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ENGINEERING MATHEMATICS 
 

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 Coal Age ^ Electric Railway Journal 
 Electrical World v Engineering News -Record 
 Railway Age Gazette " American Machinist 
 Electrical Merchandising v The Contractor 
 Engineering 8 Mining Journal ^ Power 
 Metallurgical & Chemical Engineering 
 
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MGINEEEING MATHEMATICS 
 
 A SERIES OF LECTURES DELIVERED 
 AT UNION COLLEGE 
 
 BY 
 
 CHAELES PEOTEUS STEINMETZ, A.M., Ph.D. 
 
 PAST PEBSIDENT 
 AMERICAN INSTITUTE OF ELECTKICAL ENODtEEKS 
 
 THIRD EDITION 
 REVISED AND ENLARGED 
 
 McGRAW-HILL BOOK COMPANY, Inc. 
 
 239 WEST 39TH STREET. NEW YORK 
 
 LONDON: HILL PUBLISHING CO., Ltd. 
 6 & 8 BOUVEEIE ST., E. C. 
 
 1917 
 
Copyright, 1911, 1915 and 1917, by the 
 McGkaw-Hill Book Company, Inc. 
 
PREFACE TO THIRD EDITION. 
 
 In preparing the third edition of Engineering Mathematics, 
 besides revision and correction of the previous text, considera- 
 ble new matter has been added. 
 
 The chain fraction has been recognized and discussed as a 
 convenient method of numerical representation and approxi- 
 mation; a paragraph has been devoted to the diophantic equa- 
 tions, and a section added on engineering reports, discussing 
 the different purposes for which engineering reports are made, 
 and the corresponding character and nature of the report, in 
 its bearing on the success and recognition of the engineer's 
 work. 
 
 - Charles Pbotbus Steinmetz. 
 Camp Mohawk, 
 September 1st, 1917. 
 
Cornell University 
 Library 
 
 The original of tiiis book is in 
 tine Cornell University Library. 
 
 There are no known copyright restrictions in 
 the United States on the use of the text. 
 
 http://www.archive.org/details/cu31924003954801 
 
PREFACE TO FIRST EDITION. 
 
 The following work embodies the subject-matter of a lecture 
 course which I have given to the junior and senior electrical 
 engineering students of Union University for a number of 
 years. 
 
 It is generally conceded that a fair knowledge of mathe- 
 matics is necessary to the engineer, and especially the electrical 
 engineer. For the latter, however, some branches of mathe- 
 matics are of fundamental importance, as the algebra of the 
 general number, the exponential and trigonometric series, etc., 
 which are seldom adequately treated, and often not taught at 
 all in the usual text-books of mathematics, or in the college 
 course of analytic geometry and calculus given to the engineer- 
 ing students, and, therefore, electrical engineers often possess 
 little knowledge of these subjects. As the result, an electrical 
 engineer, even if he possess a fair knowledge of mathematics, 
 may often find difficulty in dealing with problems, through lack 
 of familiarity with these branches of mathematics, which have 
 become of importance in electrical engineering, and may also 
 find difficulty in looking up information on these subjects. 
 
 In the same way the college student, when beginning the 
 study of electrical engineering theory, after completing his 
 general course of mathematics, frequently finds himself sadly 
 deficient in the knowledge of mathematical subjects, of which 
 a complete familiarity is required for effective understanding 
 of electrical engineering theory. It was this experience which 
 led me some years ago to start the course of lectures which 
 is reproduced in the following pages. I have thus attempted to 
 bring together and discuss explicitly, with numerous practical 
 applications, all those branches of mathematics which are of 
 special importance to the electrical engineer. Added thereto 
 
 vii 
 
viii PREFACE. 
 
 are a number of subjects which experience has shown me 
 to be important for the effective and expeditious execution of 
 electrical engineering calculations. Mere theoretical knowledge 
 of mathematics is not sufficient for the engineer, but it must 
 be accompanied by ability to apply it and derive results — to 
 carry out numerical calculations. It is not sufficient to know 
 how a phenomenon occurs, and how it may be calculated, but 
 very often there is a wide gap between this knowledge and the 
 ability to carry out the calculation; indeed, frequently an 
 attempt to apply the theoretical knowledge to derive numerical 
 results leads, even in simple problems, to apparently hopeless 
 complication and almost endless calculation, so that all hope 
 of getting reliable results vanishes. Thus considerable space 
 has been devoted to the discussion of methods of calculation, 
 the use of curves and their evaluation, and other kindred 
 subjects requisite for effective engineering work. 
 
 Thus the following work is not intended as a complete 
 course in mathematics, but as supplementary to the general 
 college course of mathematics, or to the general knowledge of 
 mathematics which every engineer and really every educated 
 man should possess. 
 
 In illustrating the mathematical discussion, practical 
 examples, usually taken from the field of electrical engineer- 
 ing, have been given and discussed. These are sufficiently 
 numerous that any example dealing with a phenomenon 
 with which the reader is not yet familiar may be omitted and 
 taken up at a later time. 
 
 As appendix is given a descriptive outline of the intro- 
 duction to the theory of functions, since the electrical engineer 
 should be familiar with the general relations between the 
 different functions which he meets. 
 
 In relation to " Theoretical Elements of Electrical Engineer- 
 ing," "Theory and Calculation of Alternating Current Phe- 
 nomena," and " Theory and Calculation of Transient Electric 
 Phenomena," the following work is intended as an introduction 
 and explanation of the mathematical side, and the most efficient 
 method of study, appears to me, to start with " Electrical 
 Engineering Mathematics," and after entering its third 
 chapter, to take up the reading of the first section of " Theo- 
 retical Elements," and then parallel the study of " Electrical 
 
PREFACE. IX 
 
 Engineering Mathematics," " Theoretical Elements of Electrical 
 Engineering," and " Theory and Calculation of Alternating 
 Current Phenomena," together with selected chapters from 
 "Theory and Calculation of Transient Electric Phenomena," 
 and after this, once more systematically go through all four 
 
 books. 
 
 Charles P. Steinmetz. 
 
 Schenectady, N. Y., 
 December, 1910. 
 
 PREFACE TO SECOND EDITION. 
 
 In preparing the second edition of Engineering Mathe- 
 matics, besides revision and correction of the previous text, 
 considerable new matter has been added, more particularly 
 with regard to periodic curves. In the former edition the 
 study of the wave shapes produced by various harmonics, 
 and the recognition of the harmonics from the wave shape, 
 have not been treated, since a short discussion of wave shapes 
 is given in "Alternating Current Phenomena." Since, how- 
 ever, the periodic functions are the most important in elec- 
 trical engineering, it appears necessary to consider their shape 
 more extensively, and this has been done in the new edition. 
 
 The symbolism of the general number, as applied to alter- 
 nating waves, has been changed in conformity to the decision 
 of the International Electrical Congress of Turin, a discussion 
 of the logarithmic and semi-logarithmic scale of curve plot- 
 ting given, etc. 
 
 Charles P. Steinmetz. 
 
 December, 1914. 
 
CONTENTS. 
 
 PAGE 
 
 Preface v 
 
 CHAPTER I. THE GENERAL NUMBER. 
 
 A. The System of Numbers. 
 
 1. Addition and Subtraction. Origin of numbers. Counting and 
 
 measuring. Addition. Subtraction as reverse operation of 
 addition 1 
 
 2. Limitation of subtraction. Subdivision of tiie absolute numbers 
 
 into positive and negative 2 
 
 3. Negative number a mathematical conception like the imaginary 
 
 number. Cases where the negative number has a physical 
 meaning, and cases where it has not 4 
 
 4. Multiplication and Division. Multiplication as multiple addi- 
 
 tion. Division as its reverse operation. Limitation of divi- 
 sion 6 
 
 5. The fraction as mathematical conception. Cases where it has a 
 
 physical meaning, and cases where it has not 8 
 
 6. Involution and Evolution. Involution as multiple multiplica- 
 
 tion. Evolution as its reverse operation. Negative expo- 
 nents 9 
 
 7. Multiple involution leads to no new operation 10 
 
 8. Fractional exponents 10 
 
 9. Irrational Numbers. Limitation of evolution. Endless decimal 
 
 fraction. Rationality of the irrational number 11 
 
 10. Quadrature numbers. Multiple values of roots. Square root of 
 
 negative quantity representing quadrature number, or rota- 
 tion by 90° 13 
 
 11. Comparison of positive, negative and quadrature numbers. 
 
 Reality of quadrature number. Cases where it has a physical 
 meaning, and cases where it has not 14 
 
 12. General Numbers. Representation of the plane by the general 
 
 number. Its relation to rectangular coordinates 16 
 
 13. Limitation of algebra by the general number. Roots of the unit. 
 
 Number of such roots, and their relation 18 
 
 14. The two reverse operations of involution 19 
 
 xi 
 
xii CONTENTS. 
 
 PAGE 
 
 15. Logarithmation. Relation between logarithm and exponent of 
 
 involution. Reduction to other base. Logarithm of negative 
 quantity 20 
 
 16. Quaternions. Vector calculus of space 22 
 
 17. Space rotors and their relation. Super algebraic nature of space 
 
 analysis 22 
 
 B. Algebra op the General Number of Complex Quantity. 
 
 Rectangular and Polar Coordinates 25 
 
 18. Powers of j. Ordinary or real, and quadrature or imaginary 
 
 number. Relations 25 
 
 19. Conception of general number by point of plane in rectangular 
 
 coordinates; in polar coordinates. Relation between rect- 
 angular and polar form 26 
 
 20. Addition and Subtraction. Algebraic and geometrical addition 
 
 and subtraction. Combination and resolution by parallelo- 
 gram law 28 
 
 21. Denotations 30 
 
 22. Sign of vector angle. Conjugate and associate numbers. Vec- 
 
 tor analysis 30 
 
 23. Instance of steam path of turbine 33 
 
 24. Multiplication. Multiplication in rectangular coordinates. ... 38 
 
 25. Multiplication in polar coordinates. Vector and operator 38 
 
 26. Physical meaning of result of algebraic operation. Representa- 
 
 tion of result ~. 40 
 
 27. Limitation of application of algebraic operations to physical 
 
 quantities, and of the graphical representation of the result. 
 Graphical representation of algebraic operations between 
 current, voltage and impedance 40 
 
 28. Representation of vectors and of operators 42 
 
 29. Division. Division in rectangular coordinates 42 
 
 30. Division in polar coordinates 43 
 
 31. Involution and Evolution. Use of polar coordinates 44 
 
 32. Multiple values of the result of evolution. Their location in the 
 
 plane of the general number. Polyphase and n phase systems 
 
 of numbers 45 
 
 33. The n values of Vl and their relation 46 
 
 34. Evolution in rectangular coordinates. Complexity of result ... 47 
 
 35. Reduction of products and fractions of general numbers by polar 
 
 representation. Instance 43 
 
 36. Exponential representations of general numbers. The different 
 
 forms of the general number 49 
 
 37. Instance of use of exponential form in solution of differential 
 
 equation 5q 
 
CONTENTS. xui 
 
 PKOB 
 
 38. Logarithmation. Resolution of the logarithm of a general 
 
 number 51 
 
 CHAPTER n. THE POTENTIAL SERIES AND EXPONENTIAL 
 FUNCTION. 
 
 A. General. 
 
 39. The infinite series of powers of r 52 
 
 40. Approximation by series 53 
 
 41. Alternate and one-sided approximation 54 
 
 42. Convergent and divergent series 55 
 
 43. Range of convergency. Several series of different ranges for 
 
 ' same expression 56 
 
 44 Discussion of convergency in engineering applications 57 
 
 45. Use of series for approximation of small terms. Instance of 
 
 electric circuit 58 
 
 46. Binomial theorem for development in series. Instance of in- 
 
 ductive circuit 59 
 
 47. Necessity of development in series. Instance of arc of hyperbola 60 
 
 48. Instance of numerical calculation of log (1 +x) 63 
 
 B. Differential Equations. 
 
 49. Character of most differential equations of electrical engineering. 
 
 Their typical forms 64 
 
 dy 
 
 50. -7- = y. Solution by series, by method of indeterminate co- 
 ax 
 
 efficients 65 
 
 dz 
 
 51. — = az. Solution by indeterminate coefficients 68 
 
 dx 
 
 52. Integration constant and terminal conditions 68 
 
 53. Involution of solution. Exponential function 70 
 
 54. Instance of rise of field current in direct current shunt motor . . 72 
 
 55. Evaluation of inductance, and numerical calculation 75 
 
 56. Instance of condenser discharge through resistance 76 
 
 (Py . . ^ . 
 
 57. Solution of -— = ay by indeterminate coefficients, by exponential 
 
 function 78 
 
 58. Solution by trigonometric functions 81 
 
 59. Relations between trigonometric functions and exponential func- 
 
 tions with imaginary exponent, and inversely 83 
 
 60. Instance of condenser discharge through inductance. The two 
 
 integration constants and terminal conditions 84 
 
 61. Effect of resistance on the discharge. The general differential 
 
 equation 86 
 
xiv CONTENTS. 
 
 PAGE 
 
 62. Sqlution of the general differential equation by means of the 
 
 exponential function, by the method of indeterminate 
 coefficients ■ 86 
 
 63. Instance of condenser discharge through resistance and induc- 
 
 tance. Exponential solution and evaluation of constants. . .. 88 
 
 64. Imaginary exponents of exponential functions. Reduction to 
 
 trigonometric functions. The oscillating functions 91 
 
 65. Explanation of tables of exponential functions) 92 
 
 CHAPTER III. TRIGONOMETRIC SERIES. 
 
 A. Trigonometric Functions. 
 
 66. Definition of trigonometric functions on circle and right triangle 94 
 
 67. Sign of functions in different quadrants 95 
 
 68. Relations between sin, cos, tan and cot 97 
 
 69. Negative, supplementary and complementary angles 98 
 
 70. Angles {x±7:) and (a^±:jj) 100 
 
 71. Relations between two angles, and between angle and double 
 
 angle 102 
 
 72. Differentiation and integration of trigonometric functions. 
 
 Definite integrals 10.3 
 
 73. The binomial relations 104 
 
 74. Polyphase relations 104 
 
 75. Trigonometric formulas of the triangle 105 
 
 B. Trigonometric Series. 
 
 76. Constant, transient and periodic phenomena. Univalent peri- 
 
 odic function represented by trigonometric series 106 
 
 77. Alternating sine waves and distorted waves 107 
 
 78. Evaluation of the Constants from Instantaneous Values. Cal- 
 
 culation of constant term of series 108 
 
 79. Calculation of cos-coefficients 110 
 
 80. Calculation of sin-coefficients 113 
 
 81. Instance of calculating 11th harmonic of generator wave 114 
 
 82. Discussion. Instance of complete calculation of pulsating cur- 
 
 rent wave 116 
 
 83. Alternating waves as symmetrical waves. Calculation of sym- 
 
 metrical wave 117 
 
 84. Separation of odd and even harmonics and of constant term . . . 120 
 
 85. Separation of sine and cosine components 121 
 
 86. Separation of wave into constant term and 4 component waves 122 
 
 87. Discussion of calculation 123 
 
 88. Mechanism of calculation 124 
 
CONTENTS. XV 
 
 PAGE 
 
 89. Instance of resolution of the annual temperature curve 125 
 
 90. Constants and equation of temperature wave 131 
 
 91. Discussion of temperature wave 132 
 
 C. Reduction of Trigonometric Series by Polyphase Relation. 
 
 92. Method of separating certain classes of harmonics, and its 
 
 limitation 134 
 
 93. Instance of separating the 3d and 9th harmonic of transformer 
 
 exciting current 136 
 
 D. Calculation op Trigonometric Series from other Trigono- 
 
 metric Series. 
 
 94. Instance of calculating current in long distance transmission line, 
 
 due to distorted voltage wave of generator. Line constants . . 139 
 
 95. Circuit equations, and calculation of equation of current 141 
 
 96. Effective value of current, and comparison with the current 
 
 produced by sine wave of voltage 143 
 
 97. Voltage wave of reactance in circuit of this distorted current ... 145 
 
 CHAPTER IV. MAXIMA AND MINIMA. 
 
 98. Maxima and miniina by curve plotting. Instance of magnetic 
 
 permeability. Maximum power factor of induction motor as 
 function of load 147 
 
 99. Interpolation of maximum value in method of curve plotting. 
 
 Error in case of unsymmetrical curve. Instance of efficiency 
 of steam turbine nozzle. Discussion 149 
 
 100. Mathematical method. Maximum, minimum and inflexion 
 
 point. Discussion 152 
 
 101. Instance: Speed of impulse turbine wheel for maximum 
 
 efficiency. Current in transformer for maximum efficiency. 154 
 
 102. Effect of intermediate variables. Instance: Maximum power 
 
 in resistance shunting a constant resistance in a constant cur- 
 rent circuit 155 
 
 103. Simplification of calculation by suppression of unnecessary terms, 
 
 etc. Instance 157 
 
 104. Instance: Maximum non-inductive load on inductive transmis- 
 
 sion line. Maximum current in line 158 
 
 105. Discussion of physical meaning of mathematical extremum. 
 
 Instance 160 
 
 106. Instance : External reactance giving maximum output of alter- 
 
 nator at constant external resistance and constant excitation. 
 Discussion 161 
 
 107. Maximum efficiency of alternator on non-inductive load. Dis- 
 
 ! cussion of physical limitations 162 
 
xvi CONTENTS. 
 
 PAGE 
 
 108. Fxtrema with several independent variables. Method of math 
 
 ematical calculation, and geometrical meaning ■ 
 
 109. Resistance and reactance of load to give maximum output of 
 
 transmission line, at constant supply voltage 
 
 110. Discussion of physical limitations ; • 
 
 111. Determination of extrcma by plotting curve of differential quo- 
 
 tient. Instance: Maxima of current wave of alternator of 
 distorted voltage on transmission line 
 
 112. Graphical calculation of differential curve of empirical curve, 
 
 for determining extrema 
 
 113. Instance: Maximum permeability calculation ■ •■ 170 
 
 114. Grouping of battery cells for maximum power in constant resist- 
 
 ance 
 
 115. Voltage of transformer to give maximum output at constant 
 
 loss ;•■■• 173 
 
 116. Voltage of transformer, at constant output, to give maximum 
 
 efficiency at full load, at half load 174 
 
 117. Maximum value of charging current of condenser through 
 
 inductive circuit (a) at low resistance; (b) at high resistance. 175 
 
 118. At what output is the efficiency of an induction generator a max- 
 
 imum? 177 
 
 119. Discussion of phy.sical limitations. Maximum efficiency at con- 
 
 stant current output 178 
 
 120. Method of Least Squares. Relation of number of observa- 
 
 tions to number of constants. Discussion of errors of 
 observation 179 
 
 121. Probability calculus and the minimum sum of squares of the 
 
 errors ISl 
 
 122. The differential equations of the sum of least squares 182 
 
 123. Instance: Reduction of curve of power of induction motor 
 
 running light, into the component losses. Discussion of 
 
 results 182 
 
 12.3A. Diophantic equations 186 
 
 CHAPTER V. METHODS OF APPROXIMATION, 
 
 124. Frequency of small quantities in electrical engineering problems. 
 
 Instances. Approximation by dropping terms of higher order. 187 
 
 125. Multiplication of terms with small quantities 188 
 
 126. Instance of calculation of power of direct current shunt motor . 189 
 
 127. Small quantities in denominator of fractions 190 
 
 128. Instance of calculation of induction motor current, as function 
 
 of slip 191 
 
CONTENTS. xvii 
 
 PAGE 
 
 129. Use of binomial series in approximations of powers and roots, 
 
 and in numerical calculations 193 
 
 130. Instance of calculation of current in alternating circuit of low 
 
 inductance. Instance of calculation of short circuit current 
 
 of alternator, as function of speed 195 
 
 131. Use of exponential series and logarithmic series in approxima- 
 
 tions 196 
 
 132. Approximations of trigonometric functions 198 
 
 133. McLaurin's and Taylor's series in approximations 198 
 
 134. Tabulation of various infinite series and of the approximations 
 
 derived from them 199 
 
 135. Estimation of accuracy of approximation. Application to 
 
 short circuit current of alternator 200 
 
 136. Expressions which are approximated by (1 +s) and by (1 —s) . . 201 
 
 137. Mathematical instance of approximation 203 
 
 138. Equations of the transmission ■ Line. Integration of the 
 
 differential equations 204 
 
 139. Substitution of the terminal conditions 205 
 
 140. The approximate equations of the transmission line 206 
 
 141. Numerical instance. Discussionof accuracy of approximation. 207 
 
 141A. Approximation by chain fraction 208 
 
 141B. Approximation by chain fraction 208c 
 
 CHAPTER VI. EMPIRICAL CURVES. 
 
 A. General. 
 
 142. Relation between empirical curves, empirical equations and 
 
 rational equations 209 
 
 143. Physical nature of phenomenon. Points at zero and at infinity. 
 
 Periodic or non-periodic. Constant terms. Change of curve 
 law. Scale 210 
 
 B. Non-Periodic Curves. 
 
 144. Potential Series. Instance of core-loss curve 212 
 
 145. Rational and irrational use of potential series. Instance of fan 
 
 motor torque. Limitations of potential series 214 
 
 146. Parabolic and Hyperbolic Curves. Various shapes of para- 
 
 bolas and of hyperbolas 216 
 
 147. The characteristic of parabolic and hyperbolic curves. Its use 
 
 and limitation by constant terms 223 
 
 148. The logarithmic characteristic. Its use and limitation 224 
 
 149. Exponential and Logarithmic Curves. The exponential 
 
 function 227 
 
 150. Characteristics of the exponential curve, their use and limitation 
 
 by constant term. Comparison of exponential curve and 
 hyperbola 228 
 
xviii CONTENTS. 
 
 PAGE 
 
 151. Double exponential functions. Various shapes thereof 231 
 
 152. Evaluation of Empirical Curves. General principles of 
 
 investigation of empirical curves 233 
 
 153. Instance : The volt-ampere characteristic of the tungsten lamp, 
 
 reduced to parabola with exponent 0.6. Rationalized by 
 reduction to radiation law 235 
 
 154. The volt-ampere characteristic of the magnetite arc, reduced 
 
 to hyperbola with exponent 0.5 .'.'■■■ ^"^^ 
 
 155. Change of electric current with change of circuit conditions, 
 
 reduced to double exponential function of time 241 
 
 156. Rational reduction of core-loss curve of paragraph 144, by 
 
 parabola with exponent 1.6 244 
 
 157. Reduction of magnetic characteristic, for higher densities, to 
 
 hyperbolic curve. Instance of the investigation of a hys- 
 teresis curve of silicon steel 246 
 
 C. Periodic Curves. 
 
 158. Distortion of sine wave by harmonics 255 
 
 159. Third and fifth harmonic. Peak, multiple peak, flat top and 
 
 sawtooth 255 
 
 160. Combined effect of third and fifth harmonic 263 
 
 161. Even harmonics. Unequal shape and length of half waves. 
 
 Combined second and third harmonic 266 
 
 162. Effect of high harmonics 269 
 
 163. Ripples and nodes caused by higher harmonics. Incommen- 
 
 surable waves 27] 
 
 CHAPTER vn. NUMERICAL CALCULATIONS. 
 
 164. Method of Calculation. Tabular form of calculation 275 
 
 165. Instance of transmission line regulation 277 
 
 166. Exactness op Calculation. Dsgrees of exactness: magnitude, 
 
 approximate, exact 279 
 
 167. Number of decimals 28 1 
 
 168. iNTELLiGrBiLiTT OP ENGINEERING Data. Curve plotting for 
 
 showing shape of function, and for record of numerical values 283 
 
 169. Scale of curves. Principles 286 
 
 170. Logarithmic and semi-logarithmic paper and its proper use 287 
 
 171. Completeness of record 290 
 
 171A. Engineering Reports 290 
 
 172. Reliability op Numercial Calculations. Necessity of relia- 
 
 bility in engineering calculations 293 
 
 173. Methods of checking calculations. Curve plotting 293a 
 
 174. Some frequent errors 293b 
 
 APPENDIX A. NOTES ON THE THEORY OF FUNCTIONS. 
 
 A. General Functions. 
 
 175. Implicit analytic function. Explicit analytic function. Reverse 
 
 fu°°tioQ 294 
 
CONTENTS. XIX 
 
 PAGE 
 
 176. Rational function. Integer function. Approximations by 
 
 Taylor's Theorem 295 
 
 177. Abelian integrals and Abelian functions. Logarithmic integral 
 
 and exponential functions 296 
 
 178. Trigonometric integrals and trigonometric functions. Hyperbolic 
 
 integrals and hyperbolic functions 297 
 
 179. Elliptic integrals and elliptic functions. Their double periodicity 298 
 
 180. Theta functions. Hyperelliptic integrals and functions 300 
 
 181. Elliptic functions in the motion of the pendulum and the surging 
 
 of synchronous machines 301 
 
 182. Instance of the arc of an ellipsis 301 
 
 B. Special Functions. 
 
 183. Infinite summation series. Infinite product series 302 
 
 184. Functions by integration. Instance of the propagation functions 
 
 of electric waves and impulses 303 
 
 185. Functions defined by definite integrals 305 
 
 186. Instance of the gamma function 306 
 
 C. Exponential, Trigonometric and Hyperbolic Functions. 
 
 187. Functions of real variables 306 
 
 188. Functions of imaginary variables 308 
 
 189. Functions of complex variables 308 
 
 190. Relations 309 
 
 APPENDIX B. TABLES. 
 
 Table I. Three decimal exponential functions 312 
 
 Table II. Logarithms of exponential functions 
 
 Exponential functions 313 
 
 Hyperboho functions 314 
 
 Index 315 
 
ENGINEEKING MATHEMATICS. 
 
 CHAPTER I. 
 
 THE GENERAL NUMBER. 
 
 A. THE SYSTEM OF NUMBERS. 
 Addition and Subtraction. 
 
 I. From the operation of counting and measuring arose the 
 art of figuring, arithmetic, algebra, and finally, more or less, 
 the entire structure of mathematics. 
 
 During the development of the human race throughout the 
 ages, which is repeated by every child during the first years 
 of life, the first conceptions of numerical values were vague 
 and crude: many and few, big and little, large and small. 
 Later the ability to count, that is, the knowledge of numbers, 
 developed, and last of all the ability of measuring, and even 
 up to-day, measuring is to a considerable extent done by count- 
 ing: steps, knots, etc. 
 
 From counting arose the simplest arithmetical operation — 
 addition. Thus we may count a bunch of horses: 
 
 1, 2, 3, 4, 5, 
 
 and then count a second bunch of horses, 
 
 1, 2, 3; 
 
 now put the second bunch together with the first one, into one 
 bunch, and count them. That is, after counting the horses 
 
2 ENGINEERING MATHEMATICS. 
 
 of the first bunch, we continue to count those of the second 
 
 bunch, thus : 
 
 1, 2, 3, 4, 5,-6, 7, 8; 
 
 which gives addition, 
 or, in general, 
 
 5 + 3 = 8; 
 a + b = c. 
 
 We may take away again the second bunch of horses, that 
 means, we count the entire bunch of horses, and then count 
 off those we take away thus: 
 
 1, 2, 3, 4, 5, 6, 7, 8-7, 6, 5; 
 
 which gives subtraction, 
 
 8-3-5; 
 or, in general, 
 
 c—b = a. 
 
 The reverse of putting a group of things together with 
 another group is to take a group away, therefore subtraction 
 is the reverse of addition. 
 
 2. Immediately we notice an essential difference between 
 addition and subtraction, which may be illustrated by the 
 following examples : 
 
 Addition: 5 horses +3 horses gives 8 horses, 
 Subtraction: 5 horses— 3 horses gives 2 horses. 
 Addition : 5 horses + 7 horses gives 12 horses. 
 Subtraction: 5 horses— 7 horses is impossible. 
 
 From the above it follows that we can always add, but we 
 cannot always subtract; subtraction is not always possible; 
 it is not, when the number of things which we desire to sub- 
 tract is greater than the number of things from which we 
 desire to subtract. 
 
 The same relation obtains in measuring; wc may measure 
 a distance from a starting point A (Fig. 1), for instance in steps, 
 and then measure a second distance, and get the total distance 
 from the starting point by addition: 5 steps, from A to B. 
 
THE GENERAL NUMBER. 3 
 
 then 3 steps, from B to C, gives the distance from A to C, as 
 
 8 steps. 
 
 5 steps +3 steps = 8 steps; 
 
 1234 5-678 
 
 (6 1 1 1 1 0) I 1 
 
 A B C 
 
 Fig. 1. Addition. 
 
 or, we may step off a distance, and then step back, that is, 
 subtract another distance, for instance (Fig. 2), 
 
 5 steps— 3 steps = 2 steps; 
 
 that is, going 5 steps, from A to B, and then 3 steps back, 
 from B to C, brings us to C, '2 steps away from A. 
 
 12 3 4 
 I tt . I 1- 
 
 C B 
 
 Fig. 2. Subtraction. 
 
 Trying the case of subtraction which was impossible, in the 
 example with the horses, 5 steps— 7 steps = ? We go from the 
 starting point, A, 5 steps, to £, and then step back 7 steps; 
 here we find that sometimes we can do it, sometimes we cannot 
 do it; if back of the starting point A is a stone wall, we cannot 
 step back 7 steps. If A is a chalk mark in the road, we may 
 step back beyond it, and come to in Fig. 3. In the latter case, 
 
 -> 
 
 S1O12345 
 tb I ip 1 1 1 1 *- 
 
 B 
 Fig. 3. Subtraction, Negative Result. 
 
 at C we are again 2 steps distant from the starting point, just 
 as in Fig. 2. That is, 
 
 5-3=2 (Fig. 2), 
 
 5-7 = 2 (Fig. 3). 
 
 In the case where we can subtract 7 from 5, we get the same 
 distance from the starting point as when we subtract 3 from 5, 
 
4 ENGINEERING MATHEMATICS. 
 
 but the distance AC in Fig. 3, while the same, 2 steps, as 
 in Fig. 2, is different in character, the one is toward the left, 
 the other toward the right. That means, we have two kinds 
 of distance units, those to the right and tho^-'c to the left, and 
 have to find some way to distinguish them. The distance 2 
 in Fig. 3 is toward the left of the starting point A, that is, 
 in that direction, in which we step when subtracting, and 
 it thus appears natural to distinguish it from the distance 
 2 in Fig. 2, by calhng the former -2, ^vhile we call the distance 
 AC in Fig. 2: +2, since it is in the direction from A, in which 
 we step in adding. 
 
 This leads to a subdivision of the system of absolute numbers, 
 
 12 3 
 
 J-, .^, <j, . 
 
 into two classes, positive numbers, 
 
 + 1, +2, +3, ...: 
 and negative numbers, 
 
 -1, -2, -3, ...; 
 
 and by the introduction of negative numbers, we can always 
 carry out the mathematical operation of subtraction: 
 
 and, if b is greater than c, a merely becomes a negative number. 
 
 3. We must therefore realize that the negative number and 
 the negative unit, —1, is a mathematical fiction, and not in 
 universal agreement with experience, as the absolute number 
 found in the operation of counting, and the negative number 
 does not always represent an existing condition in practical 
 experience. 
 
 In the application of numbers to the phenomena of nature, 
 we sometimes find conditions where we can give the negative 
 number a physical meaning, expressing a relation as the 
 reverse to the positive number; in other cases we cannot do 
 this. For instance, 5 horses— 7 horses = —2 horses has no 
 physical meaning. There exist no negative horses, and at the 
 best we could only express the relation by saying, 5 horses —7 
 horses is impossible, 2 horses are missing. 
 
THE GENERAL NUMBER. 5 
 
 In the same way, an illumination of 5 foot-candles, lowered 
 by 3 foot-candles, gives an illumination of 2 foot-candles, thus, 
 
 5 foot-candles— 3 foot-candles = 2 foot-candles. 
 
 If it is tried to lower the illumination of 5 foot-candles by 7 
 foot-candles, it will be found impossible; there cannot be a 
 negative illumination of 2 foot-candles; the limit is zero illumina- 
 tion, or darkness. 
 
 From a string of 5 feet length, we can cut off 3 feet, leaving 
 
 2 feet, but we cannot cut off 7 feet, leaving —2 feet of string. 
 
 In these instances, the negative number is meaningless, 
 a mere imaginary mathematical fiction. 
 
 If the temperature is 5 deg. cent, above freezing, and falls 
 
 3 deg., it will be 2 deg. cent, above freezing. If it falls 7 deg. 
 it will be 2 deg. cent, below freezing. The one case is just as 
 real physically, as the other, and in this instance' we may 
 express the relation thus: 
 
 + 5 deg. cent. —3 deg. cent. = +2 deg. cent., 
 
 +5 deg. cent. —7 deg. cent. = —2 deg. cent.; 
 
 that is, in temperature measurements by the conventional 
 temperature scale, the negative numbers have just as much 
 physical existence as the positive numbers. 
 
 The same is the case with time, we may represent future 
 time, from the present as starting point, by positive numbers, 
 and past time then will be represented by negative numbers. 
 But we may equally well represent past time by positive num- 
 bers, and future times then appear as negative numbers. In 
 this, and most other physical applications, the negative number 
 thus appears equivalent with the positive number, and inter- 
 changeable: we may choose any direction as positive, and 
 the reverse direction then is negative. Mathematically, how- 
 ever, a difference exists between the positive and the negative 
 number; the positive unit, multiphed by itself, remains a pos- 
 itive unit, but the negative unit, multiplied with itself, does 
 not remain a negative unit, but becomes positive: 
 
 ( + l)x( + l) = ( + l); 
 
 (-1)X(-1) = ( + 1), andnot =(-1). 
 
6 ENGINEERING MATHEMATICS. 
 
 Starting from 5 deg. northern latitude and going 7 deg. 
 south, brings us to 2 deg. southern latitude, which may be 
 expresses thus, 
 
 + 5 deg. latitude -7 deg. latitude = -2 deg. latitude. 
 
 Therefore, in all cases, where there are two opposite direc- 
 tions, right and left on a hne, north and south latitude, east 
 and west longitude, future and past, assets and liabilities, etc., 
 there may be apphcation of the negative number; in other cases, 
 where there is only one kind or direction, counting horses, 
 measuring illumination, etc., there is no physical meaning 
 which would be represented by a negative number. There 
 are still other cases, where a meaning may sometimes be found 
 and sometimes not; for instance, if we have 5 doUars in our 
 pocket, we cannot take away 7 dollars; if we have 5 dollars 
 in the bank, we may be able to draw out 7 dollars, or we may 
 not, depending on our credit. In the first case, 5 dollars -7 
 dollars is an impossibility, while the second case 5 dollars -7 
 dollars = 2 dollars overdraft. 
 
 In any case, however, we must realize that the negative 
 number is not a physical, but a mathematical conception, 
 which may find a physical representation, or may not, depend- 
 ing on the physical conditions to which it is applied. The 
 negative number thus is just as imaginary, and just as real, 
 depending on the case to which it is apphed, as the imaginary 
 number V— 1, and the only difference is, that we have become 
 familiar with the negative number at an earlier age, where we 
 wer.^ less critical, and thus have taken it for granted, become 
 familiar with it by use, and usually do not realize that it is 
 a mathematical conception, and not a physical reality. When 
 we first learned it, however, it was quite a step to become 
 accustomed to saying, 5— 7 =—2, and not simply, 5—7 is 
 impossible. 
 
 Multiplication and Division. 
 
 4- If we have a bunch of 4 horses, and another bunch of 4 
 horses, and still another bunch of 4 horses, and add together 
 the three bunches of 4 horses each, we get, 
 
 4 horses -I- 4 horses + 4 horses = 12 horses; 
 
THE GENERAL NUMBER. 7 
 
 or, as we express it, 
 
 3X4 horses = 12 horses. 
 
 The operation of multiple addition thus leads to the next 
 operation, mvltiplicaiion. MultipUcation is multiple addi- 
 tion, 
 
 bxa = c, 
 
 thas means 
 
 a + a + a + . . . {b terms) =c. 
 
 Just like addition, multiplication can always be carried 
 out. 
 
 Three groups of 4 horses each, give 12 horses. Inversely, if 
 we have 12 horses, and divide them into 3 equal groups, each 
 group contains 4 horses. This gives us the reverse operation 
 of multipUcation, or division, which is written, thus: 
 
 12 horses , , 
 
 ^ = 4 horses; 
 
 or, in general, 
 
 c 
 6 = «- 
 
 If we have a bunch of 12 horses, and divide it into two equal 
 groups, we get 6 horses in each group, thus: 
 
 12 horses „ . 
 ^ =D horses, 
 
 if we divide into 4 equal groups, 
 
 12 horses „ , 
 
 ;; = 3 horses. 
 
 4 
 
 If now we attempt to divide the bunch of 12 horses into 5 equal 
 groups, we find we cannot do it; if we have 2 horses in each 
 group, 2 horses are left over; if we put 3 horses in each group, 
 we do not have enough to make 5 groups; that is, 12 horses 
 divided by 5 is impossible; or, as we usually say; 12 horses 
 divided by 5 gives 2 horses and 2 horses left over, which is 
 written, 
 
 12 
 
 -r- = 2, remainder 2. 
 
8 ENGINEERING MATHEMATICS. 
 
 Thus it is seen that the reverse operation of multiplication, 
 or division, cannot always be carried out. 
 
 5. If we have 10 apples, and divide them into 3, we get 3 
 apples in each group, and one apple left over. 
 
 -15- = 3, remainder 1, 
 
 we may now cut the left-over apple into 3 equal parts, in which 
 case, 
 
 1 = 34 = 3, 
 
 In the same manner, if we have 12 apples, we can divide 
 into 5, by cutting 2 apples each into 5 equal pieces, and get 
 in each of the 5 groups, 2 apples and 2 pieces. 
 
 12 1 
 
 ^ = 2 + 2X^=21. 
 5 6 
 
 To be able to carry the operation of division through for 
 all numerical values, makes it necessary to introduce a new 
 unit, smaller than the original unit, and derived as a part of it. 
 
 Thus, if we divide a string of 10 feet length into 3 equal 
 parts, each part contains 3 feet, and 1 foot is left over. One 
 foot is made up of 12 inches, and 12 inches divided into 3 gives 
 4 inches; hence, 10 feet divided by 3 gives 3 feet 4 inches. 
 
 Division leads us to a new form of numbers: the fraction. 
 
 The fraction, however, is just as much a mathematical con- 
 ception, which sometimes may be applicable, and sometimes 
 not, as the negative number. In the above instance of 12 
 horses, divided into 5 groups, it is not apphcable. 
 
 12 horses , 
 r = 2f horses 
 
 is impossible; we cannot have fractions of horses, and what 
 we would get in this attempt would be 5 groups, each com- 
 prising 2 horses and some pieces of carcass. 
 
 Thus, the mathematical conception of the fraction is ap- 
 plicable to those physical quantities which can be divided into 
 smaller units, but is not applicable to those, which are indi- 
 visible, or individuals, as we usually call them. 
 
THE GENERAL NUMBER. 9 
 
 Involution and Evolution. 
 6. If we have a product of several equal factors, as, 
 4X4X4 = 64, 
 it is written as, 4^ = 64 ; 
 
 or, in general, a^ = c. 
 
 The operation of multiple multiplication of equal factors 
 thus leads to the next algebraic operation — involution; just as 
 the operation of multiple addition of equal terms leads to the 
 operation of multiplication. 
 
 The operation of involution, defined as multiple multiplica- 
 tion, requires the exponent h to be an integer number; 6 is the 
 number of factors. 
 
 Thus 4-3 has no immediate meaning; it would by definition 
 be 4 multiplied ( — 3) times with itself. 
 
 Dividing continuously by 4, we get, 48-j-4=45; 4'-^4 = 4*; 
 4*-^4 = 43; etc., and if this .successive division by 4 is carried 
 still further, we get the following series: 
 
 43 4X4X4 
 4 4 
 
 = 4X4 
 
 = 42 
 
 42 4X4 
 4 4 
 
 = 4 
 
 = 41 
 
 41 4 
 4 "4 
 
 = 1 
 
 = 40 
 
 40 1 
 
 1 
 
 
 4 "4 
 
 "■4 
 
 = 4-1 
 
 4 4 "^ 
 
 1 
 "4X4 
 
 = 4-2 = 1 
 * 42 
 
 4-2 I 
 4 "42 ■ ^ 
 
 1 
 
 .4-3=1 
 43 
 
 4X4x4 
 
 or, in general, 
 
 «-* = ^, 
 
 
 aO = l. 
 
10 ENGINEERING MATHEMATICS. 
 
 Thus, powers with negative exponents, as a-*, are the 
 reciprocals of the same powers with positive exponents : ^■ 
 
 7. From the definition of involution then follows, 
 
 because o'' means the product of b equal factors a, and a" the 
 product of n equal factors a, and a''Xa'' thus is a product hav- 
 ing ?>+n equal factors a. For instance, 
 
 43X42 = (4X4X4)X(4X4)=4S. 
 
 The question now arises, whether by multiple involution 
 we can reach any further mathematical operation . For instance , 
 
 (43)2 = ?, 
 may be written, 
 
 (43)2=43x43 
 
 = (4X4X4)X(4X4X4); 
 
 = 46; 
 
 and in the same manner, 
 
 (a^)" = a''"; 
 
 that is, a power a^ is raised to the n* power, by multiplying 
 its exponent. Thus also, 
 
 (a'')" = (a")''; 
 
 that is, the order of involution is immaterial. 
 
 Therefore, multiple involution leads to no further algebraic 
 operations. 
 
 8. 43 = 64; 
 
 that is, the product of 3 equal factors 4, gives 64. 
 
 Inversely, the problem may be, to resolve 64 into a product 
 of 3 equal factors. Each of the factors then will be 4. This 
 reverse operation of involution is called evolution, and is written 
 thus, 
 
 ^64 = 4; 
 or, more general, 
 
THE GENERAL NUMBER. 11 
 
 Vc thus is defined as that number a, which, raised to the power 
 h, gives c; or, in other words. 
 
 Involution thus far was defined only for integer positive 
 and negative exponents, and the question arises, whether powers 
 
 J_ n 
 
 with fractional exponents, as c** or c*, have any meaning. 
 Writing, 
 
 i\V ^x| 1 
 
 \cb ) =c ^ ^C^ = C, 
 i- . 
 
 it is seen that c^ is that number, which raised to the power b, 
 
 1 . 6/— 
 
 gives c; that is, c* is vc, and the operation of evolution thus 
 can be expressed as involution with fractional exponent, 
 
 and 
 
 1\ n 
 
 or. 
 
 ^=(ci)"=(^)". 
 
 - i 6 — 
 
 C 6 = (cm) b = Vc", 
 
 and 
 
 Obviously 
 
 then, 
 
 i>c-= 
 
 n 
 
 b 
 
 n 
 
 
 
 -^= 
 
 ^ 1 
 
 1 
 
 "Vc 
 
 Irrational Numbers. 
 
 g. Involution with integer exponents, as 4^ = 64, can always 
 be carried out. In many cases, evolution can also be carried 
 out. For instance, 
 
 VM = 4, 
 
 -^4 = 2; 
 
 while, in other cases, it cannot be carried out. For instance, 
 
 -5/2 = ?. 
 
12 ENGINEERING MATHEMATICS. 
 
 Attempting to calculate ^, we get, 
 
 ^2 = 1.4142135..., 
 
 and find, no matter how far we carry the calculation, we never 
 come to an end, but get an endless decimal fraction; that is, 
 no number exists in our system of numbers, which can express 
 -^, but we can only approximate it, and carry the approxima- 
 tion to any desired degree ; some such numbers, as it, have been 
 calculated up to several hundred decimals. 
 
 Such numbers as ^2, which cannot be expressed in any 
 finite form, but merely approximated, are called irrational 
 numbers. The name is just as wrong as the name negative 
 number, or imaginary number. There is nothing irrational 
 about -^2. If we draw a square, with 1 foot as side, the length 
 of the diagonal is -^2 feet, and the length of the diagonal of 
 a square obviously is just as rational as the length of the sides. 
 
 Irrational numbers thus are those real and existing numbers, 
 which cannot be expressed by an integer, or a fraction or finite 
 decimal fraction, but give an endless decimal fraction, which 
 does not repeat. 
 
 Endless decimal fractions frequently are met when express- 
 ing common fractions as decimals. These decimal representa- 
 tions of common fractions, however, are periodic decimals, 
 that is, the numerical values periodically repeat, and in this 
 respect are different from the irrational number, .and can, due 
 to their periodic nature, be converted into a finite common 
 
 fraction. For instance, 2.1387387 
 
 Let 
 
 then, 
 
 subtracting. 
 Hence, 
 
 x= 2.1387387. 
 1000x = 2138.7.387387. 
 999x = 2136.6 
 
 2136.6 _ 21366 1187 77 
 999 ~ 9990 ~ 555" "^555- 
 
10. It is 
 
 since, 
 
 but it also is : 
 
 since, 
 
 THE GENERAL NUMBER. 13 
 
 Quadrature Numbers. 
 
 v/+r=(+2), 
 
 (+2)x(+2) = (+4); 
 </Ta = {-2), 
 
 (-2)X(-2) = (+4). 
 
 Therefore, -^4 has two values, (+2) and (—2), and in 
 evolution we thus first strike the interesting feature, that one 
 and the same operation, with the same numerical values, gives 
 several different results. 
 
 Since all the positive and negative numbers are used up 
 as the square roots of positive numbers, the question arises, 
 What is the square root of a negative number? For instance, 
 4^^ cannot be —2, as —2 squared gives +4, nor can it be +2. 
 
 -^34= -^4x(-l)= i2-</-l, and the question thus re- 
 solves itself into : What is "^1 ? 
 
 We have derived the absolute numbers from experience, 
 for instance, by measuring distances on a line Fig. 4, from a 
 starting point A. 
 
 -5 -4 -3 -3 -1 +1 +3 +3 +4 +5 
 
 1 1 1- — ® — I — $ — I — e — I 1 1 
 
 CAB 
 
 Fig. 4. Negative and Positive Numbers. 
 
 Then we have seen that we get the same distance from A, 
 twice, once toward the right, once toward the left, and this 
 has led to the subdivision of the numbers into positive and 
 negative numbers. Choosing the positive toward the right, 
 in Fig. 4, the negative number would be toward the left (or 
 inversely, choosing the positive toward the left, would give 
 the negative toward the right). 
 
 If then we take a number, as +2, which represents a dis- 
 tance AB, and multiply by (—1), we get the distance AC= -2 
 
14 
 
 ENGINEERING MATHEMATICS. 
 
 in opposite direction from A. Inversely, if we take AC= -2, 
 and multiply by (-1), we get AB= +2; that is, multiplica- 
 tion by (-1) reverses the direction, turns it through 180 deg. 
 If we multiply +2 by V^, we get +2\/-l, a quantity 
 of which we do not yet know the meaning. Multiplying once 
 more by V~^, we get +2X\^^xV -1= -2; that is, 
 multiplying a number +2, twice by V^, gives a rotation of 
 180 deg., and multiplication by \/^ thus means rotation by 
 half of 180 deg.; or, by 90 deg., and +2v^^ thus is the dis- 
 
 OD 
 
 \90° 
 
 -®- 
 
 FiG. 5. 
 
 tance in the direction rotated 90 deg. from +2, or in quadrature 
 direction AD in Fig. 5, and such numbers as +2\/-l thus 
 are quadrature numbers, that is, represent direction not toward 
 the right, as the positive, nor toward the left, as the negative 
 numbers, but upward or downward^ 
 
 For convenience of writing, V-l is usually denoted by 
 the letter j. 
 
 II. Just as the operation of subtraction introduced in the 
 negative numbers a new kind of numbers, having a direction 
 180 deg. different, that is, in opposition to the positive num- 
 bers, so the operation of evolution introduces in the quadrature 
 number, as 2j, a new kind of number, having a direction 90 deg. 
 
THE GENERAL NUMBER. 
 
 15 
 
 different; that is, at right angles to the positive and the negative 
 numbers, as illustrated in Fig. 6. 
 
 As seen, mathematically the quadrature number is just as 
 real as the negative, physically sometimes the negative number 
 has a meaning — if two opposite directions exist — ; sometimes it 
 has no meaning — where one direction only exists. Thus also 
 the quadrature number sometimes has a physical meaning, in 
 those cases where four directions exist, and has no meaning, 
 in those physical problems where only two directions exist. 
 
 +4/ 
 + 3/ 
 +2; 
 
 -4 -3 -3 
 
 -1 
 
 +1 +2 +3 
 -3 
 
 -3j 
 
 -J 
 Fig. 6. 
 
 For instance, in problems dealing with plain geometry, as in 
 electrical engineering when discussing alternating current 
 vectors in the plane, the quadrature numbers represent the 
 vertical, the ordinary numbers the horizontal direction, and then 
 the one horizontal direction is positive, the other negative, and 
 in the same manner the one vertical direction is positive, the 
 other negative. Usually positive is chosen to the right and 
 upward, negative to the left and downward, as indicated in 
 Fig. 6. In other problems, as when dealing with time, which 
 has only two directions, past and future, the quadrature num- 
 bers are not applicable, but only the positive and negative 
 
16 
 
 ENGINEERING MATHEMATICS. 
 
 numbers. In still other problems, as when dealing with illumi- 
 nation, or with individuals, the negative numbers are not 
 applicable, but only the absolute or positive numbers. 
 
 Just as multiplication by the negative unit (—1) means 
 rotation by 180 deg., or reverse of direction, so multiplication 
 by the quadrature unit, j, means rotation by 90 deg., or change 
 from the horizontal to the vertical direction, and inversely. 
 
 General Numbers. 
 
 12. By the positive and negative numbers, all the points of 
 a line could be represented numerically as distances from a 
 chosen point A. 
 
 -a? 
 
 Fig. 7. Simple Vector Diagram. 
 
 By the addition of the quadrature numbers, all points of 
 the entire plane can now be represented as distances from 
 chosen coordinate axes x and y, that is, any point P of the 
 plane, Fig. 7, has a_horizontal distance, 05= +3, and a 
 vertical distance, BP=+2j, and therefore is given by a 
 combination of the distances, 05= +3 and BP= +2]'. For 
 convenience, the act of combining two such distances in quad- 
 rature with each other can be expressed by the plus sign, 
 and the result of combination thereby expressed by OB+BP 
 = 3+2j. 
 
THE GENERAL NUMBER. 
 
 17 
 
 Such a combination of an ordinary number and a quadra- 
 ture number is called a general number or a comiplex quantity. 
 
 The quadrature number jh thus enormously extends the 
 field of usefulness of algebra, by affording a numerical repre- 
 sentation of two-dimensional systems, as the plane, by the 
 general number a + jh. They are especially useful and impor- 
 tant in electrical engineering, as most problems of alternating 
 currents lead to vector representations in the plane, and there- 
 fore can be represented by the general number a-\-jb; that is, 
 the combination of the ordinary number or horizontal distance 
 a, and the quadrature number or vertical distance jb. 
 
 Fig. 8. Vector Diagram. 
 
 Analytically, points in the plane are represented by their 
 two coordinates: the horizontal coordinate, or abscissa x, and 
 the vertical coordinate, or ordinate y. Algebraically, in the 
 general number a+jh both coordinates are combined, a being 
 the X coordinate, jb the y coordinate. 
 
 Thus in Fig. 8, coordinates of the points are, 
 
 -Pi: a;=+3, y= +2 Pg: 2;= +3 j/= -2, 
 
 -P3: a;=-3, y=+2 P4: x= -3 y= -2, 
 
 and the points are located in the plane by the numbers: 
 Pi =3 +2/ P2 = 3-2y P3=-3+2y P4=-3-2; 
 
18 ENGINEERING MATHEMATICS. 
 
 13. Since already the square root of negative numbers has 
 extended the system of numbers by giving the quadrature 
 number, the question arises whether still further extensions 
 of the system of numbers would result from higher roots of 
 negative quantities. 
 
 For instance, 
 
 A'~r=? 
 
 The meaning of ^^^1 we find in the same manner as that 
 of f^I 
 
 A positive number a may be represented on the horizontal 
 axis as P. 
 
 Multiplying a by -^—1 gives a-^— 1, whose meaning we do 
 not yet know. Multiplying again and again by •\'—l, we get, after 
 four multiplications, o(-v'-l)*= —a; that is, in four steps we 
 have been carried from o to —a, a rotation of 180 deg., and 
 
 'v'^1 thus means a rotation of—— = 45 deg., therefore, a-v'-l 
 
 is the point Pi in Fig. 9, at distance a from the coordinate 
 center, and under angle 45 deg., which has the coordinates, 
 
 a a . 
 
 a;=— = and y = —=y, or, is represented by the general number, 
 
 -v'-l, however, may also mean a rotation by 135 deg. to P2, 
 since this, repeated four times, gives 4x135 = 540 deg., 
 or the same as 180 deg., or it may mean a rotation by 225 deg. 
 
 or by 315 deg. Thus four points exist, which represent a 4-^', 
 the points: 
 
 v2 V2 
 
 Therefore, -vZ-fis still a general number, consisting of an 
 ordinary and a quadrature number, and thus does not extend 
 our system of numbers any further. 
 
THE GENERAL NUMBER. 
 
 19 
 
 In the same manner, V + 1 can be found; it is that number, 
 which, multipUed n times with itself, gives +1. Thus it repre- 
 
 sents a rotation by — deg., or any multiple thereof; that is, 
 
 J- . • .360 , ,. . • 360 
 
 the X coordmate is cos qX — , the y coordmate sm gX — , 
 
 n n 
 
 and, 
 
 ^— 360 . . 360 
 
 + l = cos oX +] sm qX , 
 
 where q is any integer number. 
 
 Fig. 9. Vector Diagram a-ij—l. 
 
 There are therefore n different values of a\/ + l, which lie 
 equidistant on a circle with radius 1, as shown for n = 9 in 
 
 Fig. 10. 
 
 14. In the operation of addition, a + b = c, the problem is, 
 a and b being given, to find c. 
 
 The terms of addition, a and b, are interchangeable, or 
 equivalent, thus: a + b = b + a, and addition therefore has only 
 one reverse operation, subtraction; c and b being given, a is 
 found, thus: a^c-b, and c and a being given, b is found, thus: 
 j,_P_o_ Either leads to the same operation — subtraction. 
 
 The same is the case in multiplication; aXb = c. The 
 
20 
 
 ENGINEERING MATHEMATICS. 
 
 factors a and h are interchangeable or equivalent; aXh = hXa 
 
 . . c c 
 
 and the reverse operation, division, a= ,- is the same as h=—. 
 
 In involution, however, a'' = c, the two numbers a and b 
 are not interchangeable, and a^ is not equal to 6°. For instance 
 43 = 64 and 34 = 81. 
 
 Therefore, involution has two reverse operations: 
 
 (a) c and b given, a to be found. 
 
 or evolution, 
 
 b — 
 a= \ c: 
 
 Fig. 10. Points Dfetermined by v^+l. 
 
 (&) c and a given, b to be found, 
 6 = loga c; 
 
 or, logarithmation. 
 
 Logarithmation. 
 
 15. Logarithmation thus is one of the reverse operations 
 of involution, and the logarithm is the exponent of involution. 
 
 Thus a logarithmic expression may be changed to an ex- 
 ponential, and inversely, and the laws of logarithmation are 
 the laws, which the exponents obey in involution. 
 
 1. Powers of equal base are multiplied by adding the 
 exponents: a''Xa'' = a''+". Therefore, the logarithm of a 
 
THE GENERAL NUMBER. 21 
 
 product is the sum of the logarithms of the factors, thus logo cXd 
 
 = l0ga C +\0ga d. 
 
 2. A power is raised to a power by multiplying the exponents : 
 (a'')" = a*". 
 
 Therefore the logarithm of a power is the exponent times 
 the logarithm of the base, or, the number under the logarithm 
 is raised to the power n, by multiplying the logarithm by n: 
 
 loga c"=n loga c, 
 
 loga 1 = 0, because a" = 1. If the base o > 1, loga c is positive, 
 if c>l, and is negative, if c<l, but >0. The reverse is the 
 case, if a<l. Thus, the logarithm traverses all positive and 
 negative values for the positive values of c, and the logarithm 
 of a negative number thus can be neither positive nor negative. 
 
 loga (-c)=loga c+loga (-1), and the question of finding 
 the logarithms of negative numbers thus resolves itself into 
 finding the value of loga ( — 1). 
 
 There are two standard systems of logarithms one with 
 the base £ = 2.71828. . .*, and the other with the base 10 is 
 used, the former in algebraic, the latter in numerical calcula- 
 tions. Logarithms of any base a can easily be reduced to any 
 other base. 
 
 For instance, to reduce 6= loga c to the base 10: 6 = loga c 
 means, in the form of involution : a'' = c. Taking the logarithm 
 hereof gives, b logio a = logio c, hence, 
 
 , logioc logioc 
 
 o=-j ; or logo c=-, . 
 
 logio a *= logio a 
 
 Thus, regarding the logarithms of negative numbers, we need 
 to consider only logio ( — 1) or log^ ( —1). 
 
 If jx = log, ( - 1), then £'^ = - 1, 
 
 and since, as will be seen in Chapter II, 
 
 £'^ = cos x + j sin X, 
 
 it follows that, 
 
 cos x+/sin X = —1, 
 
 * Regarding e, see Chapter II, p. 71. 
 
22 ENGINEERING MATHEMATICS. 
 
 Hence, x = n, or an odd multiple thereof, and 
 
 loge(-l)=jV(2n + l), 
 
 where n is any integer number. 
 
 Thus logarithmation also leads to the quadrature number 
 j, but to no further extension of the system of numbers. 
 
 Quaternions. 
 
 i6. Addition and subtraction, multipUcation and division, 
 involution and evolution and logarithmation thus represent all 
 the algebraic operations, and the system of numbers in which 
 all these operations can be carried out under all conditions 
 is that of the general number, a+jb, comprising the ordinary 
 number a and the quadrature number jb. The number a as 
 well as b may be positive or negative, may be integer, fraction 
 or irrational. 
 
 Since by the intreduction of the quadrature number jb, 
 the application of the system of numbers was extended from the 
 line, or more general, one-dimensional quantity, to the plane, 
 or the two-dimensional quantity, the question arises, whether 
 the system of numbers could be still further extended, into 
 three dimensions, so as to represent space geometry. While 
 in electrical engineering most problems lead only to plain 
 figures, vector diagrams in the plane, occasionally space figures 
 would be advantageous if they could be expressed algebra- 
 ically. Especially in mechanics this would be of importance 
 when dealing with forces as vectors in space. 
 
 In the quaternion calculus methods have been devised to 
 deal with space problems. The quaternion calculus, however, 
 has not yet found an engineering apphcation comparable with 
 that of the general number, or, as it is frequently called, the 
 complex quantity. The reason is that the quaternion is not 
 an algebraic quantity, and the laws of algebra do not uniformly 
 apply to it. 
 
 17. With the rectangular coordinate system in the plane, 
 Fig. 11, the z axis may represent the ordinary numbers, the y 
 axis the quadrature numbers, and multipUcation by j=V—l 
 represents rotation by 90 deg. For instance, if Pi is a point 
 
THE GENERAL NUMBER. 
 
 23 
 
 a+jb=5+2j, the point P2, 90 deg. away from Pi, would 
 be: 
 
 P2 = iPi=Ka+ib) =j(3 +2j) = -2 +3j, 
 
 To extend into space, we have to add the third or z axis, 
 as shown in perspective in Fig. 12. Rotation in the plane xy, 
 by 90 deg., in the direction +x to +y, then means multiplica- 
 tion by j. In the same manner, rotation in the yz plane, by 
 90 deg., from +y to +z, would be represented by multiplica- 
 
 > + 
 
 Fig. 11. Vectors in a Plane. 
 
 tion with h, and rotation by 90 deg. in the zx plane, from +2 
 to +x would be presented by k, as indicated in Fig. 12. 
 
 AH three -of these rotors, j, h, k, would be V — 1, since each, 
 applied twice, reverses the direction, that is, represents multi- 
 plication by (—1). 
 
 As seen in Fig. 12, starting from +x, and going to +y, 
 then to +z, and then to +x, means successive multiphcation 
 by j, h and k, and since we come back to the starting point, the 
 total operation produces no change, that is, represents mul- 
 tiphcation by ( + 1). Hence, it must be, 
 
 jhk= +1. 
 
24 
 
 ENGINEERING MATHEMATICS. 
 
 Algebraically this is not possible, since each of the three quan- 
 tities is V^, and V'-ixV'^xV^== -^/^, and not 
 ( + 1). 
 
 +z 
 
 >+a; 
 
 Fig. 12. Vectors in Space, ihk=+l. 
 
 If we now proceed again from x, in positive rotation, but 
 first turn in the xz plane, we reach by multiplication with k 
 the negative z axis, —z, as seen in Fig. 13. Further multiplica- 
 
 + y 
 
 ->+« 
 
 -y 
 
 Fig. 13. Vectors in Space, hhi= —1. 
 
 tion by h brings us to +y, and multiplication by j to -x, and 
 in this case the result of the three successive rotations by 
 
THE GENERAL NUMBER. 25 
 
 90 deg., in the same direction as in Fig. 12, but in a different 
 order, is a reverse; that is, represents (-1). Therefore, 
 
 khj= —1, 
 and hence, 
 
 jhk= —khj. 
 
 Thus, in vector analysis of space, we see that the fundamental 
 law of algebra, 
 
 aXh = hXa, 
 
 does not apply, and the order of the factors of a product is 
 not immaterial, but by changing the order of the factors of the 
 product jhk, its sign was reversed. Thus common factors can- 
 not be canceled as in algebra; for instance, if in the correct ex- 
 pression, jhk = — khj, we should cancel by /, h and k, as could be 
 done in algebra, we would get +1 = —1, which is obviously wrong. 
 For this reason all the mechanisms devised for vector analysis 
 in space have proven more difficult in their appUcation, and 
 have not yet been used to any great extent in engineering 
 practice. 
 
 B. ALGEBRA OF THE GENERAL NUMBER, OR COMPLEX 
 
 QUANTITY. 
 
 Rectangular and Polar Coordinates. 
 
 i8. The general number, or complex quantity, ei+jb, is 
 the most general expression to which the laws of algebra apply. 
 It therefore can be handled in the same manner and under 
 the same rules as the ordinary number of elementary arithmetic. 
 The only feature which must be kept in mind is that j^ = — 1, and 
 where in multiplication or other operations j^ occurs, it is re- 
 placed by its value, —1. Thus, for instance, 
 
 (a + jb) (c + jd) = ac+ jad + jbc + j%d 
 = ac+ jad + jbc — bd 
 = (ac —bd) +j{ad + bc). 
 
 Hetefrom it follows that all the higher powers of j can be 
 eliminated, thus: 
 
 f- 
 
 =i, 
 
 f= 
 
 -1, f = 
 
 -i, 
 
 f= 
 
 = +1; 
 
 f= 
 
 -+i, 
 
 f= 
 
 -1, f = 
 
 -i, 
 
 f= 
 
 =+1; 
 
 f- 
 
 -+h 
 
 ■ « 1 
 
 . etc. 
 
 
 
 
26 ENGINEERING MATHEMATICS. 
 
 In distinction from the general number or complex quantity, 
 the ordinary numbers, +a and —a, are occasionally called 
 scalars, or real numbers. The general number thus consists 
 of the combination of a scalar or real number and a quadrature 
 number, or imaginary number. 
 
 Since a quadrature number cannot be equal to an ordinary 
 number it follows that, if two general numbers are equal, 
 their real components or ordinary numbers, as well as their 
 quadrature numbers or imaginary components must be equal, 
 thus, if 
 
 a+jb = c+id, 
 then, 
 
 a = c and b = d. 
 
 Every equation with general numbers thus can be resolved 
 into two equations, one containing only the ordinary numbers, 
 the other only the quadrature numbers. For instance, if 
 
 x+jy = 5-3i, 
 then, 
 
 a;=5 and y= —3. 
 
 19. The best way of getting a conception of the general 
 number, and the algebraic operations with it, is to consider 
 the general number as representing a point in the plane. Thus 
 the general number a +jb = 6+2.5]' may be considered as 
 representing a point P, in Fig. 14, which has the horizontal 
 distance from the y axis, 0A=BP = a = 6, and the vertical 
 distance from the x axis, OB = AP = b=2.5. 
 
 The total distance of the point P from the coordinate center 
 then is 
 
 0P = ^0A2+AP^ 
 
 = Va2 +62 = v/62 +2.52 = 6.5, 
 
 and the angle, which this distance OP makes with the x axis, 
 is given by 
 
 tan 6=== 
 OA 
 
 b 2.5 
 = - = —=0.417. 
 a 6 
 
THE GENERAL NUMBER. 
 
 27 
 
 Instead of representing the general number by the two 
 components, a and h, in the form a+jb, it can also be repre- 
 sented by the two quantities : the distance of the point P from 
 the center 0, 
 
 C = Va2+62; 
 
 and the angle between this distance and the x axis, 
 
 tan d=-. 
 a 
 
 Fig. 14. Rectangular and Polar Coordinates. 
 
 Then referring to Fig. 14, 
 
 a = c cos ^ and 6 = csin5, 
 
 and the general number a-\-jh thus can also be written in the 
 form, 
 
 c(cos (9+j sin d). 
 
 The form a-Vjh expresses the general number by its 
 rectangular components a and 6, and corresponds to the rect- 
 angular coordinates of analytic geometry; a is the x coordinate, 
 h the y coordinate. 
 
 The form c(cos^-f j sin ^) expresses the general number by 
 what may be called its polar components, the radius c and the 
 
28 ENGINEERING MATHEMATICS. 
 
 angle 6, and corresponds to the polar coordinates of analytic 
 geometry, c is frequently called the radius vector or scalar, 
 6 the phase angle of the general number. 
 
 While usually the rectangular form a+jb is more con- 
 venient, sometimes the polar form c(cos d +j sin d) is preferable, 
 and transformation from one form to the other therefore fre- 
 quently applied. 
 
 Addition and Subtraction. 
 
 20. If oi+j6i = 6+2.5 y is represented by the point Pi; 
 this point is reached by going the horizontal distance ai = 6 
 and the vertical distance 6i=2.5. If a2+jb2 = S+ij is repre- 
 sented by the point P2, this point is reached by going the 
 horizontal distance a2 = 3 and the vertical distance 62 = 4. 
 
 The sum of the two general numbers (ai +ihi) + (02 +^2) = 
 (6+2.5/) + (3-H4j), then is given by point Pq, which is reached 
 by going a horizontal distance equal to the sum of the hor- 
 izontal distances of Pi and P2: ao = ai +a2 = 6+3 = 9, and a 
 vertical distance equal to the sum of the vertical distances of 
 Pi and P2: 60 = 61+62 = 2.5+4 = 6.5, hence, is given by the 
 general number 
 
 ao + j6o = (ai + 02) + J (61 + 62) 
 = 9 + 6.5/. 
 
 Geometrically, point Po is derived from points Pi and P2^ 
 by the diagonal OPq of the parallelogram OP1P0P2, constructed 
 with OPi and OP2 as sides, as seen in Fig. 15. 
 
 Herefrom it follows that addition of general numbers 
 represents geometrical combination by the parallelogram law. 
 
 Inversely, if Pq represents the number 
 
 ao+y6o = 9 + 6.5y, 
 
 and Pi represents the number 
 
 Oi+j6i=6+2.5/, 
 
 the difference of these numbers will be represented by a point 
 P2, which is reached by going the difference of the horizontal 
 
THE GENERAL NUMBER. 
 
 29 
 
 distances and of the vertical distances of the points Pq and 
 Pi. P2 thus is represented by 
 
 and 
 
 «2 = ao—ai = 9—6=3, 
 ?)2 = 60-^)1 =6.5 -2.5 = 4. 
 
 Therefore, the difference of the two general numbers (ao+J^o) 
 and (tti +y6i) is given by the general number: 
 
 as seen in Fig. 15. 
 
 ^2 +y&2 = (oso -aC) +j{ho -61) 
 = 3 + 4/, 
 
 Fig. 15. Addition and Subtraction of Vectors. 
 
 This difference a2+j62 is represented by one side OP 2 of 
 the parallelogram OP1P0P2, which has OPi as the other side, 
 and OPq as the diagonal. 
 
 Subtraction of general numbers thus geometrically represents 
 the resolution of a vector OPo into two components OPi and 
 OP2, by the parallelogram law. 
 
 Herein lies the main advantage of the use of the general 
 number in engineering calculation : If the vectors are represented 
 by general numbers (complex quantities), combination and 
 resolution of vectors by the parallelogram law is carried out by 
 
30 ENGINEERING MATHEMATICS. 
 
 simple addition or subtraction of their general numerical values, 
 that is, by the simplest operation of algebra. 
 
 21. General numbers are usually denoted by capitals, and 
 their rectangular components, the ordinary number and the 
 quadrature number, by small letters, thus: 
 
 A = ai+ja2] 
 
 the distance of the point which represents the general number A 
 from the coordinate center is called the absolute value, radius 
 or scalar of the general number or complex quantity. It is 
 the vector a in the polar representation of the general number : 
 
 A = a(cos (9+ J sin d), 
 
 and is given by a = Voi^ + a-i^. 
 
 The absolute value, or scalar, of the general number is usually 
 also denoted by small letters, but sometimes by capitals, and 
 in the latter case it is distinguished from the general number by 
 using a different type for the latter, or underlining or dotting 
 it, thus: 
 
 A = ai+ja2; or A-=^ai+ja2\ov A = ai+ja2 
 or A = ai+ia2; or A=ai+/a2 
 
 a=Vai^ ^ar] or A = Va{'- +ai^, 
 
 and ai+/a2 = a(cos tf + jsin(?); 
 
 or ai+ya2 = A(cos tf +j sin (9). 
 
 22. The absolute value, or scalar, of a general number is 
 always an absolute number, and positive, that is, the sign of the 
 rectangular component is represented in the angle 6. Thus 
 referring to Fig. 16, 
 
 ^=ai+ya2 = 4+3/; 
 
 gives, a=\/ai^+a2^ = 5; 
 
 tan 61 = 1=0.75; 
 ^ = 37deg.; 
 and 4=5 (cos 37 deg. +/ sin 37 deg). 
 
The expression 
 gives 
 
 THE GENERAL NUMBER. 
 
 A = ai+ja2 = 4:-Sj 
 
 a=Vai^+a2^ = 5; 
 
 31 
 
 tan i9 = -- = -0.75; 
 4 
 
 d = -37 deg. ; or = 180 -37 = 143 deg. 
 
 Fig. 16. Representation of General Numbers. 
 
 Which of the two values of d is the correct one is seen from 
 the condition ai = a cos 6. As ai is positive, +4, it follows 
 that cos d must be positive; cos (—37 deg.) is positive, cos 143 
 deg. is negative; hence the former value is correct : 
 
 A = 5{cos( -37 deg.) +j sin( -37 deg.){ 
 = 5(cos 37 deg. — ; sin 37 deg.). 
 
 Two such general numbers as (4+3j) and (4— 3j), or, 
 in general, 
 
 (a+jb) and (a—jb), 
 
 are called conjugate numbers. Their product is an ordinary 
 and not a general number, thus: (a + jb){a~jb) = a^ +b^. 
 
32 ENGINEERING MATHEMATICS. 
 
 The expression 
 
 A=ai+ia2= -4+3; 
 
 gives 
 
 3 
 
 tan (9= --r= -0.75; 
 4 
 
 ^ = -37 deg. or = 180 -37 = 143 cleg. ; 
 
 but since ai = a cos is negative, -4, cos ^ must be negative, 
 hence, d = 143 deg. is the correct vahie, and 
 
 A = 5(cos 143 deg. +j sin 143 deg.) 
 = 5( -cos 37 deg. +j sin 37 deg.) 
 
 The expression 
 
 A = a\+ia2= -4-3y 
 
 gives 
 
 6>=37deg.; or =180+37 = 217 deg.; 
 
 but since ai = a cos 6 is negative, -4, cos d must be negative, 
 hence ^ = 217 deg. is the correct value, and, 
 
 4=5 (cos 217 deg. +j sin 217 deg.) 
 = 5( — cos 37 deg. — j sin 37 deg.) 
 
 The four general numbers, +4+3;, +4-3/, —4+3/, and 
 —4 — 3j, have the same absolute value, 5, and in their repre- 
 sentations as points in a plane have symmetrical locations in 
 the four quadrants, as shown in Fig. 16. 
 
 As the general number A = ai+ja2 finds its main use in 
 representing vectors in the plane, it very frequently is called 
 a vector quantity, and the algebra of the general number is 
 spoken of as vector analysis. 
 
 Since the general numbers A=ai+ja2 can be made to 
 represent the points of a plane, they also may be called plane 
 numbers, while the positive and negative numbers, +a and— a. 
 
THE GENERAL NUMBER. 33 
 
 may be called the linear numbers, as they represent the points 
 of a line. 
 
 Example : Steam Path in a Turbine. 
 
 23. As an example of a simple operation with general num- 
 bers one may calculate the steam path in a two-wheel stage 
 of an impulse steam turbine. 
 
 w, 
 
 »»»») 
 
 ■> +« 
 
 FiQ. 17. Path of Steam in a Two-wheel Stage of an Impulse Turbine. 
 
 Let Fig. 17 represent diagrammatically a tangential section 
 through the bucket rings of the turbine wheels. Wi and W2 
 are the two revolving wheels, moving in the direction indicated 
 by the arrows, with the velocity s = 400 feet per sec. / are 
 the stationary intermediate buckets, which turn the exhaust 
 steam from the first bucket wheel Wi, back into the direction 
 required to impinge on the second bucket wheel W2. The 
 steam jet issues from the expansion nozzle at the speed So =2200 
 
34 
 
 ENGINEERING MATHEMATICS. 
 
 feet per sec, and under the angle ^o = 20 deg., against the first 
 bucket wheel W^. 
 
 The exhaust angles of the three successive rows of buckets, 
 Wi, I, and W2, are respectively 24 deg., 30 dcg. and 45 deg. 
 These angles are calculated from the section of the bucket 
 exit required to pass the steam at its momentary velocity, 
 and from the height of the passage required to give no steam 
 eddies, in a manner which is of no interest here. 
 
 As friction coefficient in the bucket passages may be assumed 
 Ay = 0.12; that is, the exit velocity is 1— A;y=0.88 of the entrance 
 velocity of the steam in the buckets. 
 
 >+« 
 
 Fig. 18. Vector Diagram of Velocities of Steam in Turbine. 
 
 Choosing then as x-axis the direction of the tangential 
 velocity of the turbine wheels, as y-axis the axial direction, 
 the velocity of the steam supply from the expansion nozzle is 
 represented in Fig. 18 by a vector OSo of length so=2200 feet 
 per sec, making an angle (9o=20 deg. with the x-axis; hence, 
 can be expressed by the general number or vector quantity : 
 
 So = So (cos 60 +j sin ^o) 
 
 =2200 (cos 20 deg. +/sin 20 deg.) 
 = 2070 + 750/ ft. per sec 
 The velocity of the turbine wheel TFi is s = 400 feet per second, 
 and- represented in Fig. 18 by the vector OS, in horizontal 
 direction. 
 
THE GENERAL NUMBER. 35 
 
 The relative velocity with which the steam enters the bucket 
 passage of the first turbine wheel Wx thus is : 
 
 = (2070 +750j) -400 
 = 1670+750jft. per sec. 
 
 This vector is shown as OSi in Fig. 18. 
 The angle Oi, under which the steam enters the bucket 
 passage thus is given by 
 
 750 
 tan i9i=YgyQ = 0.450, as 6ii = 24.3deg. 
 
 This angle thus 'has to be given to the front edge of the 
 buckets of the turbine wheel TFi. 
 
 The absolute value of the relative velocity of steam jet 
 and turbine wheel Wi, at the entrance into the bucket passage, 
 is 
 
 Si = \/16702 + 7502 = 1830 ft. per sec. 
 
 In traversing the bucket passages the steam velocity de- 
 creases by friction etc., from the entrance value Si to the 
 exit value 
 
 S2 = si(l- Ay) = 1830X0.88 = 1610 ft. per sec, 
 
 and since the exit angle of the bucket passage has been chosen 
 as ^2 = 24 deg., the relative velocity with which the steam 
 leaves the first bucket wheel Wi is represented by a vector 
 0^ in Fig. 18, of length S2 = 1610, under angle 24 deg. The 
 steam leaves the first wheel in backward direction, as seen in 
 Fig. 17, and 24 deg. thus is the angle between the steam jet 
 and the negative x-axis; hence, ^2 = 180—24 = 156 deg. is the 
 vector angle. The relative steam velocity at the exit from 
 wheel TFi can thus be represented by the vector quantity 
 
 '?2 = S2(cos 62 +j sin 62) 
 
 = 1610 (cos 156 deg. +j sin 156 deg.) 
 = -1470+655/. 
 
 Since the velocity of the turbine wheel Wi is s = 400, the 
 velocity of the steam in space, after leaving the first turbine 
 
36 ENGINEERING MATHEMATICS. 
 
 wheel, that is, the velocity with which the steam enters the 
 intermediate /, is 
 
 = (-1470 +655/) +400 
 = -1070 + 655/, 
 
 and is represented by vector Otiz in Fig. 18. 
 The direction of this steam jet is given by 
 
 tan (?3= -Jo7Q= -0.613, 
 
 as 
 
 03 = -31.6 deg. ; or, 180 -31.6 = 148.4 deg. 
 
 The latter value is correct, as cos di is negative, and sin Oz is 
 positive. 
 
 The steam jet thus enters the intermediate under the angle 
 of 148.4 deg. ; that is, the angle 180 —148.4 =31.6 deg. in opposite 
 direction. The buckets of the intermediate / thus must be 
 curved in reverse direction to those of the wheel Wi, and must 
 be given the angle 31.6 deg. at their front edge. 
 
 The absolute value of the entrance velocity into the inter- 
 mediate / is 
 
 S3 = \/l0702 + 6552 = 1255 f^ per sec. 
 
 In passing through the bucket passages, this velocity de- 
 creases by friction, to the value : 
 
 S4=S3(1 ~kj) = 1255 XO.88 = 1105 ft. per sec, 
 
 and since the exit edge of the intermediate is given the angle: 
 ^4 = 30 deg., the exit velocity of the steam from the intermediate 
 is represented by the vector OSi in Fig. 18, of length 84 = 1105, 
 and angle ^4 = 30 deg., hence, 
 
 ^4 = 1105 (cos 30 deg. +/ sin 30 deg.) 
 = 955 + 550/ ft. per sec. 
 
 This is the velocity with which the steam jet impinges 
 on the second turbine wheel W2, and as this wheel revolves 
 
THE GENERAL NUMBER. 37 
 
 with velocity s = 400, the relative velocity, that is, the velocity 
 with which the steam enters the bucket passages of wheel W2, is, 
 
 05 = 04 — s 
 
 = (955+550j)-400 
 = 555+550/ ft. per sec; 
 
 and is represented by vector OS5 in Fig. 18. 
 The direction of this steam jet is given by 
 
 550 
 tan ds—^ = 0.990, as ds = 44.8 deg. 
 
 Therefore, the entrance edge of the buckets of the second 
 wheel W2 must be shaped under angle ^5=44.8 deg. 
 The absolute value of the entrance velocity is 
 
 S5 = V5552+ 5502 = 780 f^. pgr sec. 
 
 In traversing the bucket passages, the velocity drops from 
 the entrance value S^, to the exit value, 
 
 S6 = S5(1 -/cy) = 780X0.88 = 690 ft. per sec. 
 
 Since the exit angles of the buckets of wheel W2 has been 
 chosen as 45 deg., and the exit is in backward direction, 6q = 
 180—45=135 deg., the steam jet velocity at the exit of the 
 bucket passages of the last wheel is given by the general number 
 
 (Se =S6(cos de +] sin d^) 
 
 = 690 (cos 135 deg. +/ sin 135 deg.) 
 = -487+487J ft. per sec, 
 
 and represented by vector OSe in Fig. 18. 
 
 Since s = 400 is the wheel velocity, the velocity of the 
 steam after leaving the last wheel W2, that is, the "lost" 
 or " rejected " velocity, is 
 
 57 = ^6+5 
 
 = (-487+487/) +400 
 = -87 + 487/ ft. per sec, 
 
 and is represented by vector O^Sy in Fig. 18. 
 
38 
 
 ENGINEERING MATHEMATICS. 
 
 The direction of the exhaust steam is given by, 
 
 487 
 tan(97=— ^V-=-5.6, as 617 = 180-80 = 100 deg., 
 
 and the absolute velocity is, 
 
 S7 = \/872+ 4872 = 495 ft. per sec. 
 
 Multiplication of General Numbers. 
 
 24. If A=ai+ja2 and B^hi+jhi, are two general, or 
 plane numbers, their product is given by multiplication, thus: 
 
 AB = (ai+ya2)(6i+i&2) 
 
 = ai&i +jaih2 +ja2bi +j^a2b2, 
 and since p= —1, 
 
 AB =(«!&! -a2?'2)+y(ai&2+ a2&i), 
 
 and the product can also be represented in the plane, by a point, 
 
 C = Ci+iC2, 
 
 Ci = aibi — a2&2, 
 
 where, 
 and 
 
 C2 = aih2+a2bi. 
 
 For instance, A=2+j multiplied by 5 = 1+1.5/ gives 
 
 ci=2Xl-lXl.5 = 0.5, 
 C2 = 2X1.5 + 1X1=4; 
 
 hence. 
 
 (7 = 0.5 + 4/, 
 
 as shown in Fig. 19. 
 
 25. The geometrical relation between the factors A and B 
 and the product C is better shown by using the polar expression; 
 hence, substituting. 
 
 which gives 
 
 ai=a cos a 
 02 = a sin a 
 
 tan a 
 
 a2 
 
 ai 
 
 and 
 
 and 
 
 fcl= 
 
 -b 
 
 cos /?' 
 
 
 62 = 
 
 = h 
 b 
 
 sin ft 
 
 } 
 
 
 = V6i2 
 
 + 622 
 
 tan 
 
 ft 
 
 bo 
 ^b; 
 
 
THE GENERAL NUMBER. 
 
 39 
 
 the quantities may be written thus : 
 
 A=a(cos a+j sin a); 
 
 B = b(cos ^+/sin^), 
 and then, 
 
 C = AB = ab{cos a+/sin «)(cos /9+ j sin ^) 
 = ab i (cos a cos /? —sin a sin'/?) +/(cos a sin /? +sin a cos /?)} 
 = a6 {cos (a +13) +j sin (a +^)j ; 
 
 Fig. 19. Multiplication of Vectora 
 
 that is, two general numbers are multiplied by multiplying their 
 absolute values or vectors, a and b, and adding their phase angles 
 a and /?. 
 
 Thus, to multiply the vector quantity, A = ai+ ja2 = a (cos 
 a+j sin a)hy B = bi +jb2^b{cos p + j sin ^) the vector 0^ in Fig. 
 19, which represents the general number A, is increased by the 
 factor b = V61M-6?, and rotated by the angle /?, which is given 
 
 bv tan B=T-- 
 bi 
 
 Thus, a complex multiplier B turns the direction of the 
 
 multiphcand A, by the phase angle of the multiplier B, and 
 
 multiplies the absolute value or vector oi A, by the absolute 
 
 value of B as factor. 
 
40 ENGINEERING MATHEMATICS. 
 
 The multiplier B is occasionally called an operator, as it 
 carries out the operation of rotating the direction and changing 
 the length of the multiphcand. 
 
 26. In multiplication, division and other algebraic opera- 
 tions with the representations of physical quantities (as alter- 
 nating currents, voltages, impedances, etc.) by mathematical 
 symbols, whether ordinary numbers or general numbers, it 
 is necessary to consider whether the result of the algebraic 
 operation, for instance, the product of two factors, has a 
 physical meaning, and if it has a physical meaning, whether 
 this meaning is such that the product can be represented in 
 the same diagram as the factors. 
 
 For instance, 3x4 = 12; but 3 horses X 4 horses does not 
 give 12 horses, nor 12 horses^, but is physically meaningless. 
 However, 3 ft. X4 ft. = 12 sq.ft. Thus, if the numbers represent 
 
 (}) — I — I — ©-e — i — I — I — I — I — I — I — ©-H — '— I " 
 o A B c 
 
 Fig. 20. 
 
 horses, multiplication has no physical meaning. If they repre- 
 sent feet, the product of multiplication has a physical meaning, 
 but a meaning which differs from that of the factors. Thus, 
 if on the line in Fig. 20, OA=Z feet, 05 = 4 feet, the product, 
 12 square feet, while it has a physical meaning, carmot be 
 represented any more by a point on the same line; it is not 
 the point 0C= 12, because, if we expressed the distances OA 
 and OB in inches, 36 and 48 inches respectively, the product 
 would be 36X48 = 1728 sq.in., while the distance OC would be 
 144 inches. 
 
 27. In all mathematical operations with physical quantities 
 it therefore is necessary to consider at every step of the mathe- 
 matical operation, whether it still has a physical meaning, 
 and, if graphical representation is resorted to, whether the 
 nature of the physical meaning is such as to allow graphical 
 representation in the same diagram, or not. 
 
 An instance of this general hmitation of the application of 
 mathematics to physical quantities occurs in the representation 
 of alternating current phenomena by general numbers, or 
 complex quantities. 
 
THE GENERAL NUMBER. 
 
 41 
 
 An alternating current can be represented by a vector 01 
 in a polar diagram, Fig. 21, in which one complete revolution 
 or 360 deg. represents the time of one complete period of the 
 alternating current. This vector 01 can be represented by a 
 general number, 
 
 I=H+jl2, 
 
 where ii is thejiorizontal, i2 the vertical component of the 
 current vector 01. 
 
 Fig. 21. Current, E.M.F. and Impedance Vector Diagram. 
 
 In the same manner an alternating E.M.F. of the same fre- 
 quency can be represented by a vector OE in the same Fig. 21, 
 and denoted by a general number, 
 
 E = ei+je2. 
 
 An impedance can be represented by a general number, 
 
 Z=^r + jx, 
 
 where r is the resistance and x the reactance. 
 
 If now we have two impedances, OZi and OZ2, ^i = ri-j-jxi 
 and Z2 = r2+ix2, their product Zi Z2 can be formed mathemat- 
 ically, but it has no physical meaning. 
 
42 ENGINEERING MATHEMATICS. 
 
 If we have a current and a voltage, / = ii + jt2 and E = ei + je2, 
 the product of current and voltage is the power P of the alter- 
 nating circuit. 
 
 The product of the two general numbers 7 and E can be 
 formed mathematically, IE, and would represent a point C 
 in the vector plane Fig. M. This point C, however, and the 
 mathematical expression IE, which represents it, does not give 
 the power P of the alternating circuit, since the power P is not 
 of the same frequency as / and E, and therefore cannot be 
 represented in the same polar diagrain Fig. 21, which represents 
 I and E. 
 
 If we have a current / and an impedance Z, in Fig. 21; 
 / = i\+jt2and Z = r+ix, their product is a voltage, and as the 
 voltage is of the same frequency as the current, it can be repre- 
 sented in the same polar diagram. Fig. 21, and thus is given by 
 the mathematical product of [ and Z, 
 
 E = IZ={ii+ji2)ir-\-ji), 
 
 = {iir -i2X ) +jfer +iix). 
 
 28. Commonly, in the denotation of graphical diagrams by 
 general numbers, as the polar diagram of alternating currents, 
 those quantities, which are vectors in the polar diagram, as the 
 current, voltage, etc., are represented by dotted capitals: E, I, 
 while those general numbers, as the impedance, admittance, etc. , 
 which appear as operators, that is, as multipliers of one vector, 
 for instance the current, to get another vector, the voltage, are 
 represented algebraically by capitals without dot: Z = r-{-jx = 
 impedance, etc. 
 
 This limitation of calculation with the mathematical repre- 
 sentation of physical quantities must constantly be kept in 
 mind in all theoretical investigations. 
 
 Division of General Numbers. 
 
 2Q. The division of two general numbers, A = ai+/a2 and 
 B = bi+jb2, gives, 
 
 A ai+ja2 
 
 This fraction contains the quadrature number in the numer- 
 ator as well as in the denominator. The quadrature number 
 
THE GENERAL NUMBER. 43 
 
 can be eliminated from the denominator by multiplying numer- 
 ator and denominator by the conjugate quantity of the denom- 
 inator, hi — jb2, which gives: 
 
 (oi +ja2)(bi -,762) (aih +a2^2) +j(a2bi -0162) 
 ■~{bi+ib2){b,-jb2)' bi^+b2' 
 
 aibi+a2b2 . 02^1 — cn&2 
 bi^ + b2^ +^ bi^+b2^ ' 
 
 for instance, 
 
 ^,_ A 6+2.5/ 
 
 0=-;: 
 
 B 3 + 4/ 
 
 ■ _ (6 + 2.5/) (3 -4/) 
 (3 +4/) (3 -4/) 
 28-16.5/ 
 25 
 = 1.12-0.66/. 
 
 If desired, the quadrature number may be eliminated from 
 the numerator and left in the denominator by multiplying with 
 the conjugate number of the numerator, thus: 
 
 i. ai +/a2 
 •~B~br+]b2 
 
 (ai+ja2)(ai-ja2) 
 
 ~(&i+/&2)(ai-/a2) 
 
 ai^ + a^^ 
 
 for instance. 
 
 (ai&i +a262) +/(ai^2 -O2&1) ' 
 
 ^_^_ 6+2.5/ 
 • B 3+4/ 
 
 ' _ (6 + 2.5/) (6 -2.5/) 
 ~ (3 + 4/) (6 -2.5/) 
 42.25 
 
 28 + 16.5/ 
 
 30. Just as in multipUcation, the polar representation of 
 the general number in division is more perspicuous than any 
 other. 
 
44 ENGINEERING MATHEMATICS. 
 
 Let A = a (cos a +/ sin a) be divided by B = b{cos ,8 + ] sin /?), 
 thus: 
 
 „_i4_a(cos a+y sin a) 
 • ~B~b(cosJ+Jan^ 
 
 a(cos a+i sin a) (cos /? — / sin /?) 
 ~6(cos /3+j sin /5)(cos /?— j sin /?) 
 
 a{ (cos a cos /? + sin a sin /?) +j'(sin a cos /? —cos a sin /9) } 
 ^ i>(cos2/?+sin2|5) 
 
 = -r{cos (a — /?) +/ sin (a —/?)}. 
 
 That is, general numbers A and i? are divided by dividing 
 their vectors or absolute values, a and b, and subtracting their 
 phases or angles a and (3. 
 
 Involution and Evolution of General Numbers. 
 
 31. Since involution is multiple multiplication, and evolu- 
 tion is involution with fractional exponents, both can be resolved 
 into simple expressions by using the polar form of the general 
 number. 
 
 If, 
 
 A = ai +/a2 = a(cos a+j sin a), 
 then 
 
 C' = A" = a"(cos na+y sin na). 
 
 For instance, if 
 
 A = 3 +4/= 5 (cos 53 deg. +j sin 53 deg.); 
 then, 
 
 (7 = A4 = 54(cos 4X53 deg. +/ sin 4x53 deg.) 
 = 625(cos 212 deg. +j sin 212 deg.) 
 = 625( -cos 32 deg. -j sin 32 cleg.) 
 = 625( -0.848 -0.530 j) 
 = -529 -331 j. 
 
 If, A=ai+ja2 = a (cos a+/sin a), then 
 
 G=\/~A = A« =a" (cos - +? sin - 
 
 n/-l Oi . . a\ 
 
 = Val COS- + ? sm - ). 
 \ n ' n 
 
THE GENERAL NUMBER. 45 
 
 32. If, in the polar expression of A, we increase the phase 
 angle a by 2Tt, or by any multiple of 2k : 2qn, where q is any 
 integer number, we get the same value of A, thus: 
 
 4 = a{cos(a+2(;;r) +j sin(a +2g7r)}, 
 
 since the cosine and sine repeat after every 360 deg, or 2-k. 
 The wth root, however, is different: 
 
 r, nn: n^( a+2g;r . . a^-2qn\ 
 C=vA = Va cos — +1 sm 3_ i_ 
 
 We hereby get n different values of C, for g = 0, 1, 2. . .n— 1; 
 q=n gives again the same as 5 = 0. Since it gives 
 
 « + 2mi a 
 
 =-+2k; 
 
 n n 
 
 that is, an increase of the phase angle by 360 deg., which leaves 
 cosine and sine unchanged. 
 
 Thus, the nth root of any general number has n different 
 values, and these values have the same vector or absolute 
 
 term v^, but differ from each other by the phase angle — and 
 
 its multiples. 
 
 For instance, let 4= -529 -331/= 625 (cos 212 deg. + 
 j sin 212 deg.) then, 
 
 C= ^1= ^(cos^^^±^+/sin^i^±^) 
 
 = 5(cos53+jsin53) =3 + 4/ 
 
 = 5(cosl43+/sinl43) = 5(-cos37+/sin37)= -4 + 3/^ 
 = 5(cos 233 +/ sin 233) = 5( -cos 53-/ sin 53) = -3-4/ 
 = 5(cos 323 +/ sin 323) = 5(cos 37 -/ sin 37) =4-3/ 
 = 5(cos 413+/ sin 413) = 5(cos 53 + / sin 53) =3+4/ 
 
 The n roots of a general number A = a(cos «+/ sin a) differ 
 
 ' 2t: 
 from each other by the phase angles — , or 1/nth of 360 deg., 
 
 and since they have the same absolute value v^a, it follows, that 
 they are represented by n equidistant points of a circle with 
 radius v'^a, as shown in Fig. 22, for n = 4, and in Fig. 23 for 
 
46 
 
 ENGINEERING MATHEMATICS. 
 
 n = 9. Such a system of n equal vectors, differing in phase from 
 each other by 1/nth of 360 deg., is caMed a. polyphase system, or 
 an n-phase system. The n roots of the general number thus 
 give an ?i-phase system. 
 
 33- For instance, \/l = ? 
 
 If A = a (cos a+y sin a) = l. this means: a=l, a=0; and 
 hence, 
 
 n/- 207: . . 2q7r 
 
 vl =cos h? sm ; 
 
 n ■* n 
 
 P,=-4+3i 
 
 Pr3+4/ 
 
 Pr-S-ij 
 
 Pr^-3j 
 
 Fig. 22. Roots of a General Number, n=4. 
 
 and the n roots of the unit are 
 
 2=0 <^=l; 
 
 „ 1 360 . . 360 
 g = l cos +7sm ; 
 
 o r. 360 . . 360 
 
 q = 2 cos2x — +; sin 2x2^; 
 
 1 / ^ X 360 . . 360 
 
 q = n~l cos(n-l) — ^+;sm(n-l)-- 
 ^ n 
 
 However, 
 
 360 
 
 n 
 
 c„s,--+,-si„,-^°=(c„.'S_V,-,„ !!»)'. 
 
THE GENERAL NUMBER. 
 hence, the n roots of 1 are, 
 
 „/- / 360 . . 360\« 
 VI = I cos Vj sin — I , 
 
 47 
 
 n I 
 
 where q may be any integer number. 
 
 One of these roots is real, for g=0, and is = +1. 
 
 If n is odd, all the other roots are general, or complex 
 numbers. 
 
 If n is an even number, a second root, for 2 = 0' '^ ^^^° ^&^: 
 cos 180+ /sin 180= -1. 
 
 Fig. 23. Roots of a General Number, n=^. 
 
 If n is divisible by 4, two roots are quadrature numbers, and 
 
 are +j, for q=-^, and -], for 5 = ^- 
 
 34. Using the rectangular coordinate expression of the 
 general number, A = ai + J«2, the calculation of the roots becomes 
 more complicated. For instance, given •^ = ? 
 
 Let C'=-C4=ci+yc2; 
 
 then, squaring, 
 
 A = (ci+/c2)2; 
 
 hence, 
 
 a\ +ja2 = (ci^ -02^) +2jciC2. 
 
 Since, if two general numbers are equal, their horizontal 
 and their vertical components must be equal, it is : 
 
 ai=ci^— C2^ and a2 = 2ciC2. 
 
48 ENGINEERING MATHEMATICS. 
 
 Squaring both equations and adding them, gives, 
 
 Hence : 
 
 and since 
 
 Cl2+C22=\/ai2+a22, 
 
 then, 
 
 ci2 = K^ai' + a22+ai), 
 
 and 
 Thus 
 
 C22 = i(Vai2+a22-ai). 
 
 and 
 
 Ci='/ii^ai' + a2' + aij 
 
 and 
 
 C2 = '/iiVai2 + a22-ai}, 
 
 which is a rather comphcated expression. 
 
 35. When representing physical quantities by general 
 numbers, that is, complex quantities, at the end of the calcula- 
 -tion the final result usually appears also as a general number, 
 or as a complex of general numbers, and then has to be reduced 
 to the absolute value and the phase angle of the physical quan- 
 tity. This is most conveniently done by reducing the general 
 numbers to their polar expressions. For instance, if the result 
 of the calculation appears in the form. 
 
 (ai + ia2)(bi +jb2)^Vci + j ca 
 
 • (di+jd2)2(ei+je2) ' 
 
 by substituting 
 
 a=Va^~+a^; tana = — . 
 ai 
 
 and so on. 
 
 6 = VV + 67; tan/? = ^; 
 
 r,_a(cos a+ysin a)63(cos/? + jsin,/?)3\/c(cos ;- + jsin ;-)* 
 
 d-(cos 3+j sin 5)2e(cos e+j sin s) 
 
 a¥Vc 
 = --^\'^os{a+3[} + r/2 -2d - e)+i sin {a+3p+r/2 ~2d - e) \ 
 
THE GENERAL NUMBER. 49 
 
 Therefore, the absolute value of a fractional expression is 
 the product of the absolute values of the factors of the numer- 
 ator, divided by the product of the absolute values of the 
 factors of the denominator. 
 
 The phase angle of a fractional expression is the sum of 
 the phase angles of the factors of the numerator, minus the sum 
 of the phase angles of the factors of the denominator. 
 
 For instance. 
 
 ^ (3-4/)2(2 + 2y)-y-2.5 + 6j 
 5(4 + 3y)2\/2 
 25(cos307+/sin307)22\/2(cos45+/sin45)^(r5(cosll4+/sinll4)i 
 125(cos37+jsin37)2\/2 
 
 = 0.4^a5|cos(2x307+45+^-2x37j 
 
 / 114 
 
 +jsm (2X307 +45 +-^-2X37 
 
 = 0.4"v/a5{cos263+/sin263} 
 
 = 0.746| -0.122-0.992/) = -0.091 -0.74/. 
 
 36. As will be seen in Chapter II: 
 
 u^ u^ it* 
 
 £"=l+w+|2- + |3- + u: + . . 
 
 X^ X* X^ 1^ 
 
 cosx=l-|2 +^-^ + ^--1... 
 
 x^ x^ x^ 
 smz = x—r- + 1^—177+ -. . . 
 \6 |o \t 
 
 Herefrom follows, by substituting, x = d, u = jd, 
 cos ^+y sin d= e^', 
 and the polar expression of the complex quantity, 
 
 A = a(cos a+j sin a), 
 thus can also be written in the form. 
 
50 ENGINEERING MATHEMATICS. 
 
 where e is the base of the natural logarithms, 
 
 £ = 1+1+^ + ^ + ^ + ... =2.71828... 
 
 Since any number a can be expressed as a power of any 
 other number, one can substitute, 
 
 where ao = logea= , "^° , and the general number thus can 
 logio £ 
 
 also be written in the form, 
 
 that is the general number, or complex quantity, can be expressed 
 in the forms, 
 
 A=ai+ja2 
 = a(cos a+j sin a) 
 
 The last two, or exponential forms, are rarely used, as they 
 are less convenient for algebraic operations. They are of 
 importance, however, since solutions of differential equations 
 frequently appear in this form, and then are reduced to the 
 polar or the rectangular form. 
 
 37. For instance, the differential equation of the distribu- 
 tion of alternating current in a flat conductor, or of alternating 
 magnetic flux in a flat sheet of iron, has the form : 
 
 and is integrated by y = As~^'', where. 
 
 hence, 
 
 This expression, reduced to the polar form, is 
 
 ?/ = Ai£"'"''"^(cos cx-j sin ex) +A2£~''''(cos cx+j sin ex). 
 
THE GENERAL NUMBER. 51 
 
 Logarithmation. 
 
 38. In taking the logarithm of a general number, the ex- 
 ponential expression is most convenient, thus : 
 
 log£ (fli +ja2) =log£ a(cos a +j sin a) 
 = log£a£"' 
 = log£ a+logse'" 
 = loge a +ja; 
 
 or, if 6 = base of the logarithm, for instance, 6 = 10, it is: 
 
 log(,(ai+ja2)=logja£''' = logj a+ja logj s; 
 
 or, if b unequal lOj reduced to logio; 
 
 I / , • \ logio a . logio s 
 logio logio b 
 
 Note. In mathematics, for quadrature unit V— 1 is always 
 chosen the symbol i. Since, however, in engineering the symbol i 
 is universally used to represent electric current, for the quad- 
 rature unit the symbol j has been chosen, as the letter nearest 
 in appearance to i, and j thus is always used in engineering 
 calculations to denote the quadrature unit V — 1. 
 
CHAPTER II. 
 POTENTIAL SERIES AND EXPONENTIAL FUNCTION. 
 
 A. GENERAL. 
 
 39. An expression such as 
 
 y-ih ■ ■ (1) 
 
 represents a fraction; that is, the result of division, and hke 
 any fraction it can be calculated; that is, the fractional form 
 eliminated, by dividing the numerator by the denominator, thus: 
 
 l-x\l=l+x+x2+x^+. . . 
 1-x 
 
 + x 
 x—x^ 
 
 + X2 
 
 + x3. 
 Hence, the fraction (1) can also be expressed in the form: 
 
 y=j-— = l+x+x^+x^ + (2) 
 
 This is an infinite series of successive powers of x, or a poten- 
 tial series. 
 
 In the same manner, by dividing through, the expression 
 
 y-ih' (3) 
 
 can be reduced to the infinite series, 
 
 y^YJ^^^'~^~^^^~^^+ ~ (4) 
 
 52 
 
POTENTIAL SERIES AND EXPONENTIAL FUNCTION. 53 
 
 The infinite series (2) or (4) is another form of representa- 
 tion of the expression (1) or (3), just as the periodic decimal 
 fraction is another representation of the common fraction 
 (for instance 0.6363 = 7/11). 
 
 40. As the series contains an infinite number of terms, 
 in calculating numerical values from such a series perfect 
 exactness can never be reached; since only a finite number of 
 terms are calculated, the result can only be an approximation. 
 By taking a sufficient number of terms of tlie series, however, 
 the approximation can be made as close as desired; that is, 
 numerical values may be calculated as exactly as necessary, 
 so that for engineering purposes the infinite series (2) or (4) 
 gives just as exact numerical values as calculation by a finite 
 expression (1) or (2), provided a sufficient number of terms 
 are used. In most engineering calculations, an exactness of 
 0.1 per cent is sufficient; rarely is an exactness of 0.01 per cent 
 or even greater required, as the unavoidable variations in the 
 nature of the materials used in engineering structures, and the 
 accuracy of the measuring instruments impose a limit on the 
 exactness of the result. 
 
 For the value a; = 0.5, the expression (1) gives y = z — Tr-r = 2; 
 
 while its representation by the series (2) gives 
 
 2/ = l +0.5+0.25 +0.125 +0.0625 + 0.03125 + . (5) 
 
 and the successive approximations of the numerical values of 
 y then are : 
 
 using one term: y=l =1; error: —1 
 
 " two terms: y = l + 0.5 =1.5; " -0.5 
 
 " three terms: J/= 1 + 0.5 + 0.25 =1.75- " -0.25 
 
 " four terms: i/=l + 0.5+0.25+0.125 =1.875; " -0.125 
 
 " five terms: ;/= 1 + 0.5 + 0.25+0.125 + 0.0625 = 1.9375 " -0.0625 
 
 It is seen that the successive approximations come closer and 
 closer to the correct value, y = 2, but in this case always remain 
 below it; that is, the series (2) approaches its hmit from below, 
 as shown in Fig. 24, in which the successive approximations 
 are marked by crosses. 
 
 For the value a; = 0.5, the approach of the successive 
 approximations to the hmit is rather slow, and to get an accuracy 
 of 0.1 per cent, that is, bring the error down to less than 0.002, 
 requires a considerable number of terms. 
 
54 ENGINEERING MATHEMATICS. 
 
 For x = 0.1 the series (2) is 
 
 2/ = l +0.1 +0.01+0.001 +0.0001 + (6) 
 
 and the successive approximations thus are 
 
 l:y=l; 2:y=l.l; 3:y=l.U; 4:y=l.ni; 5:y=l.nU; 
 and as, by (1), the final or Hmiting value is 
 
 t 5 6 
 
 + 4 5 
 
 + 3 
 
 2 
 
 Fig. 24. Convergent Series with One-sided Approach. 
 
 the fourth approximation already brings the error well below 
 0.1 per cent, and sufficient accuracy thus is reached for most 
 engineering purposes by using four terms of the series. 
 41. The expression (3) gives, for a; = 0.5, the value, 
 
 ^=rTo:5=l=^-^^^^--- 
 
 Represented by series (4), it ^ves 
 
 2/ = l-0.5 + 0.25-0.125+0.0625-0.03125+- (7) 
 
 the successive approximations are; 
 
 1st: y = l =1; error: +0.333... 
 
 2d: y^l-0.5 =0.5; " -0.1666... 
 
 3d: 3/=l-0.5 + 0.25 =0.75; " +0.0833... 
 
 4th: !/- 1-0.5+0.25-0.125 =0.625; " -0.04166... 
 
 5th: i/ = l-0.5+0.25-0.125 + 0.0625 = 0.6875; " +0.020833... 
 
 As seen, the successive approximations of this series come 
 closer and closer to the correct value 2/ = 0.6666 . . . , but in this 
 case are alternately above and below the correct or limiting 
 value, that is, the series (4) approaches its limit from both sides, 
 as shown in P^ig. 25, while the series (2) approached the limit 
 from below, and still other series may approach their limit 
 from above. 
 
POTENTIAL SERIES AND EXPONENTIAL FUNCTION. 55 
 
 With such alternating approach of the series to the limit, 
 as exhibited by series (4), the limiting or final value is between 
 any two successive approximations, that is, the error of any 
 approximation is less than the difference between this and the 
 next following approximation. 
 
 Such a series thus is preferable in engineering, as it gives 
 mformation on the maximum possible error, while the series 
 with one-sided approach does not do this without special in- 
 vestigation, as the error is greater than the difference between 
 successive approximations. 
 
 42. Substituting a; =2 into the expressions (1) and (2), 
 equation (1) ^ves 
 
 +1 
 
 3 
 
 + 
 
 5 
 
 + 
 
 2 1 
 
 " 1+x 
 
 Fig. 25. Convergent Series with Alternating Approach. 
 
 while the infinite series (2) gives 
 
 j/ = l+2-l-4+8-Hl6+32-h ..; 
 
 and the successive approximations of the latter thus are 
 
 1; 3; 7; 15; 31; 63. . .; 
 
 that is, the successive approximations do not approach closer 
 and closer to a final value, but, on the contrary, get further and 
 further away from each other, and give entirely wrong results. 
 They give increasing positive values, which apparently approach 
 00 for the entire series, while the correct value of the expression, 
 by (1), is 2/= -1. 
 
 Therefore, for a; =2, the series (2) gives unreasonable results, 
 and thus cannot be used for calculating numerical values. 
 
 The same is the case with the representation (4) of the 
 expression (3) for x=2. The expression (3) gives 
 
 f/=Y^ = 0.3333 . . . ; 
 
56 ENGINEERING MATHEMATICS. 
 
 while the infinite series (4) gives 
 
 2/ = l-2 + 4-8 + 16-32+- . ., 
 
 and the successive approximations of the latter thus are 
 
 1; -1; +3; -5; +11; -21; . . .; 
 
 hence, while the successive values still are alternately above 
 and below the correct or limiting value, they do not approach 
 it with increasing closeness, but more and more diverge there- 
 from. 
 
 Such a series, in which the values derived by the calcula- 
 tion of more and more terms do not approach a final value 
 closer and closer, is called divergent, while a series is called 
 convergent if the successive approximations approach a final 
 value with increasing closeness. 
 
 43- While a finite expression, as (1) or (3), holds good for 
 all values of x, and numerical values of it can be calculated 
 whatever may be the value of the independent variable x, an 
 infinite series, as (2) and (4), frequently does not give a finite 
 result for every value of x, but only for values within a certain 
 range. For instance, in the above series, for —1 <a;< + l, 
 the series is convergent; while for values of x outside of this 
 range the series is divergent and thus useless. 
 
 When representing an expression by an infinite series, 
 it thus is necessary to determine that the series is convergent; 
 that is, approaches with increasing number of terms a finite 
 limiting value, otherwise the series cannot be used. Where 
 the series is convergent within a certain range of x, diver- 
 gent outside of this range, it can be used only in the range oj 
 convergency, but outside of this range it cannot be used for 
 deriving numerical values, but some other form of representa- 
 tion has to be found which is convergent. 
 
 This can frequently be done, and the expression thus repre- 
 sented by one series in one range and by another series in 
 
 another range. For instance, the expression (1), y = - , by 
 
 J. ~r •(/ 
 
 substituting, x = -, can be written in the form 
 
 1 u 
 
 i+i ^+"' 
 
 u 
 
POTENTIAL SERIES AND EXPONENTIAL FUNCTION. 57 
 
 and then developed into a series by dividing the numerator 
 by the denominator, which gives 
 
 y = u—u^+u^~u'^ + . . .; 
 
 or, resubstituting x, 
 
 1111 
 
 ^~x"F2+^3--4 + - ...... (8) 
 
 which is convergent for x = 2, and for a; = 2 it gives 
 
 2/ = 0.5 -0.25+0.125 -0.0625 + . . . (9) 
 With the successive approximations : 
 
 0.5; 0.25; 0.375; 0.3125..., 
 which approach the final hmiting value, 
 
 2/ = 0.333. . . 
 
 44. An infinite series can be used only if it is convergent. 
 Mathemetical methods exist for determining whether a series 
 is convergent or not. For engineering purposes, however, 
 these methods usually are unnecessary; for practical use it 
 is not sufficient that a series be convergent, but it must con- 
 verge so rapidly — that is, the successive terms of the series 
 must decrease at such a great rate — that accurate numerical 
 results are derived by the calculation of only a very few terms; 
 two or three, or perhaps three or four. This, for instance, 
 is the case with the series (2) and (4) for x = 0.1 or less. For 
 x = 0.5, the series (2) and (4) are still convergent, as seen in 
 (5) and (7), but are useless for most engineering purposes, as 
 the successive terms decrease so slowly that a large number 
 of terms have to be calculated to get accurate results, and for 
 such lengthy calculations there is no time in engineering work. 
 If, however, the successive terms of a series decrease at such 
 a rapid rate that all but the first few terms can be neglected, 
 the series is certain to be convergent. 
 
 In a series therefore, in which there is a question whether 
 it is convergent or divergent, as for instance the series 
 
 ,11111 
 
 2/ = l+2 +3 +4 +5 +6 +• • • (divergent), 
 
58 ENGINEERING MATHEMATICS. 
 
 or 
 
 .11111, r +^ 
 
 ^^^~2+3 "4'^5"6+ (convergent), 
 
 the matter of convergency is of little importance for engineer- 
 ing calculation, as the series is useless in any case; that is, does 
 not give accurate numerical results with a reasonably moderate 
 amount of calculation. 
 
 A series, to be usable for engineering work, must have 
 the successive terms decreasing at a very rapid rate, and if 
 this is the case, the series is convergent, and the mathematical 
 investigations of convergency thus usually becomes unnecessary 
 in engineering work. 
 
 45. It would rarely be advantageous to develop such simple 
 expressions as (1) and (3) into infinite series, such as (2) and 
 (4), since the calculation of numerical values from (1) and (3) 
 is simpler than from the series (2) and (4), even though very 
 few terms of the series need to be used. 
 
 The use of the series (2) or (4) instead of the expressions 
 (1) and (3) therefore is advantageous only if these series con- 
 verge so rapidly that only the first two terms are required 
 for numerical calculation, and the third term is negligible; 
 that is, for very small values of r. Thus, for .r = 0.01, accord- 
 ing to (2), 
 
 2/ = l +0.01 +0.0001 +. . . = 1 +0.01, 
 
 as the next term, 0.0001, is already less than 0.01 per cent of 
 the value of the total expression. 
 
 For very small values of x, therefore, by (1) and (2), 
 
 and by (3) and (4), 
 
 y=j~='^+^, (10) 
 
 2/=r+-^ = l-^, . . . (11) 
 
 ana tnese expressions (10) and (11) are useful and very com- 
 monly used in engineering calculation for simplifying work. 
 For instance, if 1 plus or minus a very small quantity appears 
 as factor in the denominator of an expression, it can be replaced 
 by 1 minus or plus the same small quantity as factor in the 
 numerator of the expression, and inversely. 
 
POTENTIAL SERIES AND EXPONENTIAL FUNCTION. 59 
 
 For example, if a direct-current receiving circuit, of resist- 
 ance r, is fed by a supply voltage eo over a line of low 
 resistance rg, what is the voltage e at the receiving circuit? 
 
 The total resistance is r+ro] hence, the current, i= 
 
 eo 
 
 , , r + ro' 
 
 and the voltage at the receiving circuit is 
 
 r + ro 
 If now To is small compared with r, it is 
 
 « = eo— ^^ = eo|l-y| (13) 
 
 r 
 
 As the next term of the series would be (— ) , the error 
 
 made by the simpler expression (13) is less than (— ] . Thus, 
 if ro is 3 per cent of r, which is a fair average in interior light- 
 ing circuits, ( — ) =0.03^ = 0.0009, or less than 0.1 per cent; 
 
 hence, is usually negligible. 
 
 46. If an expression in its finite form is more complicated 
 and thereby less convenient for numerical calculation, as for 
 instance if it contains roots, development into an infinite series 
 frequently simplifies the calculation. 
 
 Very convenient for development into an infinite series 
 of powers or roots, is the binomial theorem, 
 
 /1 , ^. 1, n(n-l) , , w(w-l)(n-2 ) 
 (l±u)" = l±nu-\ n^ u ± jo u^ + - 
 
 where 
 
 lm = lx2x3X. . .Xm. 
 
 (14) 
 
 Thus, for instance, in an alternating-current circuit of 
 resistance r, reactance x, and supply voltage e, the current is, 
 
60 ENGINEERING MATHEMATICS. 
 
 If this circuit is practically non-inductive, as an incandescent 
 lighting circuit; that is, if x is small compared with r, (15) 
 
 can be written in the form, 
 
 _ 1 
 e e 
 
 i = - 
 
 .+(7 
 
 -(r)1". . . . ae) 
 
 and the square root can be developed by the binomial (14), thus, 
 
 /x\2 1 
 
 M= ( -) ; n= — -, and gives 
 
 [-(r)T 
 
 '^^-m4(^-m*- <-> 
 
 In this series (17), if x = 0.lr or less; that is, the reactance 
 is not more than 10 per cent of the resistance, the third term, 
 
 a\~) , is less than 0.01 per cent; hence, neghgible, and the 
 
 series is approximated with sufficient exactness by the first 
 two terms. 
 
 1 
 
 . X. 
 1+. 
 r 
 
 '-4 iy as) 
 
 '1 \r 
 and equation (16) of the current then gives 
 
 ^-7('4©1 ™ 
 
 This expression is simpler for numerical calculations than 
 the expression (15), as it contains no square root. 
 
 47. Development into a scries may become necessary, if 
 further operations have to be carried out with an expression 
 for which the expression is not suited, or at least not well suited. 
 This is often the case \A'herc the expression has to be integrated, 
 since very few expressions can be integrated. 
 
 Expressions under an integral sign therefore very commonly 
 have to be developed into an infinite series to carry out the 
 integration. 
 
POTENTIAL SERIES AND EXPONENTIAL FUNCTION. 61 
 
 EXAMPLE 1. 
 
 Of the equilateral hyperbola (Fig. 26), 
 
 xy = a^, ... . (20) 
 
 the length L of the arc between Xi = 2a and X2 = 10a is to be 
 calculated. 
 
 An element dl of the arc is the hypothenuse of a right triangle 
 with dx and dy as cathetes. It, therefore, is, 
 
 dl = Vdx^+dy^ 
 
 -Mty^^' (21) 
 
 
 
 
 1 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 V 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 dy-' 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 xy= 
 
 ,a = 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 X, 
 
 X 
 
 
 
 
 
 
 
 
 
 
 r_. 
 
 
 
 and from (20), 
 
 Fig. 26. Equilateral Hyperbola. 
 
 dy a? 
 
 ^ X dx x^' 
 
 Substituting (22) in (21) gives, 
 
 dl=Jl + (^)dx; 
 
 hence, the length L of the arc, from xi to X2 is, 
 
 (22) 
 
 (23) 
 
62 ENGINEERING MATHEMATICS. 
 
 X . . 
 
 Substituting - = i;; that is, dx^adv, also substituting 
 
 Di = — = 2 and i)2 = — =10, 
 a a ' 
 
 gives 
 
 = U \R'^^- 
 
 The expression under the integral is inconvenient for integra- 
 tion; it is preferably developed into an infinite series, by the 
 binomial theorem (14). 
 
 Write M= -7 and 71=77, then 
 
 V* 2' 
 
 
 and 
 
 /"«! f 1 1 1 t^ 
 
 -...\dv 
 
 1 
 
 r^'( 1 i_ _i 5^_ 
 
 1 
 
 3X128X1^18 
 
 , , ._^1/1 1\ 1/1 1 
 
 6 \Vi^ V2^/ 5b \«i7 V2^ 
 
 - + 
 
 J^/l 1_\_ 
 
 ^176\i;iii V2'y '^" 
 
 and substituting the numerical values, 
 L = ai (10-2) +^(0.125-0.001) 
 
 -^(0.0078-0) +^(0.0001 -0)1 
 = al8 + 0.0207 - 0.0001 \ = 8.0206a. 
 
 As seen, in this series, only the first two terms are appreciable 
 in value, the third term less than 0.01 per cent of the total, 
 and hence negligible, therefore the series converges very 
 rapidly, and numerical values can easily be calculated by it. 
 
POTENTIAL SERIES AND EXPONENTIAL FUNCTION. 63 
 
 For Xi<2 a; that is, Vi <2, the series converges less rapidly, 
 and becomes divergent for xi<a; or, vi<l. Thus this series 
 (17) is convergent for v>l, but near this limit of convergency 
 it is of no use for engineering calculation, as it does not converge 
 with sufficient rapidity, and it becomes suitable for engineering 
 calculation only when vi approaches 2. 
 
 EXAMPLE 2. 
 
 48. log 1=0, and, therefore log (1+x) is a small quantity 
 if X is small, log (1 +x) shall therefore be developed in such 
 a series of powers of x, which permits its rapid calculation 
 without using logarithm tables. 
 
 It is 
 
 logu=J-; 
 then, substituting (l+x) for u gives, 
 
 log(l+.x)=Jp_^ (24) 
 
 From equation (4) 
 1 
 
 l+x 
 hence, substituted into (24), 
 
 = l — x + x^—x^ + . 
 
 log (1 +x) = j (1 ~x + x^ —x^ + . . .)dx 
 
 = I dx — I xdx + I x^dx — I x^dx - 
 
 x^ x^ X* , , 
 
 = a;— 2+3--J+ • • ; .... (25) 
 
 hence, if x is very small, — is negligible, and, therefore, all 
 
 terms beyond the first are negligible, thus, 
 
 log {l+x)=x; 
 
 while, if the second term is still appreciable in value, the more 
 complete, but still fairly simple expression can be used, 
 
 log {l + x) = x-^=x[l—^ 
 
64 ENGINEERING MATHEMATICS. 
 
 If instead of the natural logarithm, as used above, the 
 decimal logarithm is required, the following relation may be 
 applied : 
 
 logio a = logios logs a = 0.4343 logs a, 
 
 logio a is expressed by log^ a, and thus (19), (20) (21) assume 
 
 the form, 
 
 / .r2 j3 -f-i 
 logio (l+^)=0.4313(x--+--^+. 
 
 or, approximately, 
 
 logio(l+a:)=0.4343x; 
 
 or, more accurately, 
 
 logio(l+.T)=0.4343.r(l-| 
 
 B. DIFFERENTIAL EQUATIONS. 
 
 49. The representation by an infinite series is of special 
 value in those cases, in which no finite expression of the func- 
 tion is known, as for instance, if the relation between :r and y 
 is given bj^ a differential equation. 
 
 Differential equations are solved by separating the variables, 
 that is, bringing the terms containing the one variable, y, on 
 one side of the equation, the terms with the other variable x 
 on the other side of the equation, and then separately integrat- 
 ing both sides of the equation. Ycry rarely, however, is it 
 possible to separate the variables in this manner, and where 
 it cannot be done, usually no systematic method of solving the 
 differential equation exists, but this has to be done by trying 
 different functions, until one is found which satisfies the 
 equation. 
 
 In electrical engineering, currents and voltages are dealt 
 with as functions of time. The current and c.m.f. giving the 
 power lost in resistance are related to each other by Ohm's 
 law. Current also produces a magnetic field, and this magnetic 
 field b}' its changes generates an c.m.f.— the e.m.f. of self- 
 inductance. In this case, c.m.f. is related to the change of 
 current; that is, the differential coefficient of the current, and 
 thus also to the differential coefficient of e.m.f., since the e.m.f. 
 
POTENTIAL SERIES AND EXPONENTIAL FUNCTION. 65 
 
 is related to the current by Ohm's law. In a condenser, the 
 current and therefore, by Ohm's law, the e.m.f., depends upon 
 and is proportional to the rate of change of the e.m.f. impressed 
 upon the condenser; that is, it is proportional to the differential 
 coefficient of e.m.f. 
 
 Therefore, in circuits having resistance and inductance, 
 or resistance and capacity, a relation exists between currents 
 and e.m.fs., and their differential coefficients, and in circuits 
 having resistance, inductance and capacity, a double relation 
 of this kind exists; that is, a relation between current or e.m.f. 
 and their first and second differential coefficients. 
 
 The most common differential equations of electrical engineer- 
 ing thus are the relations between the function and its differential 
 coefficient, which in its simplest form is, 
 
 t-r, m 
 
 or 
 
 1=.., .... (27) 
 
 and where the circuit has capacity as well as inductance, the 
 second differential coefficient also enters, and the relation in 
 its simplest form is, 
 
 cPy 
 
 or 
 
 dF^-y-' (28) 
 
 S-^' (29) 
 
 and the most general form of this most common differential 
 equation of electrical engineering then is, 
 
 g+2c|+a, + 6 = (30) 
 
 The differential equations (26) and (27) can easily be inte- 
 grated by separating the variables, but not so with equations 
 (28), (29) and (30); the latter are preferably solved by trial. 
 
 SO. The general method of solution may be illustrated with 
 the equation (26) : 
 
 l=» (-) 
 
66 ENGINEERING MATHEMATICS. 
 
 To determine whether this equation can be integrated by an 
 infinite series, choose such an infinite series, and then, by sub- 
 stituting it into equation (26), ascertain \vhether it satisfies 
 the equation (26) ; that is, makes the left side equal to the right 
 side for every value of x. 
 
 Let, 
 
 y = ao+aix+a2X^ + a-sX^ + a4X'* + . . . . (31) 
 
 be an infinite series, of which the coefficients ao, a\, a^, 03. . . 
 are still unknown, and by substituting (31) into the differential 
 equation (26), determine whether such values of these coefficients 
 can be found, which make the series (31) satisfy the equation (26). 
 Differentiating (31) gives, 
 
 -^ = ai+2a2i+ 3 03^2 + 404x3 + (32) 
 
 The differential equation (26) transposed gives, 
 dy 
 
 dx 
 
 -2/ = (33) 
 
 Substituting (31) and (32) into (33), and arranging the terms 
 in the order of x, gives, 
 
 (gi - ao) + (2a2- ai)x + (803 - a2).r2 
 
 + (4a4- 03)3-3 + (5a5-a4)x*+ .=0. . (34) 
 
 If then the above series (31) is a solution of the differential 
 equation (26), the expression (34) must be an identity; that is, 
 must hold for every value of x. 
 
 If, however, it holds for every value of x, it does so also 
 for a; = 0, and in this case, all the terms except the first vanish, 
 and (34) becomes, 
 
 oi-Oo=0; or, 0i=an. . . . (35) 
 
 To make (31) a solution of the differential equation (oi-oo) 
 must therefore equal 0. This being the case, the term (ai-oo) 
 can be dropped in (34), which then becomes, 
 
 (2o2-ai)a; + (3a3-a2)a;2 + (4a4-03)x3 + (5a5-a4)a;4 + . =0; 
 
 or. 
 
 a:{(2a2-ai) + (3a3-a2)x + (4a4-a3).r2 + . . .}=0. 
 
POTENTIAL SERIES AND EXPONENTIAL FUNCTION. 67 
 
 Since this equation must hold for every value of x, the second 
 factor of the equation must be zero, since the first factor, x, is 
 not necessarily zero. This gives, 
 
 (2o2— ai) +(3a3 — a2)x + (4a4— a3)x2 + . . .=0. 
 
 As this equation holds for every value of x, it holds also for 
 x = 0. In this case, however, all terms except the first vanish, 
 and, 
 
 2a2-ai=0; (36) 
 
 hence, 
 
 ai 
 
 a2 = - 
 
 )> 
 
 and from (35), 
 
 az-- 
 
 2" 
 
 Continuing the same reasoning, 
 
 3a3 — a2 = 0, 4a4— 03=0, etc. 
 
 Therefore, if an expression of successive powers of x, such as 
 (34), is an identity, that is, holds for every value of x, then all 
 the coefficients of all the powers of x must separately be zero* 
 
 Hence, 
 
 Oi — ao=0; or ai = ao; 
 
 o A ai ao 
 
 2a2— oi = 0; or a2=-2=-^, 
 
 o n «2 Cto. 
 
 oas — a2 = U; or a3=-o" = |o; 
 
 4a4 
 
 etc.. 
 
 a3=0; or 0'i=-^ = u; 
 
 etc. 
 
 (37) 
 
 * The reader must realize the difference between an expression (34), as 
 equation in i, and as substitution product of a function; that is, as an 
 identity. 
 
 Regardless of the vahies of the coefficients, an expression (34) as equation 
 gives a number of separate values of x, the roots of the equation, which 
 make the left side of (34) equal zero, that is, solve the equation. If, however, 
 the infinite series (31) is a solution of the differential equation (26), then 
 the expression (34), which is the result of substituting (31) into (26), must 
 be correct not only for a limited number of values of x, which are the roots 
 of the equation, but for all values of x, that is, no matter what value is 
 chosen for x, the left side of (34) must always give the same result, 0, that 
 is, it must not be changed by a change of x, or in other words, it must not 
 contain x, hence all the coefficients of the powers of x must be zero. 
 
68 
 
 ENGINEERING MATHEMATICS. 
 
 Therefore, if the coefficients of tlie pcries (31) are chosen 
 by equation (37), this scries satisfies the differential equation 
 (26); that is, 
 
 \ a:2 x3 a;4 1 
 
 2/ = ao l+x+-2+^+u+-- |- • • ■ (38) 
 
 is the solution of the differential equation, 
 
 dy 
 
 dx 
 
 --y- 
 
 51. In the same manner, the differential equation (27), 
 
 dz 
 
 dx 
 
 = az. 
 
 (39) 
 
 is solved by an infinite series, 
 
 z = ao+aiX+a2X^+a3X^ + . . ., . . . (40) 
 
 and the coefficients of this scries determined by substituting 
 (40) into (39), in the same manner as clone above. This gives, 
 
 (ai — aao) +{2a2 — aai)x + {Sa3 — aa2)x^ 
 
 + {4a4-aa3)x^ + . . .=0, . (41) 
 
 and, as this equation must be an identity, all its coefficients 
 must be zero ; that is, 
 
 ai~aao = 0; or ai = aao; 
 
 a o? 
 2o2— aai=0; or a2 = «i :y = ao y; 
 
 on a a? 
 
 3a3 — aa2 = 0; or 03 = 0,20=0070"; ]■ ■ . (42) 
 
 ^ n a a^ 
 
 404—003 = 0; or 04 = 03^ = 0077; 
 
 etc., etc. 
 
 and the solution of differential equation (39) is, 
 
 f , a'^x^ a^x^ o^x* 1 
 
 z = Oo|l+ax+— +-J3-+-^ + ...|. . (43) 
 
 52. These solutions, (38) and (43), of the differential equa- 
 tions (26) and (39), are not single solutions, but each contains 
 an infinite number of solutions, as it contains an arbitrary 
 
POTENTIAL SERIES AND EXPONENTIAL FUNCTION. 69 
 
 constant ao] that is, a constant which may have any desired 
 numerical vahie. 
 
 This can easily be seen, since, if z is a solution of the dif- 
 ferential equation, 
 
 dz 
 
 d^ = ^^' 
 
 then, any multiple, or fraction of z, bz, also is a solution of the 
 differential equation; 
 
 dibz) 
 
 since the h cancels. 
 
 Such a constant, ao, which is not determined by the coeffi- 
 cients of the mathematical problem, but is left arbitrary, and 
 requires for its determinations some further condition in 
 addition to the differential equation, is called an integration 
 constant. It usually is determined by some additional require- 
 ments of the physical problem, which the differential equation 
 represents; that is, by a so-called terminal condition, as, for 
 instance, by having the value of y given for some particular 
 value of X, usually f or x = 0, or x = oo . 
 
 The differential equation, 
 
 i-r. W 
 
 thus, is solved by the function, 
 
 y = aoyo, (45) 
 
 where, 
 
 />»2 rpS 'V*4 
 
 2/o = l+x+2-+pj+jj+ ....... (46) 
 
 and the differential equation, 
 
 |-«^' (*^) 
 
 is solved by the function, 
 
 z = aoZo, (48) 
 
 where, 
 
 a^x^ aH^ a^.r* ,.„, 
 
 0o = H-fla;+^-+-rj-+-|y- + (49) 
 
70 ENGINEERING MATHEMATICS. 
 
 yo and zo thus are the simplest forms of the solutions ij and z 
 of the differential equations (26) and (39). 
 
 53. It is interesting now to determine the value of 2/". To 
 raise the infinite series (46), which represents %, to the nth 
 power, would obviously be a very complicated operation. 
 
 However, 
 
 t"^-'t ^^») 
 
 dv 
 and since from (44) w~°"2/; • (51) 
 
 by substituting (51) into (50), 
 
 t-^y-' (-^2) 
 
 hence, the same equation as (47), but with y" instead of z. 
 Hence, if y is the solution of the differential equation, 
 
 dy^ 
 
 dx y' 
 
 then z = 2/" is the solution of the differential equation (52), 
 
 dz 
 
 -j- = nz. 
 
 ax 
 
 However, the solution of this differential equation from (47), 
 (48), and (49), is 
 
 z = aoZo', 
 ZQ = l+nx+—^+^+. . . ; 
 
 that is. if 
 
 then, 
 
 2/o = l+a;+2-+73+..., 
 
 n^x^ n^3? 
 
 «o = ?/o" = l+wa;+^^+-T^ + ... ; . . . (53) 
 
 therefore the series y is raised to the nth power by multiply- 
 ing the variable x by n. 
 
POTENTIAL SERIES AND EXPONENTIAL FUNCTION. 71 
 
 Substituting now in equation (53) for n the value - gives 
 1 111 
 
 j/o^=i+i+2+j3+rf+--- ; • ■ (54) 
 
 that is, a constant numerical value. This numerical value 
 
 equals 2.7182818. . ., and is usually represented by the symbol s. 
 
 Therefore, 
 
 j_ 
 
 hence, 
 
 /ytji /v'O 'y'i 
 
 yo = e- = l+x+-+'j^ + -r^ + . . . , (.55) 
 
 and 
 
 , ,. , n^x^ n^x^ n'^x* , „, 
 
 20 = %"= («^)"=s"^ = l+nj:+—5-+-r5- +-,— + . . . ; (56) 
 
 therefore, the infinite series, which integrates above differential 
 equation, is an exponential function with the base 
 
 £ = 2.7182818 (57) 
 
 The solution of the differential equation, 
 
 l=» w 
 
 thus is, 
 
 J/ = ao^^ (59) 
 
 and the solution of the differential equation, 
 
 |-«^' ^60) 
 
 is, 
 
 J/ = ao^''^ (61) 
 
 where fflg is an integration constant. 
 
 The exponential function thus is one of the most common 
 functions met in electrical engineering problems. 
 
 The above described method of solving a problem, by assum- 
 ing a solution in a form containing a number of imknown 
 coefficients, Cg, a\, a-i. .., substituting the solution in the problem 
 and thereby determining the coefficients, is called the method 
 of indeterminate coefficients. It is one of the most convenient 
 
72 ENGINEERING MATHEMATICS. 
 
 and most frequently used methods of solving engineering 
 problems. 
 
 EXAMPLE 1. 
 
 54. In a 4-pole 500-volt 50-k\v. direct-current shunt motor, 
 the resistance of the field circuit, inclusive of field rheostat, is 
 250 ohms. Each field pole contains 4000 turns, and produces 
 at 500 volts impressed upon the field circuit, 8 megalines of 
 magnetic flux per pole. 
 
 What is the equation of the field current, and how much 
 time after closing the field switch is required for the field cur- 
 rent to reach 90 per cent of its final value? 
 
 Let r be the resistance of the field circuit, L the inductance 
 of the field circuit, and i the field current, then the voltage 
 consumed in resistance is, 
 
 fir = 'ri. 
 
 In general, in an electric circuit, the current produces a 
 magnetic field; that is, lines of magnetic flux surrounding the 
 conductor of the current; or, it is usually expressed, interlinked 
 with the current. This magnetic field changes with a change of 
 the current, and usually is proportional thereto. A change 
 of the magnetic field surrounding a conductor, however, gen- 
 erates an e.m.f. in the conductor, and this e.m.f. is proportional 
 to the rate of change of the magnetic field; hence, is pro- 
 portional to the rate of change of the current, or to 
 
 di . 
 
 -r, with a proportionality factor L, which is called the induct- 
 
 CLl' 
 
 ance of the circuit. This counter-generated e.m.f. is in oppo- 
 
 di 
 sition to the current, —L-t., and thus consumes an e.m.f., 
 
 di . . 
 +J-I-T,, which is called the e.m.f. consumed by self-inductance, 
 
 or inductance e.m.f. 
 
 Therefore, by the inductance, L, of the field circuit, a voltage 
 is consumed which is proportional to the rate of change of the 
 field current, thus, 
 
 ^ di 
 '^^^dt- 
 
POTENTIAL SERIES AND EXPONENTIAL FUNCTION. 73 
 
 Since the supply voltage, and thus the total voltage consumed 
 in the field circuit, is e = 500 volts, 
 
 . ^ di 
 e = n+Lj^; (62) 
 
 or, rearranged, 
 
 di e—ri 
 
 Substituting herein, 
 
 u = e-n; (63) 
 
 hence, 
 
 du di 
 
 gives, 
 
 dt '" 'dt' 
 
 du r 
 
 = --n (64) 
 
 This is the same differential equation as (39), with a= , 
 
 and therefore is integrated by the function, 
 
 u = ao£ L ; 
 therefore, resubstituting from (63), 
 
 e—ri = aQ£ ^ , 
 and 
 
 
 r r 
 
 This solution (65), still contains the unknown quantity ao; 
 or, the integration constant, and this is determined by know- 
 ing the current i for some particular value of the time t. 
 
 Before closing the field switch and thereby impressing the 
 voltage on the field, the field current obviously is zero. In the 
 moment of closing the field switch, the current thus is still 
 zero; that is, 
 
 i = for t = 0. . (66) 
 
74 ENGINEERING MATHEMATICS. 
 
 Substituting these values in (65) gives, 
 
 hence, 
 
 0= -; or ao = +e, 
 
 r r 
 
 ^ = 7(l-^"^') (67) 
 
 is the final solution of the differential equation (62); that is, 
 it is the value of the field current, i, as function of the time, t, 
 after closing the field switch. 
 
 After infinite time, i = oo, the current i assumes the final 
 value io, which is given by substituting i=oo into equation 
 (67), thus, 
 
 e 500 „ ,„ox 
 
 io = - = 7r,^ = 2 amperes; .... (58) 
 r /oU 
 
 hence, by substituting (68) into (67), this equation can also be 
 written, 
 
 i- = io(l-£"^') 
 
 = 2(l-rr'), (69) 
 
 where io = 2 is the final value assumed by the field current. 
 The time ti, after which the field current i has reached 90 
 per cent of its final value Iq, is given by substituting i = 0.9io 
 into (69), thus, 
 
 0.9to=?o(l-rr"), 
 and 
 
 e~^"=0.1. 
 
 Taking the logarithm of both sides, 
 r 
 
 r 
 
 J <ilog €=-1; 
 
 and 
 
 r log £ 
 
 «i=rT:^ (70) 
 
POTENTIAL SERIES AND EXPONENTIAL FUNCTION. 75 
 
 5S' The inductance L is calculated from the data given 
 in the problem. Inductance is measured by the number of 
 interlinkages of the electric circuit, with the magnetic flux 
 produced by one absolute unit of current in the circuit; that 
 is, it equals the product of magnetic flux and number of turns 
 divided by the absolute current. 
 
 A current of i^ = 2 amperes represents 0.2 absolute units, 
 since the absolute unit of current is 10 amperes. The number 
 of field turns per pole is 4000; hence, the total number of turns 
 71 = 4X4000 = 16,000. The magnetic flux at full excitation, 
 or tg = 0.2 absolute units of current, is given as <? = 8 X 10^ lines 
 of magnetic force. The inductance of the field thus is: 
 
 , n§ 16000X8X106 ^,^ ,„„ , , . „,„, 
 
 Z, =_^ = THs = 640 X 109 absolute units = 640/;., 
 
 To 0.2 ' 
 
 the practical unit of inductance, or the henry (A) being 10^ 
 absolute units. 
 
 Substituting L = 640 r = 250 and e = 500, into equation (67) 
 and (70) gives 
 
 i = 2(l- £-0-390, 
 
 and 
 
 ^^^ 250X(X4343 = ^-^^^^" ^^^^ 
 
 Therefore it takes about 6 sec. before the motor field has 
 reached 90 per cent of its final value. 
 
 The reader is advised to calculate and plot the numerical 
 values of i from equation (71), for 
 t = 0, 0.1, 0.2, 0.4, 0.6, 0.8, 1.0, 1.5, 2.0, 3, 4, 5, 6, 8, 10 sec. 
 
 This calculation is best made in the form of a table, thus; 
 
 and, 
 
 hence, 
 
 and, 
 
 logs =0.4343; 
 0.39nog, =0.1694<; 
 .-o='9' = Ar3ol694i. 
 
76 
 
 ENGINEERING MATHEMATICS. 
 
 The values of £-o-39' can also be taken directly from the 
 tables of the exponential function, at the end of the book. 
 
 I 
 
 0.1694( 
 
 — 0.1694( 
 
 ,-O.Wi 
 
 l_£-0.39i 
 
 1 = 
 
 -A'-U.1694( 
 
 0.0 
 0.1 
 0.2 
 0.4 
 0.6 
 0.8 
 etc. 
 
 
 
 0.0170 
 0.0339 
 0.0678 
 0.1016 
 0.1355 
 
 
 . 9830-1 
 0.9661-1 
 0.9322-1 
 0.8984-1 
 0.8645-1 
 
 1 
 
 0.962 
 0.925 
 0.855 
 0.791 
 0.732 
 
 
 0.038 
 0,075 
 0.145 
 0.209 
 0.268 
 
 
 0.076 
 0.150 
 0.290 
 0.418 
 0.536 
 
 
 
 
 
 
 
 
 
 
 
 EXAMPLE 2. 
 
 56. A condenser of 20 mf. capacity, is charged to a potential 
 of Cq = 10,000 volts, and tlien discharges through a resistance 
 of 2 megohms. What is the equation of the discharge current, 
 
 and after how long a time has 
 the voltage at the condenser 
 dropped to 0.1 its initial value? 
 A condenser acts as a reser- 
 voir of electric energy', similar 
 to a tank as water reservoir. 
 If in a water tank, Fig. 27, A 
 is the sectional area of the tank, 
 e, the height of water, or water 
 pressure, and water flows out 
 of the tank, then the height e 
 decreases by the flow of water; 
 that is the tank empties, and 
 the current of water, i, is proportional to the change of the 
 
 de 
 water level or height of water, — , and to the area A of the 
 
 at 
 
 tank; that is, it is, 
 
 Fig. 27. Water Reservoir. 
 
 de 
 
 (72) 
 
 The minus sign stands on the right-hand side, as for positive 
 i; that is, out-flow, the height of the water decreases; that is, 
 de is negative. 
 
POTENTIAL SERIES AND EXPONENTIAL FUNCTION. 77 
 
 In an electric reservoir, the electric pressure or voltage e 
 corresponds to the water pressure or height of the water, and 
 to the storage capacity or sectional area A of the water tank 
 corresponds the electric storage capacity of the condenser, 
 called capacity C. The current or flow out of an electric 
 condenser, thus is, 
 
 --4: ^''^ 
 
 The capacity of condenser is, 
 
 C = 20 mf = 20XlO-6 farads. 
 
 The resistance of the discharge path is, 
 
 r =2X106 ohms; 
 
 hence, the current taken by the resistance, r, is 
 
 . e 
 r 
 and thus 
 
 ,de e 
 
 ^'dt^r' 
 
 and 
 
 de 
 
 dt Cr ''■ 
 
 Therefore, from (60) (61), 
 
 and for t = 0, e = eo = 10,000 volts; hence 
 
 e = ao£ ^'•j 
 
 and 
 
 0.1 of the initial value: 
 is reached at : 
 
 10,000 = ao, 
 t_ 
 
 = 10,000£-°-°25' volts; . . (74) 
 
 e = 0.1eQ, 
 
 ii=r^ = 92sec (75) 
 
 log £ 
 
78 ENGINEERING MATHEMATICS. 
 
 The reader is advised to calculate and plot the numerical 
 values of e, from equation (74), for 
 
 t = 0; 2; 4; 6; 8; 10; 15; 20; 30; 40; CO; 80; 100; 150; 200 sec. 
 
 57. Wherever in an electric circuit, in addition to resistance, 
 inductance and capacity both occur, the relations between 
 currents and voltages lead to an equation containing the second 
 differential coefficient, as discussed above. 
 
 The simplest form of such equation is: 
 
 S-^ (^«) 
 
 To integrate this by the method of indeterminate coefficients, 
 we assume as solution of the equation (76) the infinite series, 
 
 y = ai)+aix+a2X^+a'iX^+a4,x* + (77) 
 
 in which the coefficients ao, ai, a2, 0,3, 0,4,. . . are indeterminate. 
 Differentiating (77) twice, gives 
 
 ^ = 2a2+2x3a3X+3x4a4a;2+4x5a5x3 + ... , . (78) 
 
 and substituting (77) and (78) into (76) gives the identity, 
 
 2a2+2x3a3X +3XiaiX^ +iX5asx^ + . . . 
 
 =a{ao+aix+a2X^ + a3X^ + . . .); 
 
 or, arranged in order of x, 
 
 (2a2 — aao)+x(2X3a3 — aai)+2;2(3X4a4— 002) 
 
 + x3(4x5as-aa3)+. . .=0 (79) 
 
 Since this equation (79) is an identity, the coefficients of 
 all powers of x must individually equal zero. This gives for 
 the determination of these hitherto indeterminate coefficients 
 the equations, 
 
 2a2 — aao = 0; 
 2x3a3 — aai=0; 
 3x4a4— aa2 = 0; 
 4 X Sas — aas = 0, etc. 
 
POTENTIAL SERIES AND EXPONENTIAL FUNCTION. 79 
 Therefore 
 
 aa2 Oofl^ aar, aio^ 
 
 °-t=i7m=-n-', «5=- 
 
 3X4 
 
 4 
 
 aui 
 
 aoa^ 
 
 5X6 
 
 6 
 
 aae 
 
 aptt* 
 
 4X5 |5 ' 
 
 005 aia^ 
 
 '6X7^17~' 
 
 aay aia^ 
 
 "^~7X8" |8 ' ""' 8X9 
 
 etc., etc. 
 
 Substituting these values in (77), 
 
 f ax^ a?x^ a?z^ 
 2/ = ao|l+^+Y-+-jg- + ... 
 
 r ax^ a^x^ a^x'' 1 
 
 +01X+— +—- + -- +.... (80) 
 I J-J P |7 J 
 
 In this case, two coefficients ao and a\ thus remain inde- 
 terminate, as was to be expected, as a differential equation 
 of second order must have two integration constants in its 
 most general form of solution. 
 
 Substituting into this equation, 
 
 62= a; 
 that is, 
 
 6 = \/a, (81) 
 
 S=+^^^' ^^^^ 
 
 and 
 
 62x2 6*X4 66j6 
 
 j/=ao|l+^+^+-|g- + . 
 
 ^t 
 
 63j;3 ?,5^5 67^7 
 
 6x4-73-+-^+^ + . .4. (83) 
 
80 ENGINEERING MATHEMATICS. 
 
 In this case, instead of the integration constants Oo and Oi, 
 the two new integration constants A and B cart be introduced 
 by the equations 
 
 ao = A+B and -^ = A-B; 
 hence, ^^ a^ 
 
 A=— ^ — and -8=— ^ — , 
 and, substituting these into equation (83), gives, 
 
 , r , 62x2 53x3 ¥x* 
 y=A I ^+bx+^+-.Y+~nr + - ■ ■ 
 
 1 7 ?''^' ^""^^ ^^^ 1 
 
 +5|i-5x+^2:-i3:+Tr-+---|- • (^4) 
 
 The first series, however, from (56), for n = h is e^^'^, and 
 the second series from (56), for n= —5 is e~^'^. 
 Therefore, the infinite series (83) is, 
 
 y^Ae + '^^+Bi-^''; (85) 
 
 that is, it is the sum of two exponential functions, the one with 
 a positive, the other with a negative exponent. 
 Hence, the difTerential equation, 
 
 di-^y' (76) 
 
 is integrated by the function, 
 
 y = A£ + *^+B£-'^, (86) 
 
 where, 
 
 b=\/a. . . . (87) 
 
 However, if a is a negative quantity, h = Va is imaginary, 
 and can be represented by 
 
 i = ic, (88) 
 
 where 
 
 c^=-a (89) 
 
 In this case, equation (86) assumes the form, 
 
 2/ = 4£+k^+5j-kx. ^QQ) 
 
POTENTIAL SERIES AND EXPONENTIAL FUNCTION. 81 
 
 that is, if in the differential equation (76) a is a positive quantity, 
 = +b^, this differential equation is integrated by the sum of 
 the two exponential functions (86) ; if, however, a is a negative 
 quantity, = — c^, the solution (86) appears in the form of exponen- 
 tial functions with imaginary exponents (90). 
 
 58. In the latter case, a form of the solution of differential 
 equation (76) can be derived which does not contain the 
 imaginary appearance, by turning back to equation (80), and 
 substituting therein a=—c^, which gives, 
 
 d'y 2 
 
 (91) 
 
 y = ao\ 1 
 
 fiA yZ 
 
 A 7-4 ,^6 r^ 
 
 
 
 or, writing A=aQ and B = 
 
 ai 
 
 y = A 1 
 
 ^2r2 /.4r4 
 
 C^X^ C^X* C°X' 
 
 .676 
 
 -+- 
 
 I 
 
 ■ + ■ 
 
 + B\ CX-— r-+-p- 
 
 3 
 
 + . . . (92) 
 
 The solution then is given by the sum of two infinite series, 
 thus. 
 
 and 
 
 w(cx) = l- 
 
 ^4r4 /.6,6 
 
 c-x~ C^X* C°X_ 
 ■^^"^H 16" 
 
 + ■ 
 
 v{cx)=cx- 
 
 f3.r3 g5j-5 
 
 + . 
 
 • (93) 
 
 as 
 
 y = Au{cx) +Bv{cx). 
 
 (94) 
 
 In the w-series, a change of the sign of x does not change 
 the value of u, 
 
 u{ — cx)=u{+cx) (95) 
 
 Such a function is called an even function. 
 
82 ENGINEERING MATHEMATICS. 
 
 In the ?;-series, a change of the sign of x reverses the sign 
 of V, as seen from (93): 
 
 v{ — cx) = —v{+cx) (96) 
 
 Such a function is called an odd function. 
 It can be shown that 
 
 u{cx)=Qos,cx and v{cx) = sva ex; . . . (97) 
 
 hence, 
 
 t/ = A cos cx+i? sin ex, (98) 
 
 where A and B are the integration constants, which are to be 
 determined by the terminal conditions of the physical problem. 
 Therefore, the solution of the differential equation 
 
 J-^y' (99) 
 
 has two different forms, an exponential and a trigonometric. 
 If a is positive , 
 
 ^2=+b'y, . ... (100) 
 
 it is: 
 
 y = Ae + ^='+Be-^'', .... (101) 
 
 If a is negative, 
 
 dt2--c'y, (102) 
 
 it is: 
 
 2/ = ^ cos ex +5 sin ex (103) 
 
 In the latter case, the solution (101) woukl appear as ex- 
 ponential function with imaginary exponents; 
 
 2/ = ^£+'^^+££-'" (104) 
 
 As (104) obviously must be the same function as (103), it 
 follows that exponential functions with imaginary exponents 
 must be expressible by trigonometric functions. 
 
POTENTIAL SERIES AND EXPONENTIAL FUNCTION. 83 
 
 59. The exponential functions and the trigonometric func- 
 tions, according to the preceding discussion, are expressed by 
 the infinite series, 
 
 , X2 j;4 ^6 
 
 cosx = l-2+[4-|g+-. . 
 
 /T^3 'Y-O ■v7 
 
 sin x = x—T^ +7^ — 77=- + — . . . 
 I I I 
 
 Therefore, substituting fu for x, 
 
 (105) 
 
 „-u = i+y^___j_.+_+j____j_ 
 
 ,, m2 M* U^ 
 
 W*^ M'' U' 
 
 However, the first part of this series is cos u, the latter part 
 sin u, by (105) ; that is, 
 
 £'" = cos u+j sm u. 
 
 Substituting —u for +u gives, 
 
 £-j« = cos u~j sin w. 
 Combining (106) and (107) gives. 
 
 cos M = ■ 
 
 and 
 
 c + )"_ r— J" 
 
 sm M = - 
 
 2j 
 
 Substituting in (106) to (108), jv for w, gives, 
 £'"' = cos jv+j sm jv, ] 
 £+^ = cos p — /sin jV. J 
 
 and. 
 
 (106) 
 (107) 
 
 (108) 
 
 (109) 
 
84 ENGINEERING MATHEMATICS. 
 
 Adding and subtracting gives respectively, 
 
 COS p = :^ 
 
 and 
 
 sin iv = - 
 
 '>i 
 
 (110) 
 
 By these equations, (106) to (110), exponential functions 
 with imaginary exponents can be transformed into trigono- 
 metric functions with real angles, and exponential functions 
 with real exponents into trignometric functions with imaginary 
 angles, and inversely. 
 
 Mathematically, the trigonometric functions thus do not 
 constitute a separate class of functions, but may be considered 
 as exponential functions with imaginary angles, and it can be 
 said broadly that the solution of the above differential equa- 
 tions is given by the exponential function, but that in this 
 function the exponent may be real, or may be imaginary, and 
 in the latter case, the expression is put into real form by intro- 
 ducing the trigonometric functions. 
 
 EXAMPLE 1. 
 
 6o. A condenser (as an underground high-potential, cable) 
 of 20 mf. capacity, and of a voltage of eo = 10,000, discharges 
 through an inductance of 50 mh. and of negligible resistance, 
 "What is the equation of the discharge current? 
 
 The current consumed by a condenser of capacity C and 
 potential difference e is proportional to the rate of change 
 of the potential difference, and to the capacity; hence, it is 
 
 dt' 
 current, is 
 
 C-;^, and the current from the condenser; or its discharge 
 
 '-4: ("" 
 
 The voltage consumed by an inductance L is proportional 
 to the rate of change of the current in the inductance, and to the 
 iniluctance ; hence, 
 
 ^=4 ^"2) 
 
POTENTIAL SERIES AND EXPONENTIAL FUNCTION. 85 
 Differentiating (112) gives, 
 
 de d^i 
 dt~ dfi: 
 
 
 and substituting this into (111) gives, 
 
 
 ^^dH dH 
 
 1 . 
 
 . . . (113) 
 as the differential equation of the problem. 
 
 This equation (113) is the same as (102), for c^ = 7tt, thus 
 is solved by the expression, 
 
 i^Acos — ^+-5 sin , — . . . . (114) 
 VLC VLC 
 
 and the potential difference at the condenser or at the inductance 
 is, by substituting (114) into (112), 
 
 e = 
 
 ^/ttIjScos — =^ — Asin — = i . . (115) 
 
 These equations (114) and (115) still contain two unknown 
 constants, A and B, which have to be determined by the terminal 
 conditions, that is, by the known conditions of current and 
 voltage at some particular time. 
 
 At the moment of starting the discharge; or, at the time 
 i = 0, the current is zero, and the voltage is that to which the 
 condenser is charged, that is, i = 0, and e = eo. 
 
 Substituting these values in equations (114) and (115) 
 gives, 
 
 = A and eg=^hTB; 
 
 hence 
 
 IC 
 
 B — Sq'^j J . 
 
 and, substituting for A and B the values in (114) and (115), 
 gives 
 
 jC . t 
 
 and 
 
 e-eocos^ 
 
 (116) 
 
86 ENGINEERING MATHEMATICS. 
 
 Substituting the numerical values, Cq = IOjOOO volts, C = 20 
 mf. = 20X10- 6 farads, L = 50 mh.=0.05h. gives, 
 
 ^l 
 
 ^ = 0.02 and VCL = 10-^; 
 
 hence, 
 
 t = 20G sin 1000 t and e = 10,000 cos 1000 t. 
 
 6i. The discharge thus is alternating. In reality, due to 
 the unavoidable resistance in the discharge path, the alterna- 
 tions gradually die out, that is, the discharge is oscillating. 
 
 The time of one complete period is given by, 
 
 27r 
 1000^0 =2;r; or, to=j^- 
 
 Hence the frenquency, 
 
 . 1 1000 _„ , 
 
 /= — = -^ — = 159 cycles per second. 
 
 As the circuit in addition to the inductance necessarily 
 contains resistance r, besides the voltage consumed by the 
 inductance by equation (112), voltage is consumed by the 
 resistance, thus 
 
 er-ri, (117) 
 
 and the total voltage consumed by resistance r and inductance 
 
 L, thus is 
 
 . T di 
 e = n+Lj^ (118) 
 
 Differentiating (118) gives, 
 
 de di ^ dH 
 
 jr'dt+^wr ^1^9) 
 
 and, substituting this into equation (111), gives, 
 
 ■ ^ di ^^ dH „ 
 '-^^'dt+^^dt^-'^' (120) 
 
 as the differential equation of the problem. 
 
 This differential equation is of the more general form, (30), 
 62. The more general differential equation (30). 
 
 d^y „ dy 
 
 J+2c/^ + ay + b^0, (121) 
 
POTENTIAL SERIES AND EXPONENTIAL FUNCTION. 87 
 can, by substituting, 
 
 which gives 
 
 y+- = ^, (122) 
 
 dy dz 
 dx dx' 
 
 be transformed into the somewhat simpler form, 
 
 S-^4:+--o (12^) 
 
 It may also be solved by the method of indeterminate 
 coefficients, by substituting for z an infinite series of powers of 
 X, and determining thereby the coefficients of the series. 
 
 As, however, the simpler forms of this equation were solved 
 by exponential functions, the applicabilitj' of the exponential 
 functions to this equation (123) may be directly tried, by the 
 method. of indeterminate coefficients. That is, assume as solu- 
 tion an exponential function, 
 
 z = As>'^, (124) 
 
 where A and b are unknown constants. Substituting (124) 
 into (123), if such values of A and b can be found, which make 
 the substitution product an identity, (124) is a solution of 
 the differential equation (123). 
 From (124) it follows that, 
 
 ^ = 6A£*- and ^^=bUe^, . . . (125) 
 
 and substituting (124) and (125) into (123), gives, 
 
 A£'>^{b^+2cb+a]=0 (126) 
 
 As seen, this equation is satisfied for every value of x, that 
 is, it is an identity, if the parenthesis is zero, thus, 
 
 b2+2cb+a = 0, (127) 
 
 and the value of b, calculated by the quadratic equation (127), 
 thus makes (124) a solution of (123), and leaves A still undeter- 
 mined, as integration constant. 
 
88 ENGINEERING MATHEMATICS. 
 
 From (127), 
 
 or, substituting, 
 
 Vc^-a = p, (128) 
 
 into (128), the equation becomes, 
 
 b=-c±p. ... . (129) 
 
 Hence, two values of b exist, 
 
 bi=—c + p and ?)2=— c— p, 
 and, therefore, the differential equation, 
 
 S+2.|+a.^0, . . . (130) 
 
 is solved by Ae^'''; or, by Ae''™, or, by any combination of 
 these two solutions. The most general solution is, 
 
 that is, 
 
 y = ^,j(-C + p)x_|_^„j.(-6-p)x 
 
 (J/ 
 
 As roots of a quadratic equation, bi and 62 may both be 
 real quantities, or may be complex imaginary, and in the 
 latter case, the solution (131) appears in imaginary form, and 
 has to be reduced or modified for use, so as to eliminate the 
 imaginary appearance, by the relations (106) and (107). 
 
 EXAMPLE 2. 
 
 63. Assume, in the example in paragraph 60, the discharge 
 
 circuit of the condenser of C = 20 mf. capacity, to contain, 
 
 besides the inductance, L = 0.0.5 h, the resistance, r=125 ohms. 
 
 The general equation of the problem, (120), dividing by 
 
 C L, becomes, 
 
 dH r dl i 
 
 dP^Ldt-'CL^'^ ^.132) 
 
POTENTIAL SERIES AND EXPONENTIAL FUNCTION. 89 
 This is the equation (123), for: 
 
 x = <, 2c=-7=2500: 
 1 
 
 z = i, 
 
 a = 
 
 CL 
 
 = 106 
 
 If 
 
 p = Vc^ — a, then 
 P 
 
 •2L CL 
 
 ~2L\ ^C 
 
 and, writing 
 
 ^F^ 
 
 4L 
 C" 
 
 and since 
 
 '2L' 
 
 ^ = 10 and ^ = 2500, 
 
 s = 75 and p = 750. 
 The equation of the current from (131) then is. 
 
 ■2L 
 
 -ll 
 
 . + 2l' 
 
 + Aas' 
 
 2L 
 
 (133) 
 
 (134) 
 
 (135) 
 
 (136) 
 
 (137) 
 
 This equation still contains two unknown quantities, the inte- 
 gration constants Ai and A2, which are determined by the 
 terminal condition: The values of current and of voltage at the 
 beginning of the discharge, or t = 0. 
 
 This requires the determination of the equation of the 
 voltage at the condenser terminals. This obviously is the voltage 
 consumed by resistance and inductance, and is expressed by 
 equation (118), 
 
 e = ri + L 
 
 dt' 
 
 (118) 
 
90 ENGINEERING MATHEMATICS. 
 
 di 
 hence, substituting herein the value of i and -j:, from equation 
 
 (137), gives 
 
 r + s ^ -'l^'t r-s ^ -"-±^1 
 = ^-^i£ 2L +-2-A2S 2Z, 
 
 = £ 2l' 
 
 '-±^A,e^^U'-^A,r^L I (138) 
 
 and, substituting the numerical values (133) and (136) into 
 equations (137) and (138), gives 
 
 and, 
 
 e = 100Ai£-5oo'+25A2£-20oo' 
 
 .(139) 
 
 At the moment of the beginning of the discharge, t = 0, 
 the current is zero and the voltage is 10,000; that is, 
 
 t = 0; 1 = 0; 6 = 10,000 . . .(140) 
 
 Substituting (140) into (139) gives, 
 
 = yli+A2, 10,000 = 100^1+25^2; 
 hence, 
 
 A2=-Ai; ^1 = 133.3; ^2= -133.3. . . (141) 
 
 Therefore, the current and voltage are, 
 
 i = 133.3{£-5o<«-£-2ooo'j, j 
 e = 13,333£-5°'"-3333s-2ooo' J 
 
 (142) 
 
 The reader is advised to calculate and plot the numerical 
 values of i and e, and of their two components, for. 
 
 t = 0, 0.2, 0.4, 0.6, 1, 1.2, 1.5, 2, 2.5, 3, 4, 5, BxlO'^ 
 
 sec. 
 
POTENTIAL SERIES AND EXPONENTIAL FUNCTION. 91 
 
 64. Assuming, however, that the resistance of the discharge 
 circuit is only r = 80 ohms (instead of 125 ohms, as assumed 
 above) : 
 
 4L. 
 r^— -77 in equation (134) then becomes —3600, and there- 
 fore: 
 
 s= V-3600 = 60 V"^ = 60j, 
 and 
 
 P=^=6ooy. 
 
 The equation of the current (137) thus appears in imaginary 
 form, 
 
 i=£-800'jAl£ + 600j«^_^2£-600''|. . . . (143) 
 
 The same is also true of the equation of voltage. 
 
 As it is obvious, however, physically, that a real current 
 must be coexistent with a real e.m.f., it follows that this 
 imaginary form of the expression of current and voltage is only 
 apparent, and that in reality, by substituting for the exponential 
 functions with imaginary exponents their trigononetric expres- 
 sions, the imaginary terms must eliminate, and the equation 
 (143) appear in real form. 
 
 According to equations (106) and (107), 
 
 £+600j' = cos QOOt+j sin QOOt; 1 
 
 (144) 
 
 £-6ooj( = cos 600f-j sin 600i. J 
 
 Substituting (144) into (143) gives, 
 
 i = e-^^°*\BicosQ00t+B2sm600t}, . . (145) 
 
 where Bi and B2 are combinations of the previous integration 
 constants Ai and A2 thus, 
 
 Bi = Ai+A2, and Bi^jiAi-Az). . (146) 
 
 By substituting the numerical values, the condenser e.m.f., 
 given by equation (138), then becomes, 
 
 e = £-8oo(j (40-l-30j)Ai(cos OOOf-f/sin 6000 
 
 -I- (40 - 30j)^2(cos 600i- / sin 6000 ! 
 = £-8oo«j (405i +3052)cos 600t + (40^2 -305i) sin 600i}. (147) 
 
92 ENGINEERING MATHEMATICS. 
 
 Since for i=0, i = and e = 10,000 volts (140), substituting 
 into (145) and (147), 
 
 = Bi and 10,000 = 40 5i+30 Bz. 
 Therefore, Bi = and 52 = 333 and, by (145) and (147), 
 
 i = 333£-8oo'sin600i; 1 
 
 . . (148) 
 e = 10,000£-80o' (cos 600 t + 1.33 sin GOOO-J 
 
 As seen, in this case the current i is larger, and current 
 and e.m.f. are the product of an exponential term (gradually 
 decreasing value) and a trigonometric term (alternating value) ; 
 that is, they consist of successive alternations of gradually 
 decreasing amplitude. Such functions are called oscillating 
 functions. Practically all disturbances in electric circuits 
 consist of such oscillating currents and voltages. 
 
 600^ = 277- gives, as the time of one complete period, 
 
 ^ = 1^ = 0.0105 sec; 
 bOO 
 
 and the frequency is 
 
 /=-m =95.3 cycles per sec. 
 
 In this particular case, as the resistance is relatively high, 
 the oscillations die out rather rapidly. 
 
 The reader is advised to calculate and plot the numerical 
 values of i and e, and of their exponential terms, for every 30 
 
 T T T 
 
 degrees, that is, for t = 0, t-^, 2 y^, 3 t^, etc., for the first two 
 
 periods, and also to derive the equations, and calculate and plot 
 the numerical values, for the same capacity, C = 20 mf., and 
 same inductance, L = 0.05h, but for the much lower resistance, 
 r = 20 ohms. 
 
 65. Tables of s"'"^ and e~^, for 5 decimals, and tables of 
 log £+'' and log £~^, for 6 decimals, are given at the end of 
 the book, and also a table of £"* for 3 decimals. For most 
 engineering purposes the latter is sufficient; where a higher 
 accuracy is required, the 5 decimal table may be used, and for 
 
POTENTIAL SERIES AND EXPONENTIAL FUNCTION. 93 
 
 highest accuracy interpolation by the logarithmic table may be 
 employed. For instance, 
 
 J — 13.6847 _? 
 
 From the logarithmic table, 
 
 log £-10 =5.657055, 
 log £-3 =8.697117, 
 log £-0-6 =9.739423, 
 log £-008 =9.965256, 
 log £-00047 = 9 997959^ 
 
 r interpolated, 
 between log £-0004 =9.998263, 
 added I and log £-0005 =9.997829), 
 
 log £-i3-8«*^ = 4.056810 =0.056810-6. 
 From common logarithmic tables, 
 
 j-13.6847^ 113975x10-6. 
 
 Note. In mathematics, for the base of the natural loga- 
 rithms, 2.718282 . . . , is usually chosen the symbol e. Since, 
 however, in engineering the symbol e is universally used to 
 represent voltage, for the base of natural logarithms has been 
 chosen the symbol £, as the Greek letter corresponding to e, 
 and £ is generally used in electrical engineering calculations in 
 this meaning. 
 
CHAPTER III. 
 
 TRIGONOMETRIC SERIES. 
 
 A. TRIGONOMETRIC FUNCTIONS. 
 
 66. For the engineer, and especially the electrical engineer, 
 a perfect familiarity with the trigonometric functions and 
 trigonometric formulas is almost as essential as familiarity with 
 the multiplication table. To use trigonometric methods 
 efficiently, it is not sufficient to understand trigonometric 
 formulas enough to be able to look them up when required, 
 but they must be learned by heart, and in both directions; that 
 is, an expression similar to the left side of a trigonometric for- 
 mula must immediately recall the right side, and an expression 
 similar to the right side must immediately recall the left side 
 of the formula. 
 
 Trigonometric functions are defined on the circle, and on 
 the right triangle. 
 
 Let in the circle. Fig. 28, the direction to the right and 
 upward be considered as positive, to the left and downward as 
 negative, and the angle a be counted from the positive hori- 
 zontal OA, counterclockwise as positive, clockwise as negative. 
 
 The projector s of the angle a, divided by the radius, is 
 called sin a; the projection c of the angle a, divided by the 
 radius, is called cos a. 
 
 The intercept t on the vertical tangent at the origin A, 
 divided by the radius, is calletl tan a; the intercept ct on the 
 horizontal tangent at B, or 90 deg., behmd A, divided by the 
 radius, is called cot a. 
 
 Thus, in Fig. 28, 
 
 s c 
 
 sma=-; cosa = -; 
 r r 
 
 , t ct 
 
 tana = -; cot a = — . 
 
 (1) 
 
 94 
 
TRIGONOMETRIC SERIES. 
 
 95 
 
 In the right triangle, Fig. 29, with the angles a and /3, 
 opposite respectively to the cathetes a and b, and with the 
 hypotenuse c, the trigonometric functions are : 
 
 sin a = cos /5 = 
 
 tan a = cot , 
 
 a . „ ?> 
 
 = -; cos a=sin B = — 
 
 c ^ c 
 
 a , „ b 
 
 = T-; cota=tanS = — . 
 
 a 
 
 (2) 
 
 By the right triangle, only functions of angles up to 90 deg., 
 
 or — , can be defined, while by the circle the trigonometric 
 
 functions of any angle are given. Both representations thus 
 must be so famihar to the engineer that he can see the trigo- 
 
 FiG. 28. Circular Trigonometric 
 Functions. 
 
 Fig. 29. Triangular Trigono- 
 metric Functions. 
 
 nometric functions and their variations with a change of the 
 angle, and in most cases their numerical values, from the 
 mental picture of the diagram. 
 
 67. Signs of Functions. In the first quadrant, Fig. 28, all 
 trigonometric functions are positive. 
 
 In the second quadrant. Fig. 30, the sin a is still positive, 
 as s is in the upward direction, but cos a is negative, since c 
 is toward the left, and tan a and cot a also are negative, as t 
 is downward, and d toward the left. 
 
 In the third quadrant, Fig. 31, sin a and cos « are both 
 
96 
 
 ENGINEERING MATHEMATICS. 
 
 negative: s being downward, c toward the left; but tan a and 
 cot a are again positive, as seen from t and ct in Fig. 31. 
 
 
 
 + 
 
 
 \ 
 
 ct 
 
 
 B 
 
 
 A 
 
 K 
 
 
 _o: \ 
 
 
 / 
 
 s ^ 
 
 \ 
 
 \ ^ 
 
 A 1 
 
 \ 
 
 c 
 
 
 
 
 t 
 
 Fig. 30. Second Quadrant. 
 
 Fig. 31. Third Quadrant. 
 
 In the fourth quadrant. Fig. 32, sin a is negative, as s is 
 downward, but cos a is again positive, as c is toward the right; 
 
 tan a and cot a are both 
 negative, as seen from t and 
 ct in Fig. 32. 
 
 In the fifth quadrant all 
 the trigonometric functions 
 again have the same values 
 as in the first quadrant. Fig. 
 28, that is, 360 deg., or 27z, 
 or a multiple thereof, can be 
 added to, or subtracted from 
 the angle a, without changing 
 the trigonometric functions, 
 but these functions repeat 
 after every 360 deg., or 2n; 
 
 \ ct 
 
 B 
 
 
 
 >c 
 
 1 c 
 
 A , 
 
 1 ( 
 
 
 
 t 
 
 Fig. 32. Fourth Quadrant. 
 
 that is, have 2?: or 360 deg. as their period. 
 
 SIGNS OF FUNCTIONS 
 
 Function. 
 
 Positive. 
 
 Negative. 
 
 sin a 
 COS a 
 tan a 
 cot a 
 
 1st and 2d 
 1st and 4th 
 1st and 3d 
 1st and 3d 
 
 3d and 4th quadrant 
 2d and 3d 
 2d and 4th 
 2d and 4th 
 
 (3) 
 
TRIGONOMETRIC SERIES. 
 
 97 
 
 68. Relations between sin a and cos «. Between sin a and 
 cos a the relation, 
 
 sin^ a+cos^ a = l, 
 exists; hence, 
 
 sin 
 
 .a = Vl- 
 
 cos^ a ; 
 
 cos a = Vl — sin^ a. 
 
 (4) 
 (4a) 
 
 Equation (4) is one of those which is frequently used in 
 both directions. For instance, 1 may be substituted for the 
 sum of the squares of sin a and cos a, while in other cases 
 sin^ a +cos2 a may be substituted for 1. For instance. 
 
 sin^a + cos^a /sinaX^ 
 
 I aV 
 
 cos^a cos^ a Vcosa/ 
 
 Relations between Sines and Tangents. 
 
 + I=tan2a + 1. 
 
 hence 
 
 tan a = 
 
 cot a = 
 
 cot a = 
 tana = 
 
 sm a 
 
 cos a ' 
 cos a 
 
 sin a ' 
 
 tana' 
 1 
 
 (5) 
 
 (5a) 
 
 cot «■ 
 
 As tan a and cot a are far less convenient for trigonometric 
 calculations than sin a and cos a, and therefore are less fre- 
 quently applied in trigonometric calculations, it is not neces- 
 sary to memorize the trigonometric formulas pertaining to 
 tan a and cot a, but where these functions occur, sin a and 
 and cos a are substituted for them by equations (5), and the 
 calculations carried out with the latter functions, and tan a 
 or cot a resubstituted in the final result, if the latter contains 
 
 sin a . . 
 
 , or its reciprocal. 
 
 cos a 
 
 In electrical engineering tan a or cot a frequently appears 
 
 as the starting-point of calculation of the phase of alternating 
 
 currents. For instance, if a is the phase angle of a vector 
 
98 ENGINEERING MATHEMATICS. 
 
 quantity, tan a is given as the ratio of tlie vertical component 
 
 over the horizontal component, or of the reactive component 
 
 over the power component. 
 
 In this case, if 
 
 a 
 tan « = T-, 
 
 sin « = —==, and coso:=-^ „ ; • (5&) 
 
 or, if 
 
 c 
 
 cot « =T) 
 
 sin «=—=;=, and cos a =—7===. . . (5c) 
 
 The secant functions, and versed sine functions are so 
 little used in engineering, that they are of interest only as 
 curiosities. They are defined by the following equations : 
 
 1 
 
 sec a = 
 
 cos a 
 1 
 
 cosec a- = 
 
 sm a 
 
 sin vers a = 1 — sin a, 
 
 cos vers a = 1 — cos a. 
 
 69. Negative Angles. From the circle diagram of the 
 trigonometric functions follows, as shown in Fig. 33, that when 
 changing from a positive angle, that is, counterclockwise 
 rotation, to a negative angle, that is, clockwise rotation, s, t, 
 and ct reverse their direction, but c remains the same; that is. 
 
 sin (— a) = —sin a, 
 cos (— a)= +COS a, 
 tan (— «)= —tan a. 
 cot (— «) = —cot a, ■ 
 
 (6) 
 
 cos a thus is an " even function," while the three others are 
 " odd functions." 
 
TRIGONOMETRIC SERIES. 
 
 99 
 
 Supplementary Angles. From the circle diagram of the 
 trigonometric functions follows, as shown in Fig. 34, that by 
 changing from an angle to its supplementary angle, s remains 
 in the same direction, but c, t, and ct reverse their direction, 
 and all four quantities retain the same numerical values, thus, 
 
 sin {n—a) = +sin a, -\ 
 cos (tt— a) = — cos a, 
 tan {K—a) = — tan a , 
 cot (tt— a) = — cot a. 
 
 (7) 
 
 Fig. 33. Functions of Negative 
 Angles. 
 
 Fig. 34. Functions of Supplementary 
 Angles. 
 
 Complementary Angles. Changing from an angle a to its 
 
 complementary angle 90° — a, or ^— a, as seen from Fig. 35, 
 
 the signs remain the same, but s and c, and also t and ct exchange 
 their numerical values, thus. 
 
 sin(|- 
 
 -aj= cos a-. 
 
 cos(|- 
 
 -a| =sin a, 
 
 tan(|- 
 
 -a) =C0t a, 
 
 cot(|- 
 
 -a )=tan a. 
 
 (8) 
 
100 
 
 ENGINEERING MATHEMATICS. 
 
 70. Angle («±7r). Adding, or subtracting n to an angle a, 
 gives the same numerical values of the trigonometric functions 
 
 / 
 
 + 
 
 B ct 
 
 
 ^ 
 
 1 P-^X^' 
 
 c (AY 
 
 \ 
 
 t 
 
 1 
 
 \ '* /^ 
 
 / c 
 
 '(x-n 
 
 1 
 
 A ' 
 
 Fig. 35. Functions of Complemen- Fig. 36. Functions of Angles Plus 
 tary Angles. or Minus tt. 
 
 as a, as seen in Fig. 36, but the direction of s and c is reversed, 
 
 while t and ct remain in the same direction, thus, 
 
 sin {a±7t) = —sin a, ] 
 
 cos {a±7v) = —cos a, 
 tan (a ±;r) = +tan a, 
 cot (a ±7i) = +cot «. . 
 
 (9) 
 
 FiG. 37. Functions of Angles +-H-- Fig. 38. Functions of Angles Minus —. 
 
 <x±7^j- Adding ~, or 90 dcg. to an angle a, inter- 
 changes the functions, s and c, and t and ct, and also reverses 
 
TRIGONOMETRIC SERIES. 
 
 101 
 
 the direction of the cosine, tangent, and cotangent, but leaves 
 the sine in the same direction, since the sine is positive in the 
 second quadrant, as seen in Fig. 37. 
 
 Subtracting — , or 90 deg. from angle a, interchanges the 
 
 functions, s and c, and t and ct, and also reverses the direction, 
 except that of the cosine, which remains in the same direction; 
 that is, of the same sign, as the cosine is positive in the first 
 and fourth quadrant, as seen in Fig. 38. Therefore, 
 
 = +C0S a, 
 
 
 / -\ 
 
 sin 1 
 
 
 cos 
 
 V -V 
 
 tan 
 
 
 
 / 7l\ 
 
 cot 
 
 ("+2) 
 
 sin 
 
 («-l) 
 
 
 / '^\ 
 
 cos 
 
 «-.y 
 
 
 \ -/ 
 
 tan 
 
 H) 
 
 cot 
 
 H) 
 
 (10) 
 
 . . . (11) 
 
 = — sm «, 
 
 = —cot a, 
 
 = —tan a, 
 = —cos a, 
 
 = +sin a, 
 
 = —cot a, 
 
 = —tan a. 
 
 Numerical Values. From the circle diagram. Fig. 28, etc. 
 follows the numerical values : 
 
 cot 0° = c» 
 cot 45°= 1 
 cot 90° = 
 
 cot 135°=- 1 j. . (12) 
 etc. 
 
 sin 
 
 0° = 
 
 = 
 
 sin 
 
 30° = 
 
 = i 
 
 sin 
 
 45° = 
 
 = i\ 2 
 
 sin 
 
 60° = 
 
 = iV 3 
 
 sin 
 
 90° = 
 
 = 1 
 
 sin 
 
 120° = 
 etc. 
 
 = i\/3 
 
 cos 
 
 0° = 
 
 1 
 
 
 COS 
 
 .30° = 
 
 i\- 
 
 3 
 
 cos 
 
 45° = 
 
 i\ 
 
 2 
 
 cos 
 
 60° = 
 
 i 
 
 
 cos 
 
 90° = 
 
 
 
 
 cos 
 
 120° = 
 etc. 
 
 -i 
 
 
 tan 
 
 0° = 
 
 = 
 
 tan 
 
 45° = 
 
 = 1 
 
 tan 
 
 90° = 
 
 = X 
 
 tan 135° = 
 
 = -1 
 
 
 etc. 
 
 
K 
 
 (13) 
 
 102 ENGINEERING MATHEMATICS. 
 
 71. Relations between Two Angles. The following relations 
 are developed in text-books of trigonometry : 
 
 sin (a +^) =sin a cos /? + cos a sin /5, 
 sin (a-;9)=sin a cos ^-cos a sin /5, 
 cos (a +/9) =cos a cos ^-sin a sin /?, 
 cos (a — /5)=cosa cos /?+sin a sin ^, . 
 
 Herefrom follows, by combining these equations (13) in 
 pairs : 
 
 cos a cos/3 = J[cos (a+/5) +cos (a-/?)!, ' 
 
 sin a sin ^ = JScos (a-/?)-cos (a+/5!), 
 
 sin a cos/? = Jisin (a+/5)+sin (a — /9)i, 
 
 cos a sin/3 = iisin (a+/?)-sin (a-^)}. , 
 
 By substituting ai for (a+^), and ^i for (a-/3) in these 
 equations (14), gives the equations, 
 
 (14) 
 
 sin ai+sin /?i = 
 sin ai— sin ,5i = 
 cos ai+cos/3i = 
 
 2 sm — ^ cos — 2 — 
 
 2 sm — f; — cos 
 
 2 cos — 7: — cos 
 
 cos ai — COS pi= —2 sm — ^ — sm 
 
 2 
 2 
 
 (15) 
 
 These three sets of equations are the most important trigo- 
 nometric formulas. Their memorizing can be facilitated by 
 noting that cosine functions lead to products of equal func- 
 tions, sine functions to products of unequal functions, and 
 inversely, products of equal functions resolve into cosine, 
 products of unequal functions into sine functions. Also cosine 
 functions show a reversal of the sign, thus: the cosine of a 
 sum is given by a difference of products, the cosine of a differ- 
 ence by a sum, for the reason that with increasing angle 
 the cosine function decreases, and the cosine of a sum of angles 
 thus would be less than the cosine of the single angle. 
 
TRIGONOMETRIC SERIES: 103 
 
 Double Angles. From (13) follows, by substituting a for ;9: 
 
 sin 2a = 2 sin a cos a, 
 cos 2a = cos^ a — sin^ «, 
 = 2 cos^ a— 1, 
 = 1 — 2 sin^ a. 
 Herefrom follow 
 
 (16) 
 
 sin^ a = 
 
 1 — cos 2a 
 
 and cos^ a = 
 
 1+ cos 2a 
 
 (16a) 
 
 72. Differentiation. 
 
 d ,. , 
 -J- (sm a) = + cos a, 
 
 d . . 
 
 -;- ( COS a) = — sm a. 
 da 
 
 (17) 
 
 The sign of the latter differential is negative, as with an 
 increase of angle a, the cos a decreases. 
 
 Integration. 
 
 I sin ada= —cos a, 
 
 I cos ada= +sina. 
 Herefrom follow the definite integrals : 
 
 X + 211 ^ 
 
 sin {a+a)da=0; 
 
 cos {a+a)da=0; 
 
 i sin (a +o)da:= 2 cos (c+ a); 
 
 I cos (a+a)cZa=-2sin (c+a); 
 
 Jc i 
 
 (18) 
 
 (18a) 
 
 (186) 
 
104 
 
 ENGINEERING MATHEMATICS. 
 
 
 sin arfa=0; 
 
 cos ada = ; 
 
 sin ada;= +1; 
 
 cos ada = +1. 
 
 (18c) 
 
 (18d) 
 
 73. Binomial. One of the most frequent trigonometric 
 operations in electrical engineering is the transformation of the 
 binomial, a cos a +6 sin a, into a single trigonometric function, 
 by the substitution, a = c cos p and 6 = c sin p; hence, 
 
 where 
 
 a cos a + b sin a=c cos (a — p). 
 
 c = Va^ + 62 and tanp =— ; . 
 
 or, by the transformation, a = c sin g and b = c cos q, 
 
 a cos a + ?) sin a = c sin (a+g), 
 where 
 
 c = Va^ + 62 and tang = r 
 74. Polyphase Relations. 
 
 (19) 
 (20) 
 
 (21) 
 (22) 
 
 2}cc 
 
 cos a+a±- 
 
 Lm\n 
 
 = 0, 
 
 '^r-v . / 2min\ 
 
 2j- sm ( a+a±—^;~] =0, 
 
 (23) 
 
 J 
 
 where m and n are Integer numbers. 
 
 Proof. The points on the circle which defines the trigo- 
 
 nometric function, by Fig. 28, of the angles {a + a± 
 
 2miTi\ 
 
 7' 
 
TRIGONOMETRIC SERIES. 
 
 105 
 
 are corners of a regular polygon, inscribed in the circle and 
 therefore having the center of the circle as center of gravity. 
 For instance, for n = 7, m = 2, they are shown as Pi, P2, . . . P7, 
 in Fig. 39. The cosines of these angles are the projections on 
 the vertical, the sines, the projections on the horizontal diameter, 
 and as the sum of the projections of the corners of any polygon, 
 
 p. 
 
 (} 
 
 \ 
 
 A 
 B 
 
 \ 
 
 X 
 
 ^s. 
 
 \ 
 
 V 1 
 
 { 
 
 F 
 
 / 
 
 3 
 
 
 
 
 'p* 
 
 Fig. 39. Polyphase Relations. 
 
 FJg. 40. Triangle. 
 
 on any line going through its center of gravity, is zero, both 
 sums of equation (23) are zero. 
 
 71 
 
 COS a+0±- 
 
 2m,in' 
 
 "^z" sin I a 
 
 '.+a±- 
 
 n 
 2mi: 
 
 sin a+a±- 
 
 n 
 
 2mi: 
 
 — 1 COS I a 
 
 in\ . / 
 
 — 1 sm I a 
 
 2yniTi\ n ,^ 
 
 + o± =77 COS (a— 0), 
 
 n j 2 
 
 2rnin\ n , ,, 
 
 + o± =77 cos (a — 6), 
 
 n / 2 
 
 \miTz\ I 
 
 I cos a 
 
 n f \ 
 
 + h± 
 
 SmiK 
 
 = ;y sin (a—b). 
 
 (24; 
 
 These equations are proven by substituting for the products 
 the single functions by equations (14), and substituting them 
 in equations (23). 
 
 75. Triangle. If in a triangle a, ,5, and ;- are the angles, 
 opposite respectively to the sides a, b, c, Fig. 40, then, 
 
 sin a H- sin /? -^ sin ;- = a -^ 6 -^ c, . . . 
 
 (25) 
 
IOC) ENGINEERING MATHEMATICS. 
 
 a2 + &2_c2 
 
 or 
 
 c2 = a2 + 62_2a6cos r- 
 ah sin 7- 
 
 (26) 
 
 Area = 
 
 2 
 
 c2 sin « sin ^ 
 2 sin ^ 
 
 (27) 
 
 B. TRIGONOMETRIC SERIES. 
 
 76. Engineering phenomena usually are either constant, 
 transient, or periodic. Constant, for instance, is the terminal 
 voltage of a storage-battery and the current taken from it 
 through a constant resistance. Transient phenomena occur 
 during a change in the condition of an electric circuit, as a 
 change of load; or, disturbances entering the circuit from the 
 outside or originating in it, etc. Periodic phenomena are the 
 alternating currents and voltages, pulsating currents as those 
 produced by rectifiers, the distribution of the magnetic flux 
 in the air-gap of a machine, or the distribution of voltage 
 around the commutator of the direct-current machine, the 
 motion of the piston in the steam-engine cylinder, the variation 
 of the mean daily temperature with the seasons of the year, etc. 
 
 The characteristic of a periodic function, y=f{x), is, that 
 at constant intervals of the independent -variable x, called 
 cycles or periods, the same values of the dependent variable y 
 occur. 
 
 Most periodic functions of engineering are functions of time 
 or of space, and as such have the characteristic of univalence; 
 that is, to any value of the independent variable x can corre- 
 spond only one value of the dependent variable y. In other 
 words, at any given time and given point of space, any physical 
 phenomenon can have one numerical value only, and therefore 
 must be represented by a univalent function of time and space. 
 
 Any univalent periodic function, 
 
 y=m, (1) 
 
TRIGONOMETRIC SERIES. 107 
 
 can be expressed by an infinite trigonometric series, or Fourier 
 series, of the form, 
 
 y = ao + ai cos cx+a2 cos 2cx + a3 cos 3cx+ . . . 
 
 + 6isincj+62sin2cx+?)3sin3cx + . . ; .... (2) 
 or, substituting for convenience, cx=0, this gives 
 
 i/ = ao+oicos (9+a2Cos2(9+a3Cos3^+ . 
 
 + &1 sin ^ + 62 sin 25 + 63 sin 3(?-F ; (3) 
 
 or, combining the sine and cosine functions by the binomial 
 (par. 73), 
 
 2/ = ao+ci cos ((9-^i)+C2 cos (25-/?2)+C3 cos (35-/33)+ . .1 
 = ao+cisin(5 + 7-i)+c2sin(25 + 7-2)+C3sin (SO + rs) +■ ■ .1 ' ^ ^ 
 
 where 
 
 tan jSn = 
 
 or tan rn=r~- 
 
 On 
 
 an' 
 a, 
 
 (5) 
 
 The proof hereof is given by showing that the coefficients 
 a„ and 6 „ of the series (3) can be determined from the numencal 
 values of the periodic function (1), thus, 
 
 y=f{x)=U{d). . . (6) 
 
 Since, however, the trigonometric function, and therefore 
 also the series of trigonometric functions (3) is univalent, it 
 follows that the periodic function (6), y=fo{d), must be uni- 
 valent, to be represented by a trigonometric series. 
 
 77. The most import-ant periodic functions in electrical 
 engineering are the alternating currents and e.m.fs. Usually 
 they are, in first approximation, represented by a single trigo- 
 nometric function, as: 
 
 i=io cos {6—w); 
 or, 
 
 e = eo sin {d—d); 
 
 that is, they are assumed as sine waves. 
 
108 ENGINEERING MATHEMATICS. 
 
 Theoretically, obviously this condition can never be perfectly 
 attained, and frequently the tleviation from sine shape is suffi- 
 cient to require practical consideration, especially in those cases, 
 where the electric circuit contains electrostatic capacity, as is 
 for instance, the case with long-distance transmission lines, 
 underground cable systems, high potential transformers, etc. 
 
 However, no matter how much the alternating or other 
 periodic wave differs from simple sine shape — that is, however 
 much the wave is " distorted,'' it can always be represented 
 by the trigonometric series (3). 
 
 As illustration the following applications of the trigo- 
 nometric series to engineering problems may be considered : 
 
 {A) The determination of the equation of the periodic 
 function; that is, the evolution of the constants a„ and b^ of 
 the trigonometric series, if the numerical values of the periodic 
 function are given. Thus, for instance, the wave of an 
 alternator may be t^iken by oscillograph or wave-meter, and 
 by measuring from the oscillograph, the numerical values of 
 the periodic function are derived for every 10 degrees, or every 
 5 degrees, or every degree, depending on the accuracy required. 
 The problem then is, from the numerical values of the wave, 
 to determine its equation. While the oscillograph shows the 
 shape of the wave, it obviously is not possible therefrom to 
 calculate other quantities, as from the voltage the current 
 under given circuit conditions, if the wave shape is not first 
 represented by a mathematical expression. It therefore is of 
 importance in engineering to translate the picture or the table 
 of numerical values of a periodic function into a mathematical 
 expression thereof. 
 
 (B) If one of the engineering quantities, as the e.m.f. of 
 an alternator or the magnetic flux in the air-gap of an electric 
 machine, is given as a general periodic function in the form 
 of a trigonometric series, to determine therefrom other engineer- 
 ing quantities, as the current, the generated e.m.f., etc. 
 
 A. Evaluation of the Constants of the Trigonometric Series from 
 the Instantaneous Values of the Periodic Function. 
 
 78. Assuming that the numerical values of a univalent 
 periodic function y=fo{0) are given; that is, for every value 
 of 0, the corresponding value of y is known, either by graphical 
 representation. Fig. 41; or, in tabulated form. Table I, but 
 
TRIGONOMETRIC SERIES. 
 
 109 
 
 the equation of the periodic function is not known. It can be 
 represented in the form, 
 
 2/ = ao+ai cos d + a2 cos 2d + a3 cos 3^ + . . . +a„ cos 716* + . . . 
 
 + 61 sin (9+&2sin26' + &5sin 3(9 + . . .+6„sin ntf + . . , (7) 
 
 and the problem now is, to determine the coefficients oo, oi, 
 tt2 . . . 61, 62 . . . 
 
 Fig. 41. Periodic Functions. 
 T.ABLE I. 
 
 e 
 
 !/ 
 
 g 
 
 V 
 
 9 
 
 V 
 
 « 
 
 1/ 
 
 
 
 -60 
 
 90 
 
 + 50 
 
 180 
 
 + 122 
 
 270 
 
 + 85 
 
 10 
 
 -49 
 
 100 
 
 + 61 
 
 190 
 
 + 124 
 
 280 
 
 + 65 
 
 20 
 
 -38 
 
 110 
 
 + 71 
 
 200 
 
 + 126 
 
 290 
 
 + 35 
 
 30 
 
 -26 
 
 120 
 
 + 81 
 
 210 
 
 + 125 
 
 300 
 
 + 17 
 
 40 
 
 -12 
 
 130 
 
 + 90 
 
 220 
 
 + 123 
 
 310 
 
 
 
 50 
 
 
 
 140 
 
 + 99 
 
 230 
 
 + 120 
 
 320 
 
 -13 
 
 60 
 
 + 11 
 
 150 
 
 + 107 
 
 240 
 
 + 116 
 
 330 
 
 -26 
 
 70 
 
 + 27 
 
 160 
 
 + 114 
 
 250 
 
 + 110 
 
 340 
 
 -38 
 
 80 
 
 + 39 
 
 170 
 
 + 119 
 
 260 
 
 + 100 
 
 350 
 
 -49 
 
 90 
 
 + 50 
 
 180 
 
 + 122 
 
 270 
 
 + 85 
 
 360 
 
 -60 
 
 Integrate the equation (7) between the hmits and 2n: 
 
 Jr2» r2n r^" r2n 
 
 yd6 = ao\ dO + Ux \ cos ddd + a2\ cos2ddd + ... 
 Jo Jo Jo 
 
 COS nddd + . . .+&! j sir 
 Jo 
 
 +c 
 
 sin ddO + 
 
 J^2ir 
 
 + 62 1 s\n2edd + . . .+hn\ sinnddd + .. 
 
 X2ir 
 sir 
 
 /2ir 
 
 --ao/d/ +ai/sin 
 
 + 02, 
 
 /sin 
 
 20 Z^" 
 
 2 
 
 + . 
 
 '0 / /o / ^ /o 
 
 /sin nd /^' , / „ Z^" 
 
 + a„/ / + . . .-bi/cosd/ 
 
 , /cos 20/2" , /COS nd Z^' 
 
no ENGINEERING MATHEMATICS. 
 
 All the integrals containing trigonometric functions vanish, 
 as the trigonometric function has the same value at the upper 
 limit 2n as at the lower limit 0, that is, 
 
 /- -/ = -(cos znTT— cos 0) =U 
 
 / n /o n^ 
 
 /sin nd Z^" 1 • n^ A 
 
 /• / =-(sm 2?i7r — sm 0)=0, 
 
 ' 
 and the result is 
 
 IT / /2ll 
 
 ydd^ao/o/ =2;rao; 
 
 hence 
 
 f: 
 
 1 r~' 
 
 ao = 2^| ydO. . (8) 
 
 ydd is an element of the area of the carve y, Fig. 41, and 
 
 X2iz 
 ydd thus is the area of the periodic function y, for one 
 
 period; that is, 
 
 ao = o-^> (9) 
 
 where Jl = area of the periodic function y=fo{G), for one period; 
 
 that is, from ^ = to d = 2n. 
 
 A 
 2n is the horizontal width of this area A, and 77- thus is 
 
 Zn 
 
 the area divided bj' the width of it; that is, it is the. average 
 
 height of the area A of the periodic function y; or, in other 
 
 words, it is the average value of y. Therefore, 
 
 ao = avg. {y)o^' (10) 
 
 The first coefficient, oo, thus, is the average value of the 
 instantaneous values of the periodic function y, between = Q 
 and d = 2n. 
 
 Therefore, averaging the values of y in Table I, gives the 
 first constant ao. 
 
 79. To determine the coefficient a„, multiply equation (7) 
 by cos nd, and then integrate from to 2n, for the purpose of 
 making the trigonometric functions vanish. This gives 
 
TRIGONOMETRIC SERIES. Ill 
 
 r2^ rin rin 
 
 I j/cos r2(9(i(9 = ao ) cos n(9(i(9+ai I cos n(9 cos /?d^ + 
 J^ Ji) Jo 
 
 02 I cos n^ cos 25d(9 + . . . +a„ j cos^ ?i(9d^ + . , . 
 Jo Jo 
 
 . I cosn^sin(9d^ + 62 ( 
 Ja Jo 
 
 -2i 
 
 + &! I cosnds\nddd + 'b2\ (ios,nddu.2ddd + . . . 
 
 + ?)„ I cosn^sin n^d<? + . . . 
 \ Jo 
 
 Hence, by the trigonometric equations of the preceding 
 section: ,-,. 
 
 r%i! ri^ r27, 
 
 I ycosnddd=ao\ cosn9d(?+ai I |[cos(n+l)(9+cos(n— l)(9]d(9 
 Jo Jo Jo 
 
 + a2 I i[cos (n + 2)^+cos {n-2)d]dd + . . . 
 Jo 
 r2n 
 + o„l Kl+cos2n^)di9 + . . . 
 Jo 
 
 X2it 
 i[sin (Ti + l)^-sin (n-\)S\dd 
 
 Jr2x 
 I i[sin(n + 2)^-sin(n-2)^]d9 + ... 
 
 
 + &„ I isin2n(9d(9 + . . . 
 Jo.. 
 
 All these integrals of trigonometric functions give trigo- 
 nometric functions, and therefore vanish between the limits 
 and 27r, and there only remains the JSrst term of the integral 
 multiplied with a„, which does not contain, a trigonometric 
 function, and thus remains finite : 
 
 nn /dY' 
 
 and therefore, 
 
 p. 
 
 I y cos n0dd = an7:; 
 Jo 
 
 hence 
 
 1 p' 
 
 an = - I y cos nddd (11) 
 
 ^ Jo 
 
112 
 
 ENGINEERING MATHEMATICS. 
 
 If the instantaneous values of y are multiplied with cos nd, 
 and the product y„ = y cos nd plotted as a curve, y cos nddd is 
 an element of the area of this curve, shown for n = 3 in Fig. 42, 
 
 and thus I y cos nOdd is the area of this curve ; that is. 
 
 -A 
 
 nr 
 
 (12) 
 
 Fig. 42. Curve of y cos 39. 
 where il„ is the area of the curve y cos nd, between (9=0 and 
 
 e=2n. 
 
 As 2r. is the width of this area ^„, 77^ is the aA'-erage height 
 
 of this area; that is, is the average value of y cos nd, and -A„ 
 thus is twice the average value of y cos nd ; that is, 
 
 1 
 
 o„=2 avg. (1/ cos nd)o 
 
 (13) 
 
 Fig. 43. Curve of y sin ZO. 
 
 The coefficient a„ of cos nd is derived by multiplying all 
 the instantaneous values of y by cos nd, and taking twice the 
 average of the instantaneous values of this product y cos nd. 
 
TRIGONOMETRIC SERIES. 113 
 
 80. ?)„ is determined in the analogous manner by multiply- 
 ing y by sin nd and integrating from to 27r; by the area of the 
 curve y sin nd, shown in Fig. 43, for n = 3, 
 
 /^2k rij! r2K 
 
 I y'&mnddd = ao\ sinnM/9 + ail smndcosddd 
 Jo Jo Jo 
 
 + 02 j sin n^ cos 2^^ + . +a„ I sin n^ cos n(?(i^ + . . . 
 
 X2i pin 
 
 sin nd sin ddd + 62 1 sin nd sin 2ddd + . . . 
 
 X2)r 
 sin2n(9d(9+ .. 
 
 = aoj sinn(9d» + ai I i[sin (» + l)^+sin (n-l)^]d5 
 Jo </o 
 
 + 02 i[sin(n + 2)0+sin (n-2)0]d(9 + . . . 
 
 + a„ I isin 2n&^0 + . . . 
 Jo 
 
 \hx f "i-Lcos (n-l)<?-cos (n + l)(9]d5 
 
 + &2 p^iLcos (n- 2)0 -cos (n + 2)0]d(9 + . . . 
 
 + &J Ml-cos2n0]d(? + ... 
 Jo 
 
 hencCj 
 
 6 
 
 „ = i r^sinn^dO (14) 
 
 = -^n', (15) 
 
 where A„' is the area of the curve y^ = y sin nQ. Hence, 
 
 6„ = 2 avg. (2/sinn0)o^^ • ... (16) 
 
114 
 
 ENGINEERING MATHEMATICS. 
 
 and the coefficient of sin nd thus is derived by multiplying the 
 instantaneous values of y with sin nd, and then averaging, as 
 twice the average of y sin nd. 
 
 8i. Any univalent periodic function, of which the numerical 
 values y are known, can thus be expressed numerically by the 
 equation, 
 y = aQ+ai cos 6+a2Cos 26 + . . . +a„ cos nd+ . . 
 
 + 61 sin (9+62 sin 2d + . . .+6„ sin n^ + . . . 
 
 (17) 
 
 where the coefficients a^, ai, a2, . . .h\, 62 ... , are calculated 
 as the averages : 
 
 2i 
 
 Oo = avg. {y)o , 
 ai=2avg. (j/cos 6)^^"; 
 
 2x 
 
 6i = 2avg. (ysin 6)^ ; 
 
 2>r 
 
 a2 = 2avg. (2/ cos 2(9)q""; 62 = 2 avg. fy sin 2^)q "; 
 
 a„ = 2 avg. (ycos nb 
 
 2« 
 
 6„ = 2 avg. (y sin nb 
 
 C; 
 
 (18) 
 
 Hereby any individual harmonic can be calculated, without 
 calculating the preceding harmonics. 
 
 For instance, let the generator c.m.f. wave. Fig. 44, Table 
 II, column 2, be impressed upon an underground cable system 
 
 Fig. 44. Generator e.m.f. wave 
 
 of such constants (capacity and inductance), that the natural 
 frequency of the system is G70 cycles per second, while the 
 generator frequency is 60 cycles. The natural frequency of the 
 
TRIGONOMETRIC SERIES. 
 
 115 
 
 circuit is then close to that of the Uth harmonic of the generator 
 wave, 660 cycles, and if the generator voltage contains an 
 appreciable 11th harmonic, trouble may result from a resonance 
 rise of voltage of this frequency; therefore, the 11th harmonic 
 of the generator wave is to be determined, that is, an and bn 
 calculated, but the other harmonics are of less importance. 
 
 Table II 
 
 e 
 
 1/ 
 
 cos lie 
 
 sin 115 
 
 y cos 11« 
 
 KSin 119 
 
 
 10 
 
 20 
 
 5 
 
 4 
 
 20 
 
 + 1.000 
 -0.342 
 -0.766 
 
 
 + . 940 
 -0.643 
 
 + 5.0 
 
 -1.4 
 
 -15.3 
 
 
 + 3.8 
 -12.9 
 
 30 
 40 
 50 
 
 22 
 19 
 25 
 
 + 0.866 
 + 0.174 
 -0.985 
 
 -0.500 
 + 0.985 
 -0.174 
 
 + 19.1 
 
 + 3.3 
 
 -24.6 
 
 -11:0 
 
 + 18.7 
 - 4.3 
 
 60 
 70 
 80 
 
 29 
 29 
 30 
 
 + 0.500 
 + . 643 
 -0.940 
 
 -0.866 
 + 0.766 
 + 0.342 
 
 + 14.5 
 + 18.6 
 -28.2 
 
 -25.1 
 + 22.2 
 + 10.3 
 
 90 
 
 100 
 
 110 
 
 38 
 46 
 38 
 
 
 
 + 0.940 
 - . 643 
 
 -1.000 
 + 0.342 
 + 0.766 
 
 
 + 43.3 
 -24.4 
 
 -38.0 
 + 15.7 
 + 29.2 
 
 120 
 130 
 140 
 
 41 
 50 
 32 
 
 -0.500 
 + 0.985 
 -0.174 
 
 -0.866 
 -0.174 
 + 0.985 
 
 -20.5 
 
 + 49.2 
 
 -5.6 
 
 -35.5 
 - 8.7 
 + 31.5 
 
 150 
 160 
 170 
 
 30 
 33 
 
 7 
 
 -0.866 
 + 0.766 
 + 0.342 
 
 -0.500 
 -0.643 
 + 0.940 
 
 -26.0 
 + 25.3 
 
 + 2,2 
 
 -15.0 
 -21.3 
 
 180 
 
 -5 
 
 
 
 
 
 
 
 Divided 
 
 Total 
 
 by 9 
 
 + 34.5 
 + 3.S3 = a„ 
 
 -29.8 
 -3.31 = b„ 
 
 In the third column of Table II thus are given the values 
 of cos 11^, in the fourth column sin 116, in the fifth column 
 y cos 11^, and in the sixth column y sin 11^. The former gives 
 as average +1.915, hence aii= +3.83, and the latter gives as 
 average —1.655, hence 6ii = — 3.31, and the 11th harmonic of 
 the generator wave is 
 
 an cos 11^ +bn sin 11^ = 3.83 cos 11(9-3.31 sin 11^ 
 = 5.07 cos (lie +410), 
 
116 ENGINEERING MATHEMATICS. 
 
 hence, its effective value is 
 
 5.07 
 
 —- = 3.58, 
 
 V2 
 
 while the effective value of the total generator wave, that 
 is, the square root of the mean squares of the instanta- 
 neous values y, is 
 
 e = 30.5, 
 
 thus the 11th harmonic is 11.8 per cent of the total voltage, 
 and whether such a harmonic is safe or not, can now be deter- 
 mined from the circuit constants, more particularly its resist- 
 ance. 
 
 82. In general, the successive harmonics decrease; that is, 
 with increasing n, the values of a„ and 6„ become smaller, and 
 when calculating o„ and 6„ by equation (18), for higher values 
 of n they are derived as the small averages of a number of 
 large quantities, and the calculation then becomes incon- 
 venient and less correct. 
 
 Where the entire series of coefhcients an and bn is to be 
 calculated, it thus is preferable not to use the complete periodic 
 function y, but only the residual left after subtracting the 
 harmonics which have already been calculated; that is, after 
 Uq has been calculated, it is subtracted from y, and the differ- 
 ence, ?/i =1/— flo, is used for the calculation of ai and bi. 
 
 Then ai cos ^ + ?)i sin ^ is subtracted from 1/1, and the 
 difference, 
 
 y2 = yi— (0.1 cos +bi sm d) 
 = y—{O'0+O'i COS + bi sin 6), 
 
 is used for the calculation of 02 and 62- 
 
 Then 02 cos 2d+b2 sin 26 is subtracted from 1/2, and the rest, 
 2/3, used for the calculation of a.i and 6.3, etc. 
 
 In this manner a higher accuracy is derived, and the calcu- 
 lation simplified by having the instantaneous values of the 
 function of the same magnitude as the coefficients a„ and 6^. 
 
 As illustration, is given in Table III the calculation of the 
 first three harmonics of the pulsating current, Fig. 41, Table I: 
 
TRIGONOMETRIC SERIES. 117 
 
 83. In electrical engineering, the most important periodic 
 functions are the alternating currents and voltages. Due to 
 the constructive features of alternating-current generators, 
 alternating voltages and currents are almost always symmet- 
 rical waves; that is, the periodic function consists of alternate 
 half-waves, which are the same in shape, but opposite in direc- 
 tion, or in other words, the instantaneous values from 180 deg. 
 to 360 deg. are the same numerically, but opposite in sign, 
 from the instantaneous values between to 180 deg., and each 
 cycle or period thus consists of two equal but opposite half 
 cycles, as shown in Fig. 44. In the earlier days of electrical 
 engineering, the frequency has for this reason frequently been 
 expressed by the number of half-waves or alternations. 
 
 In a symmetrical wave, those harmonics which produce a 
 difference in the shape of the positive and the negative half- 
 wave, cannot exist; that is, their coefficients a and b must be 
 zero. Only those harmonics can exist in which an increase of 
 the angle 6 by 180 deg., or n, reverses the sign of the function. 
 This is the case with cos nd and sin nd, if n is an odd number. 
 If, however, n is an even number, an increase of ^ by tt increases 
 the angle nd by 2?: or a multiple thereof, thus leaves cos nd 
 and sin nO with the same sign. The same applies to a^. There- 
 fore, symmetrical alternating waves comprise only the odd 
 harmonics, but do not contain even harmonics or a constant 
 term, and thus are represented by 
 
 i/ = ai cos ^+a3 cos 3/? + a6 cos 5/?+ .. 
 + &1 sin e+bs sin Sd+bssin5d + (19) 
 
 When calculating the coefficients a„ and bn of a symmetrical 
 wave by the expression (18), it is sufficient to average from 
 to n; that is, over one half-wave only. In the second half-wave, 
 cos nd and sin 7id have the opposite sign as in the first half-wave, 
 if n is an odd number, and since y also has the opposite sign 
 in the second half-wave, y cos nd and y sin nd in the second 
 half-wave traverses again the same values, with the same sign, 
 as in the first half-wave, and their average thus is given by 
 averaging over one half-wave only. 
 
 Therefore, a symmetrical univalent periochc function, as an 
 
118 
 
 ENGINEERING MATHEMATICS. 
 
 Table 
 
 e 
 
 V 
 
 l/i = V-Oa 
 
 ]/, coy 9 
 
 j/^ sin 9 
 
 ci = ai coaO 
 + 6i sin 6 
 
 J/a=i/o-c, 
 
 
 10 
 20 
 
 -60 
 -49 
 -38 
 
 -111 
 
 -100 
 
 -89 
 
 -111 
 
 -98 
 
 -84 
 
 
 
 -17 
 -30 
 
 -84 
 -85 
 -83 
 
 -27 
 
 -15 
 
 -6 
 
 30 
 40 
 50 
 
 -26 
 
 -12 
 
 
 
 -77 
 -63 
 -51 
 
 -67 
 -48 
 -33 
 
 -.38 
 -40 
 -39 
 
 -79 
 -72 
 -63 
 
 + 2 
 
 9 
 
 12 
 
 60 
 70 
 
 80 
 
 + 11 
 27 
 39 
 
 -40 
 -24 
 -12 
 
 -20 
 -8 
 -2 
 
 -35 
 -23 
 -12 
 
 -52 
 -40 
 -26 
 
 12 
 16 
 14 
 
 90 
 100 
 110 
 
 50 
 61 
 71 
 
 -1 
 
 + 10 
 
 20 
 
 
 -2 
 
 -7 
 
 -1 
 
 + 10 
 + 19 
 
 -11 
 
 + 4 
 
 18 
 
 10 
 6 
 
 + 2 
 
 120 
 130 
 140 
 
 81 
 90 
 99 
 
 30 
 39 
 
 48 
 
 -15 
 
 -25 
 -37 
 
 + 26 
 + 30 
 + 31 
 
 32 
 45 
 
 58 
 
 -2 
 
 -6 
 
 -10 
 
 150 
 160 
 170 
 
 107 
 114 
 119 
 
 56 
 63 
 68 
 
 -49 
 -59 
 -67 
 
 + 28 
 + 22 
 + 12 
 
 67 
 75 
 81 
 
 -11 
 -12 
 -13 
 
 180 
 190 
 200 
 
 122 
 124 
 126 
 
 71 
 73 
 75 
 
 -71 
 -72 
 -71 
 
 
 -13 
 -26 
 
 84 
 85 
 83 
 
 -13 
 -12 
 
 -8 
 
 210 
 220 
 230 
 
 125 
 123 
 120 
 
 74 
 72 
 69 
 
 -64 
 -55 
 -44 
 
 -.37 
 -47 
 -53 
 
 79 
 72 
 63 
 
 -5 
 
 
 + 6 
 
 240 
 250 
 260 
 
 116 
 110 
 100 
 
 65 
 59 
 49 
 
 -32 
 
 -20 
 
 -9 
 
 -28 
 -56 
 -48 
 
 52 
 40 
 26 
 
 13 
 19 
 23 
 
 270 
 280 
 290 
 
 85 
 65 
 35 
 
 34 
 + 14 
 -16 
 
 
 
 + 2 
 -5 
 
 -34 
 -14 
 
 + 15 
 
 11 
 
 -4 
 -18 
 
 23 
 
 18 
 
 + 2 
 
 300 
 310 
 320 
 
 + 17 
 
 
 
 -13 
 
 -34 
 -51 
 -64 
 
 -17 
 -33 ■ 
 -49 
 
 + 30 
 + 39 
 
 + 41 
 
 -.32 
 -45 
 
 -58 
 
 _2 
 -6 
 -6 
 
 330 
 340 
 350 
 
 -26 
 -38 
 -49 
 
 -75 
 
 -89 
 
 -100 
 
 -65 
 
 -84 
 -99 
 
 + 37 
 + 30 
 + 17 
 
 -67 
 -75 
 -81 
 
 -8 
 -14 
 -19 
 
 Total . . 
 Divided 
 by 36 ... 
 
 + 1826 
 
 + 50.7 = a„ 
 
 Total 
 
 Divided by 
 18 
 
 . - 1520 
 -84.4 = ai 
 
 -204 
 -11.3 = 6, 
 
 Total 
 
 Divided by 18 . . . 
 
TRIGONOMETRIC SERIES. 
 
 119 
 
 III. 
 
 
 1-2 COS 20 
 
 1/2 sin 25 
 
 c, = 02 cos 2 
 + 62siu2« 
 
 t) 
 
 V3 = Vl—ci 
 
 !/3 COS 36 
 
 J/3 sin 3« 
 
 e 
 
 
 -27 
 
 
 
 -15 
 
 -12 
 
 -12 
 
 
 
 
 
 
 -u 
 
 -5 
 
 -12 
 
 -3 
 
 -3 
 
 -1 
 
 10 
 
 
 — 5 
 
 -4 
 
 -7 
 
 + 1 
 
 
 
 + 1 
 
 20 
 
 
 + 1 
 
 + 2 
 
 -1 
 
 + 3 
 
 
 
 + 3 
 
 30 
 
 
 + 2 
 
 + 9 
 
 + 4 
 
 + 5 
 
 -2 
 
 + 4 
 
 40 
 
 
 — 2 
 
 + 12 
 
 11 
 
 + 1 
 
 -1 
 
 
 
 50 
 
 
 -6 
 
 + 10 
 
 13 
 
 -1 
 
 + 1 
 
 
 
 60 
 
 
 -12 
 
 + 10 
 
 15 
 
 + 1 
 
 -1 
 
 
 
 70 
 
 
 -13 
 
 + 5 
 
 16 
 
 _2 
 
 + 1 
 
 + 2 
 
 SO 
 
 
 -10 
 
 
 
 15 
 
 -5 
 
 
 
 + 5 
 
 90 
 
 
 -6 
 
 _2 
 
 12 
 
 -6 
 
 -3 
 
 + 5 
 
 100 
 
 
 -2 
 
 -1 
 
 7 
 
 -5 
 
 -4 
 
 + 2 
 
 110 
 
 
 + 1 
 
 + 2 
 
 + 1 
 
 -3 
 
 -3 
 
 
 
 120 
 
 
 + 1 
 
 + 6 
 
 -4 
 
 -2 
 
 -2 
 
 -1 
 
 130 
 
 
 -2 
 
 + 10 
 
 -11 
 
 + 1 
 
 
 
 + 1 
 
 140 
 
 
 -5 
 
 + 10 
 
 -13 
 
 + 2 
 
 
 
 + 2 
 
 150 
 
 
 -9 
 
 + 8 
 
 -15 
 
 + 3 
 
 -1 
 
 + 3 
 
 160 
 
 
 -12 
 
 -4 
 
 -16 
 
 + 3 
 
 -3 
 
 + 1 
 
 170 
 
 
 -13 
 
 
 
 -15 
 
 + 2 
 
 _2 
 
 
 
 180 
 
 
 -11 
 
 -4 
 
 -12 
 
 
 
 
 
 
 
 190 
 
 
 -6 
 
 -6 
 
 -7 
 
 -1 
 
 
 
 -1 
 
 200 
 
 
 — 2 
 
 -4 
 
 -1 
 
 -4 
 
 
 
 -4 
 
 210 
 
 
 
 
 
 
 + 4 
 
 -4 
 
 -2 
 
 -4 
 
 220 
 
 
 -1 
 
 + 6 
 
 11 
 
 -5 
 
 -4 
 
 -2 
 
 230 
 
 
 -6 
 
 + 11 
 
 13 
 
 
 
 
 
 
 
 240 
 
 
 -15 
 
 + 12 
 
 15 
 
 + 4 
 
 + 4 
 
 + 2 
 
 250 
 
 
 -22 
 
 + 8 
 
 16 
 
 + 7 
 
 + 3 
 
 + 6 
 
 260 
 
 
 -23 
 
 
 
 15 
 
 + 8 
 
 
 
 + 8 
 
 270 
 
 
 -17 
 
 -6 
 
 12 
 
 + 6 
 
 -3 
 
 + 5 
 
 280 
 
 
 -2 
 
 -1 
 
 7 
 
 — 5 
 
 + 4 
 
 -2 
 
 290 
 
 
 + 1 
 
 + 2 
 
 + 1 
 
 -3 
 
 + 3 
 
 
 
 300 
 
 
 + 1 
 
 + 6 
 
 -4 
 
 _2 
 
 + 2 
 
 + 1 
 
 310 
 
 
 -1 
 
 + 6 
 
 -11 
 
 + 5 
 
 — 2 
 
 -4 
 
 320 
 
 
 -4 
 
 + 7 
 
 -13 
 
 + 5 
 
 
 
 — 5 
 
 330 
 
 
 -11 
 
 + 9 
 
 -15 
 
 + 1 
 
 
 
 -1 
 
 340 
 
 
 -18 
 
 + 6 
 
 -16 
 
 -3 
 
 -3 
 
 + 1 
 
 350 
 
 
 -270 
 
 + 120 
 
 Total 
 
 -33 
 
 + 27 
 
 
 - 
 
 -15.0 = 02 
 
 + 6.7 = 62 ] 
 
 Divided by 18 
 
 -1.8 = 03 
 
 + 1.5 = 63 . 
 
 
(21) 
 
 120 ENGINEERING MATHEMATICS. 
 
 alternating voltage and current usually is, can be represented 
 by the expression, 
 
 y = ax cos O+a^ cos 3 O + a^ cos 5 + a7 cos 7(9 + . . . 
 + f>isin f^ + bssinS d+hsAn^ d+h^inl 6 +...; (20) 
 
 where, 
 
 ai = 2 avg. (y cos 6)^'; 61=2 avg. {y sin ^)o''; 
 
 as =2 avg. (j/ cos 3/9)0"; 63 =2 avg. (j/ sin 3(9)0"; 
 
 as = 2 avg. (1/ cos 50)^'; 65 = 2 avg. (?/ sin 50)^'; 
 
 07 = 2 avg. (j/ cos 7(?)o'; ?>7 = 2 avg. (1/ sin 7^)0". 
 
 84. From 180 deg. to 360 deg., the even harmonics have 
 the same, but the odd harmonics the opposite sign as from 
 to 180 deg. Therefore adding the numerical values in the 
 range from 180 deg. to 360 deg. to those in the range from 
 to 180 deg., the odd harmonics cancel, and only the even har- 
 monics remain. Inversely, by subtracting, the even harmonics 
 cancel, and the odd ones remain. 
 
 Hereby the odd and the even harmonics can be separated. 
 If y = y{0) are the numerical values of a periodic function 
 from to 180 deg., and y' = y{d+n) the numerical values of 
 the same function from 180 deg. to 360 deg., 
 
 y2{d)=l\y{d)+y{d+n)], .... (22) 
 is a periodic function containing only the even harmonics, and 
 
 ydO) = h\y{d)-y{d+n)\ . ... (23) 
 
 is a periodic function containing only the odd harmonics ; that is : 
 
 y\{d) = a\ cos d+az cos 3^+ as cos bO + . . . 
 
 + 61 sin (9 + 63 sin 3 ^ + 65 sin 5(9+ .. .; . . (24) 
 
 2/2(^) =ao+a2 cos 2(9+a4 cos 40 + . . . 
 
 + 62sin20 + 54sin46' + .. ., (25) 
 
 and the complete function is 
 
 y{0)=yx{e)+y^{0) (26) 
 
TRIGONOMETRIC SERIES. 121 
 
 By this method it is convenient to determine whether even 
 harmonics are present, and if they are present, to separate 
 them from the odd harmonics. 
 
 Before separating the even harmonics and the odd har- 
 monics, it is usually convenient to separate the constant term 
 ao from the periodic function y, by averaging the instantaneous 
 values of y from to 360 deg. The average then gives ao, 
 and subtracted from the instantaneous values of y, gives 
 
 yo{0)=yiO)-aQ (27) 
 
 as the instantaneous values of the alternating component of the 
 periodic function; that is, the component y^ contains only the 
 trigonometric functions, but not the constant term. ?/o is 
 then resolved into the odd series yi, and the even series 2/2- 
 85. The alternating wave y^ consists of the cosine components : 
 
 u{d)-=ai cos d+a2 cos 2d+a3 cos Sd+a^ cos 4(9 + , . ., (28) 
 and the sine components ; 
 
 v{d)=bi sin (? + &2 sin 2(9 + 63 sin 35+ 64 sin 45 + . ..; (29) 
 that is, 
 
 yo{e)=uie)+v(d) (.30) 
 
 The cosine functions retain the same sign for negative 
 angles {—6), as for positive angles ( + 6), while the sine functions 
 reverse their sign; that is, 
 
 u{-e)=+u{d) and v{-d) = -v{d). . . . (31) 
 
 Therefore, if the values of j/o for positive and for negative 
 angles 6 are averaged, the sine functions cancel, and only the 
 cosine functions remain, while by subtracting the values of 
 j/o for positive and for negative angles, only the sine functions 
 remain; that is, 
 
 yo(d)+yo{-d) = 2u{e); 
 
 (32) 
 
 2/o(5)-j/o(-5) = 2K5);. 
 
 hence, the cosine terms and the sine terms can be separated 
 from each other by combining the instantaneous values of y^ 
 for positive angle 6 and for negative angle (—(9), thus: 
 
 u{d) = \\y,{d)+yo{-d)],\ 
 
 (33) 
 
 K^) = i!2/o(^)-2/o(-^)!- 
 
122 ENGINEERING MATHEMATICS. 
 
 Usually, before separating the cosine and the sine terms, 
 u and V, first the constant term n^ is separated, as discussed 
 above; that is, the alternating function yo = y — ao used. If 
 the general periodic function y is used in equation (33), the 
 constant term a^ of this periodic function appears in the cosine 
 term u, thus: 
 
 u{6) =l{yid) +y{ — d)\ =ao+ai cos 6 + a2Cos20 + a3 cos 30 + . . ., 
 
 while v{d) remains the same as when using y^. 
 
 86. Before separating the alternating function y^ into the 
 cosine function u and the sine function v, it usually is more 
 convenient to resolve the alternating function ?/o into the odd 
 series yi, and the even series 2/2, as discussed in the preceding 
 paragraph, and then to separate j/i and 2/2 each into the cosine 
 and the sine terms : 
 
 uii6) = ^\yi{0)+yi{—d)}^aicosd+a3Cos3d+a5Cos5O+ 
 Vi{d) = i\yi(d)-yi(-6)\=-bismd+b3smSd+b5sm5d+ 
 U2(d) =My2(d) +y2i- 0)\ =a2Cos2d + aiCos id + . ; 1 
 V2{0)=^{y2{d)-y2(-d)\=h2sm2e+bism4d + ... J 
 
 (34) 
 
 (35) 
 
 In the odd functions mi and Vi, a change from the negative 
 angle (— 6) to the supplementary angle (tt— 6) changes the angle 
 of the trigonometric function by an odd multiple of ;: or 180 
 deg., that is, by a multiple of 2/r or 360 deg., plus 180 deg., 
 which signifies a reversal of the function, thus : 
 
 ui{0) = Myi{d)-yi(K-d)\, ] 
 
 \ . . . (36) 
 
 However, in the even functions M2 and V2 a change from the 
 negative angle (—0) to the supplementary angle (;r—i9), changes 
 the angles of the trigonometric function by an even multiple 
 of tt; that is, by a multiple of 2n or 360 deg.; hence leaves 
 the sign of the trigonometric function unchanged, thus : 
 
 U2(0)^h{y2{d)+y2{n-d)\, 1 
 
 ■ (37) 
 
 V2(d) = i{y2(0)-y2{n-e)]. 
 
TRIGONOMETRIC SERIES. 
 
 123 
 
 To avoid the possibility of a mistake, it is preferable to use 
 the relations (34) and (35^, which are the same for the odd and 
 for the even series. 
 
 87. Obviously, in the calculation of the constants a„ and 
 b„, instead of averaging from to 180 deg., the average can 
 be made from —90 deg. to +90 deg. In the cosine function 
 uid), however, the same numerical values repeated with the 
 same signs, from to —90 deg., as from to +90 deg., and 
 the multipliers cos nd also have the same signs and the same 
 numerical values from to —90 deg., as from to +90 deg. 
 In the sine function, the same numerical values repeat from 
 to —90 deg., as from to +90 deg., but with reversed signs, 
 and the multipliers sin n6 also have the same numerical values, 
 but with reversed sign, from to —90 deg., as from to +90 
 deg. The products u cos nd and v sin nd thus traverse the 
 same numerical values with the same signs, between and 
 — 90 deg., as between and +90 deg., and for deriving the 
 
 averages, it thus is sufficient to average only from to — , or 
 
 90 deg. ; that is, over one quandrant. 
 
 Therefore, by resolving the periodic function y into the 
 cosine components u and the sine components v, the calculation 
 of the constants o„ and hn is greatly simplified; that is, instead 
 of averaging over one entire period, or 360 deg., it is necessary 
 to -average over only 90 deg., thus: 
 
 ai=2 avg. (mi cos Q)^; ?)i = 2 avg. (ui sin ^^o^ ; 
 
 a2 = 2 avg. {u2 cos 2^)0^ ; 62 = 2 avg. {v^ sin 2(9)o2 
 
 as = 2 avg. {m cos 3^)0^"; 63 =2 avg. (ug sin Zd)^'^ ; I _ (35) 
 
 a4 = 2 avg. (m4 cos 4(9)o2 ; 64 = 2 avg. (i;4 sin 45)o2 
 
 05 = 2 avg. (m5 cos hff)Q^; 65 = 2 avg. (us sin hB)^ 
 etc. etc. 
 
 where mi is the cosine term of the odd function ?/i; ui the 
 cosine term of the even function 2/2; M3 is the cosine term of 
 the odd function, after subtracting the term with cos d; that is, 
 
 uz = u\ — oi cos 0, 
 
124 ENGINEERING MATHEMATICS. 
 
 analogously, Ui is the cosine term of the even function, after 
 subtracting the term cos 2d; 
 
 U4,=U2—o,2 cos 26, 
 
 and in the same manner, 
 
 u^ = U3 — az cos 3^, 
 W6 = tt4— 04 cos 40, 
 
 and so forth; Vi, V2, Vr„ V4, etc., are the corresponding sine 
 terms. 
 
 When calculating the coefficients a„ and &„ by averaging over 
 90 deg., or over 180 deg. or 360 deg., it must be kept in mind 
 that the terminal values of y respectively of u or v, that is, 
 the values for 6 = and = 90 deg. (or = 180 -deg. or 360 
 deg. respectively) are to be taken as one-half only, since they 
 are the ends of the measured area of the curves a„ cos n6 and 
 bn sin n6, which area gives as twice its average height the values 
 a„ and f)„, as discussed in the preceding. 
 
 In resolving an empirical periodic function into a trigono- 
 metric series, just as in most engineering calculations, the 
 most important part is to arrange the work so as to derive the 
 results expeditiously and rapidly, and at the same time 
 accurately. By proceeding, for instance, immediately by the 
 general method, equations (17) and (18), the work becomes so 
 extensive as to be a serious waste of time, while by the system- 
 atic resolution into simpler functions the work can be greatly 
 reduced. 
 
 88. In resolving a general periodic function y{d) into a 
 trigonometric series, the most convenient arrangement is: 
 
 1. To separate the constant term a^, by averaging all the 
 instantaneous values of y{d) from to 360 deg. (counting the 
 end values at = and at = 360 deg. one half, as discussed 
 above) : 
 
 ao=avg. |2/(0)!o^ (10) 
 
 and then subtracting a^ from y(0), gives the alternating func- 
 tion, 
 
TRIGONOMETRIC SERIES. 
 
 125 
 
 2. To resolve the general alternating function yo(d) into 
 the odd function yi{6), and the even function y2(d), 
 
 yiid) = h{yo(d)-yo(G+^)}; .... (23) 
 y2{d) = i{yo(d)+yoid+^)\ (22) 
 
 3. To resolve yi(0) gnd y2{d)) into the cosine terms u and 
 the sine terms v, 
 
 M0)=Myiid)+yi{-d)\]] 
 vi(0) = Uyi(s)~yii-0)\;\' 
 U2i0) = h{y2{0)+y2i-d)\;] 
 V2ie)=i\y2ie)-y2(-e)\.\' 
 
 4. To calculate the constants Oi, a2, as. 
 by the averages, 
 
 n 
 
 a„ = 2avg. (M„cosne)o2; 
 
 ^ . ... (<5«) 
 6„ = 2 avg.(t)„sin nd)^^^ 
 
 If the periodic function is known to contain no even har- 
 monics, that is, is a symmetrical alternating wave, steps 1 and 
 2 are omitted. 
 
 . . . (34) 
 
 . . .- (35) 
 ii, h, h- ■ ■ 
 
 Fig. 45. Mean Daily Temperature at Schenectady. 
 
 89. As illustration of the resolution of a general periodic 
 wave may be shown the resolution of the observed mean daily 
 temperatures of Schenectady throughout the year, as shown 
 in Fig. 45, up to the 7th harmonic* 
 
 * The numerical values of temperature camiot claim any great absolute 
 accuracy, as they are averaged over a relatively small number of years only, 
 and observed by instruments of only moderate accuracy. For the purpose 
 of illustrating the resolution of the empirical curve into a trigonometric 
 series, this is not essential, however. 
 
126 
 
 ENGINEERING MATHEMATICS. 
 Table IV 
 
 (1) 
 
 (2) 
 1/ 
 
 (3) 
 y — ao = yo 
 
 
 (5) 
 2/2 
 
 Jan. 
 
 
 10 
 20 
 
 - 4.2 
 
 - 4.7 
 
 - 5.2 
 
 -12.95 
 -13.45 
 -13.95 
 
 -13.10 
 -13.55 
 -13.65 
 
 + 0.15 
 + 0.10 
 -0.30 
 
 Feb. 
 
 30 
 40 
 50 
 
 - 5.4 
 
 - 3.8 
 
 - 2.6 
 
 -14.15 
 -12.55 
 -11.35 
 
 -13.55 
 -12.35 
 -11.20 
 
 -0.60 
 -0.20 
 -0.15 
 
 Mar. 
 
 60 
 70 
 80 
 
 - 1.6 
 + 0,2 
 + 1.8 
 
 -10.35 
 
 - 8.55 
 
 - 6.95 
 
 - 9.75 
 
 - 7.65 
 
 - 6.05 
 
 -0.60 
 -0.90 
 -0.90 
 
 Apr. 
 
 90 
 100 
 110 
 
 + 5.1 
 + 9.1 
 + 11.5 
 
 - 3.65 
 + 0.35 
 + 2.75 
 
 - 3.35 
 
 - 0.35 
 + 1.75 
 
 -0.30 
 + 0.70 
 + 1.00 
 
 May- 
 
 120 
 130 
 140 
 
 + 13.3 
 
 + 15.2 
 + 17.7 
 
 + 4.55 
 + 6.45 
 + 8.95 
 
 + 3 90 
 + 5.85 
 + 8.15 
 
 + 0.65 
 + 0.60 
 + 0.80 
 
 June 
 
 150 
 160 
 170 
 
 + 19.2 
 + 19.5 
 + 20.6 
 
 + 10.45 
 + 10.75 
 + 11.85 
 
 + 10.10 
 + 10.80 
 + 12.15 
 
 + 0.35 
 -0.05 
 -0.30 
 
 July 
 
 180 
 190 
 200 
 
 + 22.0 
 + 22.4 
 + 22.1 
 
 + 13.25 
 + 13.65 
 + 13.35 
 
 
 
 Aug. 
 
 210 
 220 
 230 
 
 + 21.7 
 + 20.9 
 + 19.8 
 
 + 12.95 
 + 12.15 
 + 11.05 
 
 
 
 Sept. 
 
 240 
 250 
 260 
 
 + 17.9 
 + 15.5 
 + 13.8 
 
 + 9.15 
 + 6.75 
 + 5.15 
 
 
 
 Oct. 
 
 270 
 280 
 290 
 
 + 11.8 
 + 9.8 
 + 8.0 
 
 + 3.05 
 + 1.05 
 - 0.75 
 
 
 
 Nov. 
 
 300 
 310 
 320 
 
 + 5.5 
 + 3.5 
 + 1.4 
 
 - 3.25 
 
 - 5.25 
 
 - 7.35 
 
 
 
 Dec. 
 
 330 
 340 
 350 
 
 - 1.0 
 
 - 2.1 
 
 - 3.7 
 
 - 9.75 
 -10.85 
 -12.45 
 
 
 
 Total 
 
 Divided by 36 . 
 
 315.1 
 
 8.75 = a„ 
 
 
 
 
TRIGONOMETRIC SERIES. 
 Table V. 
 
 127 
 
 (1) 
 e 
 
 (2) 
 
 (3) 
 m 
 
 (4) 
 
 VI 
 
 (5) 
 !/2 
 
 (6) 
 
 U2 
 
 (7) 
 
 Vi 
 
 -90 
 -80 
 -70 
 
 + 3.35 
 + 0.35 
 - 1.75 
 
 
 
 -0.30 
 + 0.70 
 + 1.00 
 
 + 0.65 
 + 0.60 
 + 80 
 
 
 
 
 
 
 
 -60 
 -50 
 -40 
 
 - 3.90 
 
 - 5.85 
 
 - 8.15 
 
 
 
 
 
 
 
 
 
 -3.0 
 -20 
 -10 
 
 - 
 
 + 10 
 + 20 
 
 -10.10 
 -10.80 
 -12.15 
 
 -13.10 
 -13.55 
 -13.65 
 
 
 
 + 0.35 
 -0.05 
 -0.30 
 
 + 0.15 
 + 0.10 
 -0.30 
 
 + 0.15 
 -0.10 
 -0.17 
 
 
 
 + 0.20 
 -0.12 
 
 
 
 
 
 -13.10 
 -12.85 
 -12.23 
 
 
 -0.70 
 -1.42 
 
 + 30 
 + 40 
 + 50 
 
 -13.55 
 -12.35 
 -11.20 
 
 -11.82 
 -10.25 
 - 8.53 
 
 -1.73 
 -2.10 
 -2.67 
 
 -0.60 
 -0.20 
 -0.15 
 
 -0.12 
 + 0.30 
 + 0.22 
 
 -0.47 
 -0.50 
 -0.37 
 
 + 60 
 + 70 
 + 80 
 
 - 9.75 
 
 - 7.65 
 
 - 6.05 
 
 - 6.82 
 
 - 4.70 
 
 - 2.85 
 
 -2.93 
 -2.95 
 -3.20 
 
 -0.60 
 -0.90 
 -0.90 
 
 + 0.02' 
 + 0.05 
 -0.10 
 
 -0.62 
 -0.95 
 -0.80 
 
 + 90 
 
 - 3.35 
 
 
 
 -3.35 
 
 -0.30 
 
 -0,30 
 
 
 
128 
 
 ENGINEERING MATHEMATICS. 
 
 
 
 
 -+■1 
 
 
 
 .101 
 .011 
 .022 = 0, 
 
 
 
 3 « 
 
 X 
 
 lO CM to 
 rt o o 
 
 ai 1-1 CO 
 
 CO 1-1 CO 
 IM O O 
 
 I^ Oi to 
 CO r^ r^ r^ 
 
 o o c^ ° 
 
 
 
 3 
 
 o o o 
 
 1 1 1 
 
 O O o 
 
 1 1 + 
 
 o o o 
 
 + 1 + 
 
 O O o 
 
 1 1 ) 
 
 
 
 Hm 
 
 
 
 o 
 
 00 II 
 1-1 to ^ 
 
 §83 
 
 
 
 
 X 
 
 lO lO r^ 
 
 1-1 o o 
 
 1^ w ^ 
 t^ to CO 
 C<1 o o 
 
 ^ CO CO 
 
 r^ (M oi o 
 
 O w (M ° 
 
 
 
 3 
 
 o c o 
 
 1 1 1 
 
 o o o 
 
 + 1 + 
 
 O o o 
 
 + + 1 
 
 O O o 
 
 + + + 
 
 
 gs 
 
 0.15 
 
 0.085 
 
 0.085 
 
 32 
 
 0.065 
 
 0.245 
 
 0.15 
 0.125 
 0.385 
 
 
 
 
 
 
 1 1 + 
 
 1 + 1 
 
 + + 1 
 
 
 
 CO 
 
 0.33 
 
 0.285 
 
 0.165 
 
 
 0.165 
 
 0.285 
 
 0.33 
 0.285 
 0.165 
 
 
 
 
 »-H 
 
 a 
 
 + 
 
 1 1 
 
 1 1 1 
 
 
 
 
 X 
 
 OO t^ (M 
 
 1—1 I-H 1—1 
 
 to to 
 
 O O -it 
 
 00 Tt< r~ o 
 w i-( oq ^ 
 
 e 
 
 II 
 
 CO Ttl CO 
 T)i to CO 
 
 I-H 
 
 
 o o o 
 
 o o 
 
 o o o 
 
 1-1 1-1 O 
 
 > 
 
 CO 
 
 3 
 
 + + + 
 
 + + 
 
 + + + 
 
 + + + 
 
 ^i 
 
 
 1 
 
 
 
 
 l-l 
 
 PQ 
 < 
 
 
 gg 
 
 + 0.18 
 -0.20 
 + 0.25 
 
 (M O CO 
 CO 1-1 lO 
 
 6 6 6 
 1 1 1 
 
 CO O lO 
 i-( 1— 1 lO 
 
 6 6 6° 
 
 I 1 1 
 
 
 
 <5> 
 
 00 »o 00 
 (N O Tl< 
 
 O lO o 
 lO 1-1 o 
 
 tH T)< O 
 to O CO 
 
 
 
 
 
 CO CO (N 
 I— * i-< 1-1 
 
 1 1 1 
 
 1-1 O 00 
 
 1 1 ' 
 
 CD Tl< (N O 
 1 1 1 
 
 
 
 
 <t, 
 
 -** 
 
 X 
 
 
 
 1 
 
 1 ? 
 
 
 
 3 
 
 -13.10( 
 
 -12.65 
 
 -11.50 
 
 lo CO !r> 
 
 CM 00 ** 
 
 d t^ «^ 
 
 7 ' ' 
 
 CO .-H d 
 1 1 1 
 
 -59.75 
 -6.64 
 -13.28 = 
 
 
 
 So 
 
 lO o 
 
 CO •* 
 
 to to CO 
 to to -^ 
 00 t^ to 
 
 CM rtH 
 »0 CO I-H O 
 
 
 
 
 
 
 
 O 
 
 woo 
 
 o o o 
 
 O O O 
 
 
 o- 
 > 
 
 
 
 
 '>* 3 
 
 -13.10 
 -12.85 
 -12.23 
 
 -11.82 
 
 -10.25 
 
 -8.53 
 
 CI O lO 
 
 v: t-^ GO 
 
 CD T" CM O 
 1 1 ■ 
 
 <M 
 
 ^1 
 
 
 i^^ 
 
 o o o 
 1-1 c^ 
 
 o o o 
 
 CO Tfl lO 
 
 O o o o 
 
 CD t- GO Ci 
 
 o 
 H 
 
 V 
 S 
 
 
 
TRIGONOMETRIC SERIES. 
 
 129 
 
 
 
 2 
 
 
 
 
 
 
 
 
 
 CiS 
 
 
 
 
 ^ 
 t^ 
 
 
 
 
 
 
 
 
 X 
 
 
 
 II 
 
 ^s 
 
 
 
 
 lO 
 
 
 
 
 lO ^-- 
 
 
 
 CM 
 
 o 
 
 CO 
 
 o 
 
 CO 
 
 >o 
 
 r-* 
 
 •^ 
 
 CO ^ CO 
 
 t^ 
 
 ■H 
 
 CC 
 
 .-1 to 
 
 (N 
 
 1— 1 
 
 o 
 
 I—* 
 
 o 
 
 o 
 
 O O t-i 
 
 CC 
 
 <= 
 
 o 
 
 ?1 
 
 
 o 
 
 1 
 
 o 
 
 1 
 
 o 
 
 1 
 
 o 
 
 o 
 
 + 
 
 o 
 1 
 
 o o o 
 
 + + + 
 
 e; 
 
 1 
 
 c 
 
 1 
 
 o 
 
 1 
 
 
 
 
 
 
 
 
 
 ^ ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 .^M 
 
 
 
 •o 
 
 <& 
 
 
 
 
 
 
 
 
 X 
 
 
 
 -o 
 
 ^"^ 
 
 
 
 
 lO 
 
 
 
 
 to -^ 
 
 
 ■rt 
 
 II 
 
 o a 
 
 o 
 
 01 
 
 CO 
 
 CO 
 
 to 
 
 to 
 
 ^ 
 
 N (M CD 
 
 c 
 
 ■^ 
 
 en 
 
 Cm 
 
 1 — t 
 
 tM 
 
 o 
 
 o 
 
 o 
 
 o 
 
 O O ^ 
 
 ■^ 
 
 C 
 
 o 
 
 ?i 
 
 
 o 
 
 1 
 
 O 
 1 
 
 o 
 
 + 
 
 o 
 
 1 
 
 o 
 
 + 
 
 o 
 
 + 
 
 O O O 
 1 1 1 
 
 1 
 
 c 
 
 1 
 
 o 
 1 
 
 
 
 lO 
 
 CO 
 
 I^ 
 
 lO 
 
 lO 
 
 to 
 
 i-H Trt^ CO 
 
 
 
 
 
 
 CA 
 
 1— 1 
 
 o 
 
 .— I 
 
 o 
 
 o 
 
 I— 1 O •— 1 
 
 
 
 
 ss 
 
 o 
 
 o 
 1 
 
 o 
 
 o 
 
 + 
 
 o 
 
 o 
 
 1 
 
 o 
 
 1 
 
 o o o 
 
 + 1 I 
 
 
 
 
 <to 
 
 
 r^ 
 
 IM 
 
 Tt< 
 
 (M 
 
 r>- 
 
 
 I^ CM TtH 
 
 
 
 
 CO 
 
 
 o 
 
 t-t 
 
 T— < 
 
 t— 1 
 
 o 
 
 
 O »— 1 I— 1 
 
 
 
 
 S.S 
 
 o 
 
 o 
 
 o 
 
 o 
 
 o 
 
 o 
 
 o 
 
 O O O 
 
 
 
 
 ^ 
 
 
 1 
 
 1 
 
 1 
 
 ' 
 
 1 
 
 
 + + + 
 
 
 
 
 
 
 
 
 
 
 
 
 ^,^ 
 
 
 
 
 
 
 
 
 
 
 
 
 -ti 
 
 
 
 -n 
 
 <c. 
 
 
 
 
 
 
 
 
 X 
 
 
 
 II 
 
 CO 
 
 
 
 
 
 
 
 
 
 
 
 
 o 
 
 to 
 
 ^ 
 
 1^ 
 
 CO 
 
 CO 
 
 o 
 
 03 t^ CM 
 
 u: 
 
 r~- 
 
 •* 
 
 tH 
 
 CM 
 
 ID 
 
 o 
 
 o 
 
 o o o 
 
 cc 
 
 c 
 
 »-H 
 
 S 
 
 
 o 
 
 o 
 
 o 
 
 o 
 
 o 
 
 
 o o o 
 
 c 
 
 (= 
 
 o 
 
 
 
 1 
 
 1 
 
 1 
 
 + 
 
 1 
 
 
 1 1 + 
 
 ' 
 
 1 
 
 ' 
 
 
 
 (N 
 
 on 
 
 w 
 
 CO 
 
 CM 
 
 lO 
 
 OC 00 c^ 
 
 
 
 
 ^^ 
 
 
 CC 
 
 CM 
 
 o 
 
 o 
 
 r-l 
 
 o 
 
 rH O O 
 
 
 
 
 ss 
 
 o 
 
 O 
 
 O 
 1 
 
 o 
 
 1 
 
 o 
 
 + 
 
 o 
 
 1 
 
 o 
 
 1 
 
 o o o 
 
 + + 1 
 
 
 
 
 ta 
 
 
 no 
 
 ^ 
 
 CD 
 
 CO 
 
 ira 
 
 en 
 
 CO 00 CO 
 
 
 
 
 
 
 m 
 
 1— ( 
 
 O 
 
 I— < 
 
 lO 
 
 rxi 
 
 ^ C^ CO 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 s-« 
 
 o 
 
 o 
 
 »— < 
 
 tH 
 
 (M 
 
 CM 
 
 CM 
 
 CO CO CO 
 
 
 
 
 ^ 
 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 1 1 
 
 
 
 
 
 
 
 
 
 
 
 
 ^-^ 
 
 
 
 
 
 
 
 
 
 
 
 
 -^ 
 
 
 
 M 
 
 OS 
 
 
 
 
 
 
 
 
 X 
 
 c 
 
 
 fO 
 
 d 
 
 
 tN 
 
 lO 
 
 IC 
 
 lO 
 
 O 
 
 o 
 
 o o ^-- 
 
 CO lO II 1 
 
 
 o 
 
 fN 
 
 en 
 
 CO 
 
 Tl^ 
 
 -:*< 
 
 CO 
 
 t^ LO lO 
 
 O cc CO 1 
 
 <.)H W 
 
 r-i 
 
 ■* 
 
 ai 
 
 CO 
 
 o 
 
 lO 
 
 I> ^ CO 
 
 
 CO CO 1 
 
 > 
 
 
 
 
 
 
 
 
 
 Tl 
 
 
 • 
 
 
 
 o 
 
 o 
 1 
 
 o 
 i 
 
 1-H 
 
 1 
 
 CM 
 1 
 
 CM 
 
 1 
 
 CM CO CO 
 
 1 1 1 
 
 7 
 
 1 
 
 CO 
 
 1 
 
 
 
 ■^ 
 
 ra 
 
 
 CO 
 
 CO 
 
 CO 
 
 O lO 
 
 
 
 
 
 
 
 r^ 
 
 ■* 
 
 
 ^ 
 
 CO 
 
 CO 
 
 TfH CO 
 
 
 
 
 
 CO c 
 
 o 
 
 t-H 
 
 CO 
 
 lO 
 
 CO 
 
 r^ 
 
 »J 
 
 CT C5 rt 
 
 
 
 
 
 M 
 
 
 o 
 
 o 
 
 o 
 
 o 
 
 o 
 
 o 
 
 o o 
 
 
 
 
 
 
 
 o 
 
 IN 
 
 CO 
 
 o 
 
 t^ 
 
 CO 
 
 lO o »o 
 
 
 
 CM 
 
 
 
 t^ 
 
 Tl< 
 
 t^ 
 
 f-( 
 
 CO 
 
 en 
 
 en CM CO 
 
 
 c 
 
 >> 
 
 ^^ 
 
 o 
 
 o 
 
 1 
 
 T— 1 
 
 ■ 1 
 
 CM 
 
 1 
 
 CM 
 
 1 
 
 CM 
 1 
 
 CM CO CO 
 
 1 1 1 
 
 
 > 
 
 nz 
 
 J3 
 
 1- 
 
 
 
 
 
 
 
 
 
 
 _ 
 
 
 o 
 
 o 
 
 O 
 
 o 
 
 o 
 
 o 
 
 O 
 
 o o o 
 
 oi 
 
 's 
 
 +3 
 
 
 I— ( 
 
 (N 
 
 CO 
 
 -f 
 
 •a 
 
 CO 
 
 t^ c» en 
 
 c 
 
 
 .-I 
 
 
 
 
 
 
 
 
 
 
 fr 
 
 c 
 
 S 
 
1^0 
 
 ENGINEERING MATHEMATICS. 
 
 Table VIII. 
 COSINE SERIES u^. 
 
 CD 
 
 9 
 
 (2) 
 
 U2 
 
 (3) 
 
 1/2 COS 20 
 
 (4) 
 02 COS 2 e 
 
 (B) (6) 
 m ui COS iO 
 
 (7) 
 
 (14 003 45 
 
 (8) 
 
 U6 
 
 ui cos 60 
 
 
 10 
 20 
 
 + 0.15 
 -0.10 
 -0.17 
 
 M + 0.15) 
 -0.09 
 -0.13 
 
 
 
 + 0.15 
 -0.10 
 -0.17 
 
 K + 0.15) 
 -0.08 
 -0.03 
 
 -0.16 
 -0.12 
 -0.03 
 
 + 0.31 
 + 0.02 
 -0.14 
 
 i( + 0.31) 
 + 0.01 
 + 0.07 
 
 30 
 40 
 50 
 
 -0.12 
 + 0.30 
 + 0.22 
 
 -0.06 
 + 0.05 
 -0.04 
 
 
 -0.12 
 + 0.30 
 + 0.22 
 
 + 0.06 
 -0.29 
 -0.21 
 
 + 0.08 
 + 0.15 
 + 0.15 
 
 -0.20 
 + 0.15 
 + 0.07 
 
 + 0.20 
 -0.07 
 + 0.03 
 
 60 
 70 
 80 
 90 
 
 + 0.02 
 +.0.05 
 -0.10 
 -0.30 
 
 -0.01 
 
 -0.04 
 
 + 0.09 
 
 K + 0.30) 
 
 "o" ' 
 
 + 0.02 
 + 0.05 
 -0.10 
 -0.30 
 
 -0.01 
 
 + 0.01 
 
 -0.08 
 
 K+0.30) 
 
 + 0.08 
 -0.03 
 -0.12 
 -0.16 
 
 -0.06 
 + 0.08 
 + 0.02 
 -0.14 
 
 -0.06 
 + 0.04 
 -0.01 
 
 K + 0.14) 
 
 Total 
 
 Divided by 
 
 9 
 
 Multiplied 
 
 by 2.... 
 
 -0.01 
 
 -0.001 
 
 -0.002 
 
 
 
 -0.71 
 
 -0.079 
 
 -0.158 
 
 
 
 + 0.44 
 + 0.049 
 + 0.098 
 
 
 =(h 
 
 
 
 = "3 
 
 
 
 = "« 
 
 Table IX. 
 
 SINE SERIES V,. 
 
 (1) 
 
 (2) 
 
 (3) 
 
 (4) 
 
 (5) 
 
 (6) 
 
 (7) 
 
 (8) 
 
 (9) 
 
 e 
 
 V2 
 
 V2 sin 26 
 
 fe sin 20 
 
 VI 
 
 v^ sin id 
 
 64 sin 46 
 
 1)6 
 
 ue sin GO 
 
 
 
 
 
 
 
 
 
 
 
 
 
 10 
 
 + 0.20 
 
 + 0.07 
 
 -0.20 
 
 + 0.40 
 
 + 0.26 
 
 + 0.22 
 
 + 0.18 
 
 + 0.16 
 
 20' 
 
 -0.12 
 
 -0.08 
 
 -0.39 
 
 + 0.27 
 
 + 0.27 
 
 + 0.34 
 
 -0.07 
 
 -0.07 
 
 30 
 
 -0.47 
 
 -0.41 
 
 -0.52 
 
 + 0.05 
 
 + 0.04 
 
 + 0.30 
 
 -0.25 
 
 + 
 
 40 
 
 -0.50 
 
 -0.49 
 
 -0.59 
 
 + 0.09 
 
 + 0.03 
 
 + 0.12 
 
 -0.03 
 
 + 0.03 
 
 50 
 
 -0.37 
 
 -0.36 
 
 -0.59 
 
 + 0.22 
 
 -0.08 
 
 -0.12 
 
 + 0.34 
 
 -0.30 
 
 60 
 
 -0.62 
 
 -0.54 
 
 -0.52 
 
 -0.10 
 
 + 0.09 
 
 -0.30 
 
 + 0.20 
 
 
 
 70, 
 
 -0.95 
 
 -0.61 
 
 -0.39 
 
 -0.56 
 
 + 0.55 
 
 -0.34 
 
 -0.22 
 
 -0.19 
 
 80 
 
 -0.80 
 
 -0.27 
 
 -0.20 
 
 -0.60 
 
 + 0.30 
 
 + 0.22 
 
 -0.38 
 
 -0.33 
 
 90 
 
 
 
 
 
 
 
 
 
 
 
 Total 
 
 -2.69 
 
 
 
 + 1.55 
 
 
 
 -0.70 
 
 Divided by 9 
 
 -0.30 
 
 
 
 + 0.172 
 
 
 
 -0.078 
 
 Divided by 2 
 
 -0.60 
 
 
 
 + 0.344 
 
 
 
 -0.156 
 = 6. 
 
TRIGONOMETRIC SERIES. 131 
 
 Table IV gives the resolution of the periodic temperature 
 function into the constant term ao, the odd series yi and the 
 even series j/2. 
 
 Table V gives the resolution of the series t/i and 7/2 into 
 the cosine and sine series ui, Vi, uz, t'2-- 
 
 Tables VI to IX give the resolutions of the series ui, v\, uz, 
 V2, and thereby the calculation of the constants a„ and 6„. 
 
 90. The resolution of the temperature wave, up to the 
 7th harmonic, thus gives the coefficients: 
 
 ao=+8.75; 
 
 ai = -13.28; 61= -3.33; 
 
 a2= -0.001; b2= -0.602; 
 
 03 = -0.33; 63 = -0.14; 
 
 a4= -0.154; 64= +0.386 
 
 a5= +0.014; 65= -0.090 
 
 06= +0.100; b6 = -0.154 
 
 07= -0.022; 67 =-0.082 
 
 or, transforming by the binomial, a„cosn(9+f)nsinn^ = c„cos 
 
 (n^- ^„), by substituting c„=\/a„2+6„2 andtan7-„=— gives, 
 
 ao=+8.75; 
 
 ci=-13.69; n= + 14-15°; or n=+14.15°; 
 
 c2=--0.602; 7-2=+89.9°; or ^^=+44.95°+180n; 
 
 C3=+0.359; r3=-23.0°; or ^' = -7.7+120n= + 112.3+120OT; 
 
 C4=-0.416; 7'4=-68.2°; or ^*=-17.05+90n=+72.95+90m; 
 
 C5=+0.091; 7'5=-81.15°; or ■^^=-16.23+72n= +55.77+72m; 
 
 C6=+0.184; r6=-57.0°; or ^=-9.5+60n=+50.5+60m; 
 
 C7=-0.085; 7-7=+75.0°; or y= + 10.7+51.4n, 
 where n and m may be any integer number. 
 
132 ENGINEERING MATHEMATICS. 
 
 Since to an angle rn, any multiple of 2?: or 360 deg. may 
 be added, any multiple of — may be added to the angle -^, 
 
 and thus the angle — may be made positive, etc.. 
 
 Qi. The equation of the temperature wave thus becomes: 
 
 2/ = 8.75-13.69 cos (0-14.15°)-O.6O2 cos 2(i9-44.95°) 
 -0.359 cos 3(<?- 52.3") -0.416 cos 4(^-72.95°) 
 -0.091 cos 5(^-19.77°) -0.184 cos 6(i9-20.5°) 
 -0.085 cos 7(^-10.7°); (a) 
 
 or, transformed to sine functions by the substitution, 
 cos w=— sin (w — 90°): 
 
 2/ = 8.75 + 13.69 sin ((9-104.15°) +0.602 sin 2((9-89.95°) 
 + 0.359 sin 3(^-82.3°) +0.416 sin 4((9- 95.45°) 
 +0.091 sin 5(61-109.77°) +0.184 sin 6(^-95.5°) 
 +0.085 sin 7((9- 75°). (b) 
 
 The cosine form is more convenient for some purposes, 
 the sine form for other purposes. 
 
 Substituting /? = ^-14.15°; or, 5=^-104.15°, these two 
 equations (a) and (6) can be transformed into the form, 
 
 2/ = 8.75-13.69 cos /3-0.62 cos2(/3-30.8°)-0.359 cos3(/3-38.15°) 
 
 -0.416 cos 4(/3- 58.8°) -0.091 cos 5(/?-5.6°) 
 
 -0.184 cos 6(/?- 6.35°) -0.085 cos 7(/?-48.0°), (c) 
 
 and 
 
 2/-8.75+13.69 sin a+0.602 sin 2(5+14.2°) + 0.359 sin 3(5+21.85°) 
 
 +0.416 sin 4(5+8.7°) +0.91 sin 5(5-5.6°) 
 
 +0.184 sin 6(5+8.65°) +0.085 sin 7(5+29.15°). (d) 
 
 The periodic variation of the temperature y, as expressed 
 by these equations, is a result of the periodic variation of the 
 thermomotive force; that is, the solar radiation. This latter 
 
TRIGONOMETRIC SERIES. 133 
 
 is a minimum on Dec. 22d, that is, 9 time-degrees before the 
 zero of 6, hence may be expressed approximately by: 
 
 z=c-h cos (^+9°); 
 
 or substituting /? respectively ^ for (9: 
 
 2 = c-/icos (/? +23.15°) 
 = c+/isin (.5+23.15°). 
 
 This means: the maximum of y occurs 23.15 deg. after the 
 maximum of z; in other words, the temperature lags 23.15 deg., 
 or about ^ period, behind the thermomotive force. 
 
 Near d=0, all the sine functions in (d) are increasing; that 
 is, the temperature wave rises steeply in spring. 
 
 Near 5 = 180 deg., the sine functions of the odd angles are 
 decreasing, of the even angles increasing, and the decrease of 
 the temperature wave in fall thus is smaller than the increase 
 in spring. 
 
 The fundamental wave greatly preponderates, with ampli- 
 tude ci = 13.69. 
 
 In spring, for 5 =—14.5 deg., all the higher harmonics 
 rise in the same direction, and give the sum 1.74, or 12.7 
 per cent of the fundamental. In fall, for 5=— 14.5+;r, the 
 even harmonics decrease, the odd harmonics increase the 
 steepness, and give the sum —0.67, or —4.9 per cent. 
 
 Therefore, in spring, the temperature rises 12.7 per cent 
 faster, and in autumn it falls 4.9 per cent slower than corre- 
 sponds to a sine wave, and the difTerence in the rate of tempera- 
 ture rise in spring, and temperature fall in autumn thus is 
 12.7+4.9 = 17.6 per cent. 
 
 The maximum rate of temperature rise is 90—14.5 = 75.5 
 deg. behind the temperature minimum, and 23.15+75.5 = 98.7 
 deg. behind the minimum of the thermomotive force. 
 
 As most periodic functions met by the electrical engineer 
 are symmetrical alternating functions, that is, contain only 
 the odd harmonics, in general the work of resolution into a 
 trigonometric series is very much less than in above example. 
 Where such reduction has to be carried out frequently, it is 
 advisable to memorize the trigonometric functions, from 10 
 to 10 deg., up to 3 decimals; that is, within the accuracy of 
 the slide rule, as thereby the necessity of looking up tables is 
 
134 ENGINEERING MATHEMATICS. 
 
 eliminated and the work therefore done much more expe- 
 ditiously. In general, the slide rule can be used for the calcula- 
 tions. 
 
 As an example of the simpler reduction of a symmetrical 
 alternating wave, the reader may resolve into its harmonics, 
 up to the 7th, the exciting current of the transformer, of which 
 the numerical values are given, from 10 to 10 deg. in Table X. 
 
 C. REDUCTION OF TRIGONOMETRIC SERIES BY POLY- 
 PHASE RELATION. 
 
 92. In some cases the reduction of a general periodic func- 
 tion, as a complex wave, into harmonics can be carried out 
 in a much quicker manner by the use of the polyphase equation. 
 Chapter III, Part A (23). Especially is this true if the com- 
 plete equation of the trigonometric series, which represents the 
 periodic' function, is not required, but the existence and the 
 amount of certain harmonics are to be determined, as for 
 instance whether the periodic function contain even harmonics 
 or third harmonics, and how large they may be. 
 
 This method does not give the coefficients a„, hn of the 
 individual harmonics, but derives from the numerical values 
 of the general wave the numerical values of any desired 
 haimonic. This harmonic, however, is given together with all 
 its multiples; that is, when separating the third harmonic, 
 in it appears also the 6th, 9th, 12th, etc. 
 
 In separating the even harmonics 2/2 from the general 
 wave y, in paragraph 84, by taking the average of the values 
 of y for angle 6, and the values of y for angles {0+Tt), this 
 method has already been used. 
 
 Assume that to an angle there is successively added a 
 constant quantity a, thus: d; + a; d + 2a; d + 3a; d + ia, 
 etc., until the same angle d plus a multiple of 2?! is reached; 
 
 + na = d + 2m7i:; that is, a = — — ; or, in other words, a is 
 
 1/n of a multiple of 2k. Then the sum of the cosine as well 
 as the sine functions of ail these angles is zero: 
 
 cos ^-l-cos ((9 + a)+cos (<? + 2a)+cos ((?+3a)+. . 
 
 +C0S (^ + [?i-l]a)=0; (1) 
 
TRIGONOMETRIC SERIES. 135 
 
 sin ^+sin ((?+a)+sin {d+2a) +sin (d+3a)+. . . 
 
 +sin (^+[n-l]o)=0, (2) 
 
 where 
 
 na = 2m.n (3) 
 
 These equations (1) and (2) hold for all values of a, except for 
 a = 2n, or a multiple thereof. For a=2Tt obviously all the terms 
 of equation (1) or (2) become equal, and the sums become 
 n cos d respectively n sin d. 
 
 Thus, if the series of numerical values of y is divided into 
 
 2n 
 n successive sections, each covering — degrees, and these 
 
 lb 
 
 sections added together, 
 
 2/W +2/(^+1) +.(^+2|)+,(^+3|)+... 
 
 + y(o+Jn-lf-^, . . . . . . . (4) 
 
 In this sum, all the harmonics of the wave y cancel by equations • 
 (1) and (2), except the nth harmonic and its multiples, 
 
 a„ cos nd+bn sin n(9j a2n cos 2nO+b2n sin 2nd, etc. 
 
 in the latter all the terms of the sum (4) are equal; that is, 
 the sum (4) equals n times the nth harmonic, and its multiples. 
 Therefore, the nth harmonic of the periodic function y, together i 
 with its multiples, is given by 
 
 yni6)-l 
 
 yC^) +2/(^^1) +2/(^+2^) + -+2/(^+[n-l]^)} (5) 
 
 For instance, for n = 2, 
 
 y2=i\yiO)+yi0+n)], 
 
 gives the sum of all the even harmonics; that is, gives the 
 second harmonic together with its multiples, the 4th, 6th, etc., 
 as seen in paragraph 7, and for, n = 3, 
 
 y:^-l\y(0)+y(o+'^]+y(o+~ 
 
136 
 
 ENGINEERING MATHEMATICS. 
 
 gives the third harmonic, together with its multiples, the 6th, 
 9th, etc. 
 
 This method does not give the mathematical expression 
 of the harmonics, but their numerical values. Thus, if the 
 mathematical expressions are required, each of the component 
 harmonics has to be reduced from its numerical values to 
 the mathematical equation, and the method then usually offers 
 no advantage. 
 
 It is especially suitable, however, where certain classes of 
 harmonics are desired, as the third together with its multiples. 
 In this case from the numerical values the effective value, 
 that is, the equivalent sine wave may be calculated. 
 
 93. As illustration may be investigated the separation of 
 the third harmonics from the exciting current of a transformer. 
 
 Table X 
 
 A 
 
 (1) 
 e 
 
 (2) 
 i 
 
 (3) 
 
 
 (4) 
 i 
 
 (5) 
 
 e 
 
 (6) 
 
 (7) 
 
 13 
 
 
 10 
 20 
 
 30 
 40 
 50 
 
 60 
 
 + 24.0 
 + 20.0 
 + 12 
 
 + 4 
 
 - 1.5 
 
 - 6.5 
 
 - 8.5 
 
 120 
 130 
 140 
 
 150 
 160 
 170 
 
 180 
 
 -15.1 
 -16.5 
 -18.5 
 
 -21 
 
 -22.7 
 -23.7 
 
 -24 
 
 240 
 250 
 260 
 
 270 
 280 
 290 
 
 300 
 
 + 8.5 
 + 10 
 + 11 
 
 + 12 
 + 13 
 + 14 
 
 + 15.1 
 
 + 5.8 
 + 4.5 
 + 1.5 
 
 -1.7 
 -3.7 
 -5.4 
 
 -5.8 
 
 B 
 
 30 
 
 13 
 
 39 
 
 »3 
 
 39 
 
 <3 
 
 it 
 
 
 30 
 60 
 
 + 5.8 
 + 4.5 
 
 + 1.5 
 
 120 
 150 
 180 
 
 -3.7 
 -5.4 
 -5.8 
 
 240 
 270 
 300 
 
 -1.5 
 
 + 1.7 
 + 3.7 
 
 + 0.2 
 + 0.3 
 -0.2 
 
 In table X A, are given, in columns 1, 3, 5, the angles d, 
 from 10 deg. to 10 deg., and in columns 2, 4, 6, the correspond- 
 ing values of the exciting current i, as derived by calculation 
 from the hysteresis cycle of the iron, or by measuring from the 
 
TRIGONOMETRIC SERIES. 
 
 137 
 
 photographic film of the oscillograph. Column 7 then gives 
 one-third the sum of columns 2, 4, and G, that is, the third har- 
 monic with its overtones, iz. 
 
 To find the 9th harmonic and its overtones ig, the same 
 method is now applied to iz, for angle 2,6. This is recorded 
 in Table X B. 
 
 In Fig. 46 are plotted the total exciting current i, its third 
 harmonic iz, and the 9th harmonic ig. 
 
 This method has the advantage of showing the limitation 
 of the exactness of the results resulting from the limited num- 
 
 FiG. 46. 
 
 ber of numerical values of i, on which the calculation is based. 
 Thus, in the example, Table X, in which the values of i are 
 given for every 10 deg., values of the third harmonic are derived 
 for every 30 deg., and for the 9th harmonic for every 90 deg.; 
 that is, for the latter, only two points per half wave are deter- 
 minable from the numerical data, and as the two points per half 
 wave are just sufficient to locate a sine wave, it follows that 
 within the accuracy of the given numerical values of i, the 
 9th harmonic is a sine wave, or in other words, to determine 
 whether still higher harmonics than the 9th exist, requires for 
 i more numerical values than for every 10 deg. 
 
 As further practice, the reader may separate from the gen- 
 
138 
 
 ENGINEERING MATHEMATICS. 
 
 eral wave of current, lo in Table XI, the even harmonics i2, 
 by above method, 
 
 t2 = Jh'o(^)+to(^+180deg.)!, 
 and also the sum of the odd harmonics, as the residue, 
 
 ll =10 — 12, 
 
 then from the odd harmonics z'l may be separated the third 
 harmonic and its multiples, 
 
 i:3 = ^Ki('9)+ii(^+120deg. )+ti(^+240deg.)!, 
 
 and in the same manner from {3 may be separated its third 
 harmonic; that is, ig. 
 
 Furthermore, in the sum of even harmonics, from I'a may 
 again be separated its second harmonic, ii, and its multiples, 
 and therefrom, ig, and its third harmonic, ie, and its multiples, 
 thus giving all the harmonics up to the 9th, with the exception 
 of the 5th and the 7th. These latter two would require plotting 
 the curve and taking numerical values at different intervals, 
 so as to have a number of numerical values divisible by 5 or 7. 
 
 It is further recommended to resolve this unsymmetrical 
 exciting current of Table XI into the trigonometric series by 
 calculating the coefficients a„ and b„, up to the 7th, in the man- 
 ner discussed in paragraphs 6 to 8. 
 
 Table XI 
 
 
 
 to 
 
 e 
 
 10 
 
 e 
 
 10 
 
 e 
 
 to 
 
 
 10 
 20 
 
 + 95.7 
 + 78.7 
 + 53.7 
 
 90 
 100 
 110 
 
 -26.7 
 -27.3 
 -28.1 
 
 180 
 190 
 200 
 
 -34.3 
 -27.3 
 -16.8 
 
 270 
 280 
 290 
 
 - 3.3 
 
 - 1.8 
 + 1.2 
 
 30 
 40 
 50 
 
 + 23.7 
 - 2.3 
 -16.3 
 
 120 
 130 
 140 
 
 -28.8 
 -29.3 
 -29.8 
 
 210 
 220 
 230 
 
 -11.3 
 
 - 8.3 
 
 - 7.3 
 
 300 
 310 
 320 
 
 + 4.7 
 + 10.7 
 
 + 22.7 
 
 60 
 70 
 80 
 
 -22.8 
 -24.3 
 .-25.8 
 
 150 
 160 
 170 
 
 -31 
 
 -32.6 
 
 -33.8. 
 
 240 
 250 
 260 
 
 - 6.3 
 
 - 5.3 
 
 - 4 3 
 
 330 
 340 
 350 
 
 + 41.7 
 + 65.7 
 + 85.7 
 
TRIGONOMETRIC SERIES. 139 
 
 D. CALCULATION OF TRIGONOMETRIC SERIES FROM 
 OTHER TRIGONOMETRIC SERIES. 
 
 94' An hydraulic generating station has for a long time been 
 supplying electric energy over moderate distances, from a num- 
 ber of 750-kw. 440Q-volt 60-cycle three-phase generators. The 
 station is to be increased in size by the installation of some 
 larger modern three-phase generators, and from this station 
 6000 k\v. are to be transmitted over a long distance transmis- 
 sion line at 44,000 volts. The transmission line has a length 
 of 60 miles, and consists of three wires No. OB. & S. with 5 
 ft. between the wires. 
 
 The question arises, whether during times of light load the 
 old 750-kw. generators can be used economically on the trans- 
 mission line. These old machines give an electromotive force 
 wave, which, like that of most earlier machines, differs con- 
 siderably from a sine wave, and it is to be investigated, whether, 
 due to this wave-shape distortion, the charging current of the 
 transmission line wiU be so greatly increased over the value 
 which it would have with a sine wave of voltage, as to make 
 the use of these machines on the transmission hne uneconom- 
 ical or even unsafe. 
 
 Oscillograms of these machines, resolved into a trigonomet- 
 ric series, give for the voltage between each terminal and the 
 neutral, or the Y voltage of the three-phase system, the equa- 
 tion: 
 
 e = eo{sin 61-0.12 sin (3(9-2. 3°)-0. 23 sin (5^-1.5°) 
 
 +0.13 sin (7^-6.2°)}. . (1) 
 
 In first approximation, the line capacity may be considered 
 as a condenser shunted across the middle of the line; that is, 
 half the hne resistance and half the hne reactance is in series 
 with the line capacity. 
 
 As the receiving apparatus do not utilize the higher har- 
 monics of the generator wave, when using the old generators, 
 their voltage has to be transformed up so as to give the first 
 harmonic or fundamental of 44,000 volts. 
 
 44,000 volts between the lines (or delta) gives 44,000 h- V3 = 
 25,400 volts between hne and neutral. This is the effective 
 
140 ENGINEERING MATHEMATICS. 
 
 value, and the maximum value of the fundamental voltage 
 wave thus is: 25,400 X \/2 = 36,000 volts, or 36 kv.; that is, 
 eo = 36, and 
 
 e = 36{sin (9-0.12 sin (35-2.3°)-0.23 sin (5^-1.5°) 
 
 + 0.13 sin (7^-6.2°)!, . (2) 
 
 would be the voltage supplied to the transmission line at the 
 high potential terminals of the step-up transformers. 
 
 From the wire tables, the resistance per mile of No. B. & S. 
 copper hne wire is ro=0.52 ohm. 
 
 The inductance per mile of wire is given by the formula : 
 
 Lo = 0.7415 log ^ + 0.0805mh, .... (3) 
 
 where k is the distance between the wires, and l^ the radius of 
 the wire. 
 
 In the present case, this gives Z, = 5 ft. = 60 in. 1^ = 0. 1625 in. 
 Lo = l .9655 mh., and, herefrom it follows that the reactance, at 
 /=60 cycles is 
 
 xo = 27r/Lo = . 75 ohms per mile (4) 
 
 The capacity per mile of wire is given by the formula : 
 
 Co= — p"if-; (5) 
 
 hence, in the present case, Co =0.0159 mf., and the condensive 
 reactance is derived herefrom as : 
 
 = 166000 ohms; .... (6) 
 
 ■^^ 2rfCo 
 60 miles of line then give the condensive reactance, 
 
 X 
 
 a-<. = TTj-) = 2770 ohms; 
 
 30 miles, or half the line (from the generating station to the 
 middle of the line, where the line capacity is represented by a 
 shunted condenser) give: the resistance, r = 30ro= 15.6 ohms; 
 
TRIGONOMETRIC SERIES. 141 
 
 the inductive reactance, x = 30xo = 22 . 5 ohms, and the equiva- 
 lent circuit of the line now consists of the resistance r, inductive 
 reactance x and condensive reactance x^, in series with each 
 other in the circuit of the supply voltage e. 
 
 95. If i= current in the hne (charging current) the voltage 
 consumed by the line resistance r is ri. 
 
 di 
 The voltage consumed by the inductive reactance x \s x-j^; 
 
 the voltage consumed by the condensive reactance Xc is Xc I idd, 
 
 and therefore, 
 
 di f 
 e = x-j^-\-ri-\-Xc\ idd (7) 
 
 Differentiating this equation, for the purpose of eliminating 
 the integral, gives 
 
 de dH di 
 
 dd^''dm^''do^''''''' 
 
 or [. . . . (8) 
 
 § = 22.5^ + 16.6 1 +2770. 
 
 The voltage e is given by (2), which -equation, by resolving 
 the trigonometric functions, gives 
 
 e = 36 sin (9-4.32 sin 3^-8.28 sin 50+4.64 sin 7^ 
 
 +0.18 cos 30+0.22 cos 55-0.50 cos 70; . (9) 
 
 hence, differentiating, 
 
 de 
 
 ^ = 36 cos (9-12.96 cos 30-41.4 cos 50+32.5 cos 70 
 dd 
 
 -0.54 sin 30-1.1 sin 50+3.5 sin 70. , (10) 
 
 Assuming now for the current i a tiigonometric series with 
 indeterminate coefficients, 
 
 i = Oi cos 0+a3 cos 30+ as cos 50 +07 cos 70 
 
 4 ?>i sin +6,3 sin 30 +65 sin 50 +h sin 70, . (11) 
 
142 
 
 ENGINEERING MATHEMATICS. 
 
 substitution of (10) and (11) into equation (8) must give an 
 identity, from which equations for the determination of a„ and 
 bn are derived; that is, since the product of substitution must 
 be an identity, all the factors of cos d, sin 6, cos 3d, sin 3d, 
 etc., must vanish, and this gives the eight equations: 
 
 36 =2770ai+ 15.66i- 22.5ai;l 
 =27706i- 15.6ai- 22.5?>i; 
 -12.96 = 2770a3+ 46.863- 202. Saa; 
 
 
 - 0.54 = 2770?)3- 46.8a3- 202. S&s; 
 -41.4 =2770a5+ 7865- 562. Sag; 
 
 - 1.1 =2770&5- 78a5- 56.2565; 
 32.5 =277007+109.267-1102.507; 
 
 3.5 =277067-109.207-1102.567. J 
 
 Resolved, these equations give 
 
 01= 13.12 
 61= 0.07 
 03=- 5.03 
 63 = - 0.30 
 05= -18.72 
 65=- 1.15 
 07= 19.30 
 67= 3.37 
 hence, 
 
 2 = 13.12cos6»-5.03cos3^-18.72cos5» + 19.30cos7(? ] 
 +0.07 sin (9-0. 30 sin 3^-1. 15 sin 5^+3.37 sin 7^ 
 = 13.12 cos ((9-0.3°)-5.04 cos (3^-3.3°) 
 -18.76 cos (5(9-3.6°) +19.59 cos (7^-9.9°). 
 
 (12) 
 
 (13) 
 
 (14) 
 
TRIGONOMETRIC SERIES. 143 
 
 96. The effective value of this current is given as the square 
 root of the sum of squares of the effective values of the indi- 
 vidual harmonics, thus : 
 
 '=^/2f-2:^2'■« 
 
 amp. 
 
 As the voltage between line and neutral is 25,400 effective, 
 this gives Q = 25,400X21. 6 = 540,000 volt-amperes, or 540 kv.- 
 amp. per line, thus a total of 3Q = 1620 kv.-amp. charging cur- 
 rent of the transmission line, when using the e.m.f. wave of 
 these old generators. 
 
 It thus would require a minimum of 3 of the 750-kw. 
 generators to keep the voltage on the line, even if no power 
 whatever is dehvered from the line. 
 
 If the supply voltage of the transmission line were a perfect 
 sine wave, it would, at 44,000 volts between the hnes, be given 
 
 by 
 
 ei = 36sin0, . . . (15) 
 
 which is the fundamental, or first harmonic, of equation (9). 
 
 Then the current i would also be a sine wave, and would be 
 given by 
 
 ti = oi cos d+bi sin 6, 
 
 = 13.12 cos ^+0.07 sin d, 
 = 13.12 cos (^-0.3°), 
 
 and its effective value would be 
 
 (16) 
 
 T 13.12 „ „ 
 
 /i=— 7=- = 9.3 amp (17) 
 
 This would correspond to a kv.-amp. input to the line 
 
 3Qi = 3 X 25.4 X 9.3 = 710 kv.-amp. 
 
 The distortion of the voltage wave, as given by equation (1), 
 thus increases the charging volt-amperes of the line from 710 
 
144 
 
 ENGINEERING MATHEMATICS. 
 
 kv.-amp. to 1620 kv.-amp. or 2.28 times, and while with a sine 
 wave of voltage, one of the 750-kw. generators would easily be 
 able to supply the charging current of the line, due to the 
 
 Fig. 47. 
 
 wave shape distortion, more than two generators are required. 
 It would, therefore, not be economical to use these generators 
 on the transmission Hne, if they can be used for any other 
 purposes, as short-distance distribution. 
 
 Fig. 48. 
 
 In Figs. 47 and 48 are plotted the voltage wave and thp 
 current wave, from equations (9) and (14) respectively, and 
 
TRIGONOMETRIC SERIES. 145 
 
 the numerical values, from 10 deg. to 10 deg., recorded in 
 Table XII. 
 
 In Figs. 47 and 48 the fundamental sine wave of voltage 
 and current are also shown. As seen, the distortion of current 
 is enormous, and the higher harmonics predominate over the 
 fundamental. Such waves are occasionally observed as charg- 
 ing currents of transmission lines or cable systems. 
 
 97. Assuming now that a reactive coil is inserted in series 
 with the transmission line, between the step-up transformers 
 and the line, what will be the voltage at the terminals of this 
 reactive coil, with the distorted wave of charging current 
 traversing the reactive coil, and how does it compare with the 
 voltage existing with a sine wave of charging current? 
 
 Let L= inductance, thus x = 27r/L = reactance of the coil, 
 and neglecting its resistance, the voltage at the terminals of 
 the reactive coil is given by 
 
 ^'—4 ^i«) 
 
 Substituting herein the equation of current, (11), gives 
 e' = x{ai sin d+Saa sin Sd + das sin 5d+7ar sin 76 "j 
 — bi cos d—Sbs cos 3(9— 565 cos 55— 767 cos 75 1 ; J 
 hence, substituting the numerical values (13), 
 e' = x{ 13.12 sin 6- 15.09 sin 35-93.6 sin 55+135.1 sin 75 
 -0.07 cos 5 +0.90 cos 35+5.75 cos 55-23.6 cos 75) 
 = xj 13.12 sin (5-0.3°) -15.12 sin (35-3.3°) 
 -93.8 sin (55-3.6°)+139.1 sin (75-9.9°)!. 
 This voltage gives the effective value 
 
 (19) 
 
 (20) 
 
 iJ' = a;\/i{ 13.122 + 15.122+93.82 +139.12} =119.4.r, 
 while the effective value with a sine wave would be from (17), 
 
 hence, the voltage across the reactance x has been increased 
 12.8 times by the wave distortion. 
 
146 
 
 ENGINEERING MATHEMATICS. 
 
 The instantaneous values of the voltage e' are given in the 
 last column of Table XII, and plotted in Fig. 49, for x = l. 
 As seen from Fig. 49, the fundamental wave has practically 
 
 Fig. 49. 
 
 vanished, and the voltage wave is the seventh harmonic, modi- 
 fied by the fifth harmonic. 
 
 Table XII 
 
 
 
 e 
 
 i 
 
 e' 
 
 
 
 C 
 
 i 
 
 e' 
 
 
 10 
 20 
 
 -0.10 
 
 + 2.23 
 
 3.74 
 
 + 8.67 
 + 5.30 
 - 0.86 
 
 - 17 
 + 46 
 + 3 
 
 90 
 100 
 110 
 
 27.41 
 31.77 
 40.57 
 
 - 4.15 
 4 26.19 
 + 24.99 
 
 -200 
 -106 
 + 119 
 
 30 
 
 40 
 50 
 
 7.47 
 17.35 
 31.70 
 
 + 7.39 
 + 30.39 
 
 + 38.58 
 
 + 131 
 -116 
 + 36 
 
 120 
 130 
 140 
 
 42.70 
 33.14 
 18.03 
 
 - 8.10 
 -38.79 
 -36.65 
 
 + 182 
 + 93 
 - 96 
 
 60 
 70 
 80 
 
 42.06 
 40.33 
 32.87 
 
 + 15.66 
 -19.01 
 -29.13 
 
 + 167 
 + 159 
 - 54 
 
 150 
 160 
 170 
 
 6.99 
 2.SS 
 1.97 
 
 -13.41 
 + 2.43 
 - 1.00 
 
 -138 
 - 31 
 + 54 
 
 90 
 
 27.41 
 
 - 4.15 
 
 -200 
 
 180 
 
 + 0.10 
 
 - 8.67 
 
 + 17 
 
CHAPTER IV. 
 
 MAXIMA AND MINIMA. 
 
 98. In engineering investigations the problem of determin- 
 ing the maxima and the minima, that is, the extrema of a 
 function, frequently occurs. For instance, the output of an 
 electric machine is to be found, at which its efficiency is a max- 
 imum, or, it is desired to determine that load on an induction 
 motor which gives the highest power-factor; or, that voltage 
 
 
 
 
 Y 
 
 
 
 
 
 X 
 
 -A 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 / 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 y 
 
 
 
 X 
 
 ^ 
 
 s. 
 
 
 
 
 P 
 
 
 
 
 
 
 p 
 
 
 ^ 
 
 -Q 
 
 
 
 
 
 N 
 
 \ 
 
 
 ^ 
 
 
 ^ 
 
 
 
 
 y 
 
 '5 
 
 
 / 
 
 
 
 
 
 
 
 \ 
 
 n 
 
 / 
 
 
 N 
 
 
 
 
 y 
 
 
 / 
 
 
 
 
 
 
 
 
 
 U 
 
 
 
 
 P3 
 
 
 
 / 
 
 r 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 ^. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 X 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Fig. 50. Graphic Solution of Maxima and Minima. 
 
 which makes the cost of a transmission line a minimum; or, 
 that speed of a steam turbine which gives the lowest specific 
 steam consumption, etc. 
 
 The maxima and minima of a function, y=f{x), can be found 
 by plotting the function as a curve and taking from the curve 
 the values x, y, which give a maximum or a minimum. For 
 instance, in the curve Fig. 50, maxima are at Pi and P2, minima 
 at P3 and P4. This method of determining the extrema of 
 functions is necessary, if the mathematical expression between 
 
 147 
 
148 
 
 ENGINEERING MATHEMATICS. 
 
 X and y, that is, the function y=f{x), is unknown, or if the 
 function y=f{x) is so complicated, as to make the mathematical 
 calculation of the extrema impracticable. As examples of 
 this method the following may be chosen: 
 B 
 
 16 
 
 
 
 
 
 
 
 
 ^^ 
 
 
 
 
 
 ■ 
 
 
 
 
 U 
 
 
 
 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 —^ 
 
 
 
 / 
 
 y 
 
 
 
 
 
 
 
 
 
 
 
 
 10 
 
 c 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 6 
 i2 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 u 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 A 
 
 / 
 
 
 
 
 
 
 
 
 H 
 
 
 
 
 
 
 
 
 
 ? ,. 
 
 1 
 
 ) 1 
 
 I 
 
 1 
 
 i 1 
 
 6 1 
 
 i 2 
 
 D 
 
 ^ 
 
 24 
 
 26 28 
 
 A 
 
 Fig. 51. Magnetization Curve. 
 
 Example i. Determine that magnetic density B, at which 
 the permeability ;u of a sample of iron is a maximum. The 
 relation between magnetic field intensity H, magnetic density 
 B and permeability // cannot be expressed in a mathematical 
 equation, and is therefore usually given in the form of an 
 
 1400 
 1200 
 
 1000- 
 
 -800 
 
 -600^ 
 
 4oo- 
 
 
 
 
 
 
 
 
 
 
 
 
 N 
 
 •^ 
 
 
 
 
 
 /'me 
 
 
 
 
 
 
 
 
 
 
 
 
 N 
 
 \ 
 
 
 
 
 y 
 
 // 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 \ 
 
 
 / 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 \ 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 B 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 > 
 
 \ 
 
 I 
 
 > { 
 
 ! 
 
 r \ 
 
 i ! 
 
 ) 1 
 
 1 
 
 .\ 
 
 3-lini 
 2 1 
 
 3S 
 
 3 1 
 
 1 1 
 
 5 
 
 Fig. 52. Permeability Curve. 
 
 empirical curve, relating B and E, as shown in Fig. 51. From 
 this curve, corresponding values of B and E are taken, and their 
 
 ratio, that is, the permeability ;U=—, plotted againstjBas abscissa. 
 
 This is done in Fig. 52. Fig. 52 then shows that a maximum 
 
MAXIMA AND MINIMA. 
 
 149 
 
 occurs at point /i^^^, for S=10.2 kilolines, /x = 1340, and minima 
 at the starting-point Pa, for B=0, ^ = 370, and also for B=oo, 
 where by extrapolation ;U = 1. 
 
 Example 2. Find that output of an induction motor 
 which gives the highest power-factor. While theoretically 
 an equation can be found relating output and power-factor 
 of an induction motor, the equation is too compUcated for use. 
 The most convenient way of calculating induction motors is 
 to calculate in tabular form for different values of slip s, the 
 torque, output, current, power and volt -ampere input, efficiency, 
 power-factor, etc., as is explained in " Theoretical Elements 
 of Electrical Engineering," third edition, p. 3G3. From this 
 
 
 
 
 P., 
 
 
 3 
 
 -I 
 
 p= 
 
 
 
 
 
 
 
 / 
 
 ^ 
 
 
 
 
 
 N 
 
 
 
 
 / 
 
 
 
 
 
 
 
 \ 
 
 \ 
 
 
 / 
 
 / 
 
 
 
 
 
 
 
 
 \ 
 
 k 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 20 
 
 30 
 
 30 
 
 DO 
 
 40 
 
 P 
 
 00 
 
 50 
 
 00 
 
 60 
 
 XI V 
 
 /atts 
 
 Cosg 
 0.90 
 
 0.88 
 
 0.86 
 
 0.84- 
 
 0.82 
 
 Fig. 53. Power-factor Maximum of Induction Motor. 
 
 table corresponding values of power output P and power- 
 factor cos 6 are taken and plotted in a curve. Fig. 53, and the 
 maximum derived from this curve is P = 4120, cos = 0.904. 
 
 For the purpose of determining the maximum, obviously 
 not the entire curve needs to be calculated, but only a short 
 range near the maximum. This is located by trial. Thus 
 in the present instance, P and cos d are calculated for s = 0.1 
 and s = 0.2. As the latter gives lower power-factor, the maximum 
 power-factor is below s = 0.2. Then s = 0.05 is calculated and gives 
 a higher value of cos 6 than s = 0.1; that is, the maximum is 
 below s = 0.1. Then s = 0.02 is calculated, and gives a lower 
 value of cos 6 than s = 0.05. The maximum value of cos d 
 thus lies between s = 0.02 and s = 0.1, and only the part of the 
 curve between s = 0.02 and s = 0.1 needs to be calculated for 
 the determination of the maximum of cos d, as is done in Fig. 53. 
 
 99. When determining an extremum of a function y=f{x). 
 by plotting it as a curve, the value of x, at which the extreme 
 
150 ENGINEERING MATHEMATICS. 
 
 occurs, is more or less inaccurate, since at the extreme the 
 curve is horizontal. For instance, in Fig. 53, the maximum 
 of the curve is so fiat that the value of power P, for which 
 cos d became a maximum, may be anywhere between P=4000 
 and P = 4300, within the accuracy of the curve. 
 
 In such a case, a higher accuracy can frequently be reached 
 by not attempting to locate the exact extreme, but two points 
 of the same ordinate, on each side of the extreme. Thus in 
 Fig. 53 the power Pq, at which the maximum power factor 
 cos (5 = 0.904 is reached, is somewhat uncertain. The value of 
 power-factor, somewhat below the maximum, cos (9 = 0.90, 
 is reached before the maximum, at Pi = 3400, and after the 
 maximum, at P2 = 4840. The maximum then may be calculated 
 as half-way between Pi and P2, that is, at Po = ilPi+P2! = 
 4120 watts. 
 
 This method gives usually more accurate resuhs, but is 
 based on the assumption that the curve is symmetrical on 
 both sides of the extreme, that is, falls off from the extreme 
 value at the same rate for lower as for higher values of the 
 abscissas. Where this is not the case, this method of inter- 
 polation does not give the exact maximum. 
 
 Example 3. The efSciency of a steam turbine nozzle, 
 that is, the ratio of the kinetic energy of the steam jet to the 
 energy of the steam available between the two pressures between 
 which the nozzle operates, is given in Fig. 54, as determined by 
 experiment. As abscissas are used the nozzle mouth opening, 
 that is, the widest part of the nozzle at the exhaust end, as 
 fraction of that corresponding to the exhaust pressure, while 
 the nozzle throat, that is, the narrowest part of the nozzle, is 
 assumed as constant. As ordinates are plotted the efficiencies. 
 This curve is not symmetrical, but falls off from the maximum, 
 on the sides of larger nozzle mouth, far more rapidly than on 
 the side of smaller nozzle mouth. The reason is that with 
 too large a nozzle mouth the expansion in the nozzle is carried 
 below the exhaust pressure p2, and steam eddies are produced 
 by this overexpansion. 
 
 The maximum efficiency of 94.6 per cent is found at the point 
 Po, at which the nozzle mouth corresponds to the exhaust 
 pressure. If, however, the maximum is determined as mid- 
 way between two points Pi and P2, on each side of the maximum, 
 
MAXIMA AND MINIMA. 
 
 151 
 
 at which the efficiency is the same, 93 per cent, a point Po' is 
 obtained, which lies on one side of the maximum. 
 
 With unsymmetrical curves, the method of interpolation 
 thus does not give the exact extreme. For most engineering 
 purposes this is rather an advantage. The purpose of deter- 
 mining the extreme usually is to select the most favorable 
 operating conditions. Since, however, in practice the operating 
 conditions never remain perfectly constant, but vary to some 
 extent, the most favorable operating condition in Fig. 54 is 
 not that where the average value gives the maximum efficiency 
 
 04 
 
 
 
 
 
 
 
 
 
 5«J 
 
 ^ 
 
 
 
 
 
 
 rjo 
 
 
 
 
 
 
 K 
 ^ 
 
 ^ 
 
 
 
 
 ■\ 
 
 ■<. 
 
 
 
 
 00 c 
 
 
 
 ^ 
 
 ^ 
 
 ^ 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 (D 
 (J 
 
 88-^- 
 
 Q. 
 
 8^ 
 
 
 
 ^ 
 
 
 
 
 
 
 
 1 
 
 
 
 \ 
 
 \ 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 V 
 
 qo.LlL 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 0*5 "-*■ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 iyj 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 7o 
 
 
 
 6 
 
 
 
 7 
 
 
 
 No 
 
 8 
 
 Izle 
 
 
 Oper 
 9 
 
 .ng 
 
 
 
 1 
 
 1 
 
 1 
 
 2 
 
 J 
 
 Fig. 54. Steam Turbine Nozzle Efficiency; Determination of Maximum. 
 
 (point Po), but the most favorable operating condition is that, 
 where the average efficiency during the range of pressure, occurr- 
 ing in operation, is a maximum. 
 
 If the steam pressure, and thereby the required expansion 
 ratio, that is, the theoretically correct size of nozzle mouth, 
 should vary during operation by 25 per cent from the average, 
 when choosing the maximum efficienc)'- point Po as average, 
 the efficiency during operation varies on the part of the curve 
 between Pi (91.4 per cent) and Pi (85.2 per cent), thus averaging 
 lower than by choosing the point Po'(6.25 per cent below Po) 
 as average. In the latter case, the efficiency varies on the 
 part of the curve from the Pi'(90.1 per cent) to P2'(90.1 per 
 cent). (Fig. 55.) 
 
152 
 
 EXaiXEERIXd MATHEMATICS. 
 
 Thus in apparatus design, when determining cxtrema of 
 a function y=f{.r), to scloft tliem as operating condition, 
 consideration must be gi^•en to 1h(! shape of the curve, and 
 where tlie curve is uns}-niinetrieal, the most efficient operating 
 point may not lie at tlie extreme, but on that side of it at wliich 
 the (•^u■^'e falls off slower, the more so the greater the range of 
 variation is, which may occur during operation. This is not 
 always realized. 
 
 100. If the function y=f(r) is i)lotted as a curve, Fig. 
 50, at the extremes of tlie function, tlie points Pi, P2, P3, Pi 
 of cur^-e Fig. 50, the tangent on the cur\-e is horizontal, since 
 
 
 
 
 
 
 
 
 
 Po' 
 
 . ( 
 
 £^ 
 
 
 
 
 
 
 iH 
 
 
 
 
 
 
 
 -^ 
 
 
 
 
 
 
 
 
 
 a. 
 
 
 88-S- 
 
 H 
 
 82-'^ 
 
 
 
 P,' 
 
 ^ 
 
 r' 
 
 
 
 
 
 
 
 
 \ 
 
 D^' 
 
 
 
 ^ 
 
 X 
 
 
 
 
 
 
 
 
 
 
 H 
 
 \ 
 
 
 y 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 r 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 "O 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 6 
 
 
 
 1 
 
 
 
 No 
 8 
 
 zzle 
 
 
 Dper 
 9 
 
 ing 
 1 
 
 
 
 1 
 
 1 
 
 1 
 
 2 
 
 J 
 
 Fig. 55. Steam Turbine Nozzle Efficiency; Determination of Maximum. 
 
 at the extreme the function changes from rising to decreasing 
 (maximum, Pi and P2), or from decreasing to increasing (min- 
 imum, P3 and P4), anil therefore for a moment passes through 
 the horizontal direction. 
 
 In general, the tangent of a curve, as that in Fig. 50, is the 
 line which connects two points P' and P" of the curve, which 
 are infinitely close together, and, as seen in Fig. 50, the angle 
 e, which this tangent F'P" makes with the horizontal or Z-axis, 
 thus is given by : 
 
 tan Q = 
 
 P"Q_dy 
 P'Q dx 
 
MAXIMA AND MINIMA. 153 
 
 At the extreme, the tangent on the curve is horizontal, 
 that is, 2^6 = 0, and, therefore, it follows that at an extreme 
 of the function, 
 
 y-m, (1) 
 
 l=« <^) 
 
 The reverse, however, is not necessarily the case; that is, 
 
 dv 
 if at a point x, y : -^ = 0, this point may not be an extreme; 
 
 that is, a maximum or minimum, hut may be a horizontal 
 inflection point, as points Pg and Pe are in Fig. 50. 
 
 With increasing x, when passing a maximum (Pi and P2, 
 Fig. 50), y rises, then stops rising, and then decreases again. 
 When passing a minimum (P3 and P4) y decreases, then stops 
 decreasing, and then increases again. When passing a horizontal 
 inflection point, y rises, then stops rising, and then starts rising 
 again, at P5, or y decreases, then stops decreasing, but then 
 starts decreasing again (at Pe). 
 
 The points of the function y=f{x), determined by the con- 
 
 dv 
 dition, -r = 0, thus require further investigation, whether they 
 
 represent a maximum, or a minimum, or merely a horizontal 
 inflection point. 
 
 This can be done mathematically: for increasing x, when 
 passing a maximum, tan 6 changes from positive to negative; 
 
 that is, decreases, or in other words, -t- (tan 0)<0. Since 
 
 tan (?=-/, it thus follows that at a maximum ji < 0. Inversely, 
 
 at a minimum tan 6 changes from negative to positive, hence 
 
 d d^v 
 
 increases, that is, ^ (tan (?)>0; or, ^2 > 0. When passing 
 
 a horizontal inflection point tan 6 first decreases to zero at 
 the inflection point, and then increases again; or, inversely, 
 tan 6 first increases, and then decreases again, that is, tan 6 = 
 
 — has a maximum or a minimum at the inflection point, and 
 dx 
 
 d d^y 
 
 therefore, -y- (tan d) = -f^ = at the inflection point. 
 
154 ENGINEERING MATHEMATICS. 
 
 In engineering problems the investigation, whether the 
 
 dv 
 solution of the condition of extremes, 'iz^^i- represents a 
 
 minimum, or a maximum, or an inflection point, is rarely- 
 required, but it is almost always obvious from the nature of 
 the problem whether a maximum of a mmimum occurs, or 
 neither. 
 
 For instance, if the problem is to determine the speed at 
 which the efficiency of a motor is a maximum, the solution: 
 speed =0, obviously is not a maximum but a mimimum, as at 
 zero speed the efficiency is zero. If the problem is, to find 
 the current at which the output of an alternator is a maximum, 
 the solution i = obviously is a minimum, and of the other 
 two solutions, ii and i^, the larger value, 12, again gives a 
 minimum, zero output at short-circuit current, while the inter- 
 mediate value I'l gives the maximum. 
 
 loi. The extremes of a function, therefore, are determined 
 by equating its differential quotient to zero, as is illustrated 
 by the following examples : 
 
 Example 4. In an impulse turbine, the speed of the jet 
 (steam jet or water jet) is Si. At what peripheral speed iSo is 
 the output a maximum. 
 
 The impulse force is proportional to the relative speed of 
 the jet and the rotating impulse wheel; that is, to {S1-S2). 
 The power is impulse force times speed S2; hence, 
 
 P = kS2{Sy-S2), (3) 
 
 dP 
 and is an extreme for the value of S2, given by -r^ = ; hence, 
 
 do 2 
 
 Si-2S2 = and 'S2=y; . • • (4) 
 
 that is, when the peripheral speed of the impulse wheel equals 
 half the jet velocity. 
 
 Example 5. In a transformer of constant impressed 
 e.m.f. 60 = 2300 volts; the constant loss, that is, loss which 
 is independent of the output (iron loss), is Pj = 500 watts. The 
 internal reustance (primary and secondary combined) is r = 20 
 
MAXIMA AND MINIMA. 155 
 
 ohms. At what current i is the efficiency of the transformer 
 a maximum; that is, the percentage loss, i, a minimum? 
 
 The loss is P = P,-+ri2 = 500 +20i2 (5) 
 
 The power input is Pi =ei = 2300i; .... (6) 
 
 hence, the percentage loss- is, 
 
 ;4^^^' (7) 
 
 and this is an extreme for the value of current i, given by 
 
 hence. 
 
 or. 
 
 dz-«' 
 
 (Pi+ri^)e-ei(2ri) 
 
 = 0; 
 
 Pi — ri^ = and i = .J— = 5 amperes, ... (8) 
 
 and the output is Po = ei = 11,500 watts. The loss is, P = Pj-|- 
 ri2 = 2P^ = 1000 watts; that is, the i^r loss or variable loss, is 
 equal to the constant loss P,-. The percentage loss is, 
 
 P VTr 
 ><=-p7= ^=0.087 = 8.7 percent, 
 
 and the maximum efficiency thus is, 
 
 l-;-0.913 = 91.3 per cent. 
 
 102. Usually, when the problem is given, to determine 
 those values of x for which y is an extreme, y cannot be expressed 
 directly as function of x, y=f{x), as was done in examples 
 (4) and (5), but y is expressed as function of some other quan- 
 ties, y=f(u, v . .), and then equations between u, v . . and x 
 are found from the conditions of the problem, by which expres- 
 sions of X are substituted for n, v . ., as shown in the following 
 example : 
 
 Example 6. There is a constant current I'o through a cir- 
 cuit containing a resistor of resistance ro. This resistor ro 
 
156 ENGINEERING MATHEMATICS. 
 
 is shunted by a resistor of resistance r. What must be the 
 resistance of this shunting resistor r, to make the power con- 
 sumed in r, a maximum? (Fig. 56.) 
 
 Let i be the current in the shunting resistor r. The power 
 consumed in r then is, 
 
 P = n^ (9) 
 
 The current in the resistor ro is io-i, and therefore the 
 voltage consamed by ro is roiio-i), and the voltage consumed 
 by r is ri, and as these two vohages must be equal, since both 
 
 -V\^ 
 
 Fig. 56. Shunted Resistor. 
 
 resistors are in shunt with each other, thus receive the same 
 voltage, 
 
 ri = ro(io — i), 
 
 and, herefrom, it follows that, 
 
 '-ttfJ^ ^i«) 
 
 Substituting this in equation (9) gives, 
 
 {r + Toy ^ ^ 
 
 dP 
 and this power is an extreme f or -j- = ; hence : 
 
 (r + ro)* " ' 
 
 hence, 
 
 r = ro; (12) 
 
 that is, the power consumed in r is a maximum, if the resistor 
 r of the shunt equals the resistance ro. 
 
MAXIMA AND MINIMA. 157 
 
 The current in r then is, by equation (10), 
 
 and the power is, 
 
 . ^o 
 
 p^Toio^ 
 
 4 
 
 103. If, after the function y=f{x) (the equation (11) in 
 example (6) ) has been derived, the differentiation t- = is 
 
 immediately carried out, the calculation is very frequently 
 much more complicated than necessary. It is, therefore, 
 advisable not to differentiate immediately, but first to simplify 
 the fimction y=fix). 
 
 If y is an extreme, any expression differing thereform by 
 a constant term, or a constant factor, etc., also is an extreme. 
 So also is the reciprocal of y, or its square, or square root, etc. 
 
 Thus, before differentiation, constant terms and constant 
 factors can be dropped, fractions inverted, the expression 
 raised to any power or any root thereof taken, etc. 
 
 For instance, in the preceding example, in equation (11), 
 
 rroHo'^ 
 
 (r+ro)2' 
 
 the value of r is to be found, which makes P a maximum. 
 If P is an extreme, 
 
 r 
 
 ^'^(r + ro)2' 
 
 which differs rrom P by the omission of the constant factor 
 
 roHo^, also is an extreme. 
 
 The reverse of y:, 
 
 (r + ro)2 
 2/2 = , 
 
 is also an extreme. (2/2 is a minimum, where j/i is a maximum, 
 and inversely.) 
 
 Therefore, the equation (11) can be simplified to the form : 
 
 (r+ro)2 ro2 
 
 2/2= — = r+2ro+y, 
 
158 ENGINEERING MATHEMATICS. 
 
 and, leaving out the constant term 2ro, gives the final form, 
 
 2/3 = r+^ (13) 
 
 This differentiated gives, 
 
 dr r^ ' 
 
 hence, 
 
 r = ro. 
 
 104. Example 7. From a source of constant alternating 
 e.m.f. e, power is transmitted over a line of resistance ro and 
 reactance xq into a non-inductive load. What must be the 
 resistance r of this load to give maximum power? 
 
 If i = current transmitted over the hne, the power delivered 
 at the load of resistance r is 
 
 P = ri^. (14) 
 
 The total resistance of the circuit is r+ro] the reactance 
 is Zo; hence the current is 
 
 i=— =i— =, (15) 
 
 \/(r + ro)2+.xo2 
 
 and, by substituting in equation (14), the power is 
 
 ^^(r + rof + xo^' ^^^) 
 
 if P is an extreme, by omitting e^ and inverting, 
 
 (r + roY+xa^ 
 
 2/1 = 
 
 r 
 
 r(?+Xf? 
 
 = r+2ro + 
 
 r 
 
 is also an extreme, and likewise, 
 
 , ro^+.To^ 
 J/2 = r+ , 
 
 is an extreme. 
 
MAXIMA AND MINIMA. 159 
 
 Differentiating, gives: 
 
 dr r^ ' 
 
 7- = \V+2-o2. (17) 
 
 ■V^Tierefrom follows, by substituting in equation (16), 
 
 '{ro+Vro^+Xo^f + xa' 
 
 2{ro + Vro^+Xo^y 
 
 (18) 
 
 Very often the function y=f(x) can by such algebraic 
 operations, which do not change an extreme, be simplified to 
 such an extent that differentiation becomes entirely unnecessary, 
 but the extreme is immediately seen; the following example 
 will serve to illustrate : 
 
 Example 8. In the same transmission circuit as in example 
 (7), for what value of r is the current i a maximum? 
 
 The current i is given, by equation (15), 
 
 V(r+ro)2+Xo2' 
 Dropping e and reversing, gives, 
 
 yi = V{r+ro)^+XQ^; 
 Squaring, gives, 
 
 j/2=(r+ro)2+a;o2; 
 
 dropping the constant term xo^ gives 
 
 2/3 = (r+ro)2; (19) 
 
 taking the square root gives 
 
 yi = r+ro; 
 
160 ENGINEERING MATHEMATICS. 
 
 dropping the constant term tq gives 
 
 2/5 = r; (20) 
 
 that is, the current i is an extreme, when y5 = r is. an extreme, 
 and this is the case f or r = and r = co : r = gives, 
 
 (21) 
 
 as the maximum value of the current, and r = oo gives 
 
 { = 0, 
 
 as the minimum value of the current. 
 
 With some practice, from the original equation (1), imme- 
 diately, or in very few steps, the simphfied final equation can 
 be derived. 
 
 105. In the calculation of maxima and minima of engineer- 
 ing quantities x, y, by differentiation of the function y=f(x), 
 it must be kept in mind that this method gives the values of 
 X, for which the quantity y of the mathematical equation y =f{x) 
 becomes an extreme, but whether this extreme has a physical 
 meaning in engineering or not requires further investigation; 
 that is, the range of numerical values of x and y is unlimited 
 in the mathematical equation, but may be limited in its engineer- 
 ing application. For instance, if x is a resistance, and the 
 differentiation of y='f{x) leads to negative values of x, these 
 have no engineering meaning; or, if the differentiation leads 
 to values of x, which, substituted in y=f{x), gives imaginary, or 
 negative values of y, the result also may have no engineering 
 application. In still other cases, the mathematical result 
 may give values, which are so far beyond the range of indus- 
 trially practicable numerical values as to be inapplicable. 
 For instance : 
 
 Example g. In example (8), to determine the resistance 
 r, which gives maximum current transmitted over a trans- 
 mission line, the equation (15), 
 
 \/(r + ro)2+Xo2' 
 
MAXIMA AND MINIMA. 161 
 
 immediately differentiated, gives as condition of the extremes: 
 
 *-■_ 2(r+ro) 
 
 dr 2{{r + ro)^+xo^\V{r + ro)^+xV^~ ' 
 
 hence, either r+ro = 0; (22) 
 
 or, (r+ro)2+a;o2 = oo (23) 
 
 the latter equation gives r = oo; hence i = 0, the minimum value 
 of current. 
 
 The former equation gives 
 
 r=-ro, (24) 
 
 as tne value of the resistance, which gives maximum current, 
 and the current would then be, by substituting (24) into (15), 
 
 i=- (25) 
 
 The solution (24), however, has no engineering meaning, 
 as the resistance r cannot be negative. 
 
 Hence, mathemetically, there exists no maximum value 
 of i in the range of r which can occur in engineering, that is, 
 within the range, 0< r< oo. 
 
 In such a case, where the extreme falls outside of the range 
 of numerical values, to which the engineering quantity is 
 limited, it follows that within the engineering range the quan- 
 tity continuously increases toward one limit and continuously 
 decreases toward the other limit, and that therefore the two 
 limits of the engineering range of the quantity give extremes. 
 Thus r=0 gives the maximum, r = oo the minimum of current. 
 
 io6. Example lo. An alternating-current generator, of 
 generated e.m.f. e = 2500 volts, internal resistance ro = 0.25 
 ohms, and synchronous reactance xo = 10 ohms, is loaded by 
 a circuit comprising a resistor of constant resistance r = 20 
 ohms, and a reactor of reactance x in series with the resistor 
 r. What value of reactance x gives maximum output? 
 
 If i= current of the alternator, its power output is 
 
 P = rP = 20i^; (26) 
 
162 ENGINEERING MATHEMATICS. 
 
 the total resistance is r+ro = 20.25 ohms; the total reactance 
 is x+xo = 10+x ohms, and therefore the current is 
 
 and the power output, by substituting (27) in (26), is 
 
 (27) 
 
 20X2500^ 
 
 Im- ru2) + (x + xo)2" 20.252 + (10 +a:)2 
 
 P= , .J .. = ..Z Z,^,2 - ■ ■ (28) 
 
 Simplified, this gives 
 
 2/i = (r + ro)2 + (x+xo)2; (29) 
 
 2/2 = (a;+xo)2; 
 hence, 
 
 fe=2(. + x.)=0; 
 
 and 
 
 a;= — xo= — 10 ohms; (30) 
 
 that is, a negative, or condensive reactance of 10 ohms. The 
 power output would then be, by substituting (30) into (28), 
 
 re^ 20+2500^ 
 P=7 — ; — rx-= — oA og9 watts = 305 kw. . . (31) 
 (r+ror 20.252 
 
 If, however, a condensive reactance is excluded, that is, 
 it is assumed that x >0, no mathematical extreme exists in the 
 range of the variable x, which is permissible, and the extreme 
 is at the end of the range, x = 0, and gives 
 
 P= f ^^2^ = 245 kw (32) 
 
 107. Example n. In a 500-kw. alternator, at voltage 
 e = 2500, the friction and windage loss is P„ = 6 kw., the iron 
 loss •Pt = 24 kw., the field excitation loss is Py=6 kw., and the 
 armature resistance r = 0.1 ohm. At what load is the efficiency 
 
 a maximum? 
 
MAXIMA AND MINIMA. 163 
 
 The sum of the losses is: 
 
 P = P„+P,.+P/ + n;2- 36,000 +0.11:2. _ _ (33) 
 
 The output is 
 
 Po = ei = 2500i; (34) 
 
 hence, the efficiency is 
 
 Pq ei 25OO1: 
 
 \+P~ 
 
 or, simplified, 
 
 ' Po+P ei + P„+Pi + P/ + n2 36000+2500i + 0.H2' ^^^'> 
 
 hence, 
 and. 
 
 j/i = -. -+n; 
 
 dyi P^+Pi + Pf 
 - = r- 
 
 di i^ 
 
 l P^ + Pi + Pf _ /36000 
 --yj --yJ-^^ = 600 amperes, 
 
 (36) 
 
 and the output, at which the maximum efficiency occurs, by 
 substituting (36) into (34), is 
 
 P = ei = 1500kw., 
 
 that is, at three times full load. 
 
 Therefore, this value' is of no engineering importance, but 
 means that at full load and at all practical overloads the 
 maximum efficiency is not yet reached, but the efficiency is 
 still rising. 
 
 108. Frequently in engineering calculations extremes of 
 engineering quantities are to be determined, which are func- 
 tions or two or more independent variables. For instance, 
 the maximum power is required which can be delivered over a 
 transmission line into a circuit, in which the resistance as well 
 as the reactance can be varied independently. In other 
 words, if 
 
 y=Ku,v) (37) 
 
164 ENGINEERING MATHEMATICS. 
 
 is a function of two independent variables u and v, such a 
 pair of values of m and of v is to be found, which makes y a 
 maximum, or minimum. 
 
 Choosing any value wo, of the independent variable u, 
 then a value of v can be found, which gives the maximum (or 
 minimum) A^alue of y, which can be reached for u = uo. This 
 is done by differentiating y=f{uo,v), over v, thus: 
 
 "^ = 0. (3«) 
 
 From this equation (38), a value, 
 
 v=fi{uo), (39) 
 
 is derived, which gives the maximum value of y, for the given 
 value of Mo, and by substituting (39) into (38), 
 
 y=f2M, (40) 
 
 is obtained as the equation, which relates the different extremes 
 of y, that correspond to the different values of Mq, with uq. 
 Herefrom, then, that value of Mo is found which gives the 
 maximum of the maxima, by differentiation : 
 
 1S^=« («) 
 
 Geometrically, y=f(u,v) may be represented by a surface 
 in space, with the coordinates y, u, v. y =/(Mo,y), then, represents 
 the curve of intersection of this surface with the plane Mo = 
 constant, and the differentation gives the maximum point 
 of this intersection curve. y=f 2(110) then gives the curve 
 in space, which connects all the maxima of the various inter- 
 sections with the Mo planes, and the second differentiation 
 gives the maximum of this maximum curve 2/=/2(mo), or the 
 maximum of the maxima (or more correctly, the extreme of 
 the extremes). 
 
 Inversely, it is possible first to differentiate over u, thus, 
 
 df(u,V o) 
 
 —d^-^^ (42) 
 
MAXIMA AND MINIMA. 165 
 
 and thereby get 
 
 u=f3{vo), (43) 
 
 as the value of u, which makes y a maximum for the given 
 vakie of v = vq, and substituting (43) into (42), 
 
 y=f4{vo), (44) 
 
 is obtained as the equation of the maxima, which differentiated 
 over vo, thus, 
 
 ^^« <«) 
 
 gives the maximum of the maxima. 
 
 Geometrically, this represents the consideration of the 
 intersection curves of the surface with the planes t;= constant. 
 
 However, equations (38) and (41) (respectively (42) and 
 (45)) give an extremum only, if both equations represent 
 maxima, or both minima. If one of the equations represents 
 a maximum, the other a minimum, the point is not an extre- 
 mum, but a saddle point, so called from the shape of the sur- 
 face y=f(u, v) near this point. 
 
 The working of this will be plaiq from the following example : 
 
 log. Example 12. The alternating voltage e = 30,000 is 
 impressed upon a transmission line of resistance ro==20 ohms 
 and reactance .to = 50 ohms. 
 
 What should be the resistance r and the reactance x of the 
 receiving circuit to deliver maximum power? 
 
 Let -1 = current delivered into the receiving circuit. The 
 total resistance is {r+ro); the total reactance is (x+Xo); hence, 
 the current is 
 
 i= , ' ■ ..... (46) 
 
 v(r+ro)2 + (a:+a;o)2 
 
 The power output is P = rP; (47) 
 
 hence, substituting (46), gives 
 
 re- 
 
 2 
 
 ('■+ro)2 + (a;+Xo)2 
 
 (a) For any given value of r, the reactance x, which gives 
 
 dP 
 maximum power, is derived by ^-=0. 
 
166 ENGINEERING MATHEMATICS. 
 
 P simplified, gives yi = (x+Jq)^; hence, 
 
 ^=2(x+xo)=0 and x=-xo . . . (49) 
 dx 
 
 that is, for any chosen resistance r, the power is a maximum, 
 if the reactance of the recciAang circuit is chosen equal to that 
 oftheline, but of opposite sign, that is, as condensive reactance. 
 Substituting (49) into (48) gives the maximum power 
 available for a chosen value of r, as : 
 
 -, re- 
 
 
 
 or, simplified, 
 hence, 
 
 (r+ro)2' 
 
 (50) 
 
 y2 = —^ and y3 = r+—; 
 
 |? = 1-^ and r^ro, .... (51) 
 
 and by substituting (51) into (50), the maximum power is, 
 
 ""---&. ^^2) 
 
 (b) For any given value of x, the resistance r, which gives 
 
 maximum power, is given by -r- = 0. 
 
 P simplified gives, 
 
 (r +r-o)2 + (X +Xo)2 Tq^ + (x+Xq)^ 
 2/1 = ; y2-r + ^; ; 
 
 dr r'^ 
 
 r=\/ro2 + (x+xo)2, ..... (53) 
 
 which is the value of r, that for any given value of x, gives 
 maximum power, and this maximum power by substituting 
 (53) into (48) is, 
 
 „ Vro2 + (x+xo)V 
 
 ■f^o = - 
 
 [ro + Vro2 + (x + xn)2]2 + {x+xq)^ 
 
 .... (54) 
 
 2{rQ'\-Vro^ + [x+Xoy\' 
 
MAXIMA AND MINIMA. 167 
 
 which is the maximum power that can be transmitted into a 
 
 receiving circuit of reactance x. 
 
 The value of x, which makes this maximum power Pq the 
 
 dP 
 highest maximum, is given by -^ — = 0. 
 
 Pq simplified gives 
 
 2/3 = ?-o + V?-o2 + (a;+a;o)2; 
 
 y5 = ro^ + (.x+Xoy; 
 j/6 = (x+Xo)2; 
 
 ?/7 = (x+zo); 
 
 and this value is a maximum for (a;+.'r;o)=0; that is, for 
 
 x=—xo (55) 
 
 Note. If x cannot be negative, that is, if only inductive 
 reactance is considered, x = gives the maximum power, and 
 the latter then is 
 
 e2 
 
 ■* max^TT / r, 9, > .... (00) 
 
 the same value as found in problem (7), equation (18). 
 
 Substituting (55) and (54) gives again equation (52), thus, 
 
 P = — 
 
 110. Here again, it requires consideration, whether the 
 solution is practicable within the limitation of engineering 
 constants. 
 
 With the numerical constants chosen, it would be 
 
 e2 300002 
 ^max = 47p = 80" = 1 1 ,250 kw.; 
 
 e 
 1 = ^ = 750 amperes. 
 
168 ENGINEERING MATHEMATICS. 
 
 and the voltage at the receiving end of the line would be 
 e'=ix/r2+3;2 = 750\/20- +502 = 40,400 ^qH^. 
 
 that is, the voltage at the receiving end would be far higher 
 than at the generator end, the current excessive, and the efficiency 
 of transmission only 50 per cent. This extreme case thus is 
 hardly practicable, and the conclusion would be that by the 
 use of negative reactance in the receiving circuit, an amount 
 of power could be delivered, at a sacrifice of efficiency, far 
 greater than economical transmission would permit. 
 
 In the case, where capacity was excluded from the receiv- 
 ing circuit, the maximum power was given by equation (56) as 
 
 P„, = ^''" .- ^GlOO kw. 
 
 max 
 
 2\ro + Vro- + .r'o' 
 
 III. Extremes of engineering quantities x, y, are usually 
 determined by differentiating the function, 
 
 2/=/W, (57) 
 
 and from the equation, 
 
 i">. (^« 
 
 deriving the values of x, which make y an extreme. 
 
 Occasionally, however, the equation (58) cannot be solved 
 for X, but is either of higher order in x, or a transcendental 
 equation. In this case, equation (58) may be solved by approx- 
 imation, or preferably, the function, 
 
 z=-r- (59) 
 
 ax, 
 
 is plotted as a curve, the values of x taken, at which z = 0, 
 
 that is, at which the curve intersects the A"-axis. For instance: 
 
 Example 13. The e.m.f. wavc^ of a thme-phase alternator, 
 
 as determined by oscillograph, is represented by the equation, 
 
 e = 36000jsin (9-0.12 sin (3^-2.3°)-23 sin (S/?-!. 5°) + 
 
 0.13 sin (7^-6.2°)! (60) 
 
MAXIMA AND MINIMA. 169 
 
 This alternator, connected to a long-distance transmission line, 
 gives the charging current to the hne of 
 
 1 = 13.12 cos (6i-0.3°)-5.04cos(3i9-3.3°)-18.76cos (50-3.6°) 
 + 19.59 cos (70-9.9°) .... (61) 
 
 (see Chapter III, paragraph 95). 
 
 What are the extreme values of this current, and at what 
 phase angles d do they occur? 
 
 The phase angle 6, at which the current i reaches an extreme 
 value, is given by the equation 
 
 fe-' (62) 
 
 Fia. 57. 
 
 Substituting (61) into (62) gives, 
 
 Hi 
 z=^ = -13.12 sin (0-0.3°) +15.12 sin (30-3..3°) +93.8 sin 
 au 
 
 (50-3.6°)-137.1sin (70-9.9°) = O. . . . (63) 
 
 This equation cannot be solved for d. Therefore z is 
 plotted as function of 6 by the curve, Fig. 57, and from this 
 curve the values of 6 taken at which the curve intersects the 
 zero line. They are: 
 
 = 1°; 20°; 47° 78°; 104°; 135°; 162°. 
 
170 ENGINEERING MATHEMATICS. 
 
 For these angles d, the corresponding values of i are calculated 
 by equation (61), and are: 
 
 to=+9; -1; +39; -30; +30; -42; +4 amperes. 
 
 The current thus has during each period 14 extrema, of 
 which the highest is 42 amperes. 
 
 112. In those cases, where the mathematical expression 
 
 of the function y=f{x) is not known, and the extreme values 
 
 therefore have to be determined graphically, frequently a greater 
 
 accuracy can be reached by plotting as a curve the differential 
 
 of y=f{x) and picking out the zero values instead of plotting 
 
 y=f(x), and picking out the highest and the lowest points. 
 
 If the mathematical expression of y=f{x) is not known, obvi- 
 
 dv 
 ously the equation of the differential curve z=--- (64) is usually 
 
 not known either. Approximately, however, it can fre- 
 quently be plotted from the numerical values of y=f{x), as 
 follows : 
 
 If xi, X2, X3 . . . are successive numerical values of x, 
 
 and 2/1) ?/2, 2/3 •■ • the corresponding numerical values of y, 
 
 approximate points of the differential curve z=-^ are given 
 by the corresponding values: 
 
 X2+Xx XZ+X2 Xi+Xs 
 
 as abscissas : 
 as ordinates : 
 
 y2-yi . y3-y2 . yt-ys 
 
 X-2-Xi ' X3-Z2' Xi—Xs' 
 
 113. Example 14, In the problem (1), the maximum permea- 
 bility point of a sample of iron, of which the B, H curve is given 
 as Fig. 51, was determined by taking from Fig. 51 corresponding 
 
 values of B and H, and plotting /'=77, against B in Fig. 52. 
 
 A considerable inaccuracy exists in this method, in locating 
 the value of B, at which /i is a maximum, due to the flatness 
 of the curve, Fig. 52. 
 
MAXIMA AND MINIMA. 
 
 171 
 
 The successive pairs of corresponding values of B and H, 
 as taken from Fig. 51 are given in columns 1 and 2 of Table I. 
 
 Table I. 
 
 B 
 
 Kilolinea, 
 
 H 
 
 B 
 
 J/, 
 
 B 
 
 
 
 1 
 
 2 
 
 
 1.76 
 
 2.74 
 
 370 
 570 
 730 
 
 + 200 
 160 
 
 0.5 
 1.5 
 
 3 
 4 
 5 
 
 3.47 
 4.06 
 4.59 
 
 865 
 
 985 
 
 1090 
 
 135 
 120 
 105 
 
 2.5 
 3.5 
 4.5 
 
 6 
 
 7 
 8 
 
 5.10 
 5.63 
 6.17 
 
 1175 
 1245 
 1295 
 
 85 
 70 
 60 
 
 5.5 
 6.5 
 7.5 
 
 9 
 
 10 
 11 
 
 6.77 
 7.47 
 8.33 
 
 1330 
 1340 
 1320 
 
 35 
 + 10 
 -20 
 
 8.5 
 
 9.5 
 
 10.5 
 
 12 
 13 
 14 
 
 9.60 
 11.60 
 15.10 
 
 1250 
 
 1120 
 
 930 
 
 70 
 130 
 190 
 
 11.5 
 12.5 
 13.5 
 
 15 
 
 20.7 
 
 725 
 
 205 
 
 14.5 
 
 In the third column of Table I is given the permeability, 
 /<=— , and in the fourth column the increase of permeability, 
 
 XI 
 
 per B = l, '^fjt; the last column then gives the value of B, to 
 which Jfi corresponds. 
 
 In Fig. 58, values ^n are plotted as ordinates, with B as 
 abscissas. This curve passes through zero at B=9.95. 
 
 The maximum permeability thus occurs at the approximate 
 magnetic density B=9.95 kilolines per sq.cm., and not at B= 
 10.2, as was given by the less accurate graphical determination 
 of Fig. 52, and the maximum permeability is /to = 1340. 
 
 As seen, the sharpness of the intersection of the differential 
 curve with the zero line permits a far greater accuracy than 
 feasible by the method used in instance (1). 
 
 114. As illustration of the method of determining extremes, 
 some further examples are given below: 
 
172 
 
 ENGINEERING MATHEMATICS. 
 
 Example 15. A storage battery of n = 80 cells is to be 
 connected so as to give maximum power in a constant resist- 
 ance r = 0.1 ohm. Each battery cell has the e.m.f. eo = 2.1 
 volts and the internal resistance ro = 0.02 ohm. How must 
 the cells be connected? 
 
 Assuming the cells are connected with x in parallel, hence 
 n . . 
 
 - m series. The internal resistance of the battery then is 
 
 n 
 
 -To 
 
 =—5- ohms, and the total resistance of the circuit is -^rn + r. 
 
 X X"^ x^ 
 
 Fig. 58. First Differential Quotient of B,ii Curve 
 
 The e.m.f. acting on the circuit is - eo, since - cells of e.m.f. 
 
 eo are in series. Therefore, the current delivered by the battery 
 is, 
 
 n 
 
 i = - 
 
 ■,ro+r 
 
 and the power which this current produces in the resistance 
 r, is, 
 
 rn^eo^ 
 
 ,n >. • 
 
MAXIMA AND MINIMA. 173 
 
 This is an extreme, if 
 
 nro 
 
 is an extreme, hence, 
 
 ax x^ 
 
 and 
 
 X' 
 
 ■^" = 4, 
 
 that is, ^■='\} — = 4 cells are connected in multiple, and 
 
 ^ /^ on 11 • 
 
 - = ^ — = 20 cells m series. 
 X \ro 
 
 115. Example 16, In an alternating-current transformer the 
 loss of power is limited to 900 watts by the permissible temper- 
 ature rise. The internal resistance of the transformer winding 
 (primary, plus secondary reduced to the primary) is 2 ohms, 
 and the core loss at 2000 volts impressed, is 400 watts, and 
 varies with the 1.6th power of the magnetic density and there- 
 fore of the voltage. At what impressed voltage is the output 
 of the transformer a maximum? 
 
 If e is the impressed e.m.f. and i is the current input, the 
 power input into the transformer (approximately, at non- 
 inductive load) is P = ei. 
 
 If the output is a maximum, at constant loss, the input P 
 also is a maximum. The loss of power in the winding is 
 ri2 = 2i2. 
 
 The loss of power in the iron at 2000 volts impressed is 
 400 watts, and at impressed voltage e it therefore is 
 
 . V- 
 
 X400, 
 
 \2000/ 
 and the total loss in the transformer, therefore, is 
 
 , 1-6 
 
 Pi = 2i2+400i2oy =900; 
 
174 ENGINEERING MATHEMATICS. 
 
 herefrom, it follows that, 
 
 ^^^450-200(20^ 
 and, substituting, into P'=ei: 
 
 6 
 
 P = e^450-200(2^)''. 
 Simplified, this gives, 
 
 
 y~- 
 
 = 2.25e= 
 
 , e3.6 
 
 
 
 20001-6' 
 
 and, differentiating. 
 
 
 
 
 
 dy 
 de" 
 
 = 4.5e- 
 
 3.6e2-6 
 20001-6 
 
 =0, 
 
 and 
 
 \2000/ 
 
 = 1.25. 
 
 
 
 Hence, 
 
 
 
 
 
 
 e 
 2000 
 
 = 1.15 
 
 and e = 
 
 = 2300 V( 
 
 which, substituted, gives 
 
 
 
 P = 2300 \/450- 200 XI. 25 = 32.52 kw. 
 
 116. Example 17. In a 5-kw. alternating-current transformer, 
 at 1000 volts impressed, the core loss is 60 watts, the i^r loss 
 150 watts. How must the impressed voltage be changed, 
 to give maximum efficiency, (a) At full load of 5-kw; (6) at 
 half load? 
 
 The core loss may be assumed as varying with the 1.6th 
 power of the impressed voltage. If e is the impressed voltage, 
 
 i = is the current at full load, and ii = is the current at 
 
 e e 
 
 half load, then at 1000 volts impressed, the full-load current is 
 = 5 amperes, and since the I'^r loss is 150 watts, this gives 
 
MAXIMA AND MINIMA. 175 
 
 the internal resistance of the transformer as 7- = 6 ohms, and 
 herefrom the i^r loss at impressed voltage e is respectively, 
 
 ., 150X106 ^ .^ 37.5X106 ^^ 
 n^ = ^ and ni^= ^ watts. 
 
 Since the core loss is 60 watts at 1000 volts, at the voltage e 
 it is 
 
 Viooo/ 
 
 Pi=60x(^j^QQJ watts. 
 
 The total loss, at full load, thus is 
 
 X. T. : nr. f e y-\ 150X106 
 
 and at half load it is 
 
 ■ . .^ f e \i-6 37.5X106 
 
 Simphfied, this gives 
 
 •6 
 
 ^-(lO()o)"+2.5Xl06Xe- 
 
 hence, differentiated, 
 
 eO-6 
 
 ^■^Iooo^-^-^'^^^^'>^'''=^' 
 
 6^-6 = 3.125 X 106 X 10001-6 = 3.125 X lO^o-^ ; 
 
 e3-6 = 0.78125 X 106 X lOOQi-s = 0.78125 X lO^^-^ ; 
 
 hence, e = 1373 volts for maximum efSciency at full load. 
 
 and e = 938 volts for maximum efficiency at half load. 
 
 117. Example 18. (o) Constant voltage e = 1000 is im- 
 pressed upon a condenser of capacity C = 10 mf., through a 
 reactor of inductance L = 100 mh., and a resistor of resist- 
 ance r = 40 ohms. What is the maximum value of the charg- 
 ing current? 
 
176 ENGINEERING MATHEMATICS. 
 
 (b) An additional resistor of resistance r' = 210 ohms is 
 then inserted in series, making the total resistance of the con- 
 denser charging circuit, r = 250 ohms. What is the maximum 
 value of the charging current? 
 
 The equation of the charging current of a condenser, through 
 a circuit of low resistance, is (" Transient Electric Phenomena 
 and Oscillations," p. 61) : 
 
 where 
 
 . 2e _il( . q 
 q 2L 
 
 AL 
 
 5=\/-C"'"'' 
 
 and the equation of the charging current of a condenser, through 
 a circuit of high resistance, is (" Transient Electric Phenomena 
 and Oscillations," p. 51), 
 
 s 
 where 
 
 
 4L 
 
 Substituting the numerical values gives: 
 (a) 1 = 10.2 £-200( sin 980 i; 
 
 {h) z = 6.667i £-500(_ j-20oo(j_ 
 
 Simplified and diiTercntiated, this gives : 
 
 (a) z=-^ = 4.9 cos 980i-sin 980^=0; 
 
 hence tan980< = 4.9 
 
 980< = 68.5° =1.20 
 
 1.20 +n;r 
 908 ^^^- 
 
 dio 
 
 do 
 
MAXIMA AND MINIMA. 177 
 
 hence, £+i500( = 4^ 
 
 log; 4 
 1500^=,—— = 1.38, 
 
 log s ' 
 
 f = 0.00092 sec, 
 
 and, by substituting these values of t into the equations of the 
 current, gives the maximum values: 
 
 1.20 + 7ij 
 
 (a) i=10e 4.9 =7.83 £-0-6*" = 7.83X0.53" amperes; 
 
 that is, an infinite number of maxima, of gradually decreasing 
 values: +7.83; -4.15; +.2.20; -1.17 etc. 
 
 (6) i = 6.667(£-o-^- £-1-8*) =3.16 amperes. 
 
 ii8. Example 19. In an induction generator, the fric- 
 tion losses are P/=100 kw.; the iron loss is 200 kw. at the ter- 
 minal voltage of e = 4 kv., and may be assumed as proportional 
 to the 1.6th power of the voltage; the loss in the resistance 
 of the conductors is 100 kw. at i = 3000 amperes output, and may 
 be assumed as proportional to the square of the current, and 
 the losses resulting from stray fields due to magnetic saturation 
 are 100 kw. at e=4 kv., and may in the range considered be 
 assumed as approximately proportional to the 3.2th power 
 of the voltage. Under what conditions of operation, regard- 
 ing output, voltage and current, is the efficiency a maximum? 
 
 The losses may be summarized as follows: 
 
 Friction loss, P/=100 kw.; 
 
 Iron loss, Pi+200(^) ; 
 
 i2rloss, '^'=^■^^^(3000/ ' 
 
 /g\3.2 
 
 Saturation loss, P^ = 100 (j I ; 
 
 hence the total loss is Pz^Pf+Pi+Pc+P, 
 
 -M^-<T<M4 
 
178 ENGINEERING MATHEMATICS. 
 
 The output is P = ei; hence, percentage of loss is 
 , P. 100{l+2(|)^%(3^)%(f)^"} 
 
 The efficienc}' is a maximum, if the percentage loss -^ is a 
 minimum. For any value of the voltage e, this is the case 
 
 at the current i, given by -tt = 0; hence, simplifying and differ- 
 entiating X, 
 
 di i^ ^30002 ^' 
 
 i=3000^1+2(|)"V(|)"; 
 then, substituting i in the expression of A, gives 
 
 and A is an extreme, if the simplified expression, 
 
 2/~g2+41-6gO-4 "^43-2^ 
 
 IS an extreme, at 
 
 dy 2 0.8 1.2 „ 
 de~~"e3~4i-6ei-4+4F2^ > 
 
 0.8 1.2 
 
 hence, 2+^,e^6___e3.2^0; 
 
 /e \i'^ 2 
 hence, 1^1 =--=^ and e = 5.50 Ica^, 
 
 and, by substitution the following values are obtained : /I = 0.0323; 
 efficiency 96.77 per cent; current '!: = 8000 amperes; output 
 P = 44,000 kw. 
 
 119. In all probability, this output is beyond the capacity 
 of the generator, as limited by heating. The foremost limita- 
 tion probably will be the ih heating of the conductors; that is. 
 
MAXIMA AND MINIMA. 179 
 
 the maximum permissible current will be restricted to, for 
 instance, i = 5000 amperes. 
 
 For any given value of current i, the maximum efficiency, 
 that is, minimum loss, is found by differentiating, 
 
 , 100(1+2(| )'\(34)V(|'" 
 
 A — 
 
 ei 
 over e, thus : 
 
 de 
 Simplified, X gives 
 
 hence, difTerentiated, it gives 
 
 ©■^^(l)''4(-(: 
 
 3000/ 
 
 "\/^ 
 
 e\i-6 3+J61+55 gyyj^ 
 
 V4/ 11 
 
 For 1 = 5000, this gives: 
 
 |y''' = 1.065 and e = 4.16kv.; 
 
 hence, 
 
 /I = 0.0338, Efficency 96.62 per cent. Power P=20,800 kw. 
 
 Method of Least Squares. 
 
 120. An interesting and very important application of the 
 theory of extremes is given by the method of least squares, which 
 is used to calculate the most accurate values of the constants 
 of functions from numerical observations which are more numer- 
 ous than the constants. 
 
 If y=Ax), (1) 
 
180 ENGINEERING MATHEMATICS. 
 
 is a function having the constants a, h, c . . . and the form of 
 tlie function (1) is known, for instance, 
 
 y = a + bx+cx'^, (2) 
 
 and the constants a b, c are not known, but the numerical 
 values of a number of corresponding values of x and y are given, 
 for instance, by experiment, Xi, X2, xz, x^. . . and j/i, j/2, 2/3, 2/4 • • • , 
 then from these corresponding numerical values x„ and 2/„ 
 the constants a, b, c . . . can be calculated, if the numerical 
 values, that is, the observed points of the curve, are sufficiently 
 numerous. 
 
 If less points X\ y\, X2, J/2 • ■ ■ are observed, then the equa- 
 tion (1) has constants, obviously these constants cannot be 
 calculated, as not sufficient data are available therefor. 
 
 If the number of observed points equals the number of con- 
 stants, they are just sufficient to calculate the constants. For 
 instance, in equation (2), if three corresponcUng values x\, yi; 
 X2, 2/2; ^3, 2/3 arc observed, by substituting these into equation 
 (2), three equations are obtained: 
 
 yi = a+bxi+cxi^; 
 y2 = a+bx-2+cx2~\ 
 y3 = a-3+bx+cx3^, 
 
 (3) 
 
 which are just sufficient for the calculation of the three constants 
 a, b, c. 
 
 Three observations would therefore be sufficient for deter- 
 mining three constants, if the observations were absolutely 
 correct. This, however, is not the case, but the observations 
 always contain errors of observation, that is, unavoidable inac- 
 curacies, and constants calculated by using only as many 
 observations as there are constants, are not very accurate. 
 
 Thus, in experimental work, always more observations 
 are made than just necessary for the determination of the 
 constants, for the purpose of getting a higher accuracy. Thus, 
 for instance, in astronomy, for the calculation of the orbit of 
 a comet, less than four observations are theoretically sufficient, 
 but if possible hundreds are taken, to get a greater accuracy 
 in the determination of the constants of the orbit. 
 
MAXIMA AND MINIMA. 181 
 
 If, then, for the determination of the constants a, b, c of 
 equation (2), six pairs of corresponding values of x and y were 
 determined, any three of these pairs would be sufficient to 
 give a, h, c, as seen above, but using different sets of three 
 observations, would not give the same values of a^ b, c (as it 
 should, if the observations were absolutely accurate), but 
 different values, and none of these values would have as high 
 an accuracy as can be reached from the experimental data, 
 since none of the values uses all observations. 
 
 121. If y=Ax), (1) 
 
 is a function containing the constants a,b, c . . ., which are still 
 unknown, and Xi, yi, X2, j/2; xz, y^; etc., are corresponding 
 experimental values, then, if these values Were absolutely cor- 
 rect, and the correct values of the constants a,b, c . . . chosen, 
 yi=/(xi) would be true; that is, 
 
 /(xi)-2/i = 0; 1 
 
 (5) 
 
 7(2:2) -2/2 = 0, etc. J 
 
 Due to the errors of observation, this is not the case, but 
 even if a, &, c . . . are the correct values, 
 
 yiT^fixi) etc.; . ... (6) 
 
 that is, a small difference, or error, exists, thus 
 
 /(xi)-j/i = 5i; 
 
 .... (7) 
 f{x2)-y2 = S2, etc.; j 
 
 If instead of the correct values of the constants, a, b, c . . ., 
 other values were chosen, different errors di, 82 . . . would 
 obviously result. 
 
 From probability calculation it follows, that, if the correct 
 values of the constants a, b, c . . are chosen, the sum of the 
 squares of the errors, 
 
 ai2 + V + oV + (8) 
 
 is less than for any other value of the constants a, b, c . . .; that 
 is, it is a minimum. 
 
182 ENGINEERING MATHEMATICS. 
 
 122. The problem of determining the constants a, h, c . . ., 
 thus consists in finding a set of constants, which makes the 
 sum of the squares of the errors d a minimum ; that is, 
 
 2= 2(?2 = minimum, . . . (9) 
 
 is the requirement, which gives the most accurate or most 
 probable set of values of the constants a, b, c . . . 
 
 Since by (7), S=f{x) — y, it follows from (9) as the condi- 
 tion, which gives the mpst probable value of the constants 
 a,b,c...; 
 
 2= 2]{/(x) — 2/P= minimum; . . (10) 
 
 that is, the least sum of the squares of the errors gives the most 
 probable value of the constants a, h, c . . . 
 
 To find the values oi a, b, c . ., which fulfill equation (10), 
 the differential quotients of (10) are equated to zero, and give 
 
 
 • (11) 
 
 This gives as many equations as there are constants a,b,c . . ., 
 and therefore just suffices for their calculation, and the values 
 so calculated are the most probable, that is, the most accurate 
 values. 
 
 Where extremely high accuracy is required, as for instance 
 in astronomy when calculating from observations extending 
 over a few months only, the orbit of a comet which possibly 
 lasts thousands of years, the method of least squares must be 
 used, and is frequently necessary also in engineering, to get 
 from a limited number of observations the highest accuracy 
 of the constants. 
 
 123. As instance, the method of least squares may be apphed 
 in separating from the observations of an induction motor, 
 when running light, the component losses, as friction, hysteresis, 
 etc. 
 
MAXIMA AND MINIMA. 
 
 183 
 
 In a 440-volt 50-h.p. induction motor, when running ligbt, 
 that is, without load, at various voltages, let the terminal 
 voltage e, the current input i, and the power input p be observed 
 as given in the first three columns of Table I: 
 
 Table I 
 
 e 
 
 i 
 
 p 
 
 ih- 
 
 PO 
 
 po 
 calc. 
 
 J 
 
 148 
 
 8 
 
 790 
 
 13 
 
 780 
 
 746 
 
 + 32 
 
 220 
 
 11 
 
 920 
 
 24 
 
 900 
 
 962 
 
 - 62 
 
 320 
 
 19 
 
 1500 
 
 72 
 
 1430 
 
 1382 
 
 + 48 
 
 410 
 
 23 
 
 1920 
 
 106 
 
 1810 
 
 1875 
 
 - 35 
 
 440 
 
 26 
 
 2220 
 
 135 
 
 2085 
 
 2058 
 
 + 27 
 
 473 
 
 29 
 
 2450 
 
 168 
 
 2280 
 
 2280 
 
 
 
 5S0 
 
 43 
 
 3700 
 
 370 
 
 3330 
 
 3080 
 
 + 250 
 
 640 
 
 56 
 
 5000 
 
 627 
 
 4370 
 
 3600 
 
 + 770 
 
 700 
 
 75 
 
 8000 
 
 1125 
 
 6S75 
 
 4150 
 
 +2725 
 
 The power consumed by the motor while running light 
 consists of: The friction loss, which can be assumed as con- 
 stant, a; the hysteresis loss, which is proportional to the 1.6th 
 power of the magnetic flux, and therefore of the voltage,- be'--^; 
 the eddy current losses, which are proportional to the square 
 of the magnetic flux, and therefore of the voltage, ce^ ; and the i^r 
 loss in the windings. The total power is. 
 
 p = a+be^-^+ce^+ri^ 
 
 (12) 
 
 From the resistance of the motor windings, r = 0.2 ohm, 
 and the observed values of current i, the i^r loss is calculated, 
 and tabulated in the fourth column of Table I, and subtracted 
 from p, leaving as the total mechanical and magnetic losses the 
 values of po given in the fifth column of the table, which should 
 be expressed by the equation: 
 
 p = a+be^'^+ce^. 
 
 (13) 
 
 This leaves three constants, a, b, c, to be calculated. 
 
 Plotting now in Fig. 59 with values of e as abscissas, the 
 current i and the power po give curves, which show that within 
 the voltage range of the test, a change occurs in the motor. 
 
184 
 
 ENGINEERING MATHEMATICS. 
 
 as indicated by the abrupt rise of current and of power beyond 
 473 volts. This obviously is due to beginning magnetic satura- 
 tion of the iron structure. Since with beginning saturation 
 a change of the magnetic distribution must be expected, that 
 is, an increase of the magnetic stray field and thereby increase 
 of eddy current losses, it is probable that at this point the con- 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 I 
 
 n 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 r 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 -^000 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 ) '° 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 fin 
 
 -6000 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 rj\ 
 
 -5000 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 ® 
 
 / 
 
 / 
 
 -4000 
 
 
 
 
 
 
 
 
 
 
 
 V 
 
 r 
 
 / 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 / 
 
 .> 
 
 / 
 
 
 
 -3000 
 
 
 
 
 
 
 
 
 
 / 
 
 / 
 
 / 
 
 ' 
 
 
 
 oU 
 
 
 
 
 
 
 
 
 y 
 
 / 
 
 y 
 
 y 
 
 Po 
 
 
 
 
 
 -2000 
 
 
 
 
 
 
 ^ 
 
 y 
 
 y 
 
 ^ 
 
 
 
 
 
 
 .6U 
 
 
 
 
 ■^ 
 
 ^ 
 
 -^ 
 
 ^ 
 
 
 
 
 
 
 
 
 
 -1000 
 
 
 
 ^ 
 
 y^ 
 
 ■<' 
 
 
 
 
 e= 
 
 Volt 
 
 ■> 
 
 
 
 
 lU 
 
 ^ 
 
 u 
 
 
 
 a 
 
 DO 
 
 3( 
 
 
 
 4( 
 
 » 
 
 a 
 
 » 
 
 « 
 
 
 
 7( 
 
 
 
 
 Fig. 59. Excitation Power of Induction Motor. 
 
 stants in equation (13) change, and no set of constants can be 
 expected to represent the entire range of observation. For 
 the calculation of the constants in (13), thus only the observa- 
 tions below the range of magnetic saturation can safely be used, 
 that is, up to 473 volts. 
 
 From equation (13) follows as the error of an individual 
 observation of e and po: 
 
 a = o-l-6ei-6-|-ce2-po; 
 
 (14) 
 
MAXIMA AND MIMIMA. 
 
 185 
 
 hence, 
 thus: 
 
 z=I,d2 = S{a+be^-^+ce^ -pof = immmum, (15) 
 
 dz 
 ^=2:|a+6ei-6+ce2-poi=0; 
 
 dz 
 db" 
 
 Sfa + 6ei-6 + ce2-po|ei-6=0; 
 
 dz 
 
 ■^=2Sa + 6ei-6+ce2-po}e2 = 0; 
 
 (16) 
 
 and, if n is the number of observations used (n = 6 in this 
 instance, from e = 148 to e = 473), this gives the following 
 equations: 
 
 m + 62ei-6+c2e2-Spo = 0; ' 
 
 oi:ei-6+6Ee3-2 + cSe3-6-Eei-623o = 0; ■ . . (17) 
 
 aSe2 +&Se3-6 +cSe4- I,e^po = 0. 
 
 Substituting in (17) the numerical values from Table I gives. 
 
 hence, 
 
 and 
 
 a + 11.7 b 103 + 126 c 103 = 1550 
 a +14.6 b 103+163 c 103 = 1830 
 a + 15.1 h 103 + 170 c 103 = 1880 
 
 = 540; 
 
 & = 32.5x10-3; 
 
 c = 5XlO-3, 
 
 po = 540 +0.0325 ei-6+0.005 e^. 
 
 (18) 
 
 (19) 
 
 (20) 
 
 The values of po, calculated from equation (20), are given 
 in the sixth column of Table I, and their differences from the 
 observed values in the last column. As seen, the errors are in 
 both directions from the calculated values, except for the three 
 highest voltages, in which the observed values rapidly increase 
 beyond the calculated, due probably to the appearance of a 
 
186 ENGINEERING MATHEMATICS. 
 
 loss which does not exist at lower voltages — the eddy currents 
 caused by the magnetic stray field of saturation. 
 
 This rapid divergency of the observed from the calculated 
 values at high voltages shows that a calculation of the constants, 
 based on all observations, would have led to wrong values, 
 and demonstrates the necessity, first, to critically review the 
 series of observations, before using them for deriving constants, 
 so as to exclude constant errors or unidirectional deviation. It 
 must be realized that the method of least squares gives the most 
 probable value, that is, the most accurate results derivable 
 from a series of observations, only so far as the accidental 
 errors of observations are concerned, that is, such errors which 
 follow the general law of probability. The method of least 
 squares, however, cannot eliminate constant errors, that is, 
 deviation of the observations which have the tendency to be 
 in one direction, as caused, for instance, by an instrument reading 
 too high, or too low, or the appearance of a new phenomenon 
 in a part of the observation, as an additional loss in above 
 instance, etc. Against such constant errors only a critical 
 review and study of the method and the means of observa- 
 tion can guard, that is, judgment, and not mathematical 
 formalism. 
 
 The method of least squares gives the highest accuracy 
 available with a given number of observations, but is frequently 
 very laborious, especially if a number of constants are to be cal- 
 culated. It, therefore, is mainly employed where the number of 
 observations is limited and cannot be increased at will; but where 
 it can be increased by taking some more observations — as is 
 generally the case with experimental engineering investigations 
 — the same accuracy is usually reached in a shorter time by 
 taking a few more observations and using a simpler method 
 of calculation of the constants, as the 2A-method described in 
 paragraphs 153 to 157. 
 
 Diophantic Equations. 
 
 1 23 A. — The method of least squares deals with the case, 
 when there are more equations than unknown quantities. In 
 this case, there exists no set of values of the unknown quantities, 
 which exactly satisfies the equations, and the problem is, to find 
 
MAXIMA AND MINIMA. 186a 
 
 the set of values, which most nearly satisfies the equations, that 
 is, which is the most probable. 
 
 Inversely, sometimes in engineering the case is met, when there 
 are more unknown than equations, for instance, two equations 
 with three unknown quantities. Mathematically, this gives not 
 one, but an infinite series of sets of solutions of the equations. 
 Physically however in such a case, the number of permissible 
 solutions may be limited by some condition outside of the algebra 
 of equations. Such for instance often is, in physics, engineering, 
 etc., the condition that the values of the unknown quantities 
 must be positive integer numbers. 
 
 Thus an engineering problem may lead to two equations with 
 three unknown quantities, which latter are limited by the con- 
 dition of being positive and integer, or similar requirements, 
 and in such a case, the number of solutions of the equation may 
 be finite, although there are more equations than unknown 
 quantities. 
 
 For instance: 
 
 In calculating from economic consideration, in a proposed 
 hydroelectric generating station, the number of generators, 
 exciters and step-up transformers, let: 
 
 X = number of generators 
 
 y = number of exciters 
 
 z = number of transformers 
 
 Suppose now, the physical and economic conditions of the 
 installation lead us to the equations: 
 
 8x + Zy + z = 49 (1) 
 
 2x + y + Sz = 21 (2) 
 
 These are two equations with three unknown, x, y, z; these 
 unknown however are conditioned by the physical requirement, 
 that they are integer positive numbers. 
 
 To attempt to secure a third equation would then over deter- 
 mine the problem, and give either wrong, or limited results. 
 
 Eliminating z from (1) and (2), gives: 
 
 11a; -h 4?/ = 63 (3) 
 
186& ENGINEERING MATHEMATICS. 
 
 hence: 
 
 63 - llx _ „ I 3 -3x , 
 
 2/ = 4 = 15 - 2a; + -^-- (4) 
 
 3 — 3a; 
 since y must be an integer number, — | — must also be an 
 
 integer number. Call this u, it is: 
 3 - 3a; 
 
 (5) 
 
 since x must be an integer number, ^ must also be an integer 
 
 number, that is: 
 
 M = 3y 6) 
 
 hence, substituted into (5), (4) and (2): 
 
 4 -" 
 
 
 
 3 -4m ^ 
 X- 3 -1- 
 
 - u - 
 
 u 
 "3 
 
 a; = 1 — 4y 
 2/ = 13 + lly 
 
 z == 2 - V 
 
 (7) 
 
 (7) thus are the solutions of the equations (1) (2), where v is any 
 integer number. 
 
 As seen, mathematically, there are an infinite number of 
 solutions. 
 
 Substituting now for v integer numbers: 
 
 V = +2 +1 0-1-2 
 
 X = - 7 -3 +1 +5 +9 
 
 2/=+ 35 +24 +13 +2 -9 
 
 z = +1 +2 +3+4 
 
 As seen, there are only two solutions, f or y = and v = — 1, 
 which give for x, y, and z, three integer positive values, and which 
 thus satisfy the physical restriction. 
 
 V = 0; X = 1, y = 13, z = 2 is excluded by engineering con- 
 sideration, as nobody would consider thirteen exciters with one 
 generator, and thus there remains only one applicable solution; 
 
MAXIMA AND MINIMA. 186c 
 
 X = 5 
 
 2 = 3 
 
 We thus have here the case of two equations with three un- 
 known quantities, which have only one single set of these un- 
 known quantities satisfying the problem, and thus give a definite 
 solution, though mathematically indefinite. 
 
 This type of equation has first been studied by Diophantes 
 of Alexandria. 
 
CHAPTER V. 
 METHODS OF APPROXIMATION. 
 
 124. The investigation even of apparently simple engineer- 
 ing problems frequently leads to expressions which are so 
 complicated as to make the numerical calculations of a series 
 of values very cumbersonme and almost impossible in practical 
 work. Fortunately in many such cases of engineering prob- 
 lems, and especially in the field of electrical engineering, the 
 different quantities which enter into the problem are of very 
 different magnitude. Many apparently complicated expres- 
 sions can fiequently be greatly simphfied, to such an extent as 
 to permit a quick calculation of numerical values, by neglect- 
 ing terms which are so small that their omission has no appre- 
 ciable effect on the accuracy of the result; that is, leaves the 
 result correct within the limits of accuracy required in engineer- 
 ing, which usually, depending on the nature of the problem, 
 is not greater than from 0.1 per cent to 1 per cent. 
 
 Thus, for instance, the voltage consumed by the resistance 
 of an alternating-current transformer is at full load current 
 only a small fraction of the supply voltage, and the exciting 
 current of the transformer is only a small fraction of the full 
 load current, and, therefore, the voltage consumed by the 
 exciting current in the resistance of the transformer is only 
 a small fraction of a small fraction of the supply voltage, hence, 
 it is negligible in most cases, and the transformer equations are 
 greatly simplified by omitting it. The power loss in a large 
 generator or motor is a small fraction of the input or output, 
 the drop of speed at load in an induction motor or direct- 
 current shunt motor is a small fraction of the speed, etc., and 
 the square of this fraction can in most cases be neglected, and 
 the expression simplified thereby. 
 
 Frequently, therefore, in engineering expressions con- 
 taining small quantities, the products, squares and higher 
 
 187 
 
188 ENGINEERING MATHEMATICS. 
 
 powers of such quantities may be dropped and the expression 
 thereby simphfied; or, if the quantities are not quite as small 
 as to permit the neglect of their squares, or where a high 
 accuracy is rciciuired, the first and second powers may be retained 
 and only the cul^es and higher powers dropped. 
 
 The most common method of procedure is, to resolve the 
 expression into an infinite series of successive powers of the 
 small quantity, and then retain of this series only the first 
 term, or only the first two or three terms, etc., depending on the 
 smallness of the quantity and the required accuracy. 
 
 125. The forms most frequently used in the reduction of 
 expressions containing small quantities are multiplication and 
 division, the binomial series, the exponential and the logarithmic 
 series, the sine and the cosine series, etc. 
 
 Denoting a .small quantity by s, and where several occur, 
 by Si, S2, S3 . . . the following expression holds: 
 
 (1 ± Si) (1 ± .S2) = 1 ± Si ± S2 ± S1S2, 
 
 and, since S1S2 is small compared with the small quantities 
 Si and S2, or, as usually expressed, S1S2 is a small quantity of 
 higher order (in this case of second order), it may be neglected, 
 and the expression written : 
 
 (l±Si)(l±S2) = l±Si±S2 (1) 
 
 This is one of the most useful simplifications : the multiplica- 
 tion of terms containing small quantities is replaced b}^ the 
 simple addition of the small quantities. 
 
 If the small quantities Si and S2 are not added (or subtracted) 
 to 1, but to other finite, that is, not small quantities a and h, 
 a and h can be taken out as factors, thus, 
 
 {a±s,){h±S2)=ah(\±^{l±^=ah{l±-^±'^, . (2) 
 
 where — and -r must be small quantities. 
 
 As seen, in this case, si and .S2 need not necessarilj' be abso- 
 lutely small ("luantities, but may be quite large, provided that 
 a and b are still larger in magnitude; that is, Si must be small 
 compared with a, and S2 small compared with b. For instance, 
 
METHODS OF APPROXIMATION. 189 
 
 in astronomical calculations the mass of the earth (which 
 absolutely can certainly not be considered a small quantity) 
 is neglected as small quantity compared with the mass of the 
 sun. Also in the effect of a lightning stroke on a primary 
 distribution circuit, the normal line voltage of 2200 may be 
 neglected as small compared with the voltage impressed by 
 ~ lightning, etc. 
 
 126. Example. In a direct-current shunt motor, the im- 
 pressed voltage is eo = 125 volts; the armature resistance is 
 ro = 0.02 ohm; the field resistance is ri = 50 ohms; the power 
 consumed by friction is p/=^-300 watts, and the power consumed 
 by iron loss is pi=400 watts. What is the power output of 
 the motor at io = 50, 100 and 150 amperes input? 
 
 The power produced at the armature conductors is the 
 product of the voltage e generated in the armature conductors, 
 and the current i through the armature, and the power output 
 at the motor pulley is, 
 
 ■p = ei-'Pf-'Pi- ■ ■ ■ ■ (3) 
 
 The current in the motor field is — , and the armature current 
 therefore is, 
 
 i = in- 
 
 ^ = ^o--, (4) 
 
 6n 
 
 where — is a small quantity, compared with io. 
 
 The voltage consumed by the armature resistance is roi, 
 and the voltage generated in the motor armature thus is: 
 
 e = eo — roi, (5) 
 
 where roi is a small quantity compared with eo. 
 Substituting herein for i the value (4) gives, 
 
 = eo-ro[io-:fj (6) 
 
 Since the second term of (6) is small compared with eo, 
 
 €0 
 and in this second term, the second term — is small com- 
 
 pared with io, it can be neglected as a small term of higher 
 
190 ENGINEERING MATHEMATICS. 
 
 order; that is, as small compared with a small term, and 
 expression (6) simplified to 
 
 e = eo — roio (7) 
 
 Substituting (4) and (7) into (3) gives, 
 
 p = (eo - roio) U'o -yj-Vf- Pi 
 
 = eoio{l-'-ff){l-^)-Pf-p. ... (8) 
 
 Expression (8) contains a product of two terms with small 
 quantities, which can be multiplied by equation (1), and thereby 
 gives, 
 
 p^eoto[l-—-~j-J-pf-pi 
 
 = eoio—roio^ — - — Pf-pi (9) 
 
 Substituting the numerical values gives, 
 
 p = 125i;o-0.02io2-562.5-300-400 
 = 125-io — 0.02io2— 1260 approximately; 
 
 thus, for to=50, 100, and 150 amperes; p = 4940, 11,040, and 
 17,040 watts respectively. 
 
 127. Expressions containing a small quantity in the denom- 
 inator are frequently simplified by bringing the small quantity 
 in the numerator, by division as discussed in Chapter II para- 
 graph 39, that is, by the series, 
 
 -^-— = lTx+x^Tx^+x*^x^+ .., . . (10) 
 
 which series, if x is a small quantity s, can be approximated 
 by: 
 
 1 
 
 1+s 
 
 1 
 
 1-s 
 
 = l-s: 
 
 = l+s; 
 
 (11) 
 
METHODS OF APPROXIMATION. 
 
 191 
 
 or, where a greater accuracy is required, 
 
 1 
 
 1+s 
 
 _1_ 
 1-s 
 
 = l-s+s2; 
 
 = 1+S+S2. 
 
 (12) 
 
 By the same expressions (11) and (12) a small quantity 
 contained in the numerator may be brought into the denominator 
 where this is more convenient, thus : 
 
 l+s = 
 
 1-s' 
 
 .1 — 8=^;—;—; etc. 
 1+s' 
 
 (13) 
 
 More generally then, an expression like , where s is 
 
 small compared with a, may be simplified by approximation to 
 the form, 
 
 a±s 
 
 ''i^-O^' 
 
 4^..^ 
 
 (14) 
 
 or, where a greater exactness is required, by taking in the second 
 term, 
 
 128. Example. What is the current input to an induction 
 motor, at impressed voltage eo and slip s (given as fraction of 
 synchronous speed) if tq + jxo is the impedance of the primary 
 circuit of the motor, and ri + jxi the impedance of the secondary 
 circuit of the motor at full frequency, and the exciting current 
 of the motor is neglected; assuming s to be a small quantity; 
 that is, the motor running at full speed? 
 
 Let E be the e.m.f. generated by the mutual magnetic flux, 
 that is, the magnetic flux which interlinks with primary and 
 with secondary circuit, in the primary circuit. Since the fre- 
 quency of the secondary circuit is the fraction s of the frequency 
 
192 ENGINEERING MATHEMATICS. 
 
 of the primary circuit, the generated e.m.f. of the secondary 
 circuit is sE. 
 
 Since xi is the reactance of the secondary circuit at full 
 frequency, at the fraction s of full frequency the reactance 
 of the secondary circuit is sxi, and the impedance of the sec- 
 ondary circuit at slip s, therefore, is ri+jsxi; hence the 
 secondary current is, 
 
 • ri + ]sxi 
 
 If the exciting current is neglected, the primary current 
 equals the secondary current (assuming the secondary of the 
 same number of turns as the primary, or reduced to the same 
 number of turns); hence, the current input into the motor is 
 
 7 = -^ (16) 
 
 The second term in the denominator is small compared 
 with the first term, and the expression (16) thus can be 
 approximated by 
 
 sE sE(^ .sxA 
 
 1 = —, r-=— 1-2— (17) 
 
 M^ + l-z 
 
 The voltage E generated in the primary circuit equals the 
 impressed voltage eo, minus the voltage consumed by the 
 current / in the primary impedance; ro + j'xo thus is 
 
 ^ = eo-/(ro+j>o). . . ... (18) 
 
 Substituting (17) into (18) gives 
 
 sE I s^' \ 
 
 S = e„--(ro+j>o)(^l-.7^j. . . . (19) 
 
 In expression (19), the second term on the right-hand side, 
 which is the impedance drop in the primary circuit, is small 
 
 compared with the first term eo, and in the factor ( 1 — 7 — 
 
 \ n 
 
 of this small term, the small term ]'— can thus be neglected 
 
METHODS OF APPROXIMATION. 193 
 
 as a small term of higher order, and equation (19) abbreviated 
 to 
 
 sE 
 E = eo~ — {ro + jxo) (20) 
 
 From (20) it follows that 
 
 E= :r^ ■, 
 
 and by (13), 
 
 S = eo jl-^(ro+/xo) I (21) 
 
 Substituting (21) into (17) gives 
 
 and by (1), 
 
 = — \^-s--]s—. — } (22) 
 
 If then, loo^in—jio' is the exciting current, the total 
 current input into the motor is, approximately, 
 
 =— |l+'^--F^— )+^o-po'. - . . (23) 
 
 129. One of the most important expressions used for the 
 reduction of small terms is the binomial series : 
 
 n(n— 1) „ n(n— l)(n— 2) 
 (l±x)« = l±nx+-^-2 x2± 1^ 
 
 2 -I ^ l! L -J.3 
 
 +- y-^ x4±... (24) 
 
 If x is a small term s, this gives the approximation, 
 
 (l±s)" = l±ns; ... . . (25) 
 
194 ENGINEERING MATHEMATICS. 
 
 or, using the second term also, it gives 
 
 (l±,s)" = l±ns+^^^s2 (26) 
 
 In a more general form, this expression gives 
 
 1±-) =0/(1 ±-^1; etc. . . (27) 
 
 By the binomial, higher powers of terms containing small 
 quantities, and, assuming n as a fraction, roots containing 
 small quantities, can be eliminated; for instance, 
 
 „, 1 i/ s\^ ^r-( •'-■ \ 
 
 Va±s=(a±s)« =an| 1 ±- ) =vall± — I; 
 
 1 1 1/ s\-« 1/, ns\ 
 
 =t;; l±r =T^ IT- ; 
 
 ia±sY I sY a"\ a / WX a J ' 
 
 1 , ,-i -1/ s\-i 1 / s\ 
 
 n/ ~ = (a±s) »=a " 1±-) =-;r7=(lT— ); 
 
 m 
 
 v(a±s)"> = (a±s)" =anM ±- ) =va'"l 1± — 1; etc. 
 
 One of the most common uses of the binomial series is for 
 the elimination of squares and square roots, and very fre- 
 quently it can be conveniently applied in mere numerical calcu- 
 lations; as, for instance, 
 
 (201)2 = 2002(1 +^y)' = 40,000(l +j-^ =40,400; 
 29.92 = 302 (1 _ _L j ' = 900 (1 _ ^-) = 900 - 6 = 894 ; 
 
 vmS = lOVl - 0.02 = 10(1 - 0.02) 2 = 10(1 - 0.01) = 9.99; 
 
 1 1 1 
 
 = ^ n .. ^ = 0.985; etc. 
 
 VlM (1 + 0.03)1/2 1015 
 
METHODS OF APPROXIMATION. 195 
 
 130. Example i. If r is the resistance, x the reactance of an 
 alternating-current circuit with impressed voltage e, the 
 current is 
 
 x/r^+x^ 
 
 If the reactance x is small compared with the resistance r, 
 as is the case in an incandescent lamp circuit, then. 
 
 1^ 
 
 e ( /xY] 2 
 
 ^■ = -7== = — r=== = - 1 ■ I 
 
 ^n- 
 
 
 If the resistance is small compared with the reactance, as 
 is the case in a reactive coil, then. 
 
 e e ^ J 1 /'" 
 
 Vr2+x2 / /r\2 x\ \x 
 
 X. 
 
 '^J 
 
 ''^-im (28) 
 
 x[ 2\x 
 
 Example 2. How does the short-circuit current of an 
 alternator vary with the speed, at constant field excitation? 
 
 When an alternator is short circuited, the total voltage 
 generated in its armature is consumed by the resistance and the 
 synchronous reactance of the armature. 
 
 The voltage generated in the armature at constant field 
 excitation is proportional to its speed. Therefore, if eo is the 
 voltage generated in the armature at some given speed So, 
 for instance, the rated speed of the machine, the voltage 
 generated at any other speed S is 
 
 S 
 
 -^0 
 
196 ENGINEERING MATHEMATICS. 
 
 a 
 
 or, if for convenience, the fraction -^r is denoted bj^ a, then 
 a^-^ and e = aeo, 
 
 where a is the ratio of the actual speed, to that speed at which 
 the generated voltage is eo- 
 
 If r is the resistance of the alternator armature, xq the 
 synchronous reactance at speed So, the synchronous reactance 
 at speed S is x = axn, and the current at short circuit then is 
 
 i^-^=^^-Jf^= (29) 
 
 Vr^+x^ V r2 + a^X(p 
 
 Usually r and Xq are of such magnitude that r consumes 
 at full load about 1 per cent or less of the generated voltage, 
 while the reactance voltage of Xo is of the magnitude of from 
 20 to 50 per cent. Thus r is small compared with Xq, and if 
 a is not very small, equation (29) can be approximated by 
 
 aeo eo| 1 / r V 
 
 «-roJl+ — 
 \axo/ 
 
 oJl+(- 
 
 (30) 
 
 Then if a;o = 20r, the following relations exist: 
 a= 0.2 0.5 1.0 2.0 
 
 t = -x0.9688 0.995 0.99875 0.99969 
 
 Xo 
 
 That is, the short-circuit current of an alternator is practi- 
 cally constant independent of the speed, and begins to decrease 
 only at very low speeds. 
 
 131. Exponential functions, logarithms, and trigonometric 
 functions are the ones frequently met in electrical engineering. 
 
 The exponential function is defined by the series, 
 
 ■ ^ , x^ x^ x^ x^ 
 
METHODS OF APPROXIMATION. 197 
 
 and, if a; is a small quantity, s, the exponential function, may 
 be approximated by the equation, 
 
 £±^ = l±s; (32) 
 
 or, by the more general equation, 
 
 £±'"' = l±as; . . . (33) 
 
 and, if a greater accuracy is required, the second term may 
 be included, thus, 
 
 S' 
 
 2 
 
 £±« = l±s+-, (34) 
 
 and then 
 
 2g2 
 
 j±M = ij.as+r__ (35) 
 
 fdx 
 
 The logarithm is defined by logs x= I — ; hence, 
 
 log.(l±x)=±J^|^. 
 Eesolving :j--^ into a series, by (10), and then integrating, 
 
 gives 
 
 log£(l±x)=± j {lTx+x'^^^x^ + . . .)dx 
 
 y2 'y'3 .jA 0-5 
 
 = ±.x-y±3— jig- (36) 
 
 This logarithmic series (36) leads to the approximation, 
 
 logs (lis) =±s; . ... (37) 
 
 or, including the second term, it gives 
 
 log. (lis) =±s-s2, (38) 
 
 and the more general expression is, respectively, 
 
 log. (ais)=loga(li^-)=loga+log(li^)=logai^, (39) 
 
198 ENGINEERING MATHEMATICS 
 
 and, more accurately, 
 
 s s^ 
 
 loge (a±s) = loga±----2 
 
 a a^ 
 
 (40) 
 
 Since logio iV = logio fXlogt N=0A3i3 log£ A'', equations (39) 
 and (40) may be written thus, 
 
 logio(l±s)=± 0.4343s; 
 logio (a±s) = logio a ±0.4343 - 
 
 . (41) 
 
 132. The trigonometric functions are represented by the 
 infinite series : 
 
 , x^ x^ sfi 
 cos.T = l-i2+|4-|g+. 
 
 sin X=X—-rr+T^ — '-rT+. 
 
 li li li 
 
 (42) 
 
 which when s is a small quantity, may be approximated by 
 
 coss = l and sin s = s; . . . 
 or, they may be represented in closer approximation by 
 
 (43) 
 
 cos s = l — — ; 
 
 sm s = ; 
 
 or, by the more general expressions. 
 
 (44) 
 
 cos as = 1 and cos as = 1 ■ 
 
 2q2 
 
 a-'s- 
 
 sin as = as and sin as ■- 
 
 
 (45) 
 
 133. Other functions containing small terms may frequently 
 be approximated by Taylor's series, or its special case, 
 MacLaurin's series. 
 
 MacLaurin's series is written thus: 
 
 /(■^) =/(0) +xf'(0) +p/"(0) +p/-"'(0) +. . . , . (46) 
 
METHODS OF APPROXIMATION. 199 
 
 where /', /", /'", etc., are respectively the first, second, third, 
 etc., differential quotient of/; hence, 
 
 /(s)^/(0)+s/'(0): 1 
 
 f{as)=f{0)+asf'{0). \ 
 Taylor's series is written thus. 
 
 (47^ 
 
 X .,.,,. X' 
 
 3 
 
 fib +X) =/(6) +Xf{b) + r^f"(h) +r:^f"{b) +..., . (48) 
 
 and leads to the approximations : 
 fib±s)=f(b)±sf'ib); 
 
 f{b±as)=f{b)±asf'{b). 
 
 (49) 
 
 Many of the previously discussed approximations can be 
 considered as special cases of (47) and (49). 
 
 134. As seen in the preceding, convenient equations for the 
 approximation of expressions containing small terms are 
 derived from various infinite series, which are summarized 
 below : 
 
 • ni.n—1) „ n{n—1)(n—2) „ 
 {l±x)" = l±nx+ .^ x2±- r^ -x^ + . . 
 
 /V.2 -yO -y^ 
 
 /V.2 -yO jy*^ 
 
 log£ {l±x)=±x-'-^±j-j±. . . ; 
 
 .T^2 'Y'4 ^6 
 
 cosx = l-J2+|4-|g+...; 
 
 jf'O ■po ^7 
 &iB.X = X — -nr+j^ — -r=-+. . . ; 
 II ll li 
 
 Ax) =/(0) +xf'(0) +|V(0) +Sf"'m +. ; 
 f(b±x) =f{b) ±xf'(b) +|/"(&) ±y"'(b) +... 
 
 \ m 
 
200 
 
 ENGINEERING MATHEMATICS. 
 
 The first approximations, derived by neglecting all higher 
 terms but the first power of the small quantity x = s in these 
 series, are: 
 
 (l±s)" = l±«s; 
 
 
 n(n—l) „ 
 
 L+ 2 n 
 
 £±s = l4:s; 
 
 
 . + 2 J; 
 
 oge(l±s)= ±s; 
 
 
 S2" _ 
 
 cos s = 1 ; 
 
 
 ~ 2" ' 
 
 sin s = s; 
 
 
 
 f(.s)=f(0)+sf'{0)- 
 
 ■ (.2 
 
 fib±s)=f{b)±sf 
 
 '(b); 
 
 +f /"(?>)] ; 
 
 (51) 
 
 and, in addition hereto is to be remembered the multiplication 
 rule, 
 
 (l±si)(l±S2) = l±si±S2; [±SiS2]. . . (52) 
 
 135. The accuracy of the approximation can be estimated 
 by calculating the next term beyond that which is used. 
 This term is given in brackets in the above equations (50) 
 and (51). 
 
 Thus, when calculating a scries of numerical values by 
 approximation, for the one value, for M'hich, as seen by the 
 nature of the problem, the approximation is least close, the 
 next term is calculated, and if this is less than the permissible 
 limits of accuracy, the approximation is satisfactory. 
 
 For instance, in Example 2 of paragraph 130, the approxi- 
 mate value of the short-circuit current was found in (30), as 
 
 
 1 
 
 r \2 
 axo/ 
 
METHODS OF APPROXIMATION. 201 
 
 The next term in the parenthesis of equation (30), by the 
 binomial, would have been -\ -x — s^; substituting n=— i; 
 
 s = ( — ) , the next becomes +5-( — ) • The smaller the a, the 
 \a.xo) 8 \axo/ 
 
 less exact is the approximation. 
 
 The smallest value of a, considered in paragraph 130, was 
 
 3 / J. \4 
 
 a = 0.2. For 3;o = 20r, this gives +^[ — =0.00146, as the 
 
 o \axo/ 
 
 value of the first neglected term, and in the accuracy of the 
 result this is of the magnitude of - X 0.00146, out of - X 0.9688, 
 
 Xq ' Xo 
 
 the value given in paragraph 130; that is, the approximation 
 
 gives the result correctly within ' ,„„„ =0.0015 or within one- 
 ^ 0.9680 
 
 sixth of one per cent, which is sufficiently close for all engineer- 
 ing purposes, and with larger a the values are still closer 
 approximations. 
 
 136. It is interesting to note the different expressions, 
 which are approximated by (1+s) and by (1 — s). Some of 
 them are given in the following : 
 
 1-s 
 
 = 
 
 1 
 
 
 1+s' 
 
 
 
 \\ 
 
 1 ' 
 
 H 
 
 )• 
 
 1 
 
 
 
 )"' 
 
 ("" 
 
 
 W- 
 
 ,— m 
 s 
 
202 ENGINEERING MATHEMATICS. 
 
 VrpIT; Vl-2s; 
 
 1 1 . 
 
 Vl^^' Vl+2s' 
 
 \l-s' \l+s' 
 
 1 1 . 
 
 -C/r=^' ^l+ns' 
 
 -ms 
 
 S/1 — (n— m)s' 
 etc. 
 
 
 \l+(n-m)s' 
 etc. 
 
 £»; 
 
 
 s~'; 
 
 2-e-'; 
 
 
 2-^- 
 
 1+log. (1+s); 
 
 
 l+log.(l-s); 
 
 l-loge(l-s); 
 
 
 l-loge(l+s); 
 
 l+nlog.(l+-); 
 
 l+nlog.(l--j; 
 
 l_„log,(l-i); 
 
 l-nloge(l+^); 
 
 l+log.^^_^ 
 
 J 
 
 l+loge^l+gi 
 
 1 ^«g^^|l+s 
 
 > 
 
 1 log.^^_^; 
 
 etc. 
 
 
 etc. 
 
 1+sin s; 
 
 
 1 — sin s; 
 
 . s 
 l+nsm — ; 
 n' 
 
 
 1 — ?i sm - : 
 n' 
 
METHODS OF APPROXIMATION. 
 
 203 
 
 , 1 • 
 
 1 H — sm ns] 
 
 cos V— 2s; 
 etc. 
 
 1 — sin ns; 
 
 cos V2s; 
 etc. 
 
 137. As an example may be considered the reduction to its 
 simplest form, of the expression : 
 
 2si J2g. 
 
 ■\/a^-\'(a + Si)3i4 — sin 6S2! -^lae"- cos^^/ — 
 
 — — \ (Jb 
 
 e-3«Ca+2si) 1 
 
 1 /a-S2 
 
 — alogf^ — — 
 
 v'a— 2si 
 
 then, 
 
 4— sin6s2 = 4(l — jsin6s2J =4(1 — ^82); 
 
 2f} si 
 
 ea=l+2-; 
 
 COS'' 
 
 = 1 
 
 Sl 
 
 a \ a 
 
 £-3« = ]-3s2; 
 
 = 1-2^; 
 a' 
 
 o+2si = o 1+2 
 
 a/' 
 
 -alog£^^^ = l-alog, 
 
 1+S2 
 
 -alog£^l-2j 
 
 = l-aIog£fl — -j=l+S2; 
 
 2si\ 1/2 
 
 Va-2,s,^aV2(l-=_j -a^^\l~~); 
 
 Sl 
 
204 ENGINEERING MATHEMATICS. 
 
 hence, 
 
 (ji/2xa3/4(^l+| ii) X4(l-^S2) X 
 F=- 
 
 .-x(l+2j)(.-2j) 
 
 (1-3^2) X«(l+2j)(]+S2)Xai/2^1-^) 
 
 4a3/.(l+3£l_3 ,^_2^\ 
 \ 4 a 2 a a/ 
 
 — +S2 ) 
 
 a a I 
 
 a3/2(l-3s2+2 
 
 = 4(1-1^+^-1 
 V 4 a ^2a 
 
 138. As further example may be considered the equations 
 of an alternating-current electric circuit, containing distributed 
 resistance, inductance, capacity, and shunted conductance, for 
 instance, a long-distance transmission line or an underground 
 high-potential cable. 
 
 Equations of the Transmission Line. 
 
 Let I be the distance along the line, from some starting 
 point; E, the voltage; /, the current at point I, expressed as 
 vector quantities or general numbers; Zo = ro+jxo, the line 
 impedance per unit length (for instance, per mile); Yo = go+jbo 
 = line admittance, shunted, per unit length; that is, tq is the 
 ohmic effective resistance; xo, the self-inductive reactance; 
 60, the condensive susceptance, that is, wattless charging 
 current divided by volts, and ^0 = energy component of admit- 
 tance, that is, energy component of charging current, divided 
 by volts, per unit length, as, per mile. 
 
 Considering a line element dl, the voltage, dE, consumed 
 by the impedance is Zoldl, and the current, dl, consumed by 
 the admittance is Yi,Ed.l; hence, the following relations may be 
 written : 
 
 dl 
 dl 
 
 -^=YoE (2) 
 
METHODS OF APPROXIMATION. 205 
 
 Differentiating (1), and substituting (2) therein gives 
 (PE 
 
 dP 
 
 = ZoYoE, 
 
 and from (1) it follows that, 
 
 j_J_dE 
 
 Zq dl ' 
 
 Equation (3) is integrated by 
 
 E = AeSl, 
 and (5) subbtituted in (3) gives 
 
 B=±VZoYo; 
 hence, from (5) and (4), it follows 
 
 [Yo. 
 
 Yd. 
 
 L2« 
 
 -VzoYoi: 
 
 Next assume 
 
 l = lo, the entire length of line; 
 Z = IqZo, the total line impedance ; 
 and Y=loYo, the total line admittance; 
 
 then, substituting (9) into (7) and (8), the following expressions 
 are obtained : 
 
 (3) 
 
 (4) 
 
 (5) 
 
 (6) 
 
 (7) 
 (8) 
 
 (9) 
 
 Ei = AiS + ^^Y_^_^,^.-VzY; 
 
 lY. 
 
 Iv 
 
 {Ai£ + V^-^2^-^^^! 
 
 (10) 
 
 as the voltage and current at the generator end of the line. 
 
 139. If now £0 and h respectively are the current and 
 voltage at the step-down end of the line, for 1=0, by sub- 
 stituting 1 = into (7) and (8), 
 
 Ai+A2=Eq] 
 
 Ai-A2 = Ij 
 
 Y- 
 
 (11) 
 
206 
 
 ENGINEERING MATHEMATICS. 
 
 Substituting in (10) for the exponential function, the series, 
 , ZY ZYVZY Z272 z^Y^VZY 
 
 and arranging by {A1+A2) and (Ai— A2), and substituting 
 herefor the expressions (11), gives 
 
 Ei = E 
 
 0' 
 
 ZY Z272] „{ ZY Z^Y^ 
 
 , , . Z7 i?272i f ZY Z^Y^ 
 
 7i = /o \l+~^+-,^\ +YEq\1+-j^ + 
 
 24 
 
 °|^+"6"+T20"( 
 
 (13) 
 
 When 1= —lo, that is, for Eo and /o at the generator side, and 
 El and h at the step-down side of the line, the sign of the 
 second term of equations (13) merely reverses. 
 
 140. From the foregoing, it follows that, if Z is the total 
 impedance; Y, the total shunted admittance of a transmission 
 line, 'Eo and lo, the voltage and current at one end; Ei and 7i, 
 the voltage and current at the other end of the transmission 
 line; then, 
 
 ^ ^ . ZY Z^Y^] 
 hi=Jio\ IH — 2~"' — 94 
 
 ^, . ZY Z2y2 
 
 ±ZIo{l+-^+^^ 
 
 , , ,, ZY Z^Y^, 
 
 ^^^ i ZY ZW2] 
 
 (14) 
 
 where the plus sign applies if Eq, Iq is the step-down end, 
 the minus sign, if Eq, Iq is the step-up end of the transmission 
 line. 
 
 In practically all cases, the quadratic term can be neglected, 
 and the equations simplified, thus. 
 
 Ei = Eo' 
 
 l^^Uz7ofl+f 
 
 /x=/„|i+^|±rEo{i+^}, 
 
 (15) 
 
 z^y^ 
 
 and the error made hereby is of the magnitude of less than — 
 
 24 
 
METHODS OF APPROXIMATION. 207 
 
 Except in the ease of very long lines, the second term of 
 the second term can also usually be neglected, which givcb 
 
 E, = Eo(l+^)±ZIo; 
 
 h= /o(l+^)±7?o, J 
 
 (16) 
 
 and the error made hereby is of the magnitude of less than — 
 
 of the Hne impedance voltage and line charging current. 
 
 141. Example. Assume 200 miles of 60-cycle line, on non- 
 inductive load of 60 = 100,000 volts; and io = 100 amperes. 
 The line constants, as taken from tables are 2 = 104 +140; ohms 
 and y = +0.0013/ ohms; hence, 
 
 Z7=- (0.182-0.136/); 
 
 .Bi = 100000(1-0.091+0.068]) +100(104 + 140;) 
 = 101400+20800;, in volts; 
 
 Zi = 100(l-0.091+0.068;)+0.0013;Xl000G0 
 = 91+136.8;', in amperes. 
 
 . zy 0.174X0.0013 0.226 ^ ^^„ 
 
 The error is -^ = ^ = —;5— = 0.038. 
 
 6 6 6 
 
 In El, the neglect of the second term of 0/0 = 17,400, gives 
 an error of 0.038x17,400 = 660 volts = 0.6 per cent. 
 
 In Zi, the neglect of the second term of 2/£'o = 130, gives an 
 error of 0.038x130 = 5 amperes = 3 per cent. 
 
 Although the charging current of the line is 130 per cent 
 of output current, the error in the current is only 3 per cent. 
 
 Using the equations (15), which are nearly as simple, brings 
 22„2 226^ 
 the error down to -:&■=' 94 =0.0021, or less than one-quarter 
 
 per cent. 
 
 Hence, only in extreme cases the equations (14) need to be 
 
 used. Their error would be less than ■;^ = 'i.QxlO~^, or one 
 
 three-thousandth per cent. 
 
208 ENGINEERING MATHEMATICS. 
 
 The accuracy of the preceding approximation can be esti- 
 mated by considering the physical meaning of Z and Y: Z 
 is the Hne impedance; hence ZI the impedance voltage, and 
 
 zi 
 
 u = ^, the impedance voltage of the line, as fraction of total 
 
 voltage; Y is the shunted admittance; hence YE the charging 
 
 YE . 
 
 current, and v=—j-, the charging current of the hne, as fraction 
 
 of total current. 
 
 Multiplying gives uv=ZY; that is, the constant ZY is the 
 product of impedance voltage and charging current, expressed 
 as fractions of full voltage and full current, respectively. In 
 any economically feasible power transmission, irrespective of 
 its length, both of these fractions, and especially the first, 
 must be relatively small, and their product therefore is a small 
 quantity, and its higher powers negligible. 
 
 In any economically feasible constant potential transmission 
 line the preceding approximations are therefore permissible. 
 
 Approximation by Chain Fraction. 
 
 141A. — A convenient method of approximating numerical 
 values is often afforded by the chain fraction. A chain fraction 
 is a fraction, in which the denominator contains a fraction, which 
 again in its denominator contains a fraction, etc. Thus: 
 
 2+ 1 
 
 3+ 1 
 
 1+1 
 4 
 
 Only integer chain fractions, that is, chain fractions in which 
 all numerators are unity, are of interest. 
 
 A common fraction is converted into a chain fraction thusly: 
 
APPROXIMATION BY CHAIN FRACTION. 208a 
 
 511 ^ ^ 
 
 
 1152 ■ 
 
 511 1 
 
 = 
 
 1 
 
 1152 1152 
 
 „ , 130 
 
 511 
 
 
 2 + 
 ^ 511 
 
 1 
 
 
 1 
 
 ~2 + 1 
 
 
 2 + 1 
 
 511 
 
 
 o , 121 
 
 130 
 
 
 ^+130 
 
 1 
 
 
 1 
 
 ~ 2+ 1 
 
 ~ 
 
 2 + 1 
 
 3+ 1 
 
 3+ 1 
 
 
 130 
 
 9 
 
 
 121 
 
 1+121 
 
 1 
 
 
 1 
 
 "2 + 1 
 
 
 2 + 1 
 
 3+ 1 
 
 3+ 1 
 
 
 1 + 
 
 1 r+ 1 
 
 121 13 + ^ 
 9 l"^+9 
 
 1 
 
 
 1 
 
 "2+ 1 
 
 
 2 + 1 
 
 3 + 
 
 1 
 
 3+ 1 
 
 
 1 + 
 
 1 1+1 
 13+1 13+1 
 
 ^ 2+1 
 
 4 2+4 
 
 That is, to convert a common fraction into a chain fraction, 
 the numerator is divided into the denominator, the residue 
 divided into the divisor, and so on, until no residue remains. 
 The successive quotients then are the successive denominators 
 of the chain fraction. 
 
 For instance : 
 
2086 ENGINEERING MATHEMATICS 
 
 511 ^ ^ 
 
 
 1152 
 
 * 
 
 
 
 511/1152 
 
 = 2 
 
 
 
 
 1022 
 
 
 
 
 
 130/511 
 
 = 3 
 
 
 
 
 390 
 
 
 
 
 
 121/130 
 
 = 1 
 
 
 
 
 121 
 
 
 
 
 
 9/121 = 
 
 13 
 
 
 
 
 9 
 
 
 
 
 
 31 
 
 
 
 
 
 27 
 
 
 
 
 
 4/9 
 
 = ; 
 
 
 
 
 8 
 
 
 hence: 
 
 1/4 = 4 
 
 511 
 
 1 
 
 1152 
 
 ~ 2 + 1 
 
 
 3+ 1 
 
 1 + 1 
 
 13+ 1 
 
 2+1 
 4 
 
 Inversely, the chain fraction is converted into a common 
 fraction, by rolling it up from the end : 
 
 2+i-I 
 
 
 4 4 
 
 
 1 4 
 
 
 1 ~ 9 
 
 
 2 + 4 
 
 
 13+ 1 
 
 121 
 
 1 
 
 9 
 
 2 + 4 
 
 
 1 
 
 9 
 
 13 + 1 
 
 121 
 
 ^+i 
 
 
APPROXIMATION BY CHAIN FRACTION. 208c 
 1 130 
 
 1 + 
 
 13 + 1 121 
 
 1 _ 121 
 
 ] + 1 ~ 130 
 
 13+1 
 
 ^ + 1 
 
 3 +1 _ 511 
 
 1 + 1 ~ 130 
 
 13 + 1 
 
 ^ + 1 
 
 1 _ 130 
 
 3 + 1 ~ 511 
 
 1+ 1 
 
 13+ 1 
 
 2+ 1 ^ 1152 
 
 511 
 
 13 + 1 
 
 ^^l 
 
 1 _ _51]^ 
 
 2 + 1 ~ 1152 
 
 3TI 
 
 1 + 1 
 
 13+ 1 
 
 ^^\ 
 
 The expression of the numerical value by chain fraction gives a 
 series of successive approximations. Thus the successive ap- 
 proximation of the chain fraction : 
 
208(1 ENGINEERING MATHEMATICS. 
 
 1 511 
 
 
 2 + 1 1152 
 3+1 
 
 1 + 1 
 
 13 + 1 
 
 
 
 
 ^ + \ 
 
 
 are: 
 
 
 difference: 
 
 = %: 
 
 (1)1 
 2 
 
 1 
 
 ~ 2 
 
 = .5 ... + .0564 
 
 = +12.7% 
 
 (2)1 
 
 ^-1 
 
 _ 3 
 
 7 
 
 = .42857 .. - .0150 
 
 = -3.4% 
 
 (3)1 
 2+1 
 
 4 
 9 
 
 = .44444 ... + .00086 
 
 = +.194% 
 
 3 + 
 
 (4) 1 
 
 2+ 1 
 
 = f^.= . 443548 
 
 .000028 = -.0068% 
 
 3+ 1 
 
 1 + A 
 13 
 
 (5) 1 
 
 2+ 1 
 
 114 
 
 257 =■''''''■ 
 
 + .000004 = +.0009% 
 
 3+ 1 
 
 1 + 1 
 
 13 + 
 
 (6)1 
 
 2+ 1 
 
 511 
 1152 
 
 .443576 
 
 3+ 1 
 
 1 + 1 
 
 13+ 1 
 
 ^^k 
 
APPROXIMATION BY CHAIN FRACTION. 208e 
 
 As seen, successive approximations are alternately above and 
 below the true value, and the approach to the true value is 
 extremely rapid. It is the latter feature which makes the chain 
 fraction valuable, as where it can be used, it gives very rapidly 
 converging approximations. 
 
 141B. — Chain fraction representing irrational numbers, as 
 IT, €, etc., may be endless. Thus: 
 
 3 + 1 
 
 7 + 1 
 
 = 3.14159265 ... 
 
 15 + 1 
 
 288+ 1 
 
 2 + 1 
 
 1 + 1 
 
 3 + 1 
 
 1 + 1 
 
 7 + 
 
 The first three approximations of this chain fraction of tt are : 
 
 difference : = % 
 (1)3 + ^ =3 1/7 =3.142857.. +.00127 = + .043% 
 
 [2) 3 + 1 ^g 15/106 = 3. 141 5094... -.0000832 =-.0026% 
 
 ^ + r5 
 
 ;3) 3 + 1 =3 16/113=3. 1415929... + .0000003 
 
 7+ 1 = + • 000009 % 
 
 15 + 1 
 1 
 
 As seen, the first approximation, 3 1/7, is already sufficiently 
 close for most practical purposes, and the third approximation 
 of the chain fraction is correct to the 6th decimal. 
 
 144. — Frequently irrational numbers, such as square roots, 
 can be expressed by periodic chain fractions, and the chain 
 
208/ ENGINEERING MATHEMATICS. 
 
 fraction offers a convenient way of expressing numerical values 
 containing square roots, and deriving their approximations. 
 
 For instance: 
 
 Resolve -\/6 into a chain fraction. 
 
 As the chain fraction is < 1, -\/6 has to be expressed in the form : 
 
 Ve = 2 + (V6 - 2) (1) 
 
 and the latter term: (V6 - 2), which is <1, expressed as chain 
 fraction. 
 
 To rationalize the numerator, we multiply numerator and 
 denominator by (\/6 + 2): 
 
 ^^ (V6-2)(V6 + 2) 2 1 
 
 ^^^~^^~ V6 + 2 ~-v/6 + 2~V6 + 2 
 
 thus: 
 
 \/6 = 2 + 
 
 V6 + 2 
 2 
 
 as is > 1, it is again resolved into: 
 
 V6 + 2 _ ^ |. \/6-2 
 
 __ 2 + ^ 
 
 thus: 
 
 V6=2+l 
 
 2 + ^ 
 
 continuing in the same manner: 
 
 V6-2 _ (V6-2)(\/6 + 2) _ 2 1 
 
 2 ~ 2(V6 + 2) ~ 2(a/6 + 2) ■" Ve + 2 
 
 hence: 
 
 V6 = 2 + 1 
 
 2+1 
 
 and: 
 hence: 
 
 V6 + 2 
 -v/6 + 2 = 4 + ( Ve - 2) 
 
APPROXIMATION BY CHAIN FRACTION. 208g 
 
 \/6 = 2+l 
 
 2+1 
 
 4 + (Ve - 2) 
 
 and, as the term {\/Q — 2) appeared already at (1), we are here 
 at the end of the recurring period, that is, the denominators now 
 repeat : 
 
 \/6 = 2+ 1 
 
 2+ 1 
 
 4+ 1 
 
 2+ 1 
 
 4 + 1 
 
 2+ . 
 
 a periodic chain fraction, in which the denominators 2 and 4 
 alternate. 
 
 In the same manner, 
 ■\/2 =1 + 1 with the periodic denominator 2 
 
 2r+j 
 
 2+ 1 
 
 2 + . 
 \/3 =1 + 1 with the periodic denominators 1 and 2 
 
 m 
 
 2+ 1 
 
 1+ 1 
 
 2+._ 
 
 •\/5 =2+1 with the periodic denominator 4 
 
 4T1 
 
 4+1 
 
 4+,. 
 
 This method of resolution of roots into chain fractions gives 
 a convenient way of deriving simple numerical approximations 
 of the roots, and hereby is very useful. 
 
 For instance, the third approximation of \/2is 1 ^2; with an 
 error of .2 per cent, that is, close enough for most practical 
 purposes. Thus, the diagonal of a square with 1 foot as side, 
 is very closely 1 foot 5 inches, etc. 
 
CHAPTER VI. 
 EMPIRICAL CURVES. 
 
 A. General. 
 
 142. The results of observation or tests usually are plotted 
 in a curve. Such curves, for instance, are given by the core 
 loss of an electric generator, as function of the voltage; or, 
 the current in a circuit, as function of the time, etc. When 
 plotting from numerical observations, the curves are empirical, 
 and the first and most important problem which has to be 
 solved to make such curves useful is to find equations for the 
 same, that is, find a function, y=f(x), which represents the 
 curve. As long as the equation of the curve is not known its 
 utility is very limited. ^\'hile numerical values can be taken 
 from the plotted curve, no general conclusions can be derived 
 from it, no general investigations based on it regarding the 
 conditions of efficiency, output, etc. An illustration hereof is 
 afforded by the comparison of the electric and the magnetic 
 circuit. In the electric cii'cuit, the relation between e.m.f. and 
 
 current is given by Ohm's law, i=—, and calculations are uni- 
 versally and easily made. In the magnetic circuit, however, 
 the term corresponding to the resistance, the reluctance, is not 
 a constant, and the relation between m.m.f. and magnetic flux 
 cannot be expressed by a general law, but only by an empirical 
 curve, the magnetic characteristic, and as the result, calcula- 
 tions of magnetic circuits cannot be made as conveniently and 
 as general in nature as calculations of electric circuits. 
 
 If by observation or test a number of corresponding values 
 of the independent variable x and the dependent variable y are 
 determined, the problem is to find an equation, y=f{x), which 
 represents these corresponding values: Xi, X2, Xz . . , Xn, and 
 2/1) y2, 2/3 •• • Vn, approximately, that is, within the errors of 
 observation. 
 
 209 
 
210 ENGINEERING MATHEMATICS. 
 
 The mathematical expression which represents an empirical 
 curve may be a rational equation or an empirical equation. 
 It is a rational equation if it can be derived theoretically as a 
 conclusion from some general law of nature, or as an approxima- 
 tion thereof, but it is an empirical equation if no theoretical 
 reason can be seen for the particular form of the equation. 
 For instance, when representing the . dying out of an electrical 
 current in an inductive circuit by an exponential function of 
 time, we have a rational equation: the induced voltage, and 
 therefore, by Ohm's law, the current, varies proportionally to the 
 rate of change of the current, that is, its differential quotient, 
 and as the exponential function has the characteristic of being 
 proportional to its differential quotient, the exponential function 
 thus rationally represents the dying out of the current in an 
 inductive circuit. On the other hand, the relation between the 
 loss by magnetic hysteresis and the magnetic density: W= i]B^'^, 
 is an empirical equation since no reason can be seen for this 
 law of the 1.6th power, except that it agrees with the observa- 
 tions. 
 
 A rational equation, as a deduction from a general law of 
 nature, applies universally, within the range of the observa- 
 tions as well as beyond it, while an empirical equation can with 
 certainty be relied upon only within the range of observation 
 from which it is derived, and extrapolation beyond this range 
 becomes increasingly uncertain. A rational equation there- 
 fore is far preferable to an empirical one. As regards the 
 accuracy of representing the observations, no material difference 
 exists between a rational and an empirical erjuation. An 
 empirical equation frequently represents the observations with 
 great accuracy, while inversely a rational equation usually 
 does not rigidly represent the observations, for the reason that 
 in nature the conditions on which the rational law is based are 
 rarely perfectly fulfilled. For instance, the representation of a 
 decaying current by an exponential function is based on the 
 assumption that the resistance and the inductance of the circuit 
 are constant, and capacity absent, and none of these conditions 
 can ever be perfectly satisfied, and thus a deviation occurs from 
 the theoretical condition, by what is called " secondary effects." 
 143- To derive an equation, which represents an empirical 
 curve, careful consideration shoukl first be given to the physical 
 
EMPIRICAL CURVES. 211 
 
 nature of the phenomenon which is to be expressed, since 
 thereby the number of expressions which may be tried on the 
 empirical curve is often greatly reduced. Much assistance is 
 usually given by considering the zero points of the curve and 
 the points at infinity. For instance, if the observations repre- 
 sent the core loss of a transformer or electric generator, the 
 curve must go through the origin, that is, y = for x = 0, and 
 the mathematical expression of the curve y=f(x) can contain 
 no constant term. Furthermore, in this case, with increasing x, 
 2/ must continuously increase, so that for x = GO, y = (x>. Again, 
 if the observations represent the dying out of a current as 
 function of the time, it is obvious that for a; = go, j/ = 0. In 
 representing the power consumed by a motor when running 
 without load, as function of the voltage, for x = 0, y cannot be 
 = 0, but must equal the mechanical friction, and an expression 
 like y = Ajf- cannot represent the observations, but the equation 
 must contain a constant term. 
 
 Thus, first, from the nature of the phenomenon, which is 
 represented by the empirical curve, it is determined 
 
 (a) Whether the curve is periodic or non-periodic. 
 
 (6) Whether the equation contains constant terms, that is, 
 for .T = 0, 2/5^0, and inversely, or whether the curve passes 
 through the origin: that is, 2/ = for a; = 0, or whether it is 
 hyperbolic; that is, y= oo for x = 0, or a;=oo for 2/ = 0. 
 
 (c) What values the expression reaches for oo. That is, 
 whether for a; = oo, 2/ = oo, or 2/ = 0, and inversely. 
 
 (d) Whether the curve continuously increases or decreases, or 
 reaches maxima and minima. 
 
 (e) AVhether the law of the curve may change within the 
 range of the observations, by some phenomenon appearing in 
 some observations which does not occur in the other. Thus, 
 for instance, in observations in which the magnetic density 
 enters, as core loss, excitation curve, etc., frequently the curve 
 law changes with the beginning of magnetic saturation, and in 
 this case only the data Ijclow magnetic saturation would be used 
 for deriving the theoretical equations, and the effect of magnetic 
 saturation treated as secondary phenomenon. Or, for instance, 
 when studying the excitation current of an induction motor, 
 that is, the current consumed when running light, at low 
 voltage the current may increase again with decreasing voltage. 
 
212 ENGINEERING MATHEMATICS. 
 
 instead of decreasing, as result of the friction load, when the 
 voltage is so low that the mechanical friction constitutes an 
 appreciable part of the motor output. Thus, cmijirical curves 
 can be represented by a single eciuation only when the physical 
 conditions remain constant within the range of the observations. 
 
 From the shape of the curve then frequently, with some 
 experimce, a guess can be made on the probable form of the 
 equation which may express it. In this connection, therefore, 
 it is of the greatest assistance to be familiar with the shapes of 
 the more common forms of curves, by plotting and studying 
 various forms of ecjuations y=f{x). 
 
 By changing the scale in which observations are plotted 
 the apparent shape of the curve may be modified, and it is 
 therefore desirable in plotting to use such a scale that the 
 average slope of the cuyyq is about 45 deg. A much greater or 
 much lesser slope should be avoided, since it does not show the 
 character of the curve as well. 
 
 B. Non-Periodic Curves. 
 
 144. The most common non-periodic curves are the potential 
 series, the parabolic and hyperbolic curves, and the exponential 
 and logarithmic curves. 
 
 The Potential Series. 
 
 Theoretically, any set of observations can be represented 
 exactly by a potential series of any one of the following forms: 
 
 y = ao + aix+a2.z^+asx^ + . . ; . . (1) 
 
 y = aix + a2x'~+aix^+ .;.... (2) 
 
 a 1 a2 to 
 2/ = ao+-+j,+-3 + . .; (3) 
 
 ^ = 7+.^ + ^^ + - ■ • • . (4) 
 
 if a sufficiently large number of terms are chosen. 
 
 For instance, if n corresponding numerical values of x and y 
 are given, xi, yi; x^, 1/2; ... x„, y,„ they can be represented 
 
EMPIRICAL CURVES. 
 
 213 
 
 by the series (1), when choosing as many terms as required to 
 give n constants a : 
 
 2/ = ao+aix+a2a:2 + . . .+a„_in"~i. . . . (5) 
 
 By substituting the corresponding values Xi, yi] X2, 2/2, ■ 
 into equation (5), there are obtained n equations, which de- 
 termine the n constants ao, ai, a2, . . . a„_i. 
 
 Usually, however, such representation is irrational, and 
 therefore meaningless and useless. 
 
 Table I. 
 
 e 
 100-'' 
 
 Pi = v 
 
 -0.5 
 
 + 2x 
 
 + 2.51' 
 
 -1.5x3 
 
 + 1.5i< 
 
 -2x' 
 
 + i« 
 
 0.4 
 0.6 
 0.8 
 
 0.63 
 1.36 
 2.18 
 
 -0.5 
 -0.5 
 -0.5 
 
 + 0.8 
 + 1.2 
 + 1.6 
 
 + 0.4 
 + 0.9 
 + 1.6 
 
 -0.10 
 -0.32 
 -0.77 
 
 + 0.04 
 + 0.19 
 + 0.61 
 
 - 0.02 
 
 - 0.16 
 
 - 0.65 
 
 
 + 0.05 
 + 0.26 
 
 1.0 
 1.2 
 1.4 
 
 3.00 
 3.93 
 6.22 
 
 -0.5 
 -0.5 
 -0.5 
 
 + 2.0 
 
 + 2.4 
 + 2.8 
 
 + 2.5 
 + 3.6 
 + 4.9 
 
 -1.50 
 -2.59 
 -4.12 
 
 + 1.50 
 +3.11 
 
 + 5.76 
 
 -2.00 
 - 4.98 
 -10.76 
 
 + 1.00 
 + 2.89 
 + 6.13 
 
 1.6 
 
 8.59 
 
 -0.5 
 
 +3.2 
 
 + 6.4 
 
 -6.14 
 
 + 9.83 
 
 -20.97 
 
 + 16.78 
 
 Let, for instance, the first column of Table I represent the 
 
 voltage, -jKfi^^, in hundreds of volts, and the second column 
 
 the core loss, Pi = y, in kilowatts, of an 125-Volt 100-h.p. direct- 
 current motor. Since seven sets of observations are given, 
 they can be represented by a potential series with seven con- 
 stants, thus, 
 
 y = ao+aiX+a2X^+. . .+aQX^, .... (6) 
 
 and by substituting the observations in (6), and calculating the 
 constants a from the seven equations derived in this manner, 
 there is obtained as empirical expression of the core loss of 
 the motor the equation. 
 
 2/=_0.5+2x+2.5.c2-1.5x3+1.5x*-2a;5+x6- 
 
 (7) 
 
 This expression (7), however, while exactly representing 
 the seven observations, has no physical meaning, as easily 
 seen by plotting the individual terms. In Fig. 60, y appears 
 
214 
 
 ENGINEERING MA Til EM A TI( 'S. 
 
 as the resultant of a number of large jiosUiA-e and negative 
 terms. Furthermore, if one of the observations is omitted, 
 and the potential series calculated from the remaining six 
 values, a series reaching up to x^ would )x' the result, thus, 
 
 y = ao+aix+a2X~ + . . .+a5jfi, .... (8) 
 
 16 
 
 12- 
 
 -8- 
 
 -12 
 
 -16 
 
 -SO 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 j 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 f 
 
 Ft/ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 '^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 ^ 
 
 ■^^ 
 
 ^" 
 
 
 
 
 
 
 
 
 
 
 ]l__ 
 
 ^^ 
 
 ^ 
 
 ^ 
 
 -^z 
 
 ^ 
 
 
 
 
 
 
 
 s=: 
 
 
 ^ 
 
 2 
 
 
 
 
 
 
 
 
 
 
 
 ~~ 
 
 
 =^ 
 
 ^ 
 
 N 
 
 
 -c 
 
 .5 
 
 
 
 
 
 
 
 
 
 
 
 
 
 --. 
 
 ■mL 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 s. 
 
 
 ^ 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 M--> 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 V 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 ■ 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 x = 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 3 
 
 
 
 4 
 
 
 
 6 
 
 
 
 8 
 
 1 
 
 
 
 1 
 
 2 
 
 1 
 
 1 
 
 1 
 
 8 
 
 Fig. 60. Terms of Empirical Expression of Excitation Power. 
 
 but the constants a in (S) would have entirely different numer- 
 ical values from those in (7), thus showing that the equation 
 (7) has no rational meaning. 
 
 145- The potential series (1) to (1) thus can be used to 
 re;jrescnt an empirical curve only under the followino- condi- 
 tions : 
 
 1. If the successive coefficients Oo, oi, oo, dcci-ease in 
 
 value so rapidly that within the range of observation tlie 
 higher terms become rapidly smaller and appear as mere 
 secondary terms. 
 
EMPIRICAL CURVES. 
 
 215 
 
 2. If the successive coefficients a follow a definite law, 
 indicating a convergent series which represents some other 
 function, as an exponential, trigonometric, etc. 
 
 3. If all the coefficients, a, are very small, with the exception 
 of a few of them, and only the latter ones thus need to be con- 
 sidered. 
 
 Table II. 
 
 X 
 
 V 
 
 y' 
 
 Vi 
 
 0.4 
 0.6 
 0.8 
 
 0.89 
 1.35 
 1.96 
 
 0.8S 
 1.34 
 1.94 
 
 0.01 
 0.01 
 0.02 
 
 1.0 
 1.2 
 lA 
 
 2.72 
 3.62 
 4.63 
 
 2.70 
 2.59 
 4.59 
 
 0.02 
 0.03 
 0.04 
 
 1.6 
 
 5.76 
 
 5.65 
 
 0.11 
 
 For instance, let the numbers in column 1 of Table II 
 represent the speed a: of a fan motor, as fraction of the rated 
 speed, and those in column 2 represent the torque y, that is, 
 the turning moment of the motor. These values can be 
 represented b}^ the equation, 
 
 t/ = 0.5f0.02a;+2.5.T2-0.3j-^+0.015T*-0.02a«+0.01.T6 (9) 
 
 la this case, only the constant term and the terms with 
 x2 and x^ have appreciable values, and the remaining terms 
 probably are merely the result of errors of observations, that is, 
 the approximate equation is of the form, 
 
 y = aQ+a2X^ + a3X^ (10) 
 
 Using the values of the coefficients from (9), gives 
 
 2/ = 0.5+2..5.r2-0.3i-3 (H) 
 
 The numerical values calculated from (11) are given in column 
 3 of Table II as y', and the difference between them and the 
 observations of column 2 are given in column 4, as yi. 
 
216 ENGINEERING MATHEMATICS. 
 
 The values of column 4 can now be represented by the same 
 form of equation, namely, 
 
 y, = ho^h^x^+h^x^, (12) 
 
 in which the constants io, ^2, 63 are calculated by the method 
 of least squares, as described in paragraph 120 of Chapter IV, 
 and give 
 
 2/1 = 0.031-0.093^2+0.076x3. . . . (13) 
 
 Equation (13) added to (11) gives the final approximate 
 equation of the torque, as, 
 
 2/0 = 0.531+2.407x2-0.224x3 (14) 
 
 The equation (14) probably is the approximation of a 
 rational equation, since the first term, 0.531, represents the 
 bearing friction; the second term, 2.407x2 (which is the largest), 
 represents the work done by the fan in moving the air, a 
 resistance proportional to the square of the speed, and the 
 third term approximates the decrease of the air resistance due 
 to the churning motion of the air created by the fan. 
 
 In general, the potential series is of limited usefulness; it 
 rarely has a rational meaning and is mainly used, where the 
 curve approximately follows a simple law, as a straight line, 
 to represent by small terms the deviation from this simple law, 
 that is, the secondary effects, etc. Its use, thus, is often 
 temporary, giving an empirical approximation pending the 
 derivation of a more rational law. 
 
 The Parabolic and the Hyperbolic Curves. 
 
 146. One of the most useful classes of curves in engineering 
 are those represented by the equation, 
 
 y = ax^; (15) 
 
 or, the more general eciuation, 
 
 y-h = a{x-cY (16) 
 
 Equation (16) differs from (15) only by the constant terms h 
 and c; that is, it gives a different location to the coordinate 
 
EMPIRICAL CURVES. 
 
 217 
 
 center, but the curve shape is the same, so that in discussing 
 the general shapes, only equation (15) need be considered. 
 
 If n is positive, the curves y = ax^ are 'parabolic curves, 
 passing through the origin and increasing with increasing x. 
 If n>l, 2/ increases with increasing rapidity, \in<\,y increases 
 with decreasing rapidity. 
 
 If the exponent is negative, the curves j/=aa;~"=— are 
 
 hyperbolic curves, starting from y=cc for a:=0, and decreasing 
 to 2/=0 for a;= oo. 
 
 n=l gives the straight line through the origin, n=0 and 
 n = cx) give, respectively, straight horizontal and vertical lines. 
 
 Figs. 61 to 71 give various curve shapes, corresponding to 
 different values of n. 
 
 Parabolic Curves. 
 
 
 Fig. 61. 
 
 n = 2; 
 
 y = x^; the common parabola. 
 
 Fig. 62. 
 
 n = 4; 
 
 y = x*; the biquadratic parabola. 
 
 Fig. 63. 
 
 n = 8: 
 
 y = x». 
 
 Fig. 64. 
 
 n = i; 
 
 y=Vx; again the common parabola 
 
 Fig. 65. 
 
 n=\; 
 
 2/= -^'i the biquadratic parabola. 
 
 Fig. 66. 
 
 n=i; 
 
 y = </x. 
 
 Hyperbolic 
 
 Curves. 
 
 
 Fio'. 67. n=— 1; 1/=-; the equilateral hyperbola. 
 
 Fig. 68. n=-2; 
 Fig. 69. n = 
 Fig. 70. 
 
 -4; 
 1 
 
 n= — ; 
 
 y==7^- 
 
 y=x*- 
 
 Fig. 71. n=-|- 
 
218 
 
 ENGINEERING MATHEMATICS. 
 
 
 — 
 
 
 
 
 
 ■ 
 
 
 _ 
 
 
 
 
 
 
 
 
 ~^ 
 
 
 
 
 
 
 
 
 
 
 
 
 - 
 
 
 
 
 - 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 s_ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 N 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 > 
 
 o 
 
 3 
 
 H 
 
 C 
 
 3 
 A 
 
 *■ 
 
 
 C 
 
 
 O 
 
 41 
 
 OO 
 
 6 
 
 CO 
 
 o 
 
 -*< 
 o 
 
 § 
 
 
 3 
 
 03 
 
 
 
 
 — 
 
 ■— 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ~~^ 
 
 
 --^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ' 
 
 
 -^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 -«=; 
 
 ■^ 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
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 N 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
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 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
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 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
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 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 c 
 
 =1 
 
 ; 
 
 c 
 
 D 
 
 4 
 
 <; 
 
 3 
 
 ^ 
 
 K 
 
 » 
 
 I 
 
 < 
 
 
 
 5 
 
 
 5 
 
 c 
 
 11 
 5 
 
 -1 
 
 c 
 
 5 
 
 
 
 « 
 
 o 
 
 II 
 
 i-( 
 
 ^ 
 
 CO 
 
 c5 
 
 a3 
 
 
 3 
 
 
 o 
 
 o 
 
 _o 
 
 o 
 
 o 
 
 
 /D 
 
 
 oj 
 
 
 ^ 
 
 --t* 
 
 01 
 
 o 
 
 PL, 
 
 o 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 "v 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 *v 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 S 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 -\ 
 
 s 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
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 ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
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 9 
 Si— 
 
 op 
 
 
 ^ 
 
 ^ 
 
 < 
 
 -( 
 
 c 
 
 3 
 H 
 
 4 
 
 5 
 
 <: 
 
 
 
 5 
 
 ^ 
 
 5 
 
 
 1 
 
 ^ 
 
 Si 
 
 3 
 o 
 
 o 
 
 P^ 
 
 t-l 
 
EMPIRICAL CURVES. 
 
 219 
 
 -1 4- 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^-^ 
 
 
 
 
 -1 9 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 -^ 
 
 ' 
 
 
 
 
 
 
 - 
 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 ^^ 
 
 
 
 
 
 
 
 
 
 
 u.U 
 
 
 
 
 
 
 
 
 ^ 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 U.o 
 
 
 
 
 
 y 
 
 y 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 y 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 O.G 
 
 
 / 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Ut4 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 J 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 L 
 
 
 
 2 
 
 
 
 i 
 
 
 
 6 
 
 
 
 8 
 
 1 
 
 
 
 1 
 
 ? 
 
 I 
 
 1 
 
 1 
 
 e 
 
 i 
 
 8 
 
 2 
 
 
 
 Fig. 64. Parabolic Curve. y = ^/x. 
 
 iv^- 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 — ■ 
 
 — 
 
 
 =^ 
 
 -1»0 
 
 
 
 
 
 
 
 
 
 
 
 
 -- 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 
 -^ 
 
 
 
 
 
 
 
 
 
 
 
 
 ft-R 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 /- 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 f 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 U7*i" 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 2 
 
 
 
 4 
 
 
 
 6 
 
 
 
 8 
 
 1 
 
 
 
 1 
 
 2 
 
 1 
 
 4 
 
 1 
 
 6 
 
 1 
 
 8 
 
 2 
 
 
 
 Fig. 65. Parabolic Curve. y= ijx. 
 
220 
 
 ENGINEERING MATHEMATICS. 
 
 
 
 
 
 
 
 
 
 
 
 
 _ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 -1-0- 
 
 
 
 
 J 
 
 
 ' 
 
 — 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 -^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 0^8- 
 
 /- 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 jo.Q 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 O-i 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 0.2 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 2 
 
 
 
 4 
 
 
 
 6 
 
 
 
 8 
 
 1 
 
 
 
 1 
 
 2 
 
 1 
 
 1 
 
 1 
 
 6 
 
 1 
 
 8 
 
 2 
 
 
 
 Fig. 66. Parabolic Curve. y = \^x 
 
 ■6r'J- 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 9 iJ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 4.0 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 9 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ■1 *> 
 
 
 
 \ 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 n-H 
 
 
 
 
 
 \ 
 
 ■v 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ■^ 
 
 ^v^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 fn- 
 
 
 
 
 
 
 
 
 
 ■~^ 
 
 ■--., 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ' ' 
 
 
 ■ . 
 
 
 
 
 
 
 
 
 
 
 i 
 
 
 
 8 
 
 1 
 
 2 
 
 1 
 
 6 
 
 2 
 
 
 
 2 
 
 * 
 
 2 
 
 8 
 
 3 
 
 2 
 
 3 
 
 6 
 
 4 
 
 
 
 Fig. 67. Hyperbolic Curve (Equilateral Hyperbola). y =—. 
 
EMPIRICAL CURVES. 
 
 221 
 
 a^a- 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 2.-8- 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 2-4 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 2-n- 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ■1-fi- 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ■1 9 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 J-IO 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ijiir 
 
 
 
 
 
 \ 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 r\-dr- 
 
 
 
 
 
 
 \ 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 "^ 
 
 ""^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ~r 
 
 
 
 
 
 
 
 
 
 
 
 
 
 0.4 0.8 1.2 1.6 .2.0 2.4 2.8 3.2 3.6 4.0 4.4 
 
 1 
 
 Fig. 68. Hyperbolic Curve. !/ = 
 
 Bra- 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ■ 
 
 
 
 9 R- 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 i 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ■tf;4 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 tSilJ- 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 R— 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 l-D- 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 !.<« 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Cho- 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 s 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 0:4 
 
 
 
 
 
 
 \ 
 
 s 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 -- 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 0.4 0.8 1.2 1.6 2.0 2.4 
 
 3.2 3.6 ,4.0 4.4 
 
 Fig. 69. Hyperbolic Curve, y = —^■ 
 
222 
 
 ENGINEERING MATHEMATICS. 
 
 aap 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 i.i\ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 L 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 2;0- 
 
 V 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 -t6- 
 
 \^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 -^\ 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 -1-72- 
 
 
 
 ^s 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ■0r8- 
 
 
 
 
 
 
 
 
 ""-- 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Ur4 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 4 
 
 
 
 8 
 
 1 
 
 3 
 
 1 
 
 6 
 
 2 
 
 
 
 2 
 
 i 
 
 2 
 
 8 
 
 3 
 
 2 
 
 3 
 
 6 
 
 4 
 
 
 
 i 
 
 4 
 
 Fig. 70. Hyperbolic Curve. y = — -. 
 
 Vx 
 
 pa- 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ■-5.0 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ■27l 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 T 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 i-OV 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1" 
 
 \ 
 
 s 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 ---^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 0-8- 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 n-i- 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 4 
 
 
 
 8 
 
 1 
 
 2 
 
 1 
 
 6 
 
 2 
 
 
 
 2, 
 
 4 
 
 2 
 
 8 
 
 8 
 
 2 
 
 3 
 
 6 
 
 4 
 
 
 
 4 
 
 4 
 
 Fig. 71. Hyperbolic Curve. ?/= — . 
 
 ■ijx 
 
EMPIRICAL CURVES. 223 
 
 In Fig. 72, sixteen different parabolic and hyperbolic curves 
 are drawn together on the same sheet, for the following values ■ 
 n = l, 2, 4, 8, Go_; I |, I 0: -1, -2, -4, -8; -i; -J, -j.' 
 
 147- Parabolic and hyperbohc curves may easily be recog- 
 nized by the fact that if x is changed by a constant factor, y also 
 changes by a constant factor. 
 
 Thus, in the curve y^x^, doubling the x increases the y 
 fourfold; in the curve y = x^-5^, doubling the x increases the y 
 threefold, etc.; that is, if in a curve, 
 
 fil^) 
 
 -jy-y = constant, for constant g, . . (17) 
 
 the curve is a parabolic or hyperbolic curve, y = ax^, and 
 fiqx) a{qxy 
 
 If q is nearly 1, that is, the x is changed onjy by a small 
 value, substituting g' = l+s, where s is a small quantity, from 
 equation (18), 
 
 hence, 
 
 fix-Vsx) ^, . 
 
 f{x+sx)-f(x) 
 
 fix) 
 
 = ns; (19) 
 
 that is, changing x by a small percentage s, y changes oy a pro- 
 portional small percentage ns. 
 
 Thus, parabolic and hyperbolic curves can be recognized by 
 a small percentage change of x, giving a proportional small 
 percentage change of y, and the proportionahty factor is the 
 exponent n; or, they can be recognized by doubling x and 
 seeing whether y hereby changes by a constant factor. 
 
 As illustration are shown in Fig. 73 the parabohc curves, 
 which, for a doubling of x, increase y: 2, 3, 4, 5, 6, and 8 fold. 
 
 Unfortunately, this convenient way of recognizing parabolic 
 and hyperbolic curves applies only if the curve passes through 
 the origin, that is, has no constant term. If constant terms 
 exist, as in equation (16), not x and y, but (x-c) and (y-b) 
 follow the law of proportionate increases, and the recognition 
 
224 
 
 ENGINEERING MATHEMATICS. 
 
 becomes more difficult; that is, various values of c and of h 
 are to be tried to find one which gives the proportionality. 
 
 W 
 
 Fig. 72. Parabolic and Hyperbolic Curves. y = xn. 
 148. Taking the logarithm of equation (15) gives 
 log 2/ = log ci+nlogx; . . . . 
 
 (20) 
 
EMPIRICAL CURVES. 
 
 225 
 
 that is, a straight line; hence, a parabolic or hyperbolic curve can 
 be recognized by plotting the logarithm of y against the loga- 
 rithm of X. If this gives a straight line, the curve is parabolic 
 or hyperbolic, and the slope of the logarithmic curve, tan d=n, 
 is the exponent. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 % 
 
 
 
 9-C\ 
 
 
 
 
 
 
 
 
 
 
 
 
 Co/ 
 
 II 1 
 
 
 ./ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 . 
 
 f 
 
 ] 9, 
 
 
 
 
 
 
 
 
 
 
 
 
 In 
 
 7 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 / 
 
 / 
 
 
 1-R 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 V 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 // 
 
 / 
 
 N. 
 
 y 
 
 1 4 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 7 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 // 
 
 / 
 
 / 
 
 
 
 1 9 
 
 
 
 
 
 
 
 
 
 
 
 ' 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 (/ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 // 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 //// 
 
 
 
 
 
 
 
 
 n e 
 
 
 
 
 
 
 / 
 
 A 
 
 ''/// 
 
 
 
 
 
 
 
 
 U:D 
 
 
 
 
 
 / 
 
 / 
 
 //> 
 
 '/ 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 / 
 
 'A 
 
 W 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 / 
 
 // 
 
 ^^ 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 / 
 
 // 
 
 // 
 
 y 
 
 
 
 
 
 
 
 
 
 
 6:3 
 
 / 
 
 / 
 
 ^ 
 
 ^ 
 
 /y 
 
 
 
 
 
 
 
 
 
 
 
 A 
 
 ^ 
 
 •^^ 
 
 ^ 
 
 y 
 
 
 
 
 
 
 
 
 
 
 
 
 0.2 0.4 0.6 0.8 1.0 1.2 1.4 
 
 Fig. 73. Parabolic Curves. y=xn. 
 
 1.6 
 
 This again applies only if the curve contain no constant 
 term. If constant terms exist, the logarithmic line is curved. 
 Therefore, by trying different constants c and 6, the curvature 
 of the logarithmic line changes, and by interpolation such 
 constants can be found, which make the logarithmic line straight, 
 and in this way, the constants c and b may be evaluated. If 
 only one constant exist, that is, only b or only c, the process is 
 relatively simple, but it becomes rather complicated with both 
 
226 ENGINEERING MATHEMATICS. 
 
 constants. This fact makes it all the more desirable to get 
 from the physical nature of the problem some idea on the 
 existence and the value of the constant terms. 
 Differentiating equation (20) gives : 
 
 dii dx 
 y X 
 
 that is, in a parabolic or hyperbohc curve, the percentual 
 change, or variation of y, is n times the percentual change, 
 or variation of x, if n is the exponent. 
 Herefrom follows: 
 
 dy 
 
 y 
 
 dx 
 
 that is, in a paraboUc or hyperbolic curve, the ratio of variation, 
 dy 
 
 y 
 
 m = — — , is a constant, and equals the exponent n. 
 dx 
 
 X 
 
 Or, inversely: 
 
 If in an empirical curve the ratio of variation is constant 
 the curve is — within the range, in which the ratio of variation 
 is constant — a parabolic or hyperbolic curve, which has as 
 exponent the ratio of variation. 
 
 In the range, however, in which the ratio of variation is 
 not constant, it is not the exponent, and while the empirical 
 curve might be expressed as a parabolic or hyperbolic curve 
 with changing exponent (or changing coefhcient), in this case 
 the exponent may be very different from the ratio of varia- 
 tion, and the change of exponent frequently is very much 
 smaller than the change of the ratio of variation. 
 
 This ratio of variation and exponent of the parabolic or 
 hyperbolic approximation of an empirical curve must not be 
 mistaken for each other, as has occasionally been done in 
 reducing hysteresis curves, or radiation curves. They coincide 
 
EMPIRICAL CURVES. 227 
 
 only in that range, in which exponent n and coefficient a of 
 the equation y = ax^ are perfectly constant. If this is not 
 the case, then equation (20) differentiated gives: 
 
 dy da ^ , n 
 
 — = hlogx an-i — ax, 
 
 y a X 
 
 and the ratio of variation thus is : 
 
 dy 
 
 y X da ^ dn 
 
 m = —— = n-\ \-x logx — ; 
 
 dx a X X 
 
 X 
 
 that is, the ratio of variation m differs from the exponent n. 
 
 Exponential and Logarithmic Curves. 
 
 149. A function, which is very frequently met in electrical 
 engineering, and in engineering and physics in general, is the 
 exponential function, 
 
 J/=a«"^; (21) 
 
 which may be written in the more general form, 
 
 j/-6 = a£»(^-'^^ (22) 
 
 Usually, it appears with negative exponent, that is, in the 
 form, 
 
 ?/ = a£-"^ (23) 
 
 Fig. 74 shows the curve given hy the exponential function 
 (23) for a = 1 ; n = 1 ; that is, 
 
 y=e-^, .... . . (24) 
 
 as seen, with increasing positive x, y decreases to at x= + 00, 
 and with increasing negative x, y mcreases to 00 at a; = — 00. 
 
228 
 
 ENGINEERING MATHEMATICS. 
 
 The curve, ?/=£+^, has the same shape, except that the 
 positive and the negative side (right and left) arc interchanged. 
 
 Inverted these equations (21) to (24) may also be written 
 thus, 
 
 nx = log-; 
 
 n{x—c) = \og- 
 
 1 y 
 
 nx= —log—; 
 
 x==-hgy; 
 that is, as logarithmic curves. 
 
 -iJ.O -1.6 -1.3 -0.8 -0,4 0.4 0.8 1.2 
 
 Fig. 74. Exponential Function. y = e-x. 
 
 . (25) 
 
 \1 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 1T4' 
 
 \ 
 
 
 
 
 
 
 
 
 \ 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 6;0- 
 
 \ 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 X-ti 
 
 
 \ 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 1 n 
 
 5.0- 
 
 
 \ 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 fi-ft 
 
 4tU 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 \^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 \ 
 
 
 
 
 
 
 
 s 
 
 \ 
 
 
 
 
 
 
 
 
 O-fi 
 
 d;0 
 
 
 
 
 \ 
 
 s 
 
 
 
 
 
 
 
 \ 
 
 s. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 s 
 
 
 
 
 
 
 
 \l 
 
 N 
 
 
 
 
 
 
 f\.A 
 
 ■3.0 
 
 
 
 
 
 
 \ 
 
 \ 
 
 
 
 
 
 
 
 S 
 
 \ 
 
 
 
 
 
 U.4 
 
 1 n 
 
 
 
 
 
 
 
 
 \ 
 
 ^ 
 
 
 
 
 
 
 
 
 ^^ 
 
 
 
 n '? 
 
 1<U 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 "~~ 
 
 ^~ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 — 
 
 
 
 
 
 
 
 
 
 1.6 
 
 2.0 
 
 150. The characteristic of the exponential function (21) is, 
 that an increase of x by a constant term increases (or, in (23) 
 and (24), decreases) y by a constant factor. 
 
 Thus, if an empirical curve, y=f{x), has such characteristic 
 that 
 
 fir+q) 
 ^. s = constant, for constant g, . . . (26) 
 
EMPIRICAL CURVES. 
 
 229 
 
 the curve is an exponential function, y—az^^, and the following 
 equation may be written : 
 
 f(x + q) a£"(^+g) 
 f{x) ^ ae"^ 
 
 ■ B^ . 
 
 (27) 
 
 Hereby the exponential function can easily be recognized, 
 and distinguished from the parabolic curve ; in the former a 
 constant term, in the latter a constant factor of x causes a 
 change of y by a constant factor. 
 
 As result hereof, the exponential curve with negative 
 exponent vanishes, that is, becomes negligibly small, with far 
 greater rapidity than the hyperbolic curve, and the exponential 
 
 V 
 
 
 
 
 
 
 
 
 
 
 r 
 
 
 
 
 
 
 
 
 
 -W 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 U:0 
 
 
 \ 
 
 \f~ 
 
 C 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 2.4 
 
 
 N 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 U-TJ 
 
 ( x+i.i 
 
 5) = ^ 
 
 
 
 
 ^ 
 
 V 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 fi.A. 
 
 
 
 
 
 V 
 
 N 
 
 ■s. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 U:4 
 
 
 
 
 
 
 
 ^ 
 
 ^ 
 
 
 
 
 2.4 
 
 2 
 
 
 
 
 
 
 
 {^CB.+l.SS 
 
 
 
 
 
 
 
 
 £- 
 
 »^ 
 
 --. 
 
 ~ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 -J 
 
 
 "^■^ 
 
 
 
 
 
 
 0^ 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 
 
 Fig. 75. Hyperbolic and Exponential Curves Comparison. 
 
 4.0 
 
 function with positive exponent reaches practically infinite 
 values far more rapidly than the parabolic curve. This is 
 illustrated in Fig. 75, in which are shown superimposed 
 the exponential curve, 2/=£~^, and the hyperbolic curve, 
 9 4 
 
 y = - — — , which coincides with the exponential curve 
 
 ^ (x + 1.55)2' 
 
 at x = and at x = l. 
 
 Taking the logarithm of equation (21) gives logt/=-loga + 
 nx log £, that is, log y is a hnear function of x, and plotting 
 log y against x gives a straight line. This is characteristic of 
 
230 
 
 ENGINEERING MATHEMATICS. 
 
 the exponential functions, and a convenient method of recog- 
 nizing them. 
 
 However, both of these characteristics apply only if x and y 
 contain no constant terms. With a single exponential function, 
 only the constant term of y needs consideration, as the constant 
 term of x may be eliminated. Equation (22) may be written 
 
 thus: 
 
 2/-& = a£"^^-'^> 
 
 = a£' 
 
 (28) 
 
 where A=o£~'^ is a constant. 
 
 An exponential function which contains a constant term b 
 would not give a straight line when plotting log y against x. 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 -tl 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 (1) 2/=£-a'+o.52"2* 
 
 (2) 2/=£-a+o.2^"2a' 
 (3)2/=.e-a' 
 
 (4) 2/=£-a'_0.2£~2a! 
 
 (6) 2/ = £-a5-o.8£-2a' 
 
 (7) 2/ = ra'-e-2« 
 
 (8) 2/=£-a'-i.5£-2a! 
 
 
 
 
 -hZ 
 
 \^ 
 
 {I) 
 
 
 
 
 
 
 
 
 \. 
 
 )\ 
 
 
 
 
 
 
 
 
 -to 
 
 \(a1 
 
 \ \ 
 
 
 
 
 
 
 
 
 \\ 
 
 \ 
 
 
 
 
 
 
 
 -0.8 
 
 Mv 
 
 s\ 
 
 
 
 
 
 
 
 - \. 
 
 \ 
 
 \\ 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 -0;6 
 
 
 \ 
 
 \^ 
 
 xV 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 x^ 
 
 
 ■sV. 
 
 ^ 
 
 s 
 
 
 
 
 
 
 
 
 
 
 -0;4 
 
 
 (6) 
 
 
 \ 
 
 ::;^ 
 
 ^ 
 
 X 
 
 
 
 
 
 
 
 
 
 // 
 
 ^ 
 
 ^ 
 
 "^ 
 
 ^^ 
 
 :::< 
 
 ^ 
 
 ^ 
 
 
 
 
 
 
 
 
 -0.2 
 
 , 
 
 ~~ 
 
 / 
 
 /t) 
 
 
 ^ 
 
 -— 
 
 
 ~^ 
 
 ==:^ 
 
 ^ 
 
 ^ 
 
 a2». 
 
 fe= 
 
 
 
 
 
 
 — 
 
 / 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 r 
 
 1 
 
 
 
 8 
 
 1 
 
 2 
 
 1 
 
 6 
 
 2 
 
 
 
 2 
 
 4 
 
 2 
 
 8 
 
 "^ 
 
 
 m 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 -0.2 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 -0.4 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 FiQ. 76. Exponential Functions. 
 
EMPIRICAL CURVES. 
 
 231 
 
 but would, give a curve. In this case then log (y—b) would be 
 plotted against x for various vahies of b, and by interpolation 
 that value of b found which makes the logarithmic curve a 
 straight line. 
 
 I5I. While the exponential function, when appearing singly, 
 is easily recognized, this becomes more dif&cult with com- 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 L4 
 
 i 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 (1) 2/= £-3'+0.5£~10* 
 
 (21 y=s~''' 
 
 (3) &=£-'»- 0.1 £"10^ 
 
 (4) 2/=£-*-0.5£-10"' 
 
 (5) 2/=£-a;-,e-ioa; 
 
 (6) 2/=£-«-i.6e"i''* 
 
 
 
 
 L2 
 
 ll 
 
 ) 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 LO 
 
 \A 
 
 
 
 
 
 
 
 
 
 ;^ 
 
 
 
 
 
 
 
 
 
 0.8 
 
 (U- 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 r/! 
 
 
 V 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 0.6 
 
 / 
 
 
 \ 
 
 N 
 
 
 
 
 
 
 
 
 
 
 
 
 
 f 
 
 ) 
 
 
 \ 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 0.4 
 
 
 
 
 
 s 
 
 K 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ■\ 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 
 ^_. 
 
 
 
 
 
 
 0.3 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ■ 
 
 
 
 
 
 
 
 
 4 
 
 
 
 8 
 
 1 
 
 2 
 
 1 
 
 8 
 
 2 
 
 
 
 2 
 
 4 
 
 2 
 
 8 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 -0.2 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 -0.< 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 Fig. 77. Exponential Functions. 
 
 binations of two exponential functions of different coefficients 
 in the exponent, thus, 
 
 2/ = ai£ -'"■'±02 £""''', (29) 
 
 since for the various values of ai, 0-2, c\, ca, quite a number of 
 various forms of the function appear. 
 
 As such a combination of two exponential functions fre- 
 quently appears in engineering, some of the characteristic forms 
 are plotted in Figs. 76 to 78. 
 
232 ENGINEERING MATHEMATICS. 
 
 Fig. 76 gives the following combinations of e"^ ajid £-2i. 
 
 (1) 2/= £--^+0.5^-2^; 
 
 (2) 2/=£-'+0.2£-2^; 
 
 (3) 2/=^-^; 
 
 (4) j/=£-^-0.2£-2-^; 
 
 (5) 2/=£-^-0.5e-2^ 
 
 (6) i/=£-^-0.Se-2^ 
 
 (7) y=e-- 
 
 -2x- 
 
 (8) 2/=£-^-1.5£- 
 
 -2 a- 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 / 
 
 f 
 
 \f^^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 9 C\ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 1 
 
 ^.U 
 
 
 
 cosh a; = J ]£"*■'"+£""' 
 
 
 
 
 / 
 
 / 
 
 
 
 
 
 
 
 / 
 
 
 / 
 
 
 i.o- 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 r 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 / 
 
 
 / 
 
 
 
 i.D' 
 
 
 
 
 
 
 
 
 
 . / 
 
 / 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 o^ 
 
 Y 
 
 
 / 
 
 / 
 
 
 
 
 l.i" 
 
 
 
 
 
 
 
 / 
 
 / 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 / 
 
 
 
 
 / 
 
 f 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^/ 
 
 
 
 
 
 
 
 1.0 
 
 
 
 
 
 
 
 
 7 
 
 
 
 
 
 
 
 
 -0:8' 
 
 
 
 
 
 
 / 
 
 y 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 / 
 
 
 
 
 
 
 
 
 
 -0;G 
 
 
 
 
 
 
 V 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 -0:4- 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 -0:2 
 
 z 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 2 
 
 
 
 i 
 
 06 
 
 
 
 8 
 
 1 
 
 
 
 ' 
 
 2 
 
 1 
 
 4 
 
 
 Fig. 78. Hyperbolic Functions. 
 
EMPIRICAL CURVES. 233 
 
 Fig. 77 gives the following combination of £-^ and £-1°=^: 
 
 (1) ?/=£-^+0.5£-io^; 
 
 (2) y=e-^; 
 
 (3) 2/=£-"-O.U-io^; 
 
 (4) 7/=j-^_o.5£-io^; 
 
 (5) 2/=s-^-j-10r. 
 
 (6) y=j-:^_i.5e-io^ 
 
 Fig. 78 gives the hyperbolic functions as combinations of 
 e+^and e~^ thus, 
 
 i/ = cosh x = §(£+=^ + £~^); 
 
 2/ = sinh x = J(£+^— £-^). 
 
 C. Evaluation of Empirical Curves. 
 
 152. In attempting to solve the problem of finding a mathe- 
 matical equation, y=f{x), for a series of observations or tests, 
 the corresponding values of x and y are first tabulated and 
 plotted as a curve. 
 
 From the nature of the physical problem, which is repre- 
 sented by the numerical values, there are derived as many 
 data as possible concerning the nature of the curve and of the 
 function which represents it, especially at the zero values and 
 the values at infinity. Frequently hereby the existence or 
 absence of constant terms in the equation is indicated. 
 
 The log X and log y are tabulated and curves plotted between 
 X, y, log .T, log y, and seen, whether some of these curves is a 
 straight line and thereby indicates the exponential function, or 
 the paraboUc or hyperbolic function. 
 
 If cross-section paper is available, having both coordinates 
 divided in logarithmic scale, and also cross-section paper having 
 one coordinate divided in logarithmic, the other in common 
 scale, X and y can be directly plotted on these two forms of 
 logarithmic cross-s ction paper. Usually not much is saved 
 thereby, as for the n- merical calculation of the constants the 
 logarithms still have to be tabulated. 
 
234 ENGINEERING MATHEMATICS. 
 
 If neither of the four curves: x, xj; x,\ogy; hgx,y-, logx, 
 log y is a straight line, and from the physical condition the 
 absence of a constant term is assured, the function is neither 
 an exponential nor a paraboUc or hyperbolic. If a constant 
 term is probable or possible, curves are plotted between x, 
 y—b, logx, log (y—b) for various values of b, and if hereby 
 one of the curves straightens out, then, by interpolation, 
 that value of 6 is found, which makes one of the curves a straight 
 line, and thereby gives the curve law. A convenient way of 
 doing this is: if the curve with log y (curve 0) is curved by angle 
 tto (tto being for instance the angle between the tangents at the 
 two end points of the curve, or the difference of the slopes at the 
 two end points), use a value b^, and plot the curve with log 
 iy—bi) (curve 1), and observe its curvature a^. Then inter- 
 polate a value b^, between b^ and 0, in proportion to the curva- 
 tures «! and ckq, and plot curve with log iy — bi) (curve 2), and 
 again interpolate a value 63 between 62 and either b^ or 0, which- 
 ever curve is nearer in slope to curve 2, continue until either the 
 curve with log {y — b) becomes a straight line, or an S curve and 
 in this latter case shows that the empirical curve cannot be 
 represented in this manner. 
 
 In this work, logarithmic paper is very useful, as it permits 
 plotting the curves without first looking up the logarithms, the 
 latter being done only when the last approximation of b is 
 found. In the same manner, if a constant term is suspected in 
 the X, the value ix—c) is used and curves plotted for various 
 values of c. Frequently the existence and the character of a 
 constant term is indicated by the shape of the curve; for 
 instance, if one of the curves plotted between x, y, log x, log y 
 approaches straightness for high, or for low values of the ab- 
 scissas, but curves considerably at the other end, a constant 
 term may be suspected, which becomes less appreciable at one 
 end of the range. For instance, the effect of the constant c in 
 {x—c) decreases with increase of x. 
 
 Sometimes one of the curves may be a straight line at one 
 end, but curve at the other end. This may indicate the presence 
 of a term, which vanishes for a part of the observations. In 
 this case only the observations of the range which gives a 
 straight line are used for deriving the curve law, the curve 
 calculated therefrom, and then the difference between the 
 calculated curve and the observations further investigated. 
 
EMPIRICAL CURVES. 235 
 
 Such a deviation of the curve from a straight line may also 
 indicate a change of the curve law, by the appearance of 
 secondary phenomena, as magnetic saturation, and in this case, 
 an equation may exist only for that part of the curve where the 
 secondary phenomena are not yet appreciable. The same 
 equation may then be apphed to the remaining part of the curve, 
 by assuming one of the constants, as a coefficient, or an exponent, 
 to change. Or a second equation may be derived for this part 
 of the curve and one part of the curve represented by one, the 
 other by another equation. The two equations may then over- 
 lap, and at some point the curve represented equally well by 
 either equation, or the ranges of application of the two equa^ 
 tions may be separated by a transition range, in which neither 
 applies exactly. 
 
 If neither the exponential functions nor the parabolic and 
 hj'perbolic curves satisfactorily represent the observ'ations, 
 
 X 
 
 further trials may be made by calculating and tabulating — 
 
 V X 1J 
 
 and — , and plotting curves between x, y, -, -. Also expressions 
 X y X 
 
 as x^+by^, and {x—ay+b{y—c)^, etc., may be studied. 
 
 Theoretical reasoning based on the nature of the phenomenon 
 represented by the numerical data frequently gives an indi- 
 cation of the form of the equation, which is to be expected, 
 and inversely, after a mathematical equation has been derived 
 a trial may be made to relate the equation to known laws and 
 thereby reduce it to a rational equation. 
 
 In general, the resolution of empirical data into a mathe- 
 matical expression largely depends on trial, directed by judg- 
 ment based on the shape of the curve and on a knowledge of 
 the curve shapes of various functions, and only general rules 
 can thus be given. 
 
 A number of examples may illustrate the general methods of 
 reduction of empirical data into mathematical functions. 
 
 153. Example 1. In a 118-volt tungsten filament incan- 
 descent lamp, corresponding values of the terminal voltage e 
 and the current i are observed, that is, the so-called " volt- 
 ampere characteristic " is taken, and therefrom an equation for 
 the volt-ampere characteristic is to be found. 
 
 The corresponding values of e and i are tabulated in the 
 first two columns of Table III and plotted as curve I in 
 
236 
 
 ENGINEERING MA THEM A TICS . 
 
 Fig. 79. In the thii-d and fourth column of Table III are 
 given log e and log i. In Fig. 79 then are plotted log e, i, as 
 curve II; e, log i, as curve III; log e, log i, as curve IV. 
 As seen from Fig. 79, curve IV is a straight line, that is 
 
 0.2 0.t 0.6 0.8 1.0 1.2 lA 1.6 1.8 2:0 2.2 i.i^log 6 
 
 2 
 
 , • 
 
 ) C 
 
 ) 8 
 
 ) 1 
 
 b 1! 
 
 1. 
 
 1 
 
 'f ^ 
 
 <0 200 Z 
 
 r; 
 
 
 
 
 
 
 
 
 in 
 
 ^ 
 
 
 
 / 
 
 
 
 
 
 nV^ 
 
 ^ 
 
 ^ 
 
 
 
 ( 
 
 \/ 
 
 
 
 
 
 / 
 
 o5^ 
 
 
 
 
 
 1/ 
 
 
 
 
 
 / 
 
 iV 
 
 
 
 
 
 / 
 
 IV 
 
 
 
 
 / 
 
 / 
 
 
 
 
 
 / 
 
 / 
 
 
 
 
 
 / 
 
 
 
 
 
 A 
 
 ( 
 
 
 
 
 
 / 
 
 
 
 
 
 Si 
 
 V 
 
 
 
 
 
 
 / 
 
 
 
 
 A 
 
 
 
 
 
 
 
 ' i' 
 
 / 
 
 
 
 
 / 
 
 
 
 
 y 
 
 i^ 
 
 ^ 
 
 / 
 
 
 
 
 / 
 
 
 
 ^ 
 
 ^ 
 
 ^ 
 
 
 / 
 
 f 
 
 L 
 
 
 / 
 
 < 
 
 
 ^ 
 
 /* 
 
 
 
 J 
 
 i 
 
 
 
 / 
 
 / 
 
 / 
 
 / 
 
 
 
 
 
 / 
 
 
 
 
 / 
 
 
 
 
 
 
 
 ^\ 
 
 
 
 
 / 
 
 / 
 
 
 
 
 
 ^^ 
 
 A 
 
 c 
 
 
 
 
 / 
 
 
 
 
 _^^ 
 
 
 ^ 
 
 
 
 
 
 
 1 
 
 •- 
 
 
 
 
 
 
 
 
 
 
 
 logi 
 
 
 9.6 
 
 
 9l5 
 
 
 9.1 
 
 
 9.3 
 
 
 9.2 
 
 
 9.1 
 
 i 
 
 9.0 
 
 0.45 
 
 8.9 
 
 0.40 
 
 ?.8 
 
 0.35 
 
 8.7 
 
 0.30 
 
 1.6 
 
 0.25 
 
 8.5 
 
 0.20 
 
 8.4 
 
 0.15 
 
 8.3 
 
 0.10 
 
 8.2 
 
 0.05 
 
 8.1 
 
 Fig. 79. Investigation of Volt-ampere Characteristic of Tungsten Lamp 
 
 Filament. 
 
 log i=J.+nloge; or i = oe", 
 which is a parabohc curve. 
 
 The constants a and n may now be calculated from 
 the numerical data of Table III by the method of least 
 squares, as discussed in Chapter IV, paragraph 120. "While 
 this method gives the most accurate results, it is so laborious 
 as to be seldom used in engineering; generally, values of the 
 constants a and n, sufficiently accurate for most practical 
 
EMPIRICAL CURVES. 
 
 237 
 
 purposes, are derived by the so-called " 2A method," which, 
 with proper tabular arrangement of the nimierical values, gives 
 high accuracy with a minimum of work. 
 
 Table III. 
 
 VOLT-AMPERE CHARACTERISTIC OF 118-VOLT TUNGSTEN LAMP. 
 
 e 
 
 i 
 
 log e 
 
 logi 
 
 8.211 +0.6 log c 
 
 J 
 
 2 
 
 0-0245 
 
 0-301 
 
 8-392 
 
 8-389 
 
 -0003 
 
 4 
 
 0-037 
 
 0-602 
 
 8- 568 
 
 5-572 
 
 -0-004 
 
 8 
 
 0-0568 
 
 0-903 
 
 8-754 
 
 8-753 
 
 + 0-001 
 
 16 
 
 0-0855 
 
 1-204 
 
 8-932 
 
 8-933 
 
 -0-001 
 
 25 
 
 0-1125 
 
 1-398 
 
 9-051 
 
 9-050 
 
 + 0-001 
 
 32 
 
 0-1295 
 
 1-505 
 
 9-112 
 
 9-114 
 
 -0-002 
 
 50 
 64 
 
 0-1715 
 0-200 
 
 1.699 
 1-806 
 
 9.234 
 9.301 
 
 9-230 
 9-295 
 
 + 0-004 
 + 0-006 
 
 100 
 
 0-2605 
 
 2-000 
 
 9-416 
 
 §-411 
 
 + 0-005 
 
 125 
 
 0-2965 
 
 2-097 
 
 9-472 
 
 9-469 
 
 + 0-003 
 
 150 
 
 0-3295 
 
 2-176 
 
 9-518 
 
 9-518 
 
 
 
 180 
 
 0-3635 
 
 2-255 
 
 9-561 
 
 9-564 
 
 -0-003 
 
 200 
 
 0-3865 
 
 2-301 
 
 9-587 
 
 §-592 
 
 -0-005 
 
 218 
 
 0-407 
 
 2-338 
 
 9-610 
 
 9-614 
 
 -0-004 
 
 
 n= 7-612 
 
 2-043 
 
 avg- ±0-003 
 
 
 J 
 
 7= 14-973 
 
 6-465 
 
 
 = 4-7 per cent 
 
 i= 7-361 
 
 4-422 
 
 
 4-422 
 
 n= = 
 
 7-361 
 
 O.6007~0 
 
 -6 
 
 
 
 .J'14= 22-585 
 
 8 505 
 
 
 
 
 C-6X22-585 = 
 
 = 13.561 
 
 .551 = 4-954 
 
 
 i = 8. 505-13 
 
 
 5-954-^14 = 
 
 8-211 
 
 
 
 
 ]ogi=8-2H+0-6l 
 
 og e and i = 
 
 0-01625e°-« 
 
 
 The fourteen sets of observations are divided into two 
 groups of seven each, and the sums of log e and log i formed. 
 They are indicated as S7 in Table III. 
 
 Then subtracting the two groups 27 from each other, 
 ehminates A, and dividing the two differences ^, gives the 
 exponent, n=0.6011; this is so near to 0.6 that it is reasonable 
 to assume that n=0.6, and this value then is used. 
 
 Now the sum of all the values of log e is formed, given as 
 S14 in Table III, and multipHed with 7i=0.6, and the product 
 
238 ENGINEERING MATHEMATICS 
 
 subtracted from the sum of all the \ogi. The difference A 
 then equals HA, and, divided by 14, gives 
 
 A = log a = 8.211; 
 
 hence, a = 0.01625, and the volt-ampere characteristic of this 
 timgsten lamp thus follows the equation, 
 
 log 1 = 8.211 +0.6 log e; 
 i = 0.016256° -6. 
 
 From e and i can be derived the power input p = d, and the 
 
 resistance r-- 
 
 i 
 
 p=0.01625ei-6; 
 
 ,0-4 
 
 e 
 r 
 
 0.01625' 
 
 and, eliminating e from these two expressions, gives 
 
 p = 0.01625V = 11.35r4xl0-i'', 
 
 that is, the power input varies with the fourth power of the 
 resistance. 
 
 Assuming the resistance r as proportional to the absolute 
 temperature T, and considering that the power input into the 
 lamp is radiated from it, that is, is the power of radiation P^, 
 the equation between p and r also is the equation between P^ 
 and T, thus, 
 
 P, = fcr4. 
 
 that is, the radiation is proportional to the fourth power of the 
 absolute temperature. This is the law of black body radiation, 
 and above equation of the volt-ampere characteristic of the 
 tungsten lamp thus appears as a conclusion from the radiation 
 law, that is, as a rational equation. 
 
 154. Example 2. In a magnetite arc, at constant arc length, 
 the voltage consumed by the arc, e, is observed for different 
 ■\-alues of current i. To find the equation of the volt-ampere 
 characteristic of the magnetite arc : 
 
EMPIRICAL CURVES. 
 
 239 
 
 Table IV. 
 VOLT-AMPERE CHARACTERISTIC OF MAGNETITE ARC. 
 
 e 
 
 log I 
 
 log e 
 
 (e-40l 
 
 log(e-40) 
 
 (e-30) 
 
 log (e-30) 
 
 "c 
 
 160 
 
 9. 698 
 
 2-204 
 
 120 
 
 2-079 
 
 130 
 
 2-114 
 
 158 
 
 120 
 
 000 
 
 2-079 
 
 80 
 
 1-903 
 
 90 
 
 1-954 
 
 120-4 
 
 94 
 
 0-301 
 
 1-973 
 
 54 
 
 1-732 
 
 64 
 
 1-806 
 
 94 
 
 75 
 
 0.602 
 
 1-875 
 
 35 
 
 1-644 
 
 45 
 
 1663 
 
 75-2 
 
 62 
 
 0-903 
 
 1-792 
 
 22 
 
 1-342 
 
 32 
 
 1-505 
 
 62 
 
 66 
 
 1-079 
 
 1-748 
 
 16 
 
 1-204 
 
 26 
 
 1-415 
 
 56-2 
 
 0-5 
 
 1 
 
 2 
 
 12 
 
 -2 
 + 0-4 
 
 
 + 0-2 
 
 
 + 0.2 
 
 23 = 0-000 5-874 
 
 ^■3 = 2.584 4-673 
 
 ^ = 2-584 -1-301 
 
 -1-301 „ 
 
 n= = -0-6034~-0.5 
 
 2-584 - 
 
 .2'6 = 2-584 10-447 
 
 2 - 584 X - - 5 = - 1 . 292 
 
 J= 11-739 
 11-739 + 6= l-966 = vl 
 
 log (e-30) = 1.956-0-5logt 
 
 90 4 
 e-30 =90.4t"~ "■' and e = 30+^^ 
 
 The first four columns of Table IV give i, e, logi, log e. 
 Fig. 80 gives the curves: i, e, as I; i, log e, as II; logi, e, as 
 III; logi, loge, as IV. 
 
 Neither of these curves is a straight line. Curve IV is 
 relatively the straightest, especially for high values of e. This 
 points toward the existence of a constant term. The existence 
 of a constant term in the arc voltage, the so-called " counter 
 e.m.f. of the arc " is physically probable. In Table IV thus 
 are given the values (e-40) and log (e-40), and plotted as 
 curve V. This shows the opposite curvature of IV. Thus the 
 constant term is less than 40. Estimating by interpolation, and 
 calculating in Table IV (e-30) and log (e-30), the latter, 
 plotted against log i gives the straight Ime VI. The curve law 
 
 thus is 
 
 log (e-30) = 4 +'n log I. 
 
240 
 
 ENGINEERING MA THEM A TICS. 
 
 Proceeding in Table IV in the same manner with logi 
 and log (e-30) as was done in Table III with log e and log x, 
 
 gives 
 
 n=-0.5; yi = loga = 1.956; and a = 90.4; 
 
 Fig. 80. Investigation of Volt-ampere Characteristic of Magnetite Arc. 
 
 hence 
 
 log (e-30) = 1.956-0.5 log i; 
 
 e-30 = 90.4i-o-5; 
 
 e = 30+^i 
 
 Vi 
 
EMPIRICAL CURVES. 
 
 241 
 
 which is the equation of the magnetite arc volt-ampere charac- 
 teristic. 
 
 155. Example 3. The change of current resulting from a 
 change of the conditions of an electric circuit containing resist- 
 ance, inductance, and capacity is recorded by oscillograph and 
 gives the curve reproduced as I in Fig. 81. From this curve 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 logt 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 0.-0 
 
 ' — . 
 
 N 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 ^ 
 
 N 
 
 
 
 
 
 
 
 
 
 
 
 i 
 
 
 
 \ 
 
 
 X 
 
 s 
 
 
 
 
 
 
 
 
 
 
 9 i- 
 
 
 
 \ 
 
 \ 
 
 
 ^ 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 \ 
 
 \ 
 
 
 
 
 
 
 
 
 9-fl 
 
 
 
 
 
 
 \ 
 
 
 
 -^ 
 
 k 
 
 
 
 
 
 
 
 
 
 
 r 
 
 
 \ 
 
 \ 
 
 
 
 \ 
 II 
 
 \ 
 
 
 
 
 
 
 — 1-fi 
 
 
 / 
 
 f-- 
 
 
 
 \ 
 
 
 
 
 \ 
 
 
 > 
 
 
 
 
 
 
 
 \ 
 
 
 
 \ 
 
 
 
 
 
 \ 
 
 \- 
 
 
 
 
 1 9 
 
 
 
 ni\ 
 
 
 
 
 
 \ 
 
 
 
 
 \ 
 
 \ 
 
 
 
 
 ■9:5— 
 
 
 
 \ 
 
 
 
 
 \ 
 
 \ 
 
 
 
 
 
 \ 
 
 
 -0.-8- 
 
 
 
 
 \ 
 
 
 
 
 
 I\ 
 
 '^- 
 
 
 
 
 
 \ 
 
 t 
 
 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 
 
 
 -'^i 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 -- 
 
 -_^ 
 
 
 
 
 
 4 
 
 
 
 8 
 
 1 
 
 2 
 
 t 
 1 
 
 6 
 
 2 
 
 
 
 2 
 
 4 
 
 2 
 
 8 
 
 
 Fig. 81. Investigation of Curve of Current Change in Electric Circuit. 
 
 are taken the numerical values tabulated as t and ^" in the first 
 two columns of Table V. In the third and fourth columns are 
 given log t and log i, and curves then plotted in the usual 
 manner. Of these curves only the one between I and logi 
 is shown, as II in Fig. 81, since it gives a straight line for the 
 higher values of i. For the higher values of t, therefore, 
 
 log t = ^ — ni; or, i = a£~"'; 
 that is, it is an exponential function. 
 
242 
 
 ENGINEERING MATHEMATICS. 
 
 Table V. 
 TRANSIENT CURRENT CHARACTERISTICS. 
 
 ( 
 
 i 
 
 log t 
 
 log i 
 
 
 
 2.10 
 
 — 
 
 0.322 
 
 0.1 
 
 2.48 
 
 9.000 
 
 0.394 
 
 0.2 
 
 2.66 
 
 9-301 
 
 0.425 
 
 0.4 
 
 2.58 
 
 9.602 
 
 0.412 
 
 0.8 
 
 2.00 
 1.36 
 
 5.903 
 0.079 
 
 0.301 
 
 1.2 
 
 0.134 
 
 1 .6 
 
 0.90 
 
 0.204 
 
 9.954 
 
 2.0 
 
 0.58 
 
 0.301 
 
 9.763 
 
 2.5 
 
 0.34 
 
 0.398 
 
 9.531 
 
 3.0 
 
 0.20 
 
 0.477 
 
 9.301 
 
 i' 
 
 ( 
 
 log t' 
 
 t2 
 
 ic 
 
 2.84 
 
 
 
 0.461 
 
 2.85 
 
 2.09 
 
 1.96 
 
 0.1 
 
 0.292 
 
 1.94 
 1.32 
 
 2.50 
 2.68 
 
 1.32 
 
 0.2 
 
 0.121 
 
 0.83 
 
 0.4 
 
 5.799 
 
 0.61 
 0.13 
 
 2.60 
 1 .96 
 
 0.09 
 
 0.8 
 
 g.954 
 
 
 
 — 
 
 — 
 
 0.03 
 
 1.33 
 
 0.01 
 
 — 
 
 — 
 
 0.01 
 
 0.88 
 
 
 
 — 
 
 — 
 
 — 
 
 0.58 
 
 
 
 — 
 
 
 — 
 
 0.34 
 
 
 
 ~ 
 
 — 
 
 ~ 
 
 0.20 
 
 4.94 
 4.44 
 
 3.98 
 3.21 
 
 2.09 
 
 1.36 
 0.89 
 0.58 
 
 0.34 
 0.20 
 
 -0.01 
 + 0.02 
 
 
 
 + 0.02 
 
 -0.04 
 
 -0.03 
 
 -0.02 
 
 
 
 
 
 
 J3 = 
 
 A = 
 
 4.8 
 
 4.8 
 
 3 
 
 5.5 
 
 6J 
 
 2 
 
 = 1. 
 
 = 2.75 
 
 9.851 
 9.851 
 
 - = 9.950 
 
 
 -0.1 0.753 
 -06 9.920 
 
 9.832 
 9.832 
 
 Js = 
 
 i = 1.15 
 Iog£X1.15= 0.499 
 -0.534 
 
 711= — = 
 
 0.499 
 
 10.3 
 
 10 .3Xni log £ = 
 
 = 9.416 
 
 -0.534 
 
 A =0.5-0.833 
 log 5X0.5 = 0.217 
 
 712 = 
 
 -0.833 
 
 -3.84 
 
 -1.07 
 
 S.683 
 -4.784 
 
 4= 3.467 
 3 .467 + 5 = .e93 = ^, = log ai 
 ai = 4.94 
 log 11 = 0.693-1 .07' log e 
 M=4.94£— I-"'' 
 
 tir = 4.94« 
 
 0.217 
 .^4 = 0.7 0.673 
 mlog £X0.7=-1.167 
 
 A= 1.840 
 1 .840^-4 = .460 = ^2— log ai 
 
 a2=2 .85 
 log 12= 0.460-3.84* log c 
 »2=2.85s — 3-si' 
 
 To calculate the constants a and n, the range of values is 
 used, in which the curve II is straight; that is, from t = 1.2 
 to f = 3. As these are five observations, they are grouped in two 
 pairs, the first 3, and the last 2, and then for t and logt", one- 
 third of the sum of the first 3, and one-half of the sum of the 
 last 2 are taken. Subtracting, this gives, 
 
 //< = !. 15; J log 1= -0.534. 
 
 Since, however, the equation, i^ae'^*, when logarithmated 
 gives ' 
 
 log z = log a — nt log e, 
 thus J log i = — 7z log £ J t. 
 
EMPIRICAL CURVES. 243 
 
 it is necessary to multiply Jt by log £ = 0.4343 before dividing it 
 into log i to derive the value of n. This gives n = 1.07. 
 
 Taking now the sum of all the five values of t, multiplying it 
 by log e, and subtracting this from the sum of all the five values 
 of log i, gives 5A = 3.467; hence 
 
 A = log a = 0.693, 
 a = 4.94, 
 
 and log ii = 0.693 -1.07t log s; 
 
 ii = 4.94£-i°^ 
 
 The current ii is calculated and given in the fifth column 
 of Table V, and the difference i' = d = i\—i in the sixth 
 column. As seen, from t = 1.2 upward, ij agrees with the 
 observations. Below t = 1.2, however, a difference i' remains, 
 and becomes considerable for low values of t. This difference 
 apparently is due to a second term, which vanishes for higher 
 values of t. Thus, the same method is now applied to the 
 term i'; column 8 gives hgi', and in curve III of Fig. 81 is 
 plotted logi' against t. This curve is seen to be a straight 
 line, that is, i' is an exponential function of t. 
 
 Resolving i' in the same manner, by using the first four 
 points of the curve, from i = to t = OA, gives 
 
 log i2=0.460-3.84« logs; 
 
 and, therefore, 
 
 i=ti-i2 = 4.94£-i"^'-2.85£-^-8*« 
 
 is the equation representing the current change. 
 
 The numerical values are calculated from this equation 
 and given under ic in Table V, the amount of their difference 
 from the observed values are given in the last column of this 
 
 table. 
 
 A still greater approximation may be secured by adding 
 the calculated values of t2 to the observed values of i in the 
 last five observations, and from the result derive a second 
 approximation of ii, and by means of this a second approxi- 
 mation of t2. 
 
244 
 
 ENGINEERING HI A THEM A TICS. 
 
 156. As further example may be considered the resolution 
 of the core loss curve of an electric motor, which had been 
 expressed irrationally by a potential series in paragraph 144 
 and Table I. 
 
 Table VI. 
 CORE LOSS CURVE. 
 
 e 
 Volts. 
 
 Pi kw. 
 
 log e 
 
 log Pi 
 
 1.6 log e 
 
 A=log Pi 
 -1.6 log e 
 
 Pc 
 
 </ 
 
 40 
 
 063 
 
 1.602 
 
 9.799 
 
 2.563 
 
 7. 236 
 
 0.70 
 
 -007 
 
 60 
 
 1 
 
 36 
 
 1.778 
 
 0134 
 
 2 
 
 845 
 
 
 289 
 
 1 
 
 34 
 
 + 
 
 02 
 
 80 
 
 2 
 
 18 
 
 1.903 
 
 0.338 
 
 3 
 
 045 
 
 
 293 avg. 
 
 2 
 
 12 
 
 + 
 
 06 
 
 100 
 
 3 
 
 00 
 
 2.000 
 
 0.477 
 
 3 
 
 200 
 
 
 277 7-282 
 
 3 
 
 03 
 
 -0 
 
 03 
 
 120 
 140 
 
 3 
 6 
 
 93 
 
 22 
 
 2.079 
 2.146 
 
 0.594 
 0.794 
 
 3 
 3 
 
 326 
 434 
 
 
 268 J 
 360 
 
 4 
 5 
 
 05 
 20 
 
 -0 
 
 + 1 
 
 12 
 02 
 
 160 
 
 8 
 
 69 
 
 2. 204 
 
 0934 
 
 3 
 
 526 
 
 
 403 
 
 6 
 
 43 
 
 + 2 
 
 16 
 
 
 2a 
 
 = 5.283 
 
 0271 
 
 logP 
 
 i=7. 282 + 1. 6loge 
 
 
 
 -5'3-=-3 
 
 = 1.761 
 
 0.090 
 
 P 
 
 t=l-914e^'^, in watta 
 
 
 
 
 = 4. 079 
 
 1.071 
 
 
 
 
 
 -2-2-2 
 
 = 2.0395 
 
 0.535 
 
 
 
 
 
 J 
 
 = 0.2785 
 
 0.445 
 
 
 
 
 
 r 
 
 _ 0.445 
 0.2785 
 
 = 1.598~ 
 
 16 
 
 
 
 The first two columns of Table VI give the observed values 
 of the voltage e and the core loss Pi in kilowatts. The next 
 two columns give log e and log Pj. Plotting the curves shows 
 that loge, log Pi is approximately a straight line, as seen in 
 Fig. 82, with the exception of the two highest points of the 
 curve. 
 
 Excluding therefore the last two points, the first five obser- 
 vations give a parabolic curve. 
 
 The exponent of this curve is found by Table VI as 
 n= 1.598; that is, with sufficient approximation, as n = 1.6. 
 
 To see how far the observations agree with the curve, as 
 given by the equation, 
 
 Pi=ae''^ 
 
 in the fifth column 1.6 log e is recorded, and in the sixth column, 
 ^ = log a = log Pi -1.6 loge. As seen, the first and the last 
 two values of A differ from the rest. The first value corre- 
 
EMPIRICAL CURVES. 
 
 245 
 
 sponds to such a low value of Pi as to lower the accuracy of 
 the observation. Averaging then the four middle values, 
 gives J =7.282; hence, 
 
 log Pi= 7.282 + 1.6 log e. 
 Pi =1.914ei-^' in watts. 
 
 " 
 
 1.6 
 
 1.7 
 
 1.8 
 
 1.9 
 
 2.0 
 
 2.1 
 
 2.2 
 
 
 log 
 
 Pi 
 
 
 
 
 
 lo 
 
 r 
 
 
 
 
 
 
 i 
 
 ^ 
 
 Pi 
 
 n-R 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 kw. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 9K) 
 (S 
 
 H-R 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 / 
 
 
 
 
 U.D 
 
 
 
 
 
 
 
 
 
 
 / 
 
 / 
 
 
 
 
 8H) 
 
 
 
 
 
 
 
 
 
 p> 
 
 X 
 
 <■ 
 
 
 
 
 
 
 Ur4: 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^y) 
 
 n-9- 
 
 
 
 
 
 
 \o> 
 
 '/ 
 
 
 
 
 
 
 c 
 
 ) 
 
 / 
 
 U.<5 
 
 
 
 
 
 / 
 
 /> 
 
 
 
 
 
 
 
 
 / 
 
 AU 
 
 n-fi 
 
 
 
 
 / 
 
 r 
 
 
 
 
 
 
 
 
 / 
 
 / 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 / 
 
 / 
 
 
 -5.0 
 
 Q-R 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 y 
 
 
 
 
 
 
 
 
 
 
 / 
 
 b 
 
 
 
 ±.U 
 
 
 
 
 
 
 
 
 
 
 <? 
 
 / 
 
 
 
 
 
 -3:0' 
 
 
 
 
 
 
 
 
 
 
 ;^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 < 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 y 
 
 / 
 
 
 
 
 
 
 
 
 4,'.\i 
 
 
 
 
 
 
 /^ 
 
 < 
 
 
 
 
 
 
 
 
 
 1 n 
 
 
 
 
 .'^ 
 
 X 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 •""^ 
 
 
 
 ^1 
 
 3 
 
 6 
 
 3 
 
 8 
 
 D 
 
 |=Vc 
 iSo 
 
 Its 
 12 
 
 
 
 1 
 
 fl 
 
 IfjO 
 
 Fig. 82. Investigation of Cuvres. 
 
 This equation is calculated, as Fc, and plotted in Fig. 82. 
 The observed values of Vi are marked by circles. As seen, 
 the agreement is satisfactory, with the exception of the two 
 highest values, at which apparently an additional loss appears, 
 which does not exist at lower voltages. This loss probably is 
 due to eddy currents caused by the increasing magnetic stray 
 field resulting from magnetic saturation. 
 
246 
 
 ENGINEERING MATHEMATICS. 
 
 157. As a further example may be considered the resolution 
 of the magnetic characteristic, plotted as curve I in Fig. 83, 
 and given in the first two columns of Table VII as H and B. 
 
 Table VII. 
 
 MAGNETIC CHARACTERISTIC. 
 
 H 
 
 B 
 
 kllolines 
 
 log H 
 
 log B 
 
 B 
 
 H 
 
 H 
 B 
 
 Be 
 
 J 
 
 2 
 
 30 
 
 0301 
 
 0477 
 
 15 
 
 0.687 
 
 64 
 
 + 3.4 
 
 4 
 
 8 4 
 
 
 
 602 
 
 
 
 924 
 
 2.1 
 
 0.476 
 
 9 
 
 7 
 
 + 1.3 
 
 e 
 
 8 
 
 11. 2 
 130 
 
 
 
 
 778 
 903 
 
 
 049 
 114 
 
 1867 
 1.625 
 
 0536 
 614 
 
 11 
 13 
 
 6 
 
 
 + 0.4 
 
 
 10 
 
 14. 
 
 
 000 
 
 
 146 
 
 140 
 
 0.715 
 
 13 
 
 9 
 
 -0.1 
 
 15 
 
 16 4 
 
 
 176 
 
 
 188 
 
 1033 
 
 0974 
 
 15 
 
 45 
 
 + 0.Q5 
 
 20 
 30 
 
 16.3 
 17-2 
 
 
 301 
 477 
 
 
 212 
 236 
 
 0-815 
 0573 
 
 1-227 
 
 16 
 17 
 
 3 
 3 
 
 
 + 0.1 
 
 174 
 
 40 
 
 17. 8 
 
 
 602 
 
 
 250 
 
 0.445 
 
 2.25 
 
 17 
 
 8 
 
 
 
 60 
 
 18. 5 
 
 
 778 
 
 
 267 
 
 0308 
 
 3. 25 
 
 18 
 
 4 
 
 -0-1 
 
 80 
 
 18. 8 
 
 
 903 
 
 
 274 
 
 0.235 
 
 4.25 
 
 18 
 
 8 
 
 
 
 .^4=53 
 
 
 
 
 3. 530 
 
 
 -J'4=210 
 
 
 
 
 11-49 
 
 
 ^=m 
 
 
 7.96 
 
 -- = 0.0507 = 
 
 = !> 
 
 7-96 
 
 
 28-263 
 
 
 
 
 15-020 
 
 
 
 
 26! 
 
 1X00507 = 
 
 J = 
 
 =13-334 
 
 
 
 = 1-686 
 
 
 
 
 1. 686 + 8 = 
 
 = 0-211=0 
 
 
 
 H 
 B^' 
 
 . 211 + 0. 0507H and 
 
 0.211+00507// 
 
 
 Plotting H, B, log H, log B against each other leads to no 
 results, neither does the introduction of a constant term do 
 this. Thus in the fifth and sixth columns of Table VII are 
 
 B H 
 
 calculated — and -_, and are plotted against H and against B. 
 
 Of these four curves, only the curve of -^ against H is shown 
 
 in Fig. 83, as II. This curve is a straight line with the exception 
 of the lowest values ; that is, 
 
 H 
 
 B 
 
 -a+bH. 
 
EMPIRICAL CURVES. 
 
 247 
 
 Excluding the three lowest values of the observations, as 
 not lying on the straight line, from the remaining eight values, 
 as calculated in Table VII, the following relation is derived, 
 
 B 
 
 =0.211 +0.0507 i?, 
 
 1 n 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 f 
 
 •t.U 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 /' 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 /■ 
 
 
 
 
 3:5 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 A 
 
 
 
 
 
 
 ■3;^) 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 B 
 
 
 2;5 
 
 
 
 
 
 
 
 
 / 
 
 
 
 ^ 
 
 
 
 
 4\) 
 
 
 
 
 ■ 
 
 
 
 
 
 
 
 
 
 z 
 
 r— 
 
 
 I 
 
 
 
 
 
 ir. 
 
 
 ■2:0 — ' 
 
 
 /^ 
 
 ^ 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 / 
 
 1 
 
 
 
 
 
 
 
 
 
 — 1-9- 
 
 
 -1t5 
 
 / 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 c 
 
 
 
 
 
 / 
 
 r 
 
 
 
 
 
 
 
 
 
 
 
 o 
 
 
 
 
 / 
 
 r 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 f— 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 i^ 
 
 ^ — 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 /f 
 
 ] 
 
 ) 
 
 2 
 
 3 
 
 
 3 
 
 4 
 
 .J? . 
 
 ) 
 
 i 
 
 ) 
 
 7 
 
 ) 
 
 e 
 
 ) 
 
 Fig. 83. Investigation of Magnetization Curve, 
 and herefrom, 
 
 B=-, 
 
 H 
 
 0.211 +0.0507 H 
 
 is the equation of the magnetic characteristic for values of H 
 from eight upward. 
 
 The values calculated from this equation are given as B^ 
 in Table VU. 
 
248 ENGINEERING MATHEMATICS. 
 
 TJ 
 
 The diiference between the observed values of — , and the 
 
 value given by above equation, which is appreciable up to 
 H=Q, could now be further investigated, and would be found 
 to approximately follow an exponential law. 
 
 As a final example may be considered the investigation of 
 a hysteresis curve of silicon steel, of which the numerical values 
 are given in columns 1 and 2 of Table VIII. 
 
 The first column gives the magnetic density B, in fines of 
 magnetic force per cm.^; the second column the hysteresis loss 
 iv, in ergs per cycle per kg. (specific density 7.5). The third 
 column gives log B, and the fourth column log iv. 
 
 Of the four curves between B, w, log B, log lu, only the 
 curve relating log iv to log B approximates a straight line, and 
 is given in the upper jjart of Fig. 84. This curve is not a 
 straight line throughout its entire length, but only two sections 
 of it are straight, from 5 = 50 to S =400, and from B = 1600 to 
 5=8000, but the curve bends between 500 and 1200, and above 
 8000. 
 
 Thus two empirical formulas, of the form; w = aB'^, are 
 calculated, in the usual mamier, in Table ^TII. The one 
 applies for lower densities, the other for medium densities : 
 
 Low density: £^400: w = 0.0034l52-ii 
 
 Medium density: 1600^5^8000: iv = 010mB^-^'^ 
 
 In Table VIII the values for the lower range are denoted 
 by the index 1, for the higher range by the index 2. 
 
 Neither of these empirical formulas appfies strictly to the 
 range: 400<B<1G00, and to the range 5 > 8000. They may 
 be applied within these ranges, by assuming either the coefficient 
 a as varying, or the exponent n as varying, that is, applying a 
 correction factor to a, or to n. 
 
 Thus, in the range: 400 <5< 1600, the loss may be repre- 
 sented by: 
 
 (1) An extension of the low density formula; 
 
 ■u) = ai52-" or w = 0.003415»i. 
 
 (2) An extension of the medium density formula 
 
 w = a'>B^-^ or w = 0.10965"2. 
 
EMPIRICAL CURVES. 
 
 249 
 
 by giving tables or curves of a respectively n. Such tables are 
 most conveniently given as a percentage correction. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 .M 1 
 
 I 
 
 -og 
 
 w 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ( 
 
 L^ 
 
 r 
 
 
 
 
 -6 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 > 
 
 V 
 
 if— 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 y« 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 r* 
 
 l^ 
 
 1 
 
 
 
 
 
 
 
 
 
 
 -4 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 /* 
 
 <■ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 J^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 #^ 
 
 ,4^- 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 k 
 
 
 V 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 -3 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 <^ 
 
 ^1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ff 
 
 < 
 
 p* 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 p 
 
 /' 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 A 
 
 < 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ' 
 
 -2 
 
 
 
 
 
 
 
 
 
 A 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 0- 
 
 
 
 
 
 
 
 
 <^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 i 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 M 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 — 1 
 
 
 
 
 / 
 
 Y 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 a^ 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 0- 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 — 
 
 
 
 
 
 
 
 4a 
 
 
 
 
 
 
 ^ 
 
 ^ 
 
 
 
 
 
 ^ 
 
 
 
 / 
 
 /» 
 
 
 
 
 
 
 
 ' 
 
 
 
 
 
 
 
 T ^ 
 
 
 
 
 
 
 
 sj 
 
 k. 
 
 
 / 
 
 
 
 
 
 "^ 
 
 
 
 
 
 1 
 1 
 
 
 0— 
 
 ", 
 
 
 
 1, 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 y 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 I 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 / 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 0- 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 / 
 
 
 
 \, 
 
 
 
 
 
 
 
 
 
 
 
 
 _2 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 N 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 -2 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 
 — ^ 
 
 
 
 _X 
 
 
 
 r~^ 
 
 •^ 
 
 
 
 , A 
 
 
 
 
 
 
 
 
 
 
 
 
 , 
 
 1 / 
 
 
 
 
 -2 
 
 Q 
 
 
 
 
 
 
 
 
 
 
 
 
 H 
 
 
 S, 
 
 
 y«~^ 
 
 -^ 
 
 -n 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^1 
 
 
 
 
 
 
 
 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 -1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 K 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 sj 
 
 s 
 
 
 
 
 
 
 
 V 
 
 
 
 
 
 -1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 V 
 
 >v 
 
 
 
 
 
 ^ 
 
 X' 
 
 ^ 
 
 -A- 
 
 Ic 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 '■r 
 
 tf-^ 
 
 — - 
 
 *^ 
 
 ■r-^ 
 
 
 
 
 
 ' 
 
 
 
 
 '2 
 
 
 
 
 
 -1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 L_ 
 
 Log B = 
 
 
 
 2,* 
 
 
 
 U 
 
 
 _ 
 
 
 
 
 
 3^ 
 
 
 
 
 
 
 
 
 
 
 
 
 V 
 
 
 
 
 
 Fig. 84. 
 
 The percentage correction, which is to be applied to ai and 
 02 respectively, to «i and ^2, to make the formulas apphcable 
 
250 
 
 ENGINEERING MATHEMATICS. 
 
 Table VIII. 
 HYSTERESIS OF SILICON STEEL. 
 
 
 
 
 
 Joi 
 
 Jaa 
 
 <ini 
 
 ^712 
 
 
 
 
 B 
 
 w 
 
 log J3 
 
 log w 
 
 Cl 
 
 % 
 
 02 
 
 % 
 
 ni 
 % 
 
 712 
 
 % 
 
 m 
 
 712 
 
 m 
 
 35 
 50 
 
 6.4 
 
 13 
 
 1.544 
 
 0.806 
 
 + 3.0 
 
 — 
 
 + 0.46 
 -0.03 
 
 
 
 2.120 
 
 : 
 
 2.03 
 
 1 .699 
 
 1.117 
 
 -0.2 
 
 2.109 
 
 60 
 
 19 
 
 1.778 
 
 1.279 
 
 - 1.1 
 
 — 
 
 -0.13 
 
 
 
 2.107 
 
 — 
 
 2.14 
 
 80 
 
 36 
 
 1 .903 
 
 1 .556 
 
 + 1.9 
 
 — 
 
 + 0.20 
 
 
 
 2.114 
 
 — 
 
 2 .12 
 
 100 
 
 67 
 
 2.000 
 
 1 .752 
 
 -0.2 
 
 — 
 
 -0.02 
 
 
 
 2.110 
 
 — 
 
 2.09 
 
 120 
 160 
 
 83 
 
 156 
 
 2.079 
 
 1.922 
 
 + 0.7 
 
 + 2.3 
 
 
 + 0.07 
 + 0.21 
 
 
 2.111 
 2.114 
 
 
 
 2.16 
 2.10 
 
 2.204 
 
 2.193 
 
 200 
 
 245 
 
 2.301 
 
 2.389 
 
 + 0.2 
 
 — 
 
 + 0.02 
 
 
 
 2.110 
 
 
 2.07 
 
 250 
 
 394 
 
 2.398 
 
 2.595 
 
 + 0.7 
 
 — 
 
 + 0.06 
 
 
 
 2.111 
 
 — 
 
 2.03 
 
 300 
 
 571 
 
 2.477 
 
 2.757 
 
 - 0.5 
 
 — 
 
 -0.04 
 
 
 
 2.109 
 
 
 1 .98 
 
 400 
 500 
 
 1026 
 1610 
 
 2.602 
 
 3 .011 
 
 - 2.7 
 
 -29.5 
 
 -0.22 
 
 -3.52 
 
 2.105 
 
 1.544 
 
 2.03 
 2.02 
 
 2.699 
 
 3.207 
 
 - 4.7 
 
 -0.37 
 
 2.102 
 
 600 
 
 2320 
 
 2.778 
 
 3. 366 
 
 -6.2 
 
 -24.0 
 
 -0.87 
 
 -2.68 
 
 2.092 
 
 1 .557 
 
 1 .96 
 
 800 
 
 4030 
 
 2.903 
 
 3.605 
 
 -16.5 
 
 -15.8 
 
 -1.17 
 
 -1.72 
 
 2.085 
 
 1.573 
 
 1.91 
 
 1000 
 
 6150 
 
 3.000 
 
 3.789 
 
 -15.7 
 
 -11.1 
 
 -1.14 
 
 -1.06 
 
 2.086 
 
 1.583 
 
 1 .89 
 
 1200 
 1600 
 
 8680 
 14370 
 
 3.079 
 
 3.938 
 
 -18.7 
 -26.9 
 
 -6.0 
 
 -2.02 
 
 -0.55 
 
 2.067 
 
 1.5912 
 
 1 .82 
 1.73 
 
 3.204 
 
 4.157 
 
 - 2.0 
 
 -0.18 
 
 1.5971 
 
 2000 
 
 21000 
 
 3.301 
 
 4.322 
 
 — 
 
 0.0 
 
 — 
 
 0.00 
 
 — 
 
 1 ;6000 
 
 1.67 
 
 2500 
 
 30300 
 
 3 .398 
 
 4.481 
 
 — 
 
 + 0.9 
 
 — 
 
 + .07 
 
 — 
 
 1.6011 
 
 1.62 
 
 3000 
 4000 
 
 40500 
 63400 
 
 3.477 
 
 4.607 
 
 
 + 1.2 
 -0.2 
 
 
 + 0.09 
 
 
 1.6014 
 1.5997 
 
 1.58 
 1.58 
 
 3.602 
 
 4.802 
 
 -0.02 
 
 5000 
 
 90600 
 
 3 .699 
 
 4.957 
 
 — 
 
 - 0.2 
 
 — 
 
 -0.02 
 
 — 
 
 1.5997 
 
 1.59 
 
 6000 
 
 120600 
 
 3.778 
 
 5 .082 
 
 — 
 
 -0.9 
 
 — 
 
 -0.07 
 
 — 
 
 1.5989 
 
 1.61 
 
 8000 
 10000 
 
 194100 
 282500 
 
 3.903 
 
 5.288 
 
 
 + 0.7 
 
 
 + 0.05 
 
 
 1.6008 
 
 1.66 
 1.77 
 
 4.000 
 
 6.451 
 
 + 2.6 
 
 + 0.17 
 
 1.6027 
 
 12000 
 
 397500 
 
 4.079 
 
 5.599 
 
 — 
 
 + 7.9 
 
 — 
 
 + 0.50 
 
 — 
 
 1 .6080 
 
 2.32 
 
 14000 
 
 609500 
 
 4.146 
 
 5.785 — 
 
 + 27.7 
 
 — 
 
 + 1.67 
 
 — 
 
 1.6267 
 
 2.88 
 
 18000 
 
 907500 
 
 4.204 
 
 5.958 — 
 
 + 45.9 
 
 — 
 
 + 2.85 
 
 — 
 
 1.6456 
 
 — 
 
 
 ^1.= 9.459 7.626 
 
 
 
 l5 = 
 
 = 11.982 12.945 
 
 
 
 = 2.523 6.319 
 
 
 5.319 
 
 
 
 711= = 2.11 
 
 
 
 2.523 
 
 
 
 .2'io = 21.441 20.571 
 
 
 
 2.11X21 .441 = 45.241 
 
 
 
 4=-24.670 logui=7. 533 + 2. 11 Ic 
 
 >gB 
 
 
 + 10=-2.467 „2.i, 
 
 
 
 = 7.533 =logai li> = .0034113 
 
 
 
 01 = 0.00341 
 
 
 
 -2') = 13.380 17.667 
 
 
 
 
 = 14.982 20.129 
 
 
 
 = 1.602 2.562 
 
 
 2.562 ^ 
 
 
 
 "^=T:6"52= i-ssfl^i-eo 
 
 
 
 29=28.362 37.696 
 
 
 
 1.60X28.362 
 
 J = 
 
 = 45.37! 
 
 1 
 
 ) log ui=9. 040 + 1. 60 lo 
 
 g5 
 
 — 7.68! 
 
 
 + 8=- 0.960 ,., 
 
 
 
 = 9.040 =logaj to = 0.1096i3 
 
 
 
 m = 0.1096 
 
 
EMPIRICAL CURVES. 251 
 
 to the ranges where the logarithmic curve is not a straight 
 line, are given in Table VIII as 
 
 Jai Ja2 Ani An2 
 
 « 
 
 they are calculated as follows : 
 
 Assuming n as constant, =no, then a is not constant, =ao, 
 and the ratio : 
 
 Aa a 
 a Go 
 
 is the correction factor, and it is: 
 
 w=aB'", 
 hence : 
 
 log w = log a + no log B 
 and 
 
 log a = log w -Wo log B; 
 thus: 
 
 and 
 
 log — =log a —log ao=log w -log ao —no log B, 
 ao 
 
 Aa a 
 
 l=A''log w— log ao— wologB— 1. . (1) 
 
 a ao 
 
 Assuming a as constant, =ao, then n is not constant, =no, 
 and the ratio. 
 
 An n 
 
 n no 
 
 is the correction factor, and it is 
 
 iu = oo5"; 
 
 hence 
 
 log w=\ogao+n log B, 
 
252 ENGINEERING MATHEMATICS. 
 
 and 
 
 thus 
 
 and 
 
 n log S=log w — log ao] 
 
 n n log B log w —log ao 
 no no log B no log B ' 
 
 ^w_ri log w -log gp -Tip log B 
 
 n wp wplogS 
 
 by these equations (1) and (2) the correction factors in columns 
 5 to 8 of Table VIII are calculated, by using for ap and no the 
 values of the lower range curve, in columns 5 and 7, and the 
 values of the medium range curve, in columns 6 and 8. 
 
 Thus, for instance, at B = 1000, the loss can be calculated 
 by the equation, 
 
 w=aiB"\ 
 
 by applying to ai the correction factor: 
 
 —15.7 per cent at constant: ni=2.11, that is, 
 
 ai =0.00341(1 -0.157) =0.00287; 
 
 or by applying to Wi the correction factor: 
 
 —1.14 per cent at constant: ai =0.00341, that is, 
 
 ni =2.11(1 -0.0114) =2.086. 
 
 Or the loss can be calculated by the equation, 
 ^y = a2^"^ 
 by applying to 02 the correction factor: 
 
 —11.1 per cent at constant: 712 = 1.60, that is, 
 
 02=0.1096(1 -0.111) =0.0974; 
 
EMPIRICAL CURVES. 253 
 
 or by applying to ria the correction factor: 
 
 —1.06 per cent at constant: 02=0.1096, that is, 
 
 ^2 = 1.60(1 -0.0106) =1.583; 
 
 and the loss may thus be given by either of the four ex- 
 pressions : 
 
 w = 0.0028752" = 0.00341B2°»« = 0.09745i''=0.10965i-^^^ 
 
 As seen, the variation of the exponent n, required to extend 
 the use of the parabolic equation into the range for which it 
 does not strictly apply any more, is much less than the varia- 
 tion of the coefficient a, and a far greater accuracy is thus 
 secured by considering the exponent n as constant — 1.6 for 
 medium and high values of B — and making the correction in 
 coefhcient a, outside of the range where the 1.6th power law 
 holds rigidly. 
 
 In the last column of Table VIII is recorded the ratio of 
 
 . . ^ log w 
 
 vanation, m = ~-, , as the averages each of two successive 
 
 J log B 
 
 values. As seen, m agrees with the exponent n within the 
 two ranges, where it is constant, but differs from it outside 
 of these ranges. For instance, if B changes from 1600 down- 
 ward, the ratio of variation m increases, while the exponent 
 n slightly decreases. 
 
 In Fig. 84 are shown the percentage correction of the 
 coefficients ai and a2, and also the two exponents Ui and ri2, 
 together with the ratio of variation in. 
 
 The ratio of variation ??i is very useful in calculating the 
 change of loss resulting from a small change of magnetic density, 
 as the percentual change of loss w is m times the percentual 
 (small) change of density. 
 
 As further example, the reader may reduce to empirical 
 equations the series of observations given in Table IX. This 
 table gives: 
 
 A. The candle-power L, as function of the power input p, 
 of a 40-watt tungsten filament incandescent lamp. 
 
 B. The loss of power by corona (discharge into the air), p, 
 in kw., in 1.895 km. of conductor, as function of the voltage 
 e (in kv.) between conductor and return conductor, for the 
 
254 
 
 ENGINEERING MATHEMATICS. 
 
 Table IX. 
 
 -4. Luminosity characterigtic of 40~watt tungsten incandescent lamp. 
 
 
 
 L = horizontal candlepower. 
 
 
 
 
 p = watts input. 
 
 
 
 L 
 
 P 
 
 L 
 
 P 
 
 L 
 
 P 
 
 L 
 
 P 
 
 L 
 
 P 
 
 2 
 
 12.25 
 
 20 
 
 31.64 
 
 40 
 
 44.14 
 
 128 
 
 76.77 
 
 382 
 
 135.6 
 
 i 
 
 16.33 
 
 24 
 
 34.55 
 
 44 
 
 45.42 
 
 192 
 
 95.24 
 
 460 
 
 145.2 
 
 8 
 
 21.35 
 
 28 
 
 37.29 
 
 48 
 
 47.05 
 
 256 
 
 109.0 
 
 — 
 
 — 
 
 12 
 
 25.60 
 
 32 
 
 39.26 
 
 64 
 
 54.31 
 
 291 
 
 118.2 
 
 — 
 
 — 
 
 16 
 
 28.91 
 
 36 
 
 41.47 
 
 96 
 
 65.73 
 
 320 
 
 123.1 
 
 — 
 
 
 B. Corona loss of high-voltage transmission line; at 60 cycles: 
 
 
 
 
 1895 m . length of conductor. 
 
 
 
 
 3 .10 m . distance between conductors. 
 
 
 
 
 No. 000 seven-strand cable, 1.18 cm. diameter. 
 
 
 
 
 — 13°C.; 76 .2 cm. barometer; sunshine. 
 
 
 
 
 e=kilovolts between conductors, effective. 
 
 
 
 
 p = kilowatts loss. 
 
 
 
 e 
 
 P 
 
 e 
 
 P 
 
 e 
 
 P 
 
 e 
 
 P 
 
 e 
 
 P 
 
 79.8 
 
 0.01 
 
 141 .5 
 
 0.09 
 
 181.0 
 
 l'.02 
 
 221.0 
 
 8.70 
 
 212.0 
 
 6.44 
 
 90.7 
 
 0.01 
 
 147.0 
 
 0.08 
 
 186.2 
 
 1.55 
 
 227.0 
 
 10.66 
 
 219.0 
 
 8.31 
 
 101.5 
 
 0.02 
 
 153.6 
 
 0.12 
 
 192.6 
 
 2.49 
 
 234.0 
 
 13.25 
 
 — - 
 
 
 109.5 
 
 0.03 
 
 159.0 
 
 0.16 
 
 200.6 
 
 3.77 
 
 189.0 
 
 2.10 
 
 — . 
 
 — 
 
 120.5 
 
 0.04 
 
 169.8 
 
 0.35 
 
 208.6 
 
 5.34 
 
 195.0 
 
 2.88 
 
 — 
 
 — 
 
 130.0 
 
 0.06 
 
 174.0 
 
 0.53 
 
 216.0 
 
 7.13 
 
 203.8 
 
 4.72 
 
 — 
 
 — 
 
 C. Volu] 
 
 me-pressure characteristic of dry steam at its boiling-point. 
 i=degrees C. 
 P=pressure, in kg. per cm.^ 
 F = volume, in m.^ per kg. 
 
 
 
 ( 
 
 P 
 
 V 
 
 ( 
 
 P 
 
 V 
 
 t 
 
 P 
 
 r 
 
 
 59.8 
 
 0.2 
 
 7.806 
 
 132.8 
 
 3.0 
 
 0.612 
 
 169.6 
 
 8.0 
 
 .244 
 
 
 80.9 
 
 0.5 
 
 3.297 
 
 142.8 
 
 4.0 
 
 0.467 
 
 178.9 
 
 10.0 
 
 0.197 
 
 
 99.1 
 
 1.0 
 
 1.717 
 
 151.0 
 
 5.0 
 
 0.379 
 
 186.9 
 
 12.0 
 
 .167 
 
 
 119.6 
 
 2.0 
 
 0.896 
 
 157.9 
 
 6.0 
 
 0.319 
 
 197.2 
 
 15.0 
 
 0.135 
 
 
EMPIRICAL CURVES. 255 
 
 distance of 310 cm. between the conductors, and the conductor 
 diameter of 1 . 18 cm. 
 
 C. The relation between steam pressure P, in kg. per cm.', 
 and the steam volume V, in m.^, at the boiUng-point, per kg. 
 of dry steam. 
 
 D. Periodic Curves. 
 
 is8. All periodic functions of time or distance can be ex- 
 pressed by a trigonometric series, or Fourier series, as has been 
 discussed in Chapter III, and the methods of resolution, and 
 the arrangements to carry out the work rapidly, have also 
 been discussed in Chapter III. 
 
 The resolution of a periodic function thus consists in the 
 determination of the higher harmonics, which are superimposed 
 on the fundamental wave. 
 
 As periodic functions are of the greatest importance in elec- 
 trical engineering, in the theory of alternating current pheno- 
 mena, a familiarity with the wave shapes produced by the dif- 
 ferent harmonics is desirable. This familiarity should be 
 sufficient to enable one in most cases to judge immediately from 
 the shape of the wave, as given by oscillograph, etc., on the har- 
 monics which are present or at least which predominate. 
 
 The effect of the lower harmonics, such as the third, fifth, 
 etc., (or the second, fourth, etc., where present), is to change 
 the shape of the wave, make it differ from sine shape, giving 
 such featm-es as fiat top wave, peaked wave, saw-tooth, double 
 and triple peaked, steep zero, flat zero, etc., while the high 
 harmonics do not change the shape of the wave so much, as 
 superimpose ripples on it. 
 
 Odd Lower Harmonics. 
 
 159. To elucidate the variation in shape of the alternating 
 waves caused by various lower harmonics, superimposed upon 
 the fundamental at different relative positions, that is, different 
 phase angles, in Figs. 85 and 86 are shown the effect of a third 
 harmonic, of 10 per cent and 30 per cent of the fundamental, 
 respectively. A gives the fundamental, and C D E F G the 
 waves resulting by the superposition of the triple harmonic 
 in phase with the fundamental (C), under 45 deg. lead (D), 90 
 deg. lead or quadrature (E), 135 deg. lead (F) and opposition 
 
256 
 
 ENGINEERING MATHEMATICS. 
 
 {G) . (The phase differences here are referred to the maximum 
 of the fundamental: with waves of different frequencies, the 
 phase differences naturahy change from point to point, and in 
 speaking of pliase difference, the reference point on the wave 
 
 IW Third Harmonic 
 
 Effect of Small Third Harmonic. 
 
 must thus be given. For instance, in C the third harmonic is 
 in phase with the fundamental at the maximum point of the 
 latter, but in opposition at its zero point.) 
 The equations of these waves are: 
 
 A: J/ =100 cos^ 
 C: ?y=100cos/? + 10cos3/? 
 E: 2/ = 100 cos /? + 10 cos (3/? + 90 deg.) 
 G: ?/ = 100 cos /? + 10 cos (3/? + 180 deg.) 
 = 100cos/?-10cos3/? 
 
EMPIRICAL CURVES. 
 
 C: 2/ =100 COS /? + 30 cos 3/? 
 D: 2/ =100 cos /? + 30 cos (3/? +45 deg.) 
 E: 2/ =100 cos /?+30 cos (3/3 + 90 deg.) 
 i^; 2/ =100 cos /? + 30 cos (3/? + 135 deg.) 
 G; 2/ = 100 cos ^ + 30 cos (3/? + 180 deg.) 
 =100 cos /3-30 cos 3/? 
 
 257 
 
 30 !< Third Harmonic 
 
 Fig. 86. Effect of Large Third Harmonic. 
 
 In all these waves, one cycle of the triple harmonic is given in 
 dotted lines, to indicate its relative position and intensity, and 
 the maxima of the harmonics are indicated by the arrows. 
 
258 
 
 ENGINEERING MATHEMATICS. 
 
 As seen, with the harmonic in jjhase or in opposition (C and 
 G), tlie waves are symmetrical; with the harmonic out of phase, 
 the waves are unsymmetrical, of the so-called "saw tooth" 
 type, and the saw tooth is on the rising side of the wave with a 
 lagging, on the decreasing side with a leading triple harmonic. 
 
 Third Harmonic Flat Zero & Reversal 
 
 Fig. 87. Flat Zero and Reversal by Third Harmonic. 
 
 The latter arc shown in D, E, F; the former have the same shape 
 but reversed, that is, rising and decreasing side of the wave 
 interchanged, and therefore are not shown. 
 
 The triple harmonic in phase with the fundamental, C, gives 
 a peaked wave with flat zero, and the peak and the flat zero 
 
EMPIRICAL CURVES. 
 
 259 
 
 become the more pronounced, the higher the third harmonic, 
 until finally the flat zero becomes a double reversal of volt- 
 age, as shown in Fig. 87d. 
 
 Fig. 87 shows the effect of a gradual increase of an in-phase 
 triple harmonic: a is the fundamental, b contains a 10 per 
 
 5^ Pifth Harmonic 
 
 Fig. 88. Effect of Small Fifth Harmonic. 
 
 cent, c a 38.5 per cent and d a, 50 per cent triple harmonic, as 
 given by the equations: 
 
 a: J/ =100 cos /? 
 b: 2/ = 100 cos/? + 10cos3^ 
 c: 2/=100cos/? + 38.5cos3/? 
 d: 2/ = 100cos/? + 50cos3/? 
 
260 
 
 ENGINEERING MATHEMATICS. 
 
 At c, the wave is entirely horizontal at the zero, that is, remains 
 zero for an appreciable time at the reversal. In this figure, the 
 three harmonics are showm separately in dotted lines, in their 
 relative intensities. 
 
 A triple harmonic in opposition to the fundamental (Tigs. 
 85 and 86G) is characterized by a flat top and steep zero, and 
 
 20^ Pifth Harmonic 
 
 Fig. 89. Effect of Large Fifth Harmonic. 
 
 with the increase of the third harmonic, the flat top develops 
 into a double peak (Fig. 86G), while steepness at the point of 
 reversal increases. 
 
 The simple saw tooth, produced by a triple harmonic in 
 quadratm-e with the fundamental is shown in Fig. 85E. With 
 increasing triple harmonic, the hump of the saw tooth becomes 
 
EMPIRICAL CURVES. 261 
 
 more pronounced and changes to a second and lower peak, as 
 shown in Fig. 86. This figure gives the variation of the saw- 
 tooth shape from 45 to 45 deg. phase difference : With the phase 
 of the third harmonic shifting from in-phase to 45 deg. lead, the 
 flat zero, by moving up on the wave, has formed a hump or saw 
 tooth low down on the decreasing (and with 45 deg. lag on the 
 increasing) side of the wave. At 90 deg. lead, the saw tooth has 
 moved up to the middle of the down branch of the wave, and 
 with 135 deg. lead, has moved still further up, forming practi- 
 cally a second, lower peak. With 180 deg. lead — or opposition 
 of phase — the hump, of the saw tooth has moved up to the 
 top, and formed the second peak — or the flat top, with a lower 
 third harmonic, as in Fig. 85G. 
 
 Figs. 88 and 89 give the effect of the fifth harmonic, super- 
 imposed on the fundamental, of 5 per cent in Fig. 88, and of 20 
 per cent in Fig. 89. Again A gives the fundamental sine wave, 
 C the effect of the fifth harmonic in opposition with the funda- 
 mental, E in quadratiu-e (lagging) and G in phase. One cycle 
 of the fifth harmonic is shown in dotted lines, and the maxima 
 of the harmonics indicated by the arrows. 
 
 The equations of these waves are given by: 
 
 A: 2/ =100 cos /9 
 
 C: 2/ =100 cos/? -5 cos 5/? 
 
 E: y=100 cos ^5-5 cos (5/3 -h 90 deg.) 
 
 G: 2/=100cos/?-t-5 cos 5^9 
 
 A: y=100 cos /? 
 
 C: 2/ =100 cos/? -20 cos 5,3 
 
 E: 2/ =100 cos /?-20 cos (5/?-h90 deg.) 
 
 G: 2/ =100 cos^-f-20cos5/? 
 
 In the distortion caused by the fifth harmonic (in opposi- 
 tion to the fundamental) flat top (Fig. 88C) or double peak (at 
 higher values of the harmonic. Fig. 89C), is accompanied by flat 
 zero (or, at very high values of the fifth harmonic, double rever- 
 sal at the zero, similar as in Fig. S7d), while in the distortion 
 by the third harmonic it is accompanied by sharp zero. 
 
 With the fifth harmonic in phase with the fundamental, a 
 peaked wave results with steep zero. Fig. 88^, and the transi- 
 
262 
 
 ENGINEERING MATHEMATICS. 
 
 tion from the steep zero to the peak, with larger values of the 
 fifth harmonic, then develops into two additional peaks, thus 
 giving a treble peaked wave. Fig. 88(7, with steep zero. The 
 beginning of treble peakedness is noticeable already in Fig. 
 88G, with only 5 per cent of fifth harmonic. 
 
 10^ Third Harmonic & 
 5 ^ Fifth Harmonic 
 
 Pig. 90. Third and Fifth Harmonic. 
 
 With the seventh harmonic, the treble-peaked wave would 
 be accompanied by flat zero, and a quadruple-peaked wave 
 would give steep zero (Fig. 95). 
 
 The fifth harmonic out of phase with the fundamental again 
 gives saw-tooth waves. Figs. 88 and 89£', but the saw tooth 
 
EMPIRICAL CURVES. 
 
 263 
 
 produced by the fifth harmonic contains two humps, that is, 
 is double, with one hump low down, and the other high up on 
 the curve, thereby giving the transition from the symmetrical 
 double peak C to the symmetrical treble peak G. 
 
 10 S« Third Harmonic & 
 5 ^ Fifth Harmonic 
 
 Fig. 91. Third and Fifth Harmonic. 
 
 i6o. Characteristic of the effect of the third harmonic 
 thus is: 
 
 Coincidence of peak with flat zero or double reversal, of steep 
 zero with flat top or double peak, and single hump or saw tooth, 
 
264 
 
 ENGINEERING MATHEMATICS. 
 
 While characteristic of the effect of the fifth harmonic is: 
 Coincidence of peak with steep zero, or treble peak, of flat 
 
 top or double peak with flat zero or double reversal, and double 
 
 saw-tooth. 
 
 10^ Third Harmonic & 
 5 ^ Fifth Harmonic 
 
 Fig. 92. Third and Fifth Harmonic. 
 
 By thus combining third and fifth harmonics of proper 
 values, they can be made to neutralize each other's effect in any 
 one of their characteristics, but then accentuate each other in 
 the other characteristic. 
 
 Thus peak and flat zero of the triple harmonic combined with 
 peak and steep zero of the fifth harmonic, gives a peaked wave 
 with normal sinusoidal appearance at the zero value; combin- 
 
EMPIRICAL CURVES. 265 
 
 ing the flat tops or double peaks of both harmonics, the flat 
 zero of the one neutrahzes the steep zero of the other, and we 
 get a flat top or double peak with normal zero. Or by com- 
 bining the peak of the third harmonic with the flat top of the 
 fifth we get a wave with normal top, but steep zero, and we get a 
 wave with normal top, but flat zero or double reversal, by com- 
 bining the triple harmonic peak with the fifth harmonic flat top. 
 
 Thus any of the characteristics can be produced separately 
 by the combination of the third and fifth harmonic. 
 
 By combining third and fifth harmonics out of phase with 
 fundamental — such as give single or double saw-tooth shapes, the 
 various other saw-tooth shapes are produced, and still further 
 saw-tooth shapes, by combining a symmetrical (in phase or in 
 opposition) third harmonic with an out of phase fifth, or 
 inversely. 
 
 These shapes produced by the superposition, under different 
 phase angles, of fifth and third harmonics on the fundamental, 
 and their gradual change into each other by the shifting in 
 phase of one of the harmonics, are shown in Figs. 90, 91 and 92 
 for a third harmonic of 10 per cent, and a fifth harmonic of 5 
 per cent of the fundamental. 
 
 In Fig. 90 the third harmonic is in phase, in Fig. 91 in quadra- 
 ture lagging, and in Fig. 92 in opposition with the fundamental. 
 A gives the fundamental, B the fundamental with the third har- 
 monic only, and C, D, E, F, the waves resulting from the super- 
 position of the fifth harmonic on the combination of funda- 
 mental and third harmonic, given as B. In C the fifth harmonic 
 is in opposition, in D in quadrature lagging, in E in phase, and 
 in F in quadrature leading. 
 
 We see here round tops with flat zero (Fig. 90C), nearly 
 triangular waves (Fig. 90i?), approximate half circles (Fig. 
 92£'), sine waves with a dent at the top (Fig. 92C), and vari- 
 ous different forms of saw tooth. 
 
 The equations of these waves are : 
 
 A: 1/ =100 cos/? 
 
 B: 2/=100cos/?-f-10cos3^ 
 
 C: 2/ =100 cos ^-1-10 cos 3/? -5 cos 5/? 
 
 D: 2/ =100 cos /? + 10 cos 3/? -5 cos (5/?-t-90 deg.) 
 
 E: 2/ =100 cos /? + 10 cos 3,-9 + 5 cos 5,5 
 
266 ENGINEERING MATHEMATICS. 
 
 A. 2/ =100 COS/? 
 
 B: 2/=100 cos ^-10 cos (;]:9 + !)() (Ic^.) 
 
 C: J/ =100 cos /?-10 cos (;>,? + 90 det;-.)-.") cos 5/? 
 
 D: y=100 cos ^-10 cos (3/? + 90 (lcg.)-5 cos (5/9 + 90 deg.) 
 
 E: 2/ =100 cos /9-10 cos (3/9 + 90 deg.) +5 cos 5/9 
 
 F: 2/ =100 cos /9-10 cos (3/9 + 90 deg.) + 5 cos (5/9 + 90 deg.) 
 
 A: 2/ =100 cos /9 
 
 B; 2/=100cos/9-10cos3/? 
 
 C- y=100 cos /9-10 cos 3/9-5 cos 5/9 
 
 £>.• 2/ = 100 cos /9-10 cos 3/9-5 cos (5/9 + 90 deg.) 
 
 E: 2/ =100 cos /9-10 cos 3-9 + 5 cos 5/9 
 
 Even Har.moxics. 
 
 i6i. Characteristic of the wave-shape distortioia of even har- 
 monics is that the wave is not a symmetrical wave, but the 
 two half waves have different shapes, and the characteristics 
 of the negative half wave are opposite to those of the positive. 
 This is to be expected, as an even harmonic, which is in phase 
 with the positive half wave of the fundamental, is in opposition 
 with the negative; when leading in the positive, it is lagging 
 in the negative, and inversely. 
 
 Fig. 93 shows the effect of a second harmonic of 30 per cent 
 of the fundamental A, superimposed in quadratui-e, 60 deg. 
 phase displacement, 30 deg. displacement and in phase, in 
 B, C, D and E respectively. 
 
 The equations of these waves are: 
 
 A: y =100 cos /9 and y' =30 cos (2-9-90) 
 B: 2/ =100 cos /9 + 30 cos (2/9-90) 
 C: 2/ =100 cos /9 + 30 cos (2/9-60) 
 D: 2/ =100 cos /9 + 30 cos (2./9-30) 
 E: 2/ =100 cos /9 + 30 cos 2;9 
 
 Quadrature combination (Fig. 935) gives a wave where the 
 rising side is flat, the decreasing side steep, and inversely with 
 the other half wave. C and D give a peaked wave for the one, a 
 saw tooth for the other half wave, and E, coincidence of phase 
 of fundamental and second harmonic, gives a combination of 
 one peaked half wave with one flat-top or double-peaked wave. 
 
EMPIRICAL CURVES. 
 
 267 
 
 Characteristic of C, D and E is, that the two half waves are 
 of unequal length. 
 
 In general, even harmonics, if of appreciable value are easily 
 recognized by the difference in shape, of the two half waves. 
 
 Fig. 93. Effect of Second Harmonic. 
 
 By the combination of the second harmonic with the third 
 harmonic (or the fifth), some of the features can be intensified, 
 others suppressed. 
 
 An illustration hereof is shown in Fig. 94 in the production 
 
268 
 
 ENGINEERING MATHEMATICS. 
 
 of a wave, in which the one half wave is a short high peak, the 
 other a long flat top, by the superposition of a second harmonic 
 of 4(3.5 per cent, and a third harmonic of 10 per cent both in 
 phase with the fundamental. 
 
 A gives the fundamental sine wave, B and C the second and 
 third harmonic, D the combination of fundamental and second 
 
 Second & Third Harmonic 
 
 FiQ. 94. Peak and Flat Top by Second and Third Harmonic. 
 
 harmonic, giving a double peaked negative half wave, and E the 
 addition of the third harmonic to the wave D. Thereby the 
 double peak of the negative half wave is flatted to a long flat 
 top, and the peak of the positive half wave intensified and 
 shortened, so that the positive maximum is about two and one- 
 
EMPIRICAL CURVES. 
 
 269 
 
 half times the negative maximum, and the negative half wave 
 nearly 75 per cent longer than the positive half wave. 
 The equations of these waves are given by: 
 
 A: 2/ =100 cos/? 
 
 B: y =46.5 cos 2/? 
 
 C: t/=10cos3/? 
 
 D: 2/ =100 cos /?+46.5 cos 2/? 
 
 E: 2/ =100 cos /?+46.5 cos 2/? + 10 cos 3/? 
 
 High Harmonics. 
 
 162. Comparing the effect of the fifth harmonic, Figs. 88 and 
 89, with that of the third harmonic, Figs. 85 and 86, it is seen 
 
 Fig. 95. Effect of Seventh Harmonic. 
 
 that a fifth harmonic, even if very small, is far easier distin- 
 guished, that is, merges less into the fundamental than the third 
 harmonic. Still more this is the case with the seventh har- 
 monic, as shown in Fig. 95 in phase and in opposition, of 10 per 
 cent intensity. This is to be expected: sine waves which do not 
 differ very much in frequency, such as the fundamental and 
 the second or third harmonic, merge into each other and form a 
 resultant shape, a distorted wave of characteristic appearance, 
 
270 
 
 ENGINEERING MATHEMATICS. 
 
 while sine waves of \n-y different frequencies, as the fundamen- 
 tal and its eleventh harmonic, in Fig. 96, when superimposed, 
 remain distinct from each other; the general shape of the wave 
 is the fundamental sine, and the high harmonics appear as rip- 
 ples upon the fundamental, thus giving what may be called a 
 corrugated sine wave. By counting the number of ripples per 
 
 Fig. 96. Wave in which Eleventh Harmonic Predominates. 
 
 compl(>t(^ wave, or i)cr half wave, the order of the harmonic 
 can then rapidly be determined. For instance, the wave shown 
 in Fig. 96 contains mainly the eleventh harmonic, as there are 
 eleven ripples per wave. The wave shown by the oscillogram 
 Fig. 97 shows the twenty-third harmonic, etc. 
 
 Fig. 97. C D 23510. Alternator Wave with Single High Harmonic. 
 
 \'er}' frequently high harmonics appear in pairs of nearly the 
 same freciuency and intensity, as an eleventh and a thirteenth 
 harmonic, etc. In this case, the ripples in the wave shape show 
 maxima, where the two hai-monics coincide, and nodes, where 
 the tw(j harmonics are in opposition. The presence of nodes 
 makes the counting of the number of ripples per complete wave 
 more difficult. A convenient method of procedure in this case 
 
EMPIRICAL CURVES. 271 
 
 IS, to measure the distance or space between the maxima of one 
 or a few ripples in the range where they are pronounced, and 
 count the number of nodes per cycle. For instance, in the 
 wave, Fig. 98, the space of two ripples is about 60 deg., and two 
 nodes exist per complete wave. 60 deg. for two ripples, give 
 
 Fig. 98. Wave in which Eleventh and Thirteenth Harmonics Predominate. 
 
 o 360 
 
 "fi(T "" 12 ripples per complete wave, as the average frequency 
 
 of the two existing harmonics, and since these harmonics differ 
 by 2 (since there are two nodes), their order is the eleventh and 
 the thirteenth harmonics. 
 
 163. This method of determining two similar harmonics, with 
 
 Fig. 99. C D 23512. Alternator Wave with Two Very Unequal High 
 
 Harmonics. 
 
 a little practice, becomes very convenient and useful, and may 
 frequently be used visually also, in determining the frequency 
 of hunting of synchronous machines, etc. In the phenomenon 
 of hunting, frequently two periods are superimposed, a forced 
 frequency, resulting from speed of generator, etc., and the 
 natural frequency of the machine. Counting the number of 
 impulses, /, per minute, and the number of nodes, n, gives the 
 
';>7'? 
 
 ENGINEERING MATHEMATICS. 
 
 two frequencies: /+- undf--; and as one of these frequencies 
 
 is the impressed engine frequency, this affords a check. 
 
 Where tlie two high harmonics of nearly equal order, as the 
 eleventh and the thirteenth in Fig. 98, are approximately equal 
 in intensity, at the nodes the ripples practically disappear, 
 and between the nodes the ripples give a frequency intermediate 
 between the two components: Apparently the twelfth harmonic 
 in Fig. 98. In this case the two constituents are easily deter- 
 mined: 12-1=11, and 12 + 1 =13. 
 
 A\Tiere of the two constituents one is greater than the other 
 the wave still shows nodes, but the ripples do not entirely disap, 
 pear at the nodes, but merely decrease, that is, the wave show- 
 a sine with ripples which increase and decrease along the waves 
 
 Fig. 100. CD 23511. 
 
 Alternator Wave with Two Nearly Equal High 
 Harmonics. 
 
 as shown by the oscillograms 99 and 100. In this case, one of 
 the two high frequencies is given by counting the total number 
 of ripples, but it may at first be in doubt, whether the other 
 component is higher or lower by the number of nodes. The 
 decision then is made by considering the length of the ripple at 
 the node : If the length is a maximum at the node, the secondary 
 harmonic is of higher frequency than the predominating one; 
 if the length of the ripple at the node is a minimum, the second- 
 ary frequency is lower than the predominating one. This is 
 illustrated in Fig. 101. In this figure, A and B represent the 
 tenth and twelfth harmonic of a wave, respectively; C gives 
 their superposition with the lower harmonic A predominating, 
 while B is only of half the intensity of A . D gives the superposi- 
 tion of A and B at equal intensity, and E gives the super- 
 position with the higher frequency B predominating. That is, 
 the respective equations would be : 
 
273 
 
 EMPIRICAL CURVES. 
 
 A: y=cos 10/? 
 
 B: y=cos 12/? 
 
 C: 2/=cos 10/? + 0.5 cos 12/? 
 
 Z>; y=cos 10^ + cos 12/? 
 
 £: 2/ =0.5 cos 10^ + cos 12;9 
 
 As seen, in C the half wave at the node is abnormally long, 
 showing the preponderance of the lower frequency, in E abnor- 
 
 Superposition of High Harmonics 
 
 Fig. 101. Superposition of Two High Harmonics of Various Intensities. 
 
 mally short, showing the preponderance of the higher frequency. 
 In alternating-current and voltage waves, the appearance of 
 two successive high harmonics is quite frequent. For instance, 
 if an alternating current generator contains n slots per pole, 
 this produces in the voltage wave the two harmonics of orders 
 
274 ENGINEERING MATHEMATICS. 
 
 2m -1 and 2/i + l. Such is the origm of the harmonics in the 
 oscillograms Figs. 99 and 100. 
 
 The nature of the increase and decrease of the ripples and the 
 formation of the nodes by the superposition of two adjacent 
 high harmonics is best seen by combining their expressions trig- 
 onometrically. 
 
 Thus the harmonics: 
 
 2/i =cos (2n — 1)/? 
 and 2/2 =cos (2n + l)/? 
 
 combined give the resultant : 
 '=2/1 + 2/2 
 
 2/ =2/1 + 2/2 
 =cos (2re-l)/? + cos (2?i + l)/3 
 =2 cos ^ cos 2?i/? 
 
 that is, give a wave of frequency 2n times the fundamental: 
 cos 2?i/?, but which is not constant, but varies in intensity 
 by the factor 2 cos /9. 
 
 Not infrequently wave-shape distortions are met, which are 
 not due to higher harmonics of the fundamental wave, but are 
 incommensurable therewith. In this case there are two entirely 
 unrelated frequencies. This, for instance, occurs in the second- 
 ary circuit of the single-phase induction motor; two sets of 
 currents, of the frequencies/, and (2/— /J exist (where /is the 
 primary frequency and/ the frequency of slip). Of this nature, 
 frequently, is the distortion produced by surges, oscillations, 
 arcing grounds, etc., in electric circuits; it is a combination of 
 the natural frequency of the cu'cuit with the impressed fre- 
 quency. Telephonic currents commonly show such multiple 
 frequencies, which are not harmonics of each other. 
 
CHAPTER VII. 
 NUMERICAL CALCULATIONS. 
 
 164. Engineering work leads to more or less extensive 
 numerical calculations, when applying the general theoretical 
 investigation to the specific cases which are under considera- 
 tion. Of importance in such engineering calculations are : 
 
 (a) The method of calculation. 
 
 (6) The degree of exactness required in the calculation. 
 
 (c) The intelligibility of the results. 
 
 {d) The reliabihty of the calculation. 
 
 a. Method of Calculation. 
 
 Before beginning a more extensive calculation, it is desirable 
 carefully to scrutinize and to investigate the method, to find 
 the simplest way, since frequently by a suitable method and 
 system of calculation the work can be reduced to a small frac- 
 tion of what it would otherwise be, and what appear to be 
 hopelessly complex calculations may thus be carried out 
 quickly and expeditiously by a proper arrangement of the 
 work. Indeed, the most important part of engineering work — and 
 also of other scientific work — is the determination of the method 
 of attacking the problem, whatever it may be, whether an 
 experimental investigation, or a theoretical calculation. It is 
 very rarely that important problems can be solved by a direct 
 attack, by brutally forcing a solution, and then only by wasting 
 a large amount of work unnecessarily. It is by the choice of 
 a suitable method of attack, that intricate problems are reduced 
 to simple phenomena, and then easily solved; frequently in 
 such cases requiring no solution at all, but being obvious when 
 looked at from the proper viewpoint. 
 
 Before attacking a more complicatecl problem experimentally 
 or theoretically, considerable time and study should thus first be 
 devoted to the determination of a suitable method of attack. 
 
 275 
 
276 ENGINEERING MATHEMATICS. 
 
 The next then, in cases where considerable numerical calcu- 
 lations are required, is the method of calculation. The most 
 convenient one usually is the arrangement in tabular form. 
 
 As example, consider the problem of calculating the regula- 
 tion of a 60,000-volt transmission hne, of r = 60 ohms resist- 
 ance, x = 135 ohms inductive reactance, and & = 0.0012 conden- 
 sive susceptance, for various values of non-inductive, inductive, 
 and condensive load. 
 
 Starting with the complete equations of the long-distance 
 transmission line, as given in " Theory and Calculation of 
 Transient Electric Phenomena and Oscillations," Section III, 
 paragraph 9, and considering that for every one of the various 
 power-factors, lag, and lead, a sufficient number of values 
 have to be calculated to give a curve, the amount of work 
 appears hopelessly large. 
 
 However, without loss of engineering exactness, the equa- 
 tion of the transmission line can be simplified by approxima- 
 tion, as discussed in Chapter V, paragraph 123, to the form. 
 
 ^i = ^„{l+^}+Z(o{l+^}; 
 
 (1) 
 
 where Eq, Iq are voltage and current, respectively at the step- 
 down end. El, h at the step-up end of the line; and 
 
 Z=r-t-jx=60 + 135/ is the total line impedance; 
 
 Y = g-{-jb= +0.0012] is the total shunted line admittance. 
 
 Herefrom follow the numerical values : 
 
 ZY ^ , (60+1.35])(+0.0012/) 
 "^ 2 ~^^ 2 
 
 = ] -f0.036y-0.0Sl =0.919+0.036/; 
 
 ZY 
 
 1 +-g- = 1 + 0.012/ - 0.027 = 0.973 + 0.012/; 
 
NUMERICAL CALCULATIONS. 
 
 277 
 
 ZY) 
 
 Z\ 1 +~ I =(60 + 135j)(0.973+0.012j-) 
 
 = 58.4+0.72j+131.1j-1.62 = 56.8+131.8y; 
 
 ZY 
 
 Y\ 1 +-g- J = (+0.0012j) (0.973 +0.012J) 
 
 = + 0.001168/-0.0000144 = (-0.0144+1.1687)10-3 
 
 hence, substituting in (1), the following equations may be 
 written : 
 
 El = (0.919 +0.036/)Eo + (56.8 + 131.8y)/o = A+B; 
 
 (2) 
 
 7i = (0.919+0.036j)/o - (0.0144 -1.168j)£'olO-3 = C-D. j 
 165. Now the work of calculating a series of numerical 
 values is continued in tabular form, as follows: 
 
 1. 100 PER CENT Power-factor. 
 
 ^0= 60 kv. at step-down end of line. 
 
 A = (0.919 +0.036;)Eo= 55.1 +2.2; kv. 
 
 Z)= (0.0144- 1.168;)'a 10- »=0.9 -70.1; amp. 
 
 
 
 a = ei+;ej 
 
 
 
 62 
 
 
 /g amp. 
 
 Bkv. 
 
 = A + B. 
 
 ei' + ei^=eK 
 
 e 
 
 — = tane. 
 
 4-e. 
 
 
 
 
 
 55.1+ 2.2; 
 
 3036+ 5 = 3041 
 
 55.1 
 
 + 0.040 
 
 + 2.3 
 
 20 
 
 1.1+ 2.6; 
 
 56.2+ 4.8; 
 
 3158+ 23 = 3181 
 
 56.4 
 
 + 0.085 
 
 + 4.9 
 
 40 
 
 2.3+ 5.3; 
 
 57.4+ 9.5; 
 
 3295+ 56 = 3351 
 
 57.9 
 
 + 0.131 
 
 + 7.6 
 
 60 
 
 3.4+ 7.9; 
 
 68.5 + 10.1; 
 
 3422+102 = 3524 
 
 59.4 
 
 + 0.173 
 
 + 9.9 
 
 80 
 
 4.5+10.5; 
 
 59.6 + 12.7; 
 
 3552 + 161=3713 
 
 60.9 
 
 + 0.213 
 
 +12.0 
 
 100 
 
 5.7+13.2; 
 
 60.8 + 15.4; 
 
 3697 + 237 = 3934 
 
 62.7 
 
 + 0.253 
 
 + 14.2 
 
 120 
 
 6.8 + 15.8; 
 
 61.9+18.0; 
 
 3832 + 324=4156 
 
 64.5 
 
 + 0.291 
 
 +16.3 
 
 h 
 amp. 
 
 C amp. 
 
 Ii = u+;w 
 = C-D 
 
 ii' + M' = i' 
 
 I 
 
 12 
 
 — =tant 
 
 A-i 
 
 4.i- 
 2^e, i 
 
 Power- 
 factor 
 
 
 
 20 
 40 
 60 
 80 
 100 
 120 
 
 
 
 18.4+0.7; 
 
 36.8 + 1.4; 
 55.1 + 2.2; 
 73.5 + 2.9; 
 
 91.9 + 3.6; 
 110.3 + 4.3; 
 
 -0.7-90.1; 
 
 17.5+70.8; 
 35.9+71.5; 
 54.2+72.3; 
 72.6+73.0; 
 91.0+73.9; 
 109.4+74.4; 
 
 4914 + 1 = 4915 
 
 5013+ 306= 5319 
 5112+1289= 6401 
 5227 + 2938= 8165 
 5329 + 5271=10600 
 8281 + 5432=13713 
 11969 + 5535=17504 
 
 70.1 
 
 72.9 
 
 80.0 
 
 90.4 
 
 103.0 
 
 117.1 
 
 132.3 
 
 -78 
 
 + 4.04 
 + 1.99 
 + 1.33 
 + 1.066 
 +0.811 
 +0.680 
 
 -89.1 
 + 90.9 
 +78.3 
 +63.4 
 +53.1 
 +45.2 
 + 39.1 
 4 34.1 
 
 + 88.6 
 
 + 71.4 
 + 55.9 
 +43.2 
 + 33.2 
 + 24.9 
 + 17.8 
 
 0.024 
 
 0.332 
 0.558 
 0.728 
 0.837 
 0,907 
 952 
 lead 
 
278 
 
 ENGINEERING MATHEMATICS. 
 
 
 ei=60 kv. 
 
 at step-up end of line. 
 
 
 h 
 
 amp. 
 
 Red. Factor, 
 
 
 
 
 
 
 e 
 60 
 
 amp. 
 
 
 kv. 
 
 ti 
 amp. 
 
 Power-Factor. 
 
 
 
 0.918 
 
 
 
 
 65.5 
 
 76,4 
 
 0.024 
 
 20 
 
 0.940 
 
 21.3 
 
 
 63.8 
 
 77.5 
 
 0.332 
 
 40 
 
 0.965 
 
 41 4 
 
 
 62.1 
 
 82.9 
 
 0.558 
 
 60 
 
 0.990 
 
 60.6 
 
 
 60.6 
 
 91.4 
 
 0.728 
 
 80 
 
 1.015 
 
 78 8 
 
 
 59.1 
 
 101.5 
 
 0.837 
 
 100 
 
 1.045 
 
 95.7 
 
 
 67.5 
 
 112.3 
 
 0.907 
 
 120 
 
 1.075 
 
 111.7 
 
 
 55.8 
 
 122.8 
 
 0.952 
 lead 
 
 
 Curves of t(,, ^q, 
 
 4, cos 0, plotted in Fig, 86, 
 
 
 2. 90 Per Cent Power-Factor, Lag. 
 
 cos ^ = 0.9; sin 5 = %/!- 0.92 = 0.436; 
 /o = to(cos ^-j'sin ^) ={0(0.9 -0.436j"); 
 
 £"1 = (0.919+ 0.036j>o + (56.8 + 131.8/) (0.9 -0.436j)to 
 = (0.919+ 0,036j)eo + (108.5 + 93.8j>o = A+B'\ 
 
 7i = (0.919+ 0.036/) (0.9 -0.436j>o- (0.0144 -1.168/)eolO-3 
 = (0.843 -0.366j>-o- (0.0144 -1.168y)eolO-3 = C'-Z), 
 
 and now the table is calculated in the same manner as under 1. 
 Then corresponding tables are calculated, in the same 
 manner, for power-factor, =0.8 and =0.7, respectively, lag, 
 and for power-factor =0.9, 0.8, 0.7, lead; that is, for 
 
 cos (9+jsin ^ = 0.8-0.6/; 
 0.7-0.714/; 
 0.9+0.436/; 
 0.8+0.6/; 
 0.7+0.714/. 
 
 Then curves are plotted for all seven values of power-factor, 
 from 0.7 lag to 0.7 lead. 
 
 From these curves, for a number of values of io, for instance, 
 to = 20, 40, 60, 80, 100, numerical values of ii, eo, cos d, are 
 
N UMERICAL CALC ULA TIONS. 
 
 279 
 
 taken, and plotted as curves, which, for the same voltage 
 ei = 60 at the step-up end, give ii, eo, and cos d, for the same 
 value io, that is, give the regulation of the line at constant 
 current output for varying power-factor. 
 
 b. Accuracy of Calculation. 
 
 166. Not all engineering calculations require the same 
 degree of accuracy. When calculating the efficiency of a large 
 alternator it may be of importance to determine whether it is 
 97.7 or 97.8 per cent, that is, an accuracy within one-tenth 
 per cent may be required; in other cases, as for instance, 
 when estimating the voltage which may be produced in an 
 electric circuit by a line disturbance, it may be sufficient to 
 
 — 
 
 
 
 
 
 
 
 
 
 
 
 kv. 
 
 
 
 ^- 
 
 -^ 
 
 
 
 
 
 
 > 
 
 65 
 
 
 
 «i 
 
 
 
 
 
 
 
 >■ 
 
 X 
 
 "•^.ff 
 
 
 
 
 
 
 
 
 jK' 
 
 y 
 
 ■^ 
 
 
 — 55 
 
 
 
 
 
 
 
 ^ 
 
 y 
 
 
 ..^ 
 
 ^ 
 
 " 
 
 
 
 
 
 ii^ 
 
 ^ 
 
 
 ^ 
 
 
 .-^ 
 
 
 
 
 
 
 
 
 
 ft-^ 
 
 X 
 
 
 
 
 
 
 
 
 
 ) 
 
 Ao 
 
 
 
 
 
 
 
 
 
 
 / 
 
 / 
 
 
 
 
 
 
 
 
 
 
 / 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 2 
 
 D 
 
 i 
 
 3 
 
 ti 
 
 J 
 
 8 
 
 3 
 
 K 
 
 K) 
 
 4< 
 
 s 
 
 ■< 
 
 120 
 
 1001.00 
 
 80 0.80 
 
 0.10 
 
 0.20 
 
 Fig. 102. Transmission Line Characteristics. 
 
 determine whether this voltage would be limited to double 
 the normal circuit voltage, or whether it might be 5 or 10 
 times the normal voltage. 
 
 In general, according to the degree of accuracy, engineering 
 calculations may be roughly divided into three classes : 
 
280 ENGINEERINO MATHEMATICS. 
 
 (a) Estimation of the magnitude of an effect; that is, 
 determining approximate numerical values within 25, 50, or 
 100 per cent. Very frequently such very rough approximation 
 is sufficient, and is all that can be expected or calculated. 
 For instance, when investigating the short-circuit current of an 
 electric generating system, it is of importance to know whether 
 this current is 3 or 4 times normal current, or whether it is 
 40 to 50 times normal current, but it is immaterial whether 
 it is 45 to 46 or 50 times normal. In studying lightning 
 phenomena, and, in general, abnormal voltages in electric 
 systems, calculating the discharge capacity of lightning arres- 
 ters, etc., the magnitude of the quantity is often suflBcient. In 
 calculating the critical speed of turbine alternators, or the 
 natural period of oscillation of s3Tichronous machines, the 
 same applies, since it is of importance only to see that these 
 speeds are sufficiently remote from the normal operating speed 
 to give no trouble in operation. 
 
 (b) Approximate calculation, requiring an accuracy of one 
 or a few per cent only; a large part of engineering calcu- 
 lations fall in this class, especially calculations in the realm of 
 design. Although, frequently, a higher accuracy could be 
 reached in the calculation proper, it would be of no value, 
 since the data on which the calculations are based are sus- 
 ceptible to variations beyond control, due to variation in the 
 material, in the mechanical dimensions, etc. 
 
 Thus, for instance, the exciting current of induction motors 
 may vary by several per cent, due to variations of the length 
 of air gap, so small as to be beyond the limits of constructive 
 accuracj', and a calculation exact to a fraction of one per cent, 
 while theoretically possible, thus would be practically useless. 
 The calculation of the ampere-turns required for the shunt 
 field excitation, or for the series field of a direct-current 
 generator needs only moderate exactness, as variations in the 
 magnetic material, in the speed regulation of the driving 
 power, etc., produce differences amounting to several per 
 cent. 
 
 (c) Exact engineering calculations, as, for instance, the 
 calculations of the efficiency of apparatus, the regulation of 
 transformers, the characteristic curves of induction motors, 
 etc. These are determined with an accuracy frequently amount- 
 ing to one-tenth of one per cent and even greater. 
 
NUMERICAL CALCULATIONS. 281 
 
 Even for most exact engineering calculations, the accuracy 
 of the slide rule is usually sufficient, if intelligently used, that 
 is, used so as to get the greatest accuracy. For accurate calcu- 
 lations, preferably the glass slide should not be used, but the 
 result interpolated by the eye. 
 
 _ Thereby an accuracy within \ per cent can easily be main- 
 tained. 
 
 For most engineering calculations, logarithmic tables are 
 sufficient for three decimals, if intelUgently used, and as such 
 tables can be contained on a single page, their use makes the 
 calculation very much more expeditious than tables of more 
 decimals. The same apphes to trigonometric tables: tables 
 of the trigonometric functions (not their logarithms) of three 
 decimals I find most convenient for most cases, given from 
 degree to degree, and using decimal fractions of the degrees 
 (not minutes and seconds).* 
 
 Expedition in engineering calculations thus requires the use 
 of tools of no higher accuracy than required in the result, and 
 such are the slide rules, and the three decimal logarithmic and 
 trigonometric tables. The use of these, however, make it 
 neccessary to guard in the calculation against a loss of accuracy. 
 
 Such loss of accuracy occurs in subtracting or dividing two 
 terms which are nearly equal, in some logarithmic operations, 
 solution of equation, etc,, and in such cases either a higher 
 accuracy of calculation must be employed — seven decimal 
 logarithmic tables, etc. — or the operation, which lowers the 
 accuracy, avoided. The latter can usually be done. For 
 instance, in dividing 297 by 283 by the slide rule, the proper 
 way is to divide 297-283 = 14 by 283, and add the result 
 to 1. 
 
 It is in the methods of calculation that experience and judg- 
 ment and skill in efficiency of arrangement of numerical calcu- 
 lations is most marked. 
 
 167. While the calculations are unsatisfactory, if not carried 
 out with the degree of exactness which is feasible and desirable, 
 it is equally wrong to give numerical values with a number of 
 
 * This obviously does not apply to some classes of engineering work, in 
 which a much higher accuracy of trigonometric functions is required, aa 
 trigonometric surveying, etc. 
 
282 ENGINEERING MATHEMATICS. 
 
 ciphers greater than the method or the purpose of the calcula- 
 tion warrants. For instance, if in the design of a direct-current 
 generator, the calculated field ampere-turns are given as 9738, 
 such a numerical value destroys the confidence in the work of 
 the <!alculator or designer, as it implies an accuracy greater 
 than possible, and thereby shows a lack of judgment. 
 
 The number of ciphers in which the result of calculation is 
 given should signify the exactness, In this respect two 
 systems are in use: 
 
 (a) Numerical values are given with one more decimal 
 than warranted by the probable error of the result; that is, 
 the decimal before the last is correct, but the last decimal may 
 be wrong by several units. This method is usually employed 
 in astronomy, physics, etc. 
 
 (b) Numerical values are given with as many decimals as 
 the accuracy of the calculation warrants; that is, the last 
 decimal is probably correct within half a unit. For instance, 
 an efficiency of 86 per cent means an efficiency between 85.5 
 and 86.5 per cent; an efficiency of 97.3 per cent means an 
 efficiency between 97.25 and 97.35 per cent, etc. This system 
 is generally used in engineering calculations. To get accuracy 
 of the last decimal of the result, the calculations then must 
 be carried out for one more decimal than given in the result. 
 For instance, when calculating the efficiency by adding the 
 various percentages of losses, data like the following may be 
 given : 
 
 Core loss 2.73 per cent 
 
 i^r 1.06 
 
 Friction 0.93 
 
 Total 4.72 
 
 Efficiency 100-4.72 = 95.38 
 
 Approximately 95.4 " 
 
 It is obvious that throughout the same calculation the 
 same degree of accuracy must be observed. 
 It follows herefrom that the values : 
 
 2i; 2.5; 2.50; 2.500, 
 
NUMERICAL CALCULATIONS. 283 
 
 while mathematically equal, are not equal in their meaning as 
 an engineering result : 
 
 2.5 means between 2.45 and 2.55; 
 2.50 means between 2.495 and 2.505; 
 2.500 means between 2.4995 and 2.5005; 
 
 while 2^ gives no clue to the accuracy of the value. 
 
 Thus it is not permissible to add zeros, or drop zeros at 
 the end of numerical values, nor is it permissible, for instance, 
 to replace fractions as 1/16 by 0.0625, without changing the 
 meaning of the numerical value, as regards its accuracy. 
 This is not always realized, and especially in the reduction of 
 common fractions to decimals an unjustified laxness exists 
 which impairs the reliability of the results. For instance, if 
 in an arc lamp the arc length, for which the mechanism is 
 adjusted, is stated to be 0.8125 inch, such a statement is 
 ridiculous, as no arc lamp mechanism can control for one-tenth 
 as great an accuracy as implied in this numerical value: the 
 value is an unjustified translation from 13/16 inch. 
 
 The principle thus should be adhered to, that all calcula- 
 tions are carried out for one decimal more than the exactness 
 required or feasible, and in the result the last decimal dropped ; 
 that is, the result given so that the last decimal is probably 
 correct within half a unit. 
 
 c. Intelligibility of Engineering Data and Engineering Reports. 
 
 1 68. In engineering calculations the value of the results 
 mainly depends on the information derived from them, that is, 
 on their intelligibility. To make the numerical results and 
 their meaning as inteUigible as possible, it thus is desirable, 
 whenever a series of values are calculated, to carefully arrange 
 them in tables and plot them in a curve or in curves. The 
 latter is necessary, since for most engineers the plotted curve 
 gives a much better conception of the shape and the variation 
 of a quantity than numerical tables. 
 
 Even where only a single point is required, as the core 
 loss at full load, or the excitation of an electric generator at 
 rated voltage, it is generally preferable to calculate a few 
 
284 
 
 ENGINEERING MATHEMATICS. 
 
 
 
 
 
 
 
 
 
 
 V 
 
 jIts 
 
 
 . 
 
 
 — — 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 "DOU 
 
 
 
 
 
 
 
 
 
 
 
 -iAfl 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 "oUU 
 
 
 
 
 
 
 
 
 
 
 
 jtVVl 
 
 
 
 
 
 
 
 
 
 
 
 AN 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 100 
 
 
 
 
 2 
 
 
 
 4 
 
 
 
 6 
 
 
 
 8 
 
 1 
 
 
 
 Fig. 103. Compounding Curve. 
 
 points near the desired value, so as to get at least a short piece 
 of curve including the desired point. 
 
 The main advantage, and foremost purpose of curve plotting 
 thus is to show the shape of the function, and thereby give 
 a clearer conception of it ; 
 but for recording numerical 
 values, and deriving numer- 
 ical values from it, the plotted 
 curve is inferior to the table, 
 due to the limited accuracy 
 possible in a plotted curve, 
 and the further inaccuracy 
 resulting when drawing a 
 curve through the plotted cal- 
 culated points. To some 
 extent, the numerical values 
 as taken from a plotted curve, 
 depend on the particular 
 kind of curve rule used in 
 plotting the curve. 
 
 In general, curves are used for two different purposes, and 
 on the purpose for which the curve is plotted, should depend 
 the method of plotting, as the scale, the zero values, etc. 
 
 TVTien curves are used to 
 illustrate the shape of the 
 function, so as to show how 
 much and in what manner a 
 quantity varies as function 
 of another, large divisions of 
 inconspicuous cross-section- 
 ing are desirable, but it is 
 essential that the cross- 
 sectioning should extend to 
 the zero values of the func- 
 tion, even if the numerical 
 values do not extend so 
 far, since otherwise a wrong 
 impression would be con- 
 ferred. As illustrations are plotted in Figs. 103 and 104, the 
 compounding curve of a direct-current generator. The arrange- 
 
 
 
 
 
 
 
 
 
 
 V 
 
 3ltS 
 
 
 
 
 
 
 
 
 -— 
 
 
 
 "OOU 
 
 
 
 
 /' 
 
 ^ 
 
 
 
 
 
 
 
 
 
 ^ 
 
 y 
 
 
 
 
 
 
 
 ■DdU 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 /■ 
 
 
 
 
 
 
 
 
 
 
 ■510 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ■490 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ■ii'O 
 
 
 
 
 2 
 
 
 
 i 
 
 
 
 6 
 
 
 
 8 
 
 1 
 
 
 
 Fig. 104. Compounding Curve. 
 
NUMERICAL CALCULATIONS. 
 
 285 
 
 ment in Fig. 103 is correct; it shows the relative variation 
 of voltage as function of the load. Fig. 104, in which the 
 cross-sectioning does not begin at the scale zero, confers the 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 _^ 
 
 
 
 
 
 
 
 — 
 
 
 
 
 
 
 
 
 
 
 
 
 
 /^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 16 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 , 
 
 12 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 8 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ± 
 
 
 
 
 
 
 
 
 cj- 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 2 
 
 ) 
 
 3 
 
 f * 
 
 ' 
 
 50 
 
 60 
 
 7 
 
 ) 
 
 
 Fig. 105. Curve Plotted to show Characteristic Shape. 
 
 Fig. 106. Curve Plotted for Use as Design Data. 
 
 wrong impression that the variation of voltage is far greater 
 than it really is. 
 
 When curves are used to record numerical values and 
 derive them from the curve, as, for instance, is commonly the 
 
286 
 
 ENGINEERING MATHEMATICS. 
 
 case with magnetization curves, it is unnecessary to have the zero 
 of the function coincide with the zero of the cross-sectioning, but 
 rather preferable not to have it so, if thereby a better scale of 
 the curve can be secured. It is desirable, however, to use suffi- 
 ciently small cross-sectioning to make it possible to take numer- 
 ical values from the curve with good accuracy. This is illus- 
 trated by Figs. 105 and 106. Both show the magnetic charac- 
 teristic of soft steel, for the range above 5 =8000, in which it is 
 usually employed. Fig. 105 shows the proper way of plotting for 
 showing the shape of the function, Fig. 106 the proper way of 
 plotting for use of the curve to derive numerical values therefrom. 
 
 K 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \\ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 \ 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 \ 
 
 ^, 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 \ 
 
 II 
 
 \ 
 
 \, 
 
 
 
 
 
 
 
 
 
 
 
 
 
 X 
 
 n\ 
 
 
 s 
 
 K 
 
 
 
 
 
 
 
 
 
 
 
 
 
 V 
 
 N 
 
 ^ 
 
 
 \ 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 X 
 
 ::^ 
 
 ^ 
 
 HI 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 ^ 
 
 
 
 
 
 " 
 
 — 
 
 ■ — 
 
 , 
 
 
 
 
 Fig. 107. Same Function Plotted to Different Scales; I is correct. 
 
 169. Curves should be plotted in such a manner as to show 
 the quantity which they represent, and its variation, as well as 
 possible. Two features are desirable herefor: 
 
 1. To use such a scale that the average slope of the curve, 
 or at least of the more important part of it, does not differ 
 much from 45 deg. Hereby variations of curvature are best 
 shown. To illustrate this, the exponential function 2/=e~^ is 
 plotted in three different scales, as curves I, II, III, in Fig. 107. 
 Curve I has the proper scale. 
 
 2. To use such a scale, that the total range of ordinates is 
 not much different from the total range of abscissas. Thus 
 when plotting the power-factor of an induction motor, in 
 Fig. 108, curve I is preferable to curves II or III. 
 
NUMERICAL CALCULATIONS. 
 
 287 
 
 These two requirements frequently are at variance with 
 each other, and then a compromise has to be made between 
 them, that is, such a scale chosen that the total ranges of the 
 two coordinates do not differ much, and at the same time 
 the average slope of the curve is not far from 45 deg. This 
 usually leads to a somewhat rectangular area covered by the 
 curve, as shown, for instance, by curve I, in Fig. 107. 
 
 It is obvious that, where the inherent nature of the curve 
 is incompatible with 45 degree slope, this rule does not apply. 
 Such for instance is the case with instrument calibration curves, 
 which inherently are essentially horizontal lines, with curves 
 like the slip of induction motors, etc. 
 
 As regards to th? magnitude of the scale of plotting, the larger 
 the scale, the plainer obviously is the curve. It must be kept in 
 mind, however, that it would be wrong to use a scale which is 
 materially larger than the accuracy of the values plotted. 
 
 Thus for instance, in 
 plotting the calibration 
 curve of an instrument, 
 if the accuracy of the 
 calibration is not greater 
 than .05 per cent, it 
 would be wrong to use 
 .01 per cent as the unit 
 of ordinate scale. 
 
 In curve plotting, a 
 scale should be used 
 which is easily read. 
 Hence, only full scale, 
 double scale, and half 
 scale should be used. 
 Triple scale and one- 
 third scale are practi- 
 cally unreadable, and should therefore never be used. Quadruple 
 scale and quarter scale are difficult to read and therefore unde- 
 sirable, and are generally unnecessary, since quadruple scale is 
 not much different from half scale with a ten times smaller unit, 
 and quarter scale not much different from double scale of a ten 
 times larger unit. 
 
 170. In plotting a curve to show a relation y=f{x), in gen- 
 eral X and y should be plotted directly, on ordinary coordinate 
 
 
 
 
 
 
 
 
 
 
 
 
 
 y 
 
 /' 
 
 
 
 -^ 
 
 1 
 
 
 ■s~ 
 
 
 / 
 
 / 
 
 / 
 
 y 
 
 
 
 
 
 
 
 / 
 
 A 
 
 / 
 
 
 
 
 
 
 
 
 / 
 
 / 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 III 
 
 — 
 
 
 
 ^ 
 
 
 
 
 f 
 
 ^ 
 
 ^ 
 
 ^ 
 
 
 
 
 
 
 
 / 
 
 k 
 
 
 
 
 
 
 
 
 V 
 
 r 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 Fig. 108. Same Function Plotted to Dif- 
 ferent Scales; I is Correct. 
 
288 ENGINEERING MATHEMATICS. 
 
 paper, but not log x, or y^, or logarithmic paper used, etc., as 
 this would not show the shape of the relation y=f{x). Using 
 for instance semi-logarithmic paper, that is, with logarith- 
 mic abscissae and ordinary ordinates, the plotted curve would 
 show the shape of the relation y=f (log x), etc. The use of 
 logarithmic paper, or the use of y'^, or -y/a; as coordinate, etc., is 
 justified only where the purpose is to show the relation between 
 y and log x, or between y^ and x, or between y and ^/x, etc., as is 
 the case when investigating the equation of an empir cal curve, 
 or when intending to show some particular feature of the relation 
 y=f{x). Thus for instance when plotting the power p consumed 
 by corona in a high potential transmission hne, as function of the 
 line voltage e, by using y/p as ordinate, a straight line results. 
 Also where some particular function of one of the coordinates, 
 as log X, gives a more rational relation, it may be used instead 
 of X. Thus for instance in radiation curves, or when expressing 
 velocity as function of wave length or frequency, or expressing 
 attenuation of a wireless wave, etc., the log of wave length or 
 frequency, that is, the geometric scale (as used in the theory of 
 sound, with the octave as unit) is more rational and therefore 
 preferable. 
 
 Sometimes the values of a relationship extend over such a 
 wide range as to make it impossible to represent all of them in 
 one curve, and then a number of curves may have to be used, 
 with different scales. In such cases, the logarithmic scale often 
 brings all values within one curve without improperly crowding, 
 and especially where the purpose of curve plotting is not so 
 much to show the shape of the relation, as to record for the pur- 
 pose of taking numerical values from the curve, the latter ar- 
 rangement, that is, the use of logarithmic or semi-logarithmic 
 paper may be desirable. Thus the magnetic characteristic of 
 iron is used over a range of field intensities from very few am- 
 pere turns per cm. in transformers, to thousands of ampere 
 turns, in tooth densities of railway motors, and the magnetic 
 characteristic thus is either represented by three curves with 
 different scales, of ratios 1-HlO-^lOO, as shown in Fig. 109, or 
 the log of field intensity used as abscissae, that is, semi-logarith- 
 mic paper, with logarithmic scale as abscissae, and regular scale 
 as ordinates, as shown in Fig. 110. 
 
 It must be realized that the logarithmic or geometrical scale 
 
NUMERICAL CALCULATIONS. 
 
 289 
 
 B 
 
 21 
 
 4o » 
 
 )0 41 
 
 9 
 
 M 9 
 
 1 
 
 po a 
 
 » 900 IC 
 
 1 
 
 X) 1 
 
 00 12 
 
 » MOOMWl 
 
 IGOO 1700 IfOO 1900 2000 2100 
 
 21001 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 axxx 
 
 
 
 ^^ 
 
 ^ 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1900( 
 
 
 / 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 18001 
 
 / 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 . 
 
 -- 
 
 — 
 
 ^^ 
 
 1-2/ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 — 
 
 —' 
 
 
 
 
 
 
 
 leJ 
 
 
 
 
 
 
 
 
 
 ■ 
 
 
 
 
 
 
 
 
 
 
 
 
 imx) 
 
 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 i-ttm 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 — 
 
 -= 
 
 
 13000/ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ■ 
 
 
 
 
 
 ^ 
 
 J 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 JIOOC 
 
 
 
 
 / 
 
 ^ 
 
 
 
 
 
 M 
 
 agn. 
 
 ;tic 
 
 Characteristic 
 
 of 
 
 
 
 
 
 looa 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 S: 
 
 lico 
 
 n Sieel 
 
 
 
 
 
 
 
 9000 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 8000 
 
 2 
 
 /.. 
 
 1 4 
 
 -) 5 
 
 ) 6 
 
 ) 7 
 
 ) a 
 
 } ! 
 
 3 1 
 
 r 
 
 l: 
 
 i; 
 
 1 
 
 1; 
 
 a 160 no ISO i90 200 210 
 
 
 f- 1 i 
 
 ^. \ \ W i li il ,1 , 
 
 1 1 1 1 1 1 1 
 L 15 16 17 18 19 20 21 
 
 FiQ. 109. 
 
 B 
 
 23000 
 
 2200(1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 M;u 
 
 net 
 
 i(! Oha 
 
 rac 
 
 eristic 
 
 of 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 r-' 
 
 .' 
 
 2100a 
 
 
 
 1 Silicon Ste 
 ' Semilogarithmic 
 
 Scale) 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 -^ 
 
 
 
 20000, 
 13000. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 ^ 
 
 
 
 
 
 
 
 
 ISOOO- 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 
 
 
 
 
 
 
 
 UOOO- 
 
 
 
 - 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 / 
 
 
 
 
 
 
 
 
 
 
 IGOOQ- 
 
 
 
 
 
 
 
 
 
 
 
 
 
 X 
 
 ,/ 
 
 
 
 
 
 
 
 
 
 
 
 
 15000- 
 
 
 
 
 
 
 
 
 
 
 
 -^ 
 
 
 " 
 
 >' 
 
 
 
 
 
 
 
 
 
 
 
 
 
 14000- 
 J3000- 
 12000 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 11000. 
 10000. 
 900ft/ 
 
 
 ? 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 ■ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 yftOOO 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 *> •■ 
 
 
 5 
 
 f 
 
 ' 
 
 8 
 
 10 1 
 
 2 
 
 10 
 
 20 3 
 
 ) 4 
 
 5 
 
 D6 
 
 ) 
 
 sp. 
 
 iio§ 
 
 ^ 
 
 1 
 
 1 ? 
 
 ^ ^ 
 
 y 
 
 3§ 
 
 i 
 
 |g 
 
 s S 
 
 to 
 
 Fig. 110. 
 
290 ENGINEERING MATHEMATICS. 
 
 — in which equal divisions represent not equal values of the 
 quantity, but equal fractions of the quantity — is somewhat 
 less easy to read than common scale. However, as it is the same 
 scale as the slide rule, this is not a serious objection. 
 
 A disadvantage of the logarithmic scale is that it cannot 
 extend down to zero, and relations in which the entire range 
 down to zero requires consideration, thus are not well suited 
 for the use of logarithmic scale. 
 
 171. Any engineering calculation on which it is worth 
 while to devote any time, is worth being recorded with suffi- 
 cient completeness to be generally intelligible. Very often in 
 making calculations the data on which the calculation is based, 
 the subject and the purpose of the calculation are given incom- 
 pletely or not at all, since they are familiar to the calculator at 
 the time of calculation. The calculation thus would be unin- 
 telligible to any other engineer, and usually becomes unintelli- 
 gible even to the calculator in a few weeks. 
 
 In addition to the name and the date, all calculations should 
 be accompanied by a complete record of the object and purpose 
 of the calculation, the apparatus, the assumptions made, the 
 data used, reference to other calculations or data employed, 
 etc., in short, they should include all the information required 
 to make the calculation intelligible to another engineer without 
 further information besides that contained in the calculations, 
 or in the references given therein. The small amount of time 
 and work required to do this is negligible compared with the 
 increased utility of the calculation. 
 
 Tables and curves belonging to the calculation should in 
 the same way be completely identified with it and contain 
 sufficient data to be intelligible. 
 
 1 71 A. Engineering investigations evidently are of no value, 
 unless they can be communicated to those to whom they are of 
 interest. Thus the engineering report is an essential and im- 
 portant part of the work. If therefore occasionally an engineer 
 or scientist is met, who is so much interested in the investigating 
 work, that he hates to "waste" the time of making proper and 
 complete reports, this is a very foohsh attitude, since in general 
 it destroys the value of the work. 
 
 As practically every engineering investigation is of interest 
 and importance to different classes of people, as a rule not one, 
 
NUMERICAL CALCULATIONS. 291 
 
 but several reports must be written to make the most use of 
 the work; the scientific record of the research would be of no 
 more value to the financial interests considering the industrial 
 development of the work than the report to the financial or 
 administrative body would be of value to the scientist, who 
 considers repeating and continuing the investigation. 
 
 In general thus three classes of engineering reports can be 
 distinguished, and all three reports should be made with every 
 engineering investigation, to get best use of it. 
 
 (a) The scientific record of the investigation. This must be 
 so complete as to enable another investigator to completely check 
 up, repeat and further extend the investigation. It thus must 
 contain the original observations, the method of work, apparatus 
 and facilities, calibrations, information on the limits of accuracy 
 and reliabihty, sources of error, methods of calculation, etc., etc. 
 
 It thus is a lengthy report, and as such will be read by very few, - 
 if any, except other competent investigators, but is necessary 
 as the record of the work, since without such report, the work 
 would be lost, as the conclusions and results could not be checked 
 up if required. 
 
 This report appeals only to men of the same character as the 
 one who made the investigation, and is essentially for record 
 and file. 
 
 {h) The general engineering report. It should be very much 
 shorter than the scientific report, should be essentially of the 
 nature of a syllabus thereof, avoid as much as possible complex 
 mathematical and theoretical considerations, but give all the 
 engineering results of the investigation, in as plain language as 
 possible. It would be addressed to administrative engineers, 
 that is, men who as engineers are capable of understanding the 
 engineering results and discussion, but have neither time nor 
 famiharity to follow in detail through the investigation, and are 
 not interested in such things as the original readings, the discus- 
 sion of methods, accuracy, etc., but are interested only in the 
 results. 
 
 This is the report which would be read by most of the men 
 interested in the matter. It would in general be the form in 
 which the investigation is communicated to engineering societies 
 as paper, with the scientific report relegated into an appendix of 
 the paper. 
 
292 ENGINEERING MATHEMATICS. 
 
 (c) The general report. This should give the results, that is, 
 explain what the matter is about, in plain and practically non- 
 technical language, addressed to laymen, that is, non-engineers. 
 In other words, it should be understood by any intelligent non- 
 technical man. 
 
 Such general report would be materially shorter than the 
 general engineering report, as it would omit all details, and 
 merely deal with the general problem, purpose and solution. 
 
 In general, it is advisable to combine all three reports, by 
 having the scientific record preceded by the general engineering 
 report, and the latter preceded by the general report. Roughly, 
 the general report would usually have a length of 20 to 40 per 
 cent of the general engineering report, the latter a length of 10 
 to 25 per cent of the complete scientific record. 
 
 The bearing of the three classes of reports may be understood 
 by illustration on an investigation which appears of commercial 
 utility, and therefore is submitted for industrial development to 
 a manufacturing corporation; the financial and general adminis- 
 trative powers of the corporations, to whom the investigation is 
 submitted, would read the general report and if the matter 
 appears to them of sufficient interest for further consideration, 
 refer it to the engineering department. The general report thus 
 must be written for, and intelligible to the non-engineering 
 administrative heads of the organization. The administrative 
 engineers of the engineering department then peruse the general 
 engineering report, and this report thust must be an engineering 
 report, but general and not require the knowledge of the specialist 
 in the particular field. If then the conclusion derived by the 
 administrative engineers from the reading of the general engineer- 
 ing report is to the effect that the matter is worth further con- 
 sideration, then they refer it to the specialists in the field covered 
 by the investigation, and to the latter finally the scientific record 
 of the investigation appeals and is studied in making final report 
 on the work. 
 
 Inversely, where nothing but a lengthy scientific report is 
 submitted, as a rule it will be referred to the engineering depart- 
 ment, and the general engineer, even if he could wade through 
 the lengthy report, rarely has immediately time to do so, thus 
 lays it aside to study sometime at his leisure — and very often 
 this time never comes, and the entire matter drops, for lack of 
 proper representation. 
 
NUMERICAL CALCULATIONS. 293 
 
 Thus it is of the utmost importance for the engineer and the 
 scientist, to be able to present the results of his worlf not only by 
 elaborate and lengthy scientific report, but also by report of 
 moderate length, intelligible without dif&culty to the general 
 engineer, and by short statement intelligible to the non-engineer. 
 
 d. Reliability of Ntxmerical Calculations. 
 
 172. The most important and essential requirement of 
 numerical engineering calculations is their absolute reliability. 
 When making a calculation, the most brilliant ability, theo- 
 retical knowledge and practical experience of an engineer are 
 made useless, and even worse than useless, by a single error in 
 an important calculation. 
 
 Reliability of the numerical calculation is of vastly greater 
 importance in engineering than in any other field. In pure 
 mathematics an error in the numerical calculation of an 
 example which illustrates a general proposition, does not detract 
 from the interest and value of the latter, which is the main 
 purpose; in physics, the general law which is the subject of 
 the investigation remains true, and the investigation of interest 
 and use, even if in the numerical illustration of the law an 
 error is made. With the most brilhant engineering design, 
 however, if in the numerical calculation of a single structural 
 member an error has been made, and its strength thereby calcu- 
 lated wrong, the rotor of the machine flies to pieces by centrifugal 
 forces, or the bridge collapses, and with it the reputation of the 
 engineer. The essential difference between engineering and 
 purely scientific caclulations is the rapid check on the correct- 
 ness of the calculation, which is usually afforded by the per- 
 formance of the calculated structure — but too late to correct 
 errors. 
 
 Thus rapidity of calculation, while by itself useful, is of no 
 value whatever compared with rehability — that is, correctness. 
 
 One of the first and most important requirements to secure 
 rehability is neatness and care in the execution of the calcula- 
 tion. If the calculation is made on any kind of a sheet of 
 paper, with lead pencil, with frequent striking out and correct- 
 ing of figures, etc., it is practically hopeless to expect correct 
 results from any more extensive calculations. Thus the work 
 
293a ENGINEERING MATHEMATICS. 
 
 should be done with pen and ink, on white ruled paper; if 
 changes have to be made, they should preferably be made by 
 erasing, and not by striking out. In general, the appearance of 
 the work is one of the best indications of its rehability. The 
 arrangement in tabular form, where a series of values are calcu- 
 lated, offers considerable assistance in improving the reliability. 
 
 173. Essential in all extensive calculations is a complete 
 system of checking the results, to insure correctness. 
 
 One way is to have the same calculation made independently 
 by two different calculators, and then compare the results. 
 Another way is to have a few points of the calculation checked 
 by somebody else. Neither way is satisfactory, as it is not 
 always possible for an engineer to have the assistance of another 
 engineer to check his work, and besides this, an engineer should 
 and must be able to make numerical calculations so that he can 
 absolutely rely on their correctness without somebody else 
 assisting him. 
 
 In any more important calculations every operation thus 
 should be performed twice, preferably in a different manner. 
 Thus, when multiplying or dividing by the slide rule, the multi- 
 plication or division should be repeated mentally, approxi- 
 mately, as check; when adding a colmnn of figures, it should be 
 added first downward, then as check upward, etc. 
 
 Where an exact calculation is required, first the magnitude 
 of the quantity should be estimated, if not already known, 
 then an approximate calculation made, which can frequently 
 be done mentally, and then the exact calculation ; or, inversely, 
 after the exact calculation, the result may be checked by an 
 approximate mental calculation. 
 
 Where a series of values is to be calculated, it is advisable 
 first to calculate a few individual points, and then, entirely 
 independently, calculate in tabular form the series of values, 
 and then use the previously calculated values as check. Or, 
 inversely, after calculating the series of values a few points 
 should independently be calculated as check. 
 
 When a series of values is calculated, it is usually easier to 
 secure reliability than when calculating a single value, since 
 in the former case the different values check each other. There- 
 fore it is always advisable to calculate a number of values, 
 that is, a short curve branch, even if only a single point is 
 
NUMERICAL CALCULATIONS. 2936 
 
 required. After calculating a series of values, they are plotted 
 as a curve to see whether they give a smooth curve. If the 
 entire curve is irregular, the calculation should be thrown away, 
 and the entire work done anew, and if this happens repeatedly 
 with the same calculator, the calculator is advised to find 
 another position more in agreement with his mental capacity. 
 If a single point of the curve appears irregular, this points to 
 an error in its calculation, and the calculation of the point is 
 checked; if the error is not found, this point is calculated 
 entirely separately, since it is much more difficult to find an 
 error which has been made than it is to avoid making an 
 error. 
 
 174. Some of the most frequent numerical errors are: 
 
 1. The decimal error, that is, a misplaced decimal point. 
 This should not be possible in the final result, since the magni- 
 tude of the latter should by judgment or approximate calcula- 
 tion be known sufficiently to exclude a mistake by a factor 10. 
 However, under a square root or higher root, in the exponent 
 of a decreasing exponential function, etc., a decimal error may 
 occur without affecting the result so much as to be immediately 
 noticed. The same is the case if the decimal error occurs in a 
 term which is relatively small compared with the other terms, 
 and thereby does not affect the result very much. For instance, 
 in the calculation of the induction motor characteristics, the 
 quantity ri^+s^Xi^ appears, and for small values of the slip s, 
 the second term s^Xi^ is small compared with ri^, so that a 
 decimal error in it would affect the total value sufficiently to 
 make it seriously wrong, but not sufiiciently to be obvious. 
 
 2. Omission of the factor or divisor 2. 
 
 3. Error in the sign, that is, using the plus sign instead of 
 the minus sign, and inversely. Here again, the danger is 
 especially great, if the quantity on which the wrong sign is 
 used combines with a larger quantity, and so does not affect 
 the result sufficiently to become obvious. 
 
 4. Omitting entire terms of smaller magnitude, etc. 
 
APPENDIX A. 
 
 NOTES ON THE THEORY OF FUNCTIONS. 
 
 A. General Functions. 
 
 175. The most general algebraic expression of powers of 
 X and y, 
 
 ■P'(^)2/) = (000+0012; + 002^^ + . • .) + (aio+aiix + ai2x2+. . .)i/ 
 + (020+0212;+ 022^2 +. . .)2/2+. . . 
 + (a„o+a„ix+o„2a;^+. • . )2/" = 0) ••••(!) 
 
 is the imflicit analytic function. It relates y and x so that to 
 every value of x there correspond n values of y, and to every 
 value of y there correspond m values of x, if m is the exponent 
 of the highest power of a; in (1). 
 
 Assuming expression (1) solved for y (which usually carmot 
 be carried out in final form, as it requires the solution of an 
 equation of the nth order in y, with coefficients which are 
 expressions of x), the explicit analytic function, 
 
 y=m, (2) 
 
 is obtained. Inversely, solving the implicit function (1) for 
 X, that is, from the explicit function (2), expressing x as 
 function of y, gives the reverse function of (2); that is 
 
 ^=/i(2/) (3) 
 
 In the general algebraic function, in its implicit form (1), 
 or the explicit form (2), or the reverse function (3), x and y 
 are assumed as general numbers; that is, as complex quan- 
 tities; thus, 
 
 x = xi-\-jx2; \ 
 
 y=yi+jy2, J 
 
 and likewise are the coefficients Ooo, aoi 
 
 (4) 
 
 294 
 
APPENDIX A. 295 
 
 If all the coefficients a are real, and x is real, the corre- 
 sponding n values of y are either real, or pairs of conjugate 
 complex imaginary quantities: yi +J2/2 and 2/1-/2/2. 
 
 176- For n = l, the implicit function (1), solved for y, gives 
 the rational function, 
 
 aoo + aoiX + ao2X^ + . . . 
 '^~aio + anX + ai2X^ + .. ' ^^ 
 
 and if in this function (5) the denominator contains no x, the 
 integer function, 
 
 y = a^+aix+a2X^+. . .+a,nX'^, , . . (6) 
 
 is obtained. 
 
 For n=2, the implicit function (1) can be solved for y as a, 
 quadratic equation, and thereby gives 
 
 — ( ai(i+Oiii + at;a:' + ...)J:: 
 
 that is, the explicit form (2) of equation (1) contains in this 
 case a square root. 
 
 For n>2, the explicit form y=f{x) either becomes very 
 complicated, for n = 3 and n = 4, or cannot be produced in 
 finite form, as it requires the solution of an equation of more 
 than the fourth order. Nevertheless, y is still a function of 
 X, and can as such be calculated by approximation, etc.. 
 
 To find the value yi, which by function (1) corresponds to 
 x = xi, Taylor's theorem offers a rapid approximation. Sub- 
 stituting xi in function (1) gives an expression which is of 
 the nth order in y, thus: F(xiy), and the problem now is to 
 find a value 2/1, which makes F{xi,yi)=0. 
 
 However, 
 
 w ^ wr , ,. dF(M, y) h^ d?F{x,,y) 
 
 F{Xi,yx) = F(x,,y)+h -^ — +J2 dhf +" " " ' " ^^^ 
 
 where h = yi — y is the difference between the correct value 2/1 
 and any chosen value y. 
 
296 
 
 ENGINEERING MATHEMATICS. 
 
 Neglecting the higher orders of the small quantity h, in 
 (8), and considering that F{xi,yi)=0, gives 
 
 h=- 
 
 dF{x,,yy 
 dy 
 
 (y) 
 
 and herefrom is obtained yi=y+h, as first approximation. 
 Using this value of yi as y in (9) gives a second approximation, 
 which usually is sufficiently close. 
 
 177. New functions are defined by the integrals of the 
 analytic functions (1) or (2), and by their reverse functions. 
 They are called Abelian integrals and Ahelian functions. 
 
 Thus in the most general case (1), the explicit function 
 corresponding to (1) being 
 
 the integral, 
 
 y=f{^), ■ ■ 
 
 ^= ) f{x)dx, 
 
 (2) 
 
 then is the general Abelian integral, and its reverse function, 
 
 x = 4>{z), 
 
 the general Abelian function. 
 
 (a) In the case, n = l, function (2) gives the rational function 
 C5), and its special case, the integer function (6). 
 
 Function (6) can be integrated by powers of x. (5) can be 
 resolved into partial fractions, and thereby leads to integrals 
 of the following forms : 
 
 x«^dx; 
 
 (1) /. 
 
 (10) 
 
APPENDIX A. 297 
 
 Integrals (10), (1), and (3) integrated give rational functions, 
 (10), (2) gives the logarithmic function log (x-a), and (10), (4) 
 the arc function arc tan x. 
 
 As the arc functions are logarithmic functions with complex 
 imaginary argument, this case of the integral of the rational 
 function thus leads to the logarithmic function, or the loga- 
 rithmic integral, which in its simplest form is 
 
 /dx , , ^ 
 
 — = logx, . . . . (11) 
 
 and gives as its reverse function the exponential function, 
 
 (12) 
 
 It is expressed by the infinite series. 
 
 Z^ gS ^4 
 
 £^=l+Z+p-+Tg+|^+ (13) 
 
 as seen in Chapter II, paragraph 53. 
 
 178. b. In the case, n=2, function (2) appears as the expres- 
 sion (7), which contams a square root of some power of x. Its 
 first part is a rational function, and as such has already been 
 discussed in a. There remains thus the integral function. 
 
 =/^ 
 
 +i,.i+S«^ + ..^+l,^._j_. ^j^, 
 
 co + CiX + CiX-+ . 
 
 This expression (14) leads to a series of important functions. 
 (1) Forp = lor2, 
 
 =/: 
 
 V5o+ 61.x +62x2 , ,_. 
 
 ax. . . . . (15) 
 
 Cn+Cix + C22;^+. 
 
 By substitution, resolution into partial fractions, and 
 separation of rational functions, this integral (11) can be 
 reduced to the standard form, 
 
 '" (16) 
 
 ■/. 
 
 In the case of the minus sign, this gives 
 dx 
 
 r d: 
 
 "JvT- 
 
 arc sin x, . . . (17) 
 
298 
 
 ENGINEERING MATHEMATICