # 2 / J2 . ^/^ . . . BJ*, results whicli are known as De Moivre's Property of the Circle. 4. Resolution of z- n 2a n z n cos u + a 2n into factors. Since 2 cos u = i u + i~ u , z in _ 2a n z n cos u + a? n = (z n - i u a n ) (z n -i~ ") 4A+u 4A+it = { ( n a) n }{z n -(i "")"}, X being any integer. Each of these factors may, as in the last article, be resolved 4A+W 4A+ into n factors of the form z i n , z i n a respectively by giving X any n consecutive integral values, and for the same value of X in each, 4 b _ 4x + tt 4X+M (z i n a)(z-i n a) = z z - 2az cos - + a 2 . 7i T2 276 SERIES OF FACTORS Hence, putting X successively = 0, 1, 2 . . ., n-l, z 2 ' 1 2a n z n cos u + a 2 ' 1 = (z 2 2az cos - + a 2 ) ( 2 2az cos + a 2 ) n n 4w 1 +u 2az cos, - + a l ). n And, putting u + 2 for u, and therefore cosw for cos u, z- n + 2a n z H cos u + a 2n = (z 2 2az cos - - + a 2 ) n / 2 , 9\ / o , o, (s 2 - 2tts cos - -- + a 2 ) . . . (a 2 - 22 cos - - + a 2 ). ?t The geometrical interpretation of these formulae gives De Moivre's properties of the circle. 5. Factor Series for sine and cosine. In the formula B v B. 2 and (7 P C. 2 of 2, put z = i u and a = i~ u : then, since - a = 2i sin w, 2 + a = 2 cos w, 2" - a 11 = 2i sin ?m, z tl + a n = 2 cos >m, *- 2ct2cos +a z = i^ + i- 2u '-2cos - =2 c = sin \ o 2A ., \ in- -- sin-w ) n J 4X + 2 4X + 2 z n 2az cos ..... + a 2 = t 2 " + i ~ 2lt 2 cos n n = 2 ( cos 2u cos ) = 2 2 (sin 2 Bin*tt ] \ n ) \ n ) we have / 2 \ / 4 2i . sin nu 21 . sin u . 2 2 ( sin 2 sin 2 w ) . 2 2 ( sin 2 sin% ) \ n J \ n J 2 2 sin 2 - sin , n odd \ w / / TC-2 \ 2* I sin z sin-w ) . 2 cos u, n even, n / sin u sin - n SERIES OF FACTORS 277 2 cos nu = 2 2 ( sin 2 sin 2 w ) . 2 2 ( sin 2 sin 2 ) \ n / \ n ) 2 2 ( sin 2 sin 2 w } . 2 cos M, n odd V n ) 2 2 (sin 2 sin 2 w) , n even \ n / whence, writing - for u, 2" " l . ( sin 2 - sin 2 - ) ( sin 2 sin 2 - ) \ n n/ \ n n/ '/ n-1 . z u\ 1 1 sin 2 sin 2 - I , n odd I \ n n/ / . n 2 . u\ u I sin 2 sin 2 - I cos - , n even ! \ n n/ n ( 1 \ / . 3 u\ cosw-=2"- 1 sin 2 sin 2 -I sm 2 sm 2 - \ n n/ \ n n/ i / n 2 w\ u , , / S { n 2 _ s in 2 - ) cos - , n odd \ n n/ n (^l J iy\ sin 2 sin 2 - ) , n even n n/ sin u . . , , Making u in these formula?, since - = n ultimately, sin - when u = 0, n f O M 1 sin- 2 - >n odd 24 w -.sin 2 - . . . ( n n . n 2 sin 2 , n even w 2 2 , w odd I sin 2 , w odd 13 >* 1 =2"->. sin 2 -.sin 2 -... ?i *i n 1 1 sin- , n even t n 278 whence SERIES OF FACTORS (. ,W 1 sin 2 - n 1 _nunt_nd / . u\ 1 ,u\ w - 1 sin u sin 2 - \ sin 2 - \ sin 2 ^ . T 1 n\ / . u 1 . . n sin - n \ nf sin 2 -1 I 71 / w ^ / sin 2 - , n odd cos u = sin 2 - \ / sin 2 -1 ^ i- sin 2 - / 1 sin 2 - ri ^ -^ I cos - , // even , . , n 2 i n sin 2 , n o u sin 2 - 1 I cos - , n odd. <>- 2 sin z Now, suppose n to increase without limit, then . u sin - ,, . u n IT It. n sin - = u It. = ?/..- , n u n and It, . u sin - n u P P sin - n so that, if we can prove that the products on the right-hand side of these equations are convergent, when the number of factors becomes infinite, we have sin u SERIES OF FACTORS 279 Let m be an even number less than n (n being odd) but greater u I '>n than , so that sin - sin is less than 1, and let R denote the n n ni product of all the factors in the expression for sin u/n sin - after the factor 1 - \ Then * R is less than 1, but greater than 1 - sin 2 - ( cosec 2 h cosec 2 - - + ...+ cosec 2 n \ n n n Now sin u/u, as u increases from to 1 , continually decreases 1 from 7T/2 to 1, so that for any acute angle measured by L , 7T 211 sin u < - n and > u, and therefore cosec u> and < - , so that, substituting for the cosecants in the expression above the re- ciprocals of their arguments, still more is 'n t ( + 2)*~ + (m + 4) 2 + > . U TT U and, since sin -<--.-, still more is n n _j 7 -^ 1 a,Z J _ ." 1- + ' -> J ~ 4 * 1 /_ . o\9 / i A\2 4 * This proof is given by Schliimilch as due to Prof. Schroter (Compendium 1 , o ?n 9 9 71- M- - , 8 m where e is some quantity between and 1. Hence sin* - 2 >in 2 - m n sin - n and, if e l denote the value of c, when n is increased without limit so that, the last factor becoming 1 when n is increased without limit, sir- u 2 2 z as we were led to expect above. SERIES OF FACTORS 281 It is plain that the same reasoning would apply to the series for cos u, the only difference being that m must be taken to be an odd number, and also that the difference of the cases, when n is even and when odd, disappears, when n is infinite, since then u cos - = 1 . n If denote the circular measure of the angle, since = U L , a the formula? become sin0 . cos 0= 1 - 1 402 \ / 4*02 \ - r- 1 - r- . . . This last is the form, in which these series are usually written, as, in analysis generally, it is most convenient to use circular measure. These formulae have been proved on the supposition that u is scalar, but they are also valid, when u is a complex number or vector. For the formulae above for a finite value of n are true without restriction as to the value of u from the laws of Algebra, or the principle of the per- manence of equivalent forms, and it is only necessary to show that the deduction from them, when M becomes infinite, also holds good. Let u x + iy, then . n . x iy . x fiv\ x , y . . x . , y sin - = sin - . cos - + cos - . sin I )= sin - cosh - 4- i cos - sinn - n n n n \n J n n n n and if p be the modulus of sin -, n p 2 = sin 2 - cosh 2 ^ + cos 2 - sinh 2 ^ = sin 2 f + sinh 2 ^, n it n n n n and as we have seen in Art. 1, the product is certainly finite, if it is convergent in the extreme case, when the modulus p 2 is substituted for sin 2 -. Then by the foregoing reasoning, n R > 1 - n 2 ( sin 2 - + sinh 2 -^) . ~ \ n nj 2 282 SERIES OF FACTORS Now sin -<- and sinh^>~, but the limits of n sin - and of n n n n 11 11 sinh - are respectively - .r and - y, so that, when n is very large M 35 Sy compared with x and y, 2 (sin 2 - \ n where /, is a small quantity, which may be either positive or negative depending on the values of x and y, and which vanishes, when n becomes infinite, so that the limiting value of /?, when n is infinite, may in this case be written as 1 fj^ '- J , and the infinite product has there- 8 m fore a finite modulus. The versor of the product is also definite. For, if sin 2 (: r - p'm i m 1 " 1 ^~ 1 sin 1 - n and sm 2 - cosh 2 - - cos 2 - smh 2 - M i n ?* p 'TO COS W TO = 1 - ^ ' 2 sin - cos ' . sinh - cosh - n n n 11 p m smi% = Whence, after reduction sin . sinh n n 2m 2.r , 2y cos cos cosh n n n which reduces, when 11 = oc , to and since, when m is taken large enough, cos v m is positive, the prime value of r m lies between - 1 L and -\-l L , and therefore tan r m > -r m numerically, so that the series SERIES OF FACTORS rm+2 + l'ro+4 + Tm-fG + . is numerically less than 2 283 a convergent series. Hence the versor has a definite value, and therefore, as we have proved the tensor or modulus of the infinite product to be finite, the infinite product has a definite finite limit or is convergent. 6. Geometrical Illustration. The argument of the last article may be well illustrated by a Geometrical Construction. Let IP n be a given arc of the unit circle and IP its nth part, (in the figure n = 7), and let the whole circumference be divided into n equal arcs, IA V -4jJ.>, A A. t> . . . . : then, as shown in Art. 3, 284 SERIES OF FACTORS P and P n being on the circumference, taking the case of n being an odd number, we have the following relations of the chords, = /P x ^P . A n ^P x A.J> . A n - -"TO-' -tlm-nt . A 7" 1S *' A m P- SERIES OF FACTORS 285 Hence the relation above between the chords drawn from P and / to the A's and the chord IP n of the arc 2u becomes in the limit sin u TT /. 2 \ /. M 2 \ /. u 2 \ The corresponding relations for TP n and the chords drawn to B v B.,, . . . may be shown in like manner to give the series for cos u. 7. Factor series for sink and cosh. In the formulae established in Art. 3 above, put iu for u, then since cos iu = cosh u, sin iu = i sinh u, we have n odd, n even, SERIES OF FACTORS u \ /, u ' 2 \ + - j (1 + - 2 j . . . act iV- 2 or, using circular measure, since u = - , i /i v \/i u \ / i *-l/ \ i / 1 + 1 f l + _ ...ad in j. These series might, of course, be deduced independently from the formula} in Art. 2 above, by putting z rj H , and a = rj~ M , reducing as in the case of the sine and cosine, and with a similar proof of the convergency of the infinite products. 8. Tlie Factor Series for sine and cosine are periodic. The periodic character of the series for sine and cosine may be easily shewn as follows. The formula gives sin Now ( _ /, u + m\ / \ n ) \ n n - 'in n arid putting t = 2 and n = 2, 4, 6, . . . successively, t u 2 A \ 4 sin ( + 2) = o SERIES OF FACTORS 287 -*\(l-*\ 2V \ 4V' = - sin u, whence sin (u + 4) = sin (u + 2) = sin u, &c. The student will in like manner easily verify the periodicity of the cosine series, the change of the sine series into the cosine series, itc. 9. Series for lo 1 . ^ o By means of these series, the numerical values of log-sin u, log-sinhw, &c. may be found directly, without calculating the values of sin M, sinh w, &c. themselves. The series for log-sine and log-sinh are rapidly convergent, if u < |, so that they are suitable for the calculation of their values for angles less than 45, whence the values for angles greater than 45 can be obtained. Differentiating * the expressions for log e sin u and log e sinh u, * Or, what is the same thing, taking the limit, when h vanishes, of the equation loge sin (u + h) - loge sin u _ loge (u + h)- loge u h h ---S 4 - SERIES OF FACTORS 289 have cot = - - 2 (, u + ,9. w 3 + S R M S + . . .) which satisfy the ordinary test of convergency and therefore are quantitatively valid, if u 2 < 4, or u is between 2 and 4-2. If u = 2 or 2, the series for cot u is divergent, and that for coth u fluctuating. Also differentiating the expressions for log e cos u and log e cosh u, we have ^ tan u = 2 (,S" 2 M + 5' 4 3 + ' 5 + . . .) | tanh u = 2 ('., ?t - + S 4 if 2 + S G u l + . . . , and lt. M=0 (- ~ ) = 4: U \ U / 2 putting u = 0, = /S' 2 . o 10. WaHiss Theorem. In the series for , putting u = 1 , u whence TT 2 2 4 2 6 2 2 1.3'3.5'5.7' '(2p-l)(2p + l) ultimately, when p oo a result which is known as Wallis's Theorem. Hence, when p is a large number, 2. 4. 6... 2p / 7 1.3. 5. ..(2^-1) since | may be neglected in comparison with p, when p is large. [The following deduction of an important theorem, first given by Stirling, for an approximation to the value of [n, when n is a large number, is taken from De Morgan. It does not profess to be a valid proof, but it is one which may give the student some confidence in the formula, pending the more difficiilt and complete proofs given in works on higher analysis. The theorem is that, when n is a large number, 1 .2.3... n or [^ \^7rn .(-} approximately. \eJ Let [ = ". ) 2 = ^/TTJJ . < (2) nearly, when n is large, or dividing by 27r, hence, equating the coefficients of u z in these two series, 1 /7T\ 2 7T 2 1 _! 1 . _ 7T 2 22 + 42 + 52 + a in J- ~ 24 Multiplying by 2 2 , 1 1 _!_ p + 22 + 32 whence The same results might be in like manner obtained from the series for cos u or from those for cosh u or sirth u. SERIES OF FACTORS 293 12. Relations between Bernoulli s numbers and sums of inverse powers of the natural numbers. Comparing the formulae obtained in Art. 9 for cotw (after 2x changing to circular measure, or putting for w) with that in 7T Chap. IX. Art. 18, namely : 1 2 3 2 5 2 7 cot x - S., x - 8, x 3 S.. a; 5 . X 7T 2 ' 7T 4 7T ' with cotu^ 1 _ ] h -T2 X _ ^ 94a . 3 _^ 20 .o_ x -2 ' [4 ~ [6 ' we tind, by equating coefficients of like powers, whence, using the values of B. n B ... as found in Chap. IX. 17, 7r' 2 ^4 (5 ^S ^10 2 ~24' ' 4 ~1440' ~ 60480' 8 ~ 241 9200' 10 "~ 95800320 '" or *Sj= -41 12336, S 4 = -0676452, S 6 = -0158960 AS' S = -0039222, AS' 10 = -00097753 . . . Hence, if S r = - + - + - + . .. = '2 r S r 2 2 = 1-644934, S 4 = 1-082323, 2 = 1-018944, 2 8 - 1 -004084, 2 10 = 1 -0009904 1 1 1 and, it S r = lr + g r + f + . . . S 2 = 1-233700, ^=1-014678, *S" = 1 -003048 ^ = 1-000162, AS" 10 = 1-00001 29. 294 SERIES OF FACTORS 13. Limits of Convergence/ of the series dependent on Bernoulli's numbers. Since but 2 2 /-+2/22-< 1 for any finite value of r, and ultimately = 1, when r = oo, therefore Szr+z/Szr < i f r an y finite value of r, and ultimately = , when r = GO . Also $. 2r = 2 whence-^ = - . ? so that ^ ^- ' (27T) 2 if r is finite and = nearly, if r is large. TT Hence Bernoulli's numbers form a series of decreasing fractions as far as H 6 inclusive, after which they increase continually without limit. In the expansion of ~ ^ (Chap. IX. 18), the coefficient of 6 ~~"~ JL the term involving x 2r - l is -^, and the ratio of the next f ollow- 2r If.-. |2 r x 2 ing term to this = -=~- . ^ x z and is therefore less than - 2r |2r+2 47T 2 , x 2 if r is finite, and ultimately = - , if r = oc . Hence the series 47T 2 is absolutely convergent if x lies between 2?r and + 2?r. SERIES OF FACTORS 295 In the series for coth x and cot x, the ratio of the term con- volving x- r+] - to the preceding term 7^- '2r 2- r +- x 2 . , x 2 = ultimately, when r - oc , Jj 2t 7T so that these series are both convergent for values of x between TT and + TT. The same is seen in like manner to be true for the series for cosech x and cosec x. Those for tanh x and tan x however are convergent only between the limits and + % 296 EXAMPLES EXAMPLES. Prove the following formulae : A A /) W" " tr T . cos -.cos --.cos-... aclinf. (2) and sin '- (n odd) (3) sin ?. sin 1 sin ?... n n n \ . n 2 , .. -i sin . (n even) "" 2 and log-sin 10 -f log-sin 20 + log-sin 30 -f ... + log-sin 80 = log 3 - 8 log 2. (4) If n is an odd number, log-cos ? + log-cos f + ... + log-cos IJ^ 1 = - 'i.7 J . log 2. ? + log- cos f -f ... + log-cos 'Li- 1 = '~ l (5) sin nu = 2-i sin u . sin ( M -f ? V sin f w -f V / V and (2\ / 4\ / 9 f u-\- - ) . tan ( M + - ) . . . tan ( u -f ^ / \ ;?