Text extracted from opening pages of book: HIGHER ALGEBRA BY S. BARNARD, M. A. FORMERLY ASSISTANT MASTER AT RUGBY SCHOOL, LATE FELLOW AND LECTURER AT EMMANUEL COLLEGE, CAMBRIDGE AND J. M. CHILD, B. A., B. Sc. FORMERLY LECTURER IN MATHEMATICS IN THE UNIVERSITY OF' MANCHESTER LATE HEAD OF MATHEMATICAL DEPARTMENT, TECHNICAL COLLEGE, DERBY FORMERLY SCHOLAR AT JESUS COLLEGE, CAMBRIDGE LON-DON MACMILLAN fcf'CO LTD * v NEW YORK ST MARTIN * S PRESS 1959 This book is copyright in all countries which are signatories to the Berne Convention First Edition 1936 Reprinted 1947, 949> I952> * 955, 1959 MACMILLAN AND COMPANY LIMITED London Bombay Calcutta Madras Melbourne THE MACMILLAN COMPANY OF CANADA LIMITED Toronto ST MARTIN'S PRESS INC New York PRINTED IN GREAT BRITAIN BY LOWE AND BRYDONE ( PRINTERS) LIMITED, LONDON, N. W. IO CONTENTS ix IjHAPTER EXEKCISE XV ( 128). Minors, Expansion in Terms of Second Minors ( 132, 133). Product of Two Iteterminants ( 134). Rectangular Arrays ( 135). Reciprocal Deteyrrtlilnts, Two Methods of Expansion ( 136, 137). Use of Double Suffix, Symmetric and Skew-symmetric Determinants, Pfaffian ( 138-143), ExERtad XVI ( 143) X. SYSTEMS OF EQUATIONS. Definitions, Equivalent Systems ( 149, 150). Linear Equations in Two Unknowns, Line at Infinity ( 150-152). Linear Equations in Three Unknowns, Equation to a Plane, Plane at Infinity ( 153-157). EXEKCISE XVII ( 158). Systems of Equations of any Degree, Methods of Solution for Special Types ( 160-164). EXERCISE XVIII ( 164). XL RECIPROCAL AND BINOMIAL EQUATIONS. Reduction of Reciprocal Equations ( 168-170). The Equation x n - 1= 0, Special Roots ( 170, 171). The Equation x n - A = 0 ( 172). The Equation a 17 - 1 == 0, Regular17-sided Polygon ( 173-176). EXERCISE XIX ( 177). AND BIQUADRATIC EQUATIONS. The Cubic Equation ( roots a, jS, y), Equation whose Roots are ( - y) 2, etc., Value of J, Character of Roots ( 179, 180). Cardan's Solution, Trigonometrical Solution, the Functions a - f eo/? - f-\> V> a-f a> 2 4-a> y ( 180, 181). Cubic as Sum of Two Cubes, the Hessftfh ( 182, 183). Tschirnhausen's Transformation ( 186). EXERCISE XX ( 184). The Biquadratic Equation ( roots a, y, 8) ( 186). The Functions A= y ] aS, etc., the Functions /, J, J, Reducing Cubic, Character of Roots ( 187-189). Ferrari's Solution and Deductions ( 189-191). Descartes' Solution ( 191). Conditions for Four Real Roots ( 192-ty). Transformation into Reciprocal Form ( 194). Tschirnhausen's Trans formation ( 195). EXERCISE XXI ( 197). OP IRRATIONALS. Sections of the System of Rationals, Dedekind's Definition ( 200, 201). Equality and Inequality ( 202). Use of Sequences in defining a Real Number, Endless Decimals ( 203, 204). The Fundamental Operations of Arithmetic, Powers, Roots and Surds ( 204-209). Irrational Indices, Logarithms ( 209, 210). Definitions, Interval, Steadily Increasing Functions ( 210). Sections of the System of Real Numbers, the Continuum ( 211, 212). Ratio and Proportion, Euclid's Definition ( 212, 213). EXERCISE XXII ( 214). x CONTENTS CHAPTER XIV/ INEQUALITIES. Weierstrass' Inequalities ( 216). Elementary Methods ( 210, 217) For n Numbers a l9 a 2 a > \* JACJJ n n n ( a* -!)/* ( a - I)/*, , ( 219). xa x ~ l ( a-b)$ a x - b x xb x ~ l ( a - 6), ( 219). ( l+ x) n l+ nx, ( 220). Arithmetic and Geometric Means ( 221, 222). - - V n and Extension ( 223). Maxima and Minima ( 223, 224). EXERCISE XXIII ( 224). XV. SEQUENCESAND LIMITS. Definitions, Theorems, Monotone Sequences ( 228-232). E* ponential Inequalities and Limits, l\ m / i\ n / l\-m / 1 \ ~ n 1) >(!+-) and ( 1--) n, m/ \ n/ \ mj \ nj / 1 \ n / l\ w lim ( 1-f-= lim( l--) = e, ( 232,233). n _ > 00 V nj \ nj EXERCISE XXIV ( 233). General Principle of Convergence ( 235-237). Bounds of a Sequent Limits of Inde termination ( 237-240). Theorems: ( 1) Increasing Sequence ( u n ), where u n - u n l 0 and u n+ l lu n -* l, then u n n -* L ( 3) If lim u n l, then lim ( U

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