/ \ ? = ( 1) 2 . tan H#, if n is odd. /fl % , sin 2 a 9/1 tan 2 a\ j , , W A ~ . , - = cos 2 a ( 1 - I, and hence show that (n being sin 2 /3 \ tan 2 /3/ odd), f tan 2 -\ X ^an2 u \ / ^\^ H I 2 \ f sin M = ( cos - ) . ?itan - 1 r 1 V "/ V tan 2 f / V tan 2 - n X \ / tan 2 " 1 EXAMPLES 297 (7) Frum Ex. 6, show that, if u > 2, sin w

P n ( cos - j : whence sin u P^. - and tan = n v 1 - i (2ra - 1 + ) (2 + 1 - u) (9) (1 + sec )( 1 + sec I) ( 1 + sec ) ... (l + sec = tan a . cot * 1 + secl 1 + secl 1 + secl and TT = 2 sec - . sec - . sec - . sec ...ad inf. 2 4 o ID (10) (l +4sinh2i t Yl +4 sinh 2 ."Yl + 4 sinh 2 ^Y.. ad inf. = coshd. (11) From the formulae of 4, prove that (/ ' 2 ' \ T ""* \ I l S ' n 2 \ 1 5T-:. X ~ _L / sin2 ^ \ / H 1 --^)! 1 \ 2;i / \ 2/t 1 - . ,2(2- !) + . sin 2 i : Why cannot it be inferred from this, by taking n = co , that (l - V 298 EXAMPLES Prove that the true result is cos v + cos u = 2 cos 2 - . (l - "" \. (l - } . 2 \ (2 + ) 2 / V (2 - ) 2 / , . sinh ( + t?) /, , \ /, , 2 + tA A , 2H+j sinh u r / V r 2 2 + 2 A 4 2 + u*" 2 2 + 2 4 2 + i 2OT 3 /10N sinh sinh tt /, . v' 2 2 \ /-, , if- - -\ /, . r 2 - ?i 2 \ \"-&) ~ == I ^ ~r i> i) ) \ ^ T ~ " / I T T" . j , s sinh a ' - = /T/.\ T. 1. o 1.9 i i , , (16) cosh o + cosh = 2 cosh- - . 1 + 2 l + M -j- !LlL l W i _j_ L_ ^ \ ... (17) COSM + sin a ( 1 -j- LLJJL!!? J ( l -f J ^j~\ \ 1.3/v 5.7/ / ,. 4(l-)\ I 1 ~^rrrJ" and ft _ 4 (1 + M )\ /i _ 4** (1 { M )\ f-i _ ^ w (1 H~ M )\ ~\ ~TT"3 A 5 . 7~ / \ 9 . 11 )" Hence verify the formula, cos 2w = cos 2 u - sin 2 u. (18) By the factor series for sine and cosine, verify the formula sin 2w = 2 sin u cos u. EXAMPLES 299 (19) Prove that, if n is a prime number, - _ _ 2 * - 1 2 3 - 1 ' 3* - 1 ' 5 2 - 1 ' 7 2 - 1 ' II 2 - 1 " = 1 4- l . 4- l + ... = ? 2 (Crofton), 1-' "^ 2 2 ^ 3 2 6 v (20) ! + 1 + l H = '* = ( 1 4 J Vl + _L_ ^ . ; H + 2*+3 4 ^ 90 V 2* - 1/V ^3 4 - I/ ( 1 -j- _ - )..., n a prime number. V 4 I/ (l - -. )..., n a prime number. \ a 4- I/ (23) _ 3 (l + ^I t ) (l + j^) (l + y^, 7 14-- j ..., n a prime number. CHAPTER XT SERIES OF PARTIAL FRACTIONS 1. Resolution of \jz n \ into the sum of n fractions. We have seen (Chap. IX. 2) that z 11 - 1 can be resolved into 4A n factors of the form z i n for n consecutive integral values of A : hence its reciprocal may be expressed as the sum of 11 A 4 / fractions of the form , if a = i"(the prime value of l n z -a A where ^A is & quantity (independent of z) to be determined. We have then 1 A A l A^ A- h T ~ "T T T' - -. I t * T^ . i^ ... T^ _ z n 1 z 1 z - a z or- z a n ~ Multiplying by z a A , and putting z = a*, we obtain A i = limit -- -, when z = a A , but z aA z n ^l n, * . 1 a A a A therefore A^= = = , SERIES OF PARTIAL FRACTIONS 301 n 1 a a 2 a" 1 so that = + - - + - -.+.,,+. r. " 1 z 1 z a. z a" z a"" 1 Another form may be given to this result by grouping terms in pairs as follows : a n-p a -p Since a" = 1, - - = - , ana we may write z-a n ~P z a~i> a Z a z a n-l o 2 a- a = + ...+ 2 z a 2 a ' a - z a" 1 z a " 2 a a- a i i . T T -z a z a* n-l 2 n-l 2 (n odd). J_ -1 n-2 ,-1 -2 Z a' 1 z a.-- a~ L a' -, + 5+- + n-2 "T" S+l (TI even). Farther, since " C 3S H 2 2 - 22 COS - + 1 w w -! 2(cos 1) 2(2cos 1) 1 \ n / \ n / / 2 (n-l) .\ 2(scos- -1) (n odd). 302 SERIES OF PARTIAL FRACTIONS 4 Yzcos 1) 2(2cos--l| V n ) \ n ) 9 2 _+_ S.+ _L_ MHBBBBHMHMHBBHH|M|B|H ^^ HBB , ^^^^ ^ B^^^^^^B^B^^^^ -J- w - 1 2 1 4 8 2 2 - 22 COS - + 1 C 2 2? COS - + 1 i w 2 ?i - 2 2 ( z cos 1 (w even). tt which are scalar formulae,* if z is scalar. 2. Resolution of l/z n +l into the sum of n fractions. The corresponding formulae for l/z n +l (the details of the proof, or the deduction of the formulae from the last Article, being left to the student) are, if a = i 2 /" so that a" = - 1, Z n +l ~ Z-a z-a 3 a ' a z a* z a w ~ 1 -f - - (nodd). ~ l ~ 3 -(- 2 ) % + 1 even). a' ~ ,-i : o- 1 s-a- 3 ""- a -(-i*' * These results, as most of the formulae of this chapter, may readily be obtained from the factor formulae of the preceding chapter by a slight trans- formation after taking logarithms and differentiating. SERIES OF PARTIAL FRACTIONS 303 9N 2 ( 1 - z cos n 2\ / 6\ 2(l-zcos-) 2(l-2cos-) V / V / + +. . . ^ +1 no 2 , P i o 6 , 2 1 2z cos I- s 1 "z cos - -f z* it n / 2(n-2)\ 2 1 scos \ n ) 1 , 2(n-2). J + g - + -l< l>0dd >- 2 1 -2 cos -^ 2(l cos - n/ \ TO ^ / 2(n-l)\ 2 ( 1 3 COS - ' ) V n / 3. C'o< M a'i tow u expressed as the mean qf n cotangents. In the series obtained in 1 , put s = i n ; then z n 1 i~ u 1 *"* 2 sin cos (1 + w) i sin (1 + w) 1 1 , - - ^ .-cot u. 2 sin M 22 it-2 a 1 1 i 1 . 1 /u 2\ ,^- a = r~ : 2^4- = jT2 -^ = - 2"" ?t 2 c H^r~ / ) n n u+2 1 , 1 fu + : i. -cot "2V n 304 SERIES OF PARTIAL FRACTIONS with similar reductions for the other terms ; and it is evident that the sum of the projects on both sides of the equation -, and equating the trajects we have 2 u /2 w\ 4 u II COt 11 = COt COt ( I COt . . . n. \ n / n cot . 2 + u 4 + u + cot + cot n n n - lu n l +u + cot - n (n odd). u , 2u 4 u cot - cot cot n n n cot 4 + M x + cot h cot + cot n - 2 + u . u ; - tan n (n even). A similar reduction from the series in 2, or the substitution of 1 u for u in the result just obtained, gives 1 u 3 u ntanw = cot - I- cot --- + n n cot l+u cot 3 + u +cot n-2-wl + ... cot u + tan n , jjv. ( odd). 1-u 3 u nlu^ = cot - - + cot -+...+ cot n n 1 +u 3+u - cot - -- cot - ... cot n n n -i +wl <" even >- SERIES OF PARTIAL FRACTIONS 305 Also since sm (u + v - u v) sin 2y cot (u v) - cot (u + v) = . . . . - --r -v , sin (u + v) sin (u v) sin z w sirrv the series for cot u, tan M may be reduced to . Ztt . Zu sin sin Ml H n cot u = - cot - it n n . ., 2 u n . 4 M sin z sin 4 - sm- - sin- - n u n n . 2u sin 7i s^^ ^ (rt odd). n . w sin- , -, , sin sin , u 1 ?4 1 n = cot n >t n . , 2 t n ' . , 4 sin- - sin' 1 - sin- - sin 2 n n n n . 2u sin - 1 n 1 u tan - (u even). . . - 2 M w rt v sin 2 sm 2 - n n . 2 , 2i ! sin - sin - & n tan M = - + - + . . . w . ., 1 . 9 M . ., 3 . ?t sm^ ear - sin- sin 2 - n n n n sin ' n u + ~ + - tan - (w odd). n . n 2 u n w v sin^ sin- 2 - w w 306 SERIES OF PARTIAL FRACTIONS , 111 , 'lu sin sin 1 n 1 n tan u - ., + - . + . . . n . 1 . - n 3 . ,,u HOT: sin" - sin' 2 sm~ n n n 'it Sill n H , (>ieven). n . .-! . sin" 2 sin' 2 4. Geometrical Interpretation. The reasoning of the preceding article is well illustrated by a geometrical interpretation. Let OZ, a radius of the unit circle, = z n or i" u so that fOZ=2u L , then the chord /2T= z' 1 1 : so also if OZ (> = z or i'% IZ = z- 1. Then if the arcs Z Q l, 12, 23 ... are taken each equal to 4/n L and Z Y, 1'2', 2'3' . . . also each equal to 4/V L but measured in the contrary sense, 48 _4 _8 s~\-t *w /So "^ *1 /^T ' *^ /^O' " " c/ J. ^= ^/i . c/^ ^?< tnd \j x - ^''Z- j \j A -- &ii ... so that 48 _4 . Hence the equation of 1 asserts that * i + /i iWti 1 \4 /^ /^ \71 /!'/ V/2 727 or that IZ is the vector harmonic mean of the n vector chords f rf T-\ ft' ff) TC)' J^Q, 1 I, 1 L , IZ, 1A , . . . if n is an odd number, or of ~~7n 2\ ~7n^~2\' if n is even. SERIES OF PARTIAL FRACTIONS 307 (The diagram is drawn for the case of n = 5. ) x 2 308 SERIES OF PARTIAL FRACTIONS Now if Iz is the reciprocal of IZ or Iz = I///?, is found by constructing a triangle OIz on the opposite side of OI similar to &IO : hence OIz is isosceles and z is a point on the line which bisects 01 at right angles and the angle OIz OIK= 1 u, so that 1 r 1 -" 1 . 1 Similarly if Iz , Iz v Iz- v Iz. 2 , Iz-. 2 . . . are the reciprocals respectively of IZ , II, II', 12, I'2' . . ., the points ~ , z v s_ p z. 2 , z- 2 . . . all lie on the line bisecting 01 at right angles, which is the inverse of the circle with respect to / as the centre and 10 as the radius of inversion, and 1 u + 2A .- ! ji~ 1 1 . u + 2A A == ^ cosec ^ ^ . - cot 2 n 22 n where X may have any one of the n values 0, 1, 1, 2, 2 ... Hence Iz is the vector mean of the vectors Iz^, Iz v Iz-\. . . and, M being the middle point of 01, Mz is the (arithmetic) mean of Mz , Mz\, Mz-i ... or cotw is the mean of the n cotangents, u u + 2 u-2 u + 4: u4: cot-, cot , cot , cot , cot , . . . n n n n n It should be observed that z z v z Z-i, z-fiy z -\z~-> all* 2 L subtend at / angles = n Also if m v m. 2 , m. A . . ., are respectively the middle points of so that Iz 1 +Iz~ l = 2Im } , lz. 2 + Iz~. 2 = 2Im.,, . . . Jz is the mean of Iz , 2Im lt 2fm. 2 , . . . and Mz that of J/s , 2Mm v 2Mm. 2 , . . . and it is easy to show that / 2 L \ * ( In what sense does z ( p- 1 in the diagram subtend the angle - ? ) \ n ] SERIES OF PARTIAL FRACTIONS 309 . u 11 2r 2r sin - . cos - sin . cos n n n n .Vm,. = -i z r m r = m r z_, = - i n "' . , . 9 . . . 9 sin- sin- sin 2 sin 1 * n n n n If u 2r, the chord Ir vanishes, and therefore lz- r is infinite, as is also Iz. The corresponding interpretation for the series for 1/2" + 1 may be left to the student as an exercise. 5. cot u and tanu as the sum of an infinite series of fractions. The series for cot ^t in 3 may be written thus (taking n as an odd number) : 1 tu u\ 1 /2 - u ' 2 - u\ cot u = - . I - cot - . I cot } ... u \n n/ 2 -u \ n n / 1 /nlu n 1 - u\ ... - - cot n- I u\ n n / 1 /2 + u 2 + u\ + -z ( - .cot -) + ... 2+u \ n n / 1 /n-l+u i _________ / . 1 -4- u \ n 1 /n-l+u n-l+u\ cot - -l n ) , 2u n sin 1 tu u n 1 tu u\ cot u = - [ - cot - ) n\n n ., ( . 2\ 2 / . u ( n sin - ) I n sin - \ n) \ n n . sin n / n l\ 2 ( . u [ n . sin - ) (n sin - V n J \ n 310 SERIES OF PARTIAL FRACTIONS Suppose n to be increased without limit, then, the angles being measured in right angles, . I2ru 2ru] , \1ru / 2ru] 2 cot / tan - '/==-, I n n I [ n n \ IT as long as r is finite, so that for any finite number of terms, the successive terms of the first sei'ies above become 21 2121 2121 7T U 7T 2 U TT 2 + u' 7T 4 ' 7T 4- + ?f Also, when r is so large as to be comparable with n, 2ru 2ru 2r 2r 2r / 2r 2 - . cot - - = . cot = ' tan < - , n n n n n/ n ir (2r being less than n), so that the terms after the r th are less than those of 1 1 1 1 T ;r * T 2r- 2r + u 2r+lu 2r+l+n an alternating series of evanescent terms. Hence we conclude that, for all values of u, except only for u = or a multiple of 2 L , in which case cot u and one of the fractions on the right-hand side become infinite, the equation 2 U 1 1 1 1 1 1 ) cotu = -{-- + n - + r - + fl-T'+---i TT m 2 - u 2 + u 4 u 4 + u D - u o + u is quantitatively true, the infinite series being convergent. - By combining the pairs of terms, or by proceeding to the limit, when n = oo , from the second form above, we have _2|1 2u 2u 2u \ in which the infinite series may be easily shown to be absolutely convergent. The proof we leave to the student. SERIES OF PARTIAL FRACTIONS 311 T'sing circular measure, and putting = - u, these series 3 become 11 - -- _l _ 10 20 ~ ~ 7T 2 - 6* ~ 2-V - Putting 1 - u for u, or - for in the series for cot u or A cot 0, we have, for all values of u except < cot - + cot ! I 4- ( - 1} cot EXAMPLES 317 ; , .sin/y 8 (sin 2/j 2sin4u 3sin6/> ) Lf. : - =: \ - _ J- - / sm u TT 12-' - u 2 4- - w 2 6'- - it- / cospu _ 4 (cos p 3 cos 3/; 5 cos 5/; _ ) cos = ^\ri^ " 3 2 -w 2 5 2 - 2 ~ ' '/ , sin^w_4 fsinp sin3y; sin 5y> ) ' "cos u ~n U \l - M 2 " S 2 ^^ + 5 s -i^~" ' 'j 8 I 1 a- 9. sech 2 = 1(- L ~" + "--"-_ . "--"- 4. 7T 2 1(1 4 "V (3 2 4 2 ) a (o 2 4 w 2 ) 2 ) 10. If j) is an integer <, .1 1 .3 '3 . 2-l 2-l sm- cos p - sm-cos p - sin cos p cos^0 _ ra n n , cos nd 1 3 2a-l cos cos - cos^- cos- cos 6 cos n n n _L = ! _L _A_ 2 ^ - 1 2 ^ 2V 4 1 ^ 4V 4 1 T 6V 4 1 "* 112 2 2 6> + 1 2 TT ' 4 1 3V 4 1 5V 4 1 1 - 1 4 4 (-4 * X j_ \ ;-! 2 ^7rl2^4 2 4 1 ^ 8 2 4 1^12*4 1 ") 1 2 (1 _1 1 1 1 ) i, 2 -! ~ 8 fe S*+ 1 4 2 4 l + 6 2 4 1 8 2 + 1 + ' / 13. From the formula cot u =- ( cot - 4 cot -J deduce that, if X \ 21 2t J 2n-l , eot= JL{cot_" 42"" " I cot ~ _ ' '-' - cot "' " I 4- cot I .u -m-if *2r+K 2--tA = {cot 4 2 lew cot -l-j- 2/// I 2/u i \ 2? 2w / and thence the fraction series for cot. (Schroter, Zeitsclirift, 1868.) CHAPTER XII THE general theory of functions of a complex variable forms one of the most important of the more recent developments of Mathematics, whether regarded with reference to abstract theory or to its applications to Mathematical Physics. This theory in its general aspect lies beyond the scope of an elementary Avork like the present ; but it may prove useful to the student, as introductory to the Avider theory, to consider here a feAV general properties of that particular class of functions of a complex variable, knoAvn as Rational and Integral Functions. 1. Rational and Integral Functions Definition, Notation, &c. Let z denote a vector or complex number, whose tensor or modulus is r and versor i u , and whose project is x and traject y, so that z ^ ri u ^ x + iy : and let w=fz ^z n + a^z 11 ' 1 + a.. 2 i^z n -' 2 + . . . +,,_i i*"- 1 + o"i a where a v a. 2 . . , a n are tensors or moduli, and i a i . . i an versors, independent of z, and n is a finite positive integer : then/2 is said to be a rational and integral function of z of the rt th degree. Let p, i" be the tensor or modulus and versor of fz, and A", Y its project and traject respectively, then w=fz~pi v =X + iY and we have the folloAving identities : RATIONAL AND INTEGRAL FUNCTIONS 319 A ~r" co&nu + a l r n ~ l cos(n lu -fetj) + a. 2 r n - cos (n 2u + 14 '-.'.. + H cos a,, }" /" sin IM ^ ttj'/ 1 '^ 1 sin (n I u + a x ) + a./-"" 1 sin (?i 2 + a 2 ) . . . -f sin a ,, ; as also, 1,,, 2,,, 3,,, ...(*)... denoting the coefficients of the expansion of (1 + #)". AN'lu-nce A", F can be expressed, if necessary, in terms of x, y. 2. fz expi-essed in powers of z z . Let z be a particular value of z : then, supposing for simpli- city the versor factors i a to be included in the a coefficients, /'., _ f-y ._ /-, > n\ i _ /^n-1 " n-l\ i ^n-2 _ ,, n-2 /* y*0 \* ' ~ z ) + a l \ z ~ Z ) ^ a -2\ Z Z + a n . l (z z ) = (z - Zo){(z n - 1 + zz n -' 2 + z 2 z n - 3 + . . . + So"- 1 ) + j (z n ~' 2 + z z n - 3 + . . . + n " 2 ) + + a n -. 2 (z + z ) + a n _!>} = (* *o)/i* suppose, where f^z is a rational and integi-al function of the n 1 th degree. Arranging the expression for ^2 abov r e according to powers of z, in which it is obvious that the coefficient of the term involving .j" " r is obtained from that of the preceding term by multiplying it by 2 and adding the term a r -i- Thus it appears that/^ is the quotient, andy*2 the remainder 320 RATIONAL AND INTEGRAL FUNCTIONS when./':; is divided by z c , and a simple mode of calculating these has been found. In like manner if f.,z,f. & z,. . .f n z denote respectively the quotients, when f^f^z, . . .f n -\~ are divided by z z u , so that they are functions of the degrees n 2, n 3, ... 1 , re- spectively, we have fz = (s ~ )/ r :: + fz f\~ = ( s s )/ 2 3 + /i~o fn -\Z = (z Z )f n Z + fn - l^'o fe = 1, whence, multiplying both sides of the successive equations by 1, z s , (z ) 2 , . . . (z ~ )' 1 respectively, and omitting like terms on opposite sides, we have or, if we put h for z ^ , f(z, + h] =fz [3. The f unctions *yi^' ,/ 2 ^ , . . . may be obtained in their general form by putting z + h for 3 in /s, expanding each term by the Binomial Theorem, and arranging the resulting expression according to powers of /. Hence it will be found that The previous investigation, however, suggests a far simpler mode of calculating the values of / 1 ^ >./-> 5; o in any particular case than by * The student, familiar with the elements of the Differential Calculus, will also r ec g n i se /i~n>/2 ;:; oj ... as the values for ~ = ~ of the successive divided ( jiff ere ntial coefficients or derived functions of fz : or that f^ t} = fz ot />f p"~ "n. f ., J "n Jro = f72' /3 ~ F.T. 3' ' ' ' 321 substituting in the general form, especially when the coefficients are numerical. This will be best shown, in an example. 1 * 4- -4- -I- o o j? - -5 = 2 , for the corresponding point P Q , AP = MJ O orfz . Let q be a point, r near to ;> , such that p^q = h = ki", and therefore oq = z + h = r i" + ki K , then Q corresponding to q, if f^ does not vanish, AQ =/( + /*) = fz +f l z .h very nearly, when h is small, and AP = fz , therefore P Q Q = f^ . ki K very nearly. Hence, if q moves round a small circle, centre p and radius k, Q moves round a corre- sponding curve very nearly a small circle, centre P and radius Y 2 324 RATIONAL AND INTEGRAL FUNCTIONS = k . mod (j \z^), and one complete circuit of q determines one complete circuit of Q. Iff^Q vanishes, but f 2 z does not, P Q =fz^Q k 2 i 2 * nearly, and then .an increase of K by 4, giving an increase of 2/c by 8, one complete circuit of q determines two complete circuits of Q. So generally if / a s ,/ 2 2; , , . ./ TO _iz all vanish, but / m z does not, P()Q frnZo k m i mic , and one complete circuit of q determines m complete circuits of Q. Hence it follows that for every point on the w plane there is a corresponding point on the z plane. For if there were any region on the w plane, the points in which had no corresponding points in the z plane, P being taken on or close to the boundary of such region, Q in moving round P , while q moves round p , would pass into that region, and thus there would be points in it corresponding to points in the z plane, contrary to the hypothesis. Thus for any assumed value of w or/2, there is certainly a value of z. Now for the origin A w = 0, and as we have seen there must be some point in the z plane corresponding to A, therefore there is some value of z, for which fz = Q; and this proves that " Every rational and integral equation has a root." Let then z l denote such a root, so that/^ = 0, then fz contains the factor z - z v and the other factor, the quotient of fz divided by z z v will be a rational and integral function of the n -1 th degree. For a like reason this last function will have a factor z 2 and its second factor will be of the n 2 degree : and this again will have a factor z - z s and another factor of the n 3 ' degree, and so on. Hence finally fz must be the product of n factors of the form zz v or fz = (z- zj (z - z. 2 ) . . . (z - zj, from which it is evident that fz vanishes, for the n values z v z 2 . . . z and for these only. Hence, understanding that when RATIONAL AND INTEGRAL FUNCTIONS 325 two or more of these values are equal, each still counts as one root, "Every rational and integral equation of the n l degree has n roots." Let OA V OA.i . . . OA n be the n vectors in the z plane cor- responding to the values z lt z. 2 , . . . z n , then A v Ac . . . A H are the n points on the z plane corresponding to the origin A on the w plane. They are termed the radical points for the f unction fz. In the case where two or more of the roots are equal, the cor- responding number of radical points coincide, and the point of coincidence must be regarded as haviug a weight equal to the number of coincident points. Also for any point P in the w plane such that AP = w , the equation fz = w being of the n degree in z has n roots, so that to any point on the w plane there correspond n points on the z plane, the radical points of the f unction fz w . 5. Expression of w in rector factors. Let p be any point in the z plane and Op = z, then z - z l = 0p OA 1 = and similarly and therefore . A. 2 p . whence the points A v A. 2 , . . . A n being given, the value of w or AP corresponding to any point p may be determined, and it is 326 RATIONAL AND INTEGRAL FUNCTIONS to be observed that this determination is quite independent of the origin 0, which may therefore be changed arbitrarily without altering w, though as such change alters z, it also alters the form of fz in respect of its coefficients. It will often be convenient to assume the origin to be the mean point of the n radical points, in which case . . .+OA n or z l +z. 2 + . . .+s n = 0. In this case the coefficient of z n ~ l in fz, being (as appears by multiplying out the factors s z v z z. 2 . . .) z l + z. 2 + . . . + z n , will vanish, or the equation fz = will be deprived of its second term. Let Atf = Pl i u \ A 2 p = p, i v - . . . A n p = PR i u ", then we have ] + V; + . . . + W = P* = Pi P-2 P n * whence p = Pl p. 2 . . . p n and v = u l + u., ... +u n or the tensor of AP or w is the product of the tensors of A l p l . . . A n p, and its inclination the sum of their inclinations. 6. Tests for the number of roots within given limits. Cauchy's Theorem. Take any closed curve in the z plane, arid let the point p describe this contour always in the positive sense, which we shall take to be that in which a man advancing would have the inside of the contour on his left, as shown by the arrow-heads in the figures. Then leaving out of account the variations of p l p.,...p and p, consider the variations of u^ u. } . . . u and the consequent variation of v. First, suppose that the contour includes none of the radical points A v A. 2 . . . Then it is plain that in a complete cii-cuit the amounts of positive and negative rotation of A l p about A l exactly balance one another, so that at the end u^ remains unchanged. The same will for like reasons be true of M.,, ., . . . u n , and therefore also for v, which is equal to their sum RATIONAL AND INTEGRAL FUNCTIONS 327 Next, suppose the contour to include the radical point A l and none of the rest. Then at the end of a complete circuit of p, the balance of positive and negative rotations of A^p about A v will give an increase of u^ by 4 L , while u. 2 M 3 . . . u n remain unchanged : therefore v will have increased by 4 L , or AP will have made one complete revolution round A. 328 RATIONAL AND INTEGRAL FUNCTIONS Again suppose the contour to include two radical points A v A. 2 and no others. Then by like reasoning both u^ and u. 2 are increased by 4 L and therefore v by eight right angles, or AP will have made two complete revolutions about A. The same will be true, however near A l is to A,, and therefore also if A^ and A 2 coincide, as they do when the two roots z r , z.-, are equal.* Lastly, similar reasoning proves generally that if the contour includes m radical points, m being the sum of the weights of the points where several are coincident, for one complete circuit of p there will be an increase of v by 4m L , or AP will make m com- plete revolutions about A- Hence the number of complete revolutions of AP about A, when p makes one complete circuit round any closed curve in the z plane, is the same as, and so determines, the number of radical points within that contour. Y From the identity pi = X + iY in 1, we have tan v = , so that JL if the value of Y/X were determined for each point on the contour the changes of v would be completely determined. Now if, as * This may also be seen from the consideration that in this case fz has the factor (z-Zj) 2 and therefore both/;^ and/^ vanish, so that / (z x + h) =/ 2 z l h 2 -f/s z l h s + . . . - =/ 2 z 1 7t 2 very nearly, when h is small, so that for one complete circuit of a small contour round A l there are two complete revolutions of AP about A. Then it may be shown that for any finite contour including no other radical points, the number of complete revo- lutions is the same. RATIONAL AND INTEGRAL FUNCTIONS 329 v increases or AP revolves in the positive sense, tan v changes sign through 0, it changes from to + ; while if, as v decreases or AP revolves in the negative sense, tan v changes sign through 0, it changes from + to , and in one complete revolution it makes in either case two such changes. Hence in making one complete revolution irregularly, that is, where the forward rotation exceeds the backward rotation by one whole revolution, the number of changes of sign through of tanv from -- to + must exceed those from + to by two : and so in making in complete revolutions the first must exceed the other by 2m. Hence if in one complete circuit of p, YjX * changes sign by the vanishing of Y k times from to + and I times from + to \ (kl) will be the number of complete revolutions of AP and therefore the number of radical points within the contour described by p. This resulo is due to Cauchy, but the proof above wilt (it is believed) be found easier and more direct than that given by him. 7. Conjugate Functions. 1. A function of a complex variable x + iy may always be reduced to the form X + iY, where X, Y are scalar functions of the scalars x, y. The functions X, Y have important properties in relation to one another, from which they are called Conjugate Functions, and the theory of such functions has wide and im- portant applications in the investigations of Mathematical Physics. This is not the place to treat of the Theory of Conjugate Functions generally, but some of its principal proposi- tions may here be exemplified in the case of Rational and Integral Functions. We have seen that by taking for the origin in the z plane the mean point of the n radical points, fz is changed as to its co- efficients, so that the coefficient a l vanishes while w is unaltered hence with this origin we have from 1, * If there be no radical point on the contour, .Y and Y cannot both vanish for the same position of p. 330 RATIONAL AND INTEGRAL FUNCTIONS X = r n cos nu 4- 2 r n ~ 2 cos (n 2u + a ) + . . . + cos a,, T = r n sin nu + a. 2 r n ~ 2 sin (n 2w + a ) + . . . + a n sin a n . Consider the curves on the z planecorresponding to the lines parallel to the axes on the w planes, for which the conjugate functions X, Y are constant. Then since the values of X, Y are unique for each point on the plane, no two curves of the series X = X , or of the series Y = Y , can intersect. But the curves corresponding to A' = 0, Y = respectively, must determine by their intersec- tion the n radical points and no other ; and so the curves cor- responding to A' = AT , Y = YQ must determine by their intersec- tion the n points corresponding to the extremity of the vector A' + iY . Hence, remembering that we are dealing only with scalar values of x, y and X, Y, any curve of the system X = const, intersects any curve of the system Y = const., though both are of the n ih degree, in n points and n only, reckoning 2, 3 ... coincident points as 2, 3 ... distinct points. A Iso for the curve X = X, since cos nu = -I cos (n 2 w + a,) . . n cos a n + A.* \ A' _*. _ r" when r is very large, cos nu is very small, and ultimately cos nu 2A. + 1 L = 0, when r GO , or u = - , where A is an integer ; hence n the curve has its ultimate direction, when r becomes infinite, that determined by one or o 2n-l 2n + 4n 1 3 determined by one or other of the series of n angles, -,-... n n . right angles. Farther, since r cosnu = -- 2 cos (n- 2 u -f a,) ... - -^-_ cos a w + -- , , fjf Mil 1 ft*7l 1 Lt r -oo (r cos nu) = 0, from which it is to be inferred that these ultimate directions all pass through the origin. Hence the curves of the system X = A' have n rectilinear asymptotes all passing through the mean point of the n radical points, each inclined to the preceding at an angle of 2/n L , and the first making the angle l/rc L with the prime vector. RATIONAL AND INTEGRAL FUNCTIONS 331 Like reasoning proves that the curves of the system Y = Y have n rectilinear asymptotes all passing through the mean point of the n radical points, each inclined to the preceding at an angle of 2/n L , and the first coincident with the prime vector. In fact the asymptotes of either system are the bisectors of the angles between those of the other, as in the figure (for the case n = 5) where the continuous lines are the asymptotes of the Y system, and the broken lines those of the A" system. FIG. [The figure is drau n for the case, where fz = s + z 3 - 2. The continuous lines represent the curve Y = and its asymptotes, the prime vector axis OAi being reckoned as one of its branches. The clotted lines represent the curve X = and its asymptotes. Their intersec- tions A lt AfrAy, AH A 6 are the 5 radical points. The student should verify these results.] Now consider the curves in the neighbourhood of the ordinary point J)Q, that is, a point for which f^^ does not vanish. We '332 RATIONAL AND INTEGRAL FUNCTIONS have seen that, if a circle, rad. k, is described about j9 as a centre, the corresponding curve on the w plane, when k is very small, is a circle, rad. k mod. f^z^, about P Q as a centre, and one circuit of the one circle corresponds to one circuit of the other, so that the angles between corresponding radii in the two circles are equal. Hence since the lines X = X Q , Y Y on the w plane intersect at PQ at right angles, the corresponding lines on the z plane passing through the point j5 intersect also at right angles. There- fore the curves X = JT , Y = Y Q on the z plane, form systems orthogonal to one another, or any curve of the one system cuts any curve of the other at right angles. It results from this that if the w plane is divided up into elementary squares by the straight lines X = Jf , }' = Y drawn at small equal intervals, the z plane will be divided by the corresponding curves into elementary areas which are ultimately squares, when the intervals are sufficiently small, and the area of any element in the w plane is to the corresponding element in the z plane in the ratio of (mod. f^z^f to 1 . There are n 1 points, for which y^ = 0, and at these points, as we have seen above, the circles about P and p correspond, but in such a way that one circuit of the circle about p Q cor- responds to two about P , so that the angle between two radii of the p circle is only half that between the corresponding radii of the P circle. Hence the curve corresponding to A" = A r has RATIONAL AND INTEGRAL FUNCTIONS 333 two branches passing through j} at right angles to one another, and that to Y = Y has also two branches bisecting the angles between the former. If two of these n 1 points coincide, at this point f it z vanishes as well as f^, and then one circuit of the circle about p Q cor- responds to three about P , so that the angle between two radii of the p circle is only one-third of that between the correspond- ing radii of the P circle, and both A' = AT and T = YQ have three branches passing through the point. 334 RATIONAL AND INTEGRAL FUNCTIONS So generally if r of these n 1 points coincide, at such a point f^fzz, . . ,f r z all vanish, and there are r + 1 branches of both curves X = Jf and Y = Y passing through the point po. The foregoing form the principal general properties of the curves representing the Conjugate Functions derived from a rational and integral function of a complex variable. Subject to these, the forms of the curves are infinitely varied according to the different values of the coefficients a- 2 , a s . . . a n . We will illustrate this by the case of a function of the third degree or .cubic function. '8. Discussion of a Cubic Function. We will limit the discussion to the case of a cubic function whose coefficients are scalar, and as we have seen that there is no loss of generality by assuming the coefficient of the second term to be zero, since this only amounts to referring the curve systems to the mean of the radical points as origin, we may write w fz s 3 + az + b, where a, b are scalars. Hence the values of the conjugate functions X, Y are X = x 3 3xy 2 + ax + b Y = 3x z y y 3 + ay or X -b = x (x* - 3y* + a) Y=y (3a; 2 - y* + a), and we have to consider the characters of the curve systems A" = A" , Y F for different values of a and b. We observe that the asymptotes of either system are the same for all the curves of that system, and aie independent of the values of a and b. They are for the X system the axis of y, and two lines through the origin inclined to it at angles of 60 RATIONAL AND INTEGRAL FUNCTIONS 335 on either side, and for the Y system the axis of x and two lines through the origin inclined to it at angles of 60 on either side, so that the asymptotes of one system bisect the angles between the asymptotes of the other. Also the Y system is a system of curves of the third degree, which is completely determinate when a is given, and the same may be said of the X' system, if we put X' for X b. In the particular case of Y 0, the curve of the third degree becomes the axis of x, and an hyperbola whose asymptotes are the inclined asymptotes of the Y system ; and in that of X' = or X = b, the curve becomes the axis of y and an hyperbola, whose asymptotes are the inclined asymptotes of the X system. These form guiding curves or curved axes of the two systems. Farther, since f^z = 3s 2 + a,f^z will vanish, if z z = - ; that 3 is, if a is positive, for the two points x = 0, y \/ -; but, if 9 = V -, y = 0. is negative, for the two points x = V -, y = 0. These, o then, will be double points on the curves of both systems, which pass through them, and there are only two such points in either system. The curves through the double points will form further guiding curves of the system. In the special case, where a = 0, the two double points merge in one and produce a triple point at the origin. I. Suppose a to be positive, and put 3k? for a. Then the double points are x = 0, y = k, and the curves passing through them are X = b, Y = 2 3 . The curve A" = b or x(x 2 3y 2 + 3k' 1 ) = 0, has for its three branches the axis of y, and the two branches of the hyperbola, whose vertices are the points + k (B, B' in the figure) on that axis, the semi-axes being k and k \/3, as shown in the figure by the continuous lines. If A'< b, it will be easily seen that the equation x(x z 3y 2 + 3k' 2 ) = a negative constant defines a curve which has three branches, such as those shown by the broken lines (P v P 2 , P 3 ) in the 336 RATIONAL AND INTEGRAL FUNCTIONS figure ; while, if X>b, the curve will have three branches, such as those shown by the dotted line (Q v $ 2 > Q s ] in the figure. FIG. 1. The continuous change from one form to the other, as X passes through the value b, is evident also from the figure. The curve T = 2^ or y (3 2 - y* + 3/fc 2 ) = 2#* reduces to the form (y - k) z (y + 2A) = 0. (y _ Jf \ / 3^ \ I __) = limit ( 9 ,) = 1, X / y=k\y + ^ ; / the ___ X two branches passing through the double point B (x = 0, y = make angles of 45 with the axes. Also where the curve cuts the o asymptotes, y = -k. Hence the curve consists of three branches, o RATIONAL AND INTEGRAL FUNCTIONS 337 as shown by the continuous lines through ]3, B l in Fig. 2, in which OJ^ == 2k. /a\- If Y<2k 3 , that is, <2( - )-, it will be easily seen that the curve \o/ FIG. 2. has three branches, such as those shown by the broken lines (2\, P 2 , P 3 ) in Fig. 2 : while if Y>2k 3 the three branches will be such as those shown by the dotted lines (Q lt Q 2 , Q.^) in the z 338 RATIONAL AND INTEGRAL FUNCTIONS figure. The continuous transition from one form to the other as Y passes through the value 2& 3 is also evident from the figure. The curve Y = - 2k s and the neighbouring curves, would be represented by the same diagram turned upside down, corre- sponding to the change of y into - y. In the next diagram, Fig. 3, are shown the guiding curves through the double point B for both systems ; also the hyper- FIG. 3. bola, vertices C, C', such that OC = OC' = k \/3 = \/a, which, with the x axis, represents the curve Y = 0. The broken lines represent curves of the X system,, the hyperbola, vertices B, B' , such that OB = OB' Jc, which, with the y axis, represents the curve X = b, and the three branches through A v A 2 , A 3 corre- sponding to X = 0, b being supposed positive. The parts shaded by horizontal lines indicate the regions in which X is negative RATIONAL AND INTEGRAL FUNCTIONS 339 and those shaded by vertical lines the regions in which Y is negative. The points A v A. 2 , A, A , where the bounding curves of these regions intersect, are the three radical points, and hence it is obvious that in this case, where a and b are both positive, the equation z 3 + az + b = has one negative and two complex roots. The case, where b is negative, would be represented by turn- ing the diagram through half a revolution about the y axis, corresponding to the change of x into a;. In this case there is one positive and two complex roots. Hence, when a is positive, the roots of the equation 2 3 + az + b = are of the form - 2a, a + ift, a ifi, where a is of the same sign as b. II. Suppose a = 0. Then fz = s 3 + b. After the preceding discussion the diagram needs no explanation. It is useful as FIG. 4. 340 RATIONAL AND INTEGRAL FUNCTIONS showing the continuous transition from the case of a positive to that of a negative. It is drawn for the case of b positive, and OA V OA 2 , OA 3 have equal tensors and are the three cube roots of b. Observe the triple point in both systems at 0, arising from the coincidence of B, B' with 0, with which also C, C' and B l coincide. III. Suppose a to be negative and put 3& 2 for a. Then the double points are x = k, y 0, and the curves passing through them are T= 0, X- b = + 2& 3 . The curve X = b - 2k* or x (a; 2 - 3y 2 - 3k 2 ) = - 2 3 reduces to the form (x + 2k) (x - A;) 2 - 3xy z = 0. FIG. 5. RATIONAL AND INTEGRAL FUNCTIONS 341 A similar discussion to that above for Y=0, when a is positive, leads for X b 2k 3 to the diagram (Fig. 5), which is in fact the same as Fig. 2 turned back through a right angle. The broken lines (P v P 2 , P 3 ) correspond to X > b 2^, and the dotted lines (Q lt Q. 2 , Q. 3 ) to X < b 2%?. If the figure is reversed by turning about the y axis, the curves correspond to the cases A' > = or < b + 2t?. The curve Y= is the hyperbola, vertices C, C' (Fig. 6) and the x axis, (7, C' being thus double points. The broken lines (P v P 9 , P 3 ) correspond to Y positive, and the dotted lines (Q v <2 2 > #3) to y negative. FIG. 6. In the following diagram (Fig. 7), both the curves are shown for the case, in which b = 2P or b = 0, 7=0 342 RATIONAL AND INTEGRAL FUNCTIONS In this case the double point at C is a double radical point, since there both X and Y 0, the other radical point being C v such that OC^= - WC, where the third branch of A~=0 cuts the x axis. Hence the three roots of fz = are in this case scalar, two of them being equal. The different regions shaded horizontally and vertically show those for which A", Y respectively are nega- tive. The hyperbola, vertices B, B', together with the y axis, represent the curve A' = b. FIG. 7. If b > 2k 3 or < - 2k 3 , so that i - J + ( - j is positive, the curve for A' = passes into the negative region from C as shown in Fig. 5, and the two radical points which coincided with C (or with C') leave the x axis and lie on the hyperbola through C (or RATIONAL AND INTEGRAL FUNCTIONS - 343 C'). so that the two roots become complex of the form a + i(3, a - i/3. (&\2 /\ 3 -} + (-) is negative, the curve '"/ \3/ for A' = passes into the positive regions of A 7 " and so intersects the x axis in three points as P lf P. 2 , P 3 in Fig. 5. The three radical points are therefore on the x axis, and the three roots are scalar. Hence the well-known condition that the three roots of /J\2 /a\ 3 a cubic are all scalar, only if f J + f J is negative. 9. The Binomial Function z n l. If w = z n l, the relation between the variations of z and w may be very simply expressed as follows. (The proof, and diagrammatic illustrations for particular values of n, are left to the student.) To a circle described on the z plane about the origin as centre corresponds a circle on the w plane, the vector of whose centre is 1, in such a manner that to one complete circuit of the former correspond n complete circuits of the latter, any radius of the one inclined at the angle u to the prime vector corresponding to a radius of the other inclined at the angle nu. Also the ratio of the radii of any two such circles on the w plane is the w-plicate ratio of those of the corresponding circles on the z plane. For other forms of w, to a circle similarly described on the z plane corresponds a curve on the w plane formed by the combi- nation of two or more circular motions of greater or less com- plexity according to the number of the coefficients of the terms offz. RICHARD CLAY AND SONS, LIMITED, LONDON AND BUNGAV. r UNIVERSITY OF CALIFORNIA LIBRARY Los Angeles This book is DUE on the last date stamped below. 5DL9 JUN 1 8 1959 MAY 3 1960 JUL 2 7 196g 20 1965 Form L9-17wt-8,'55(B3339s4)441 H33a ring 4 A 000210452 9 JUL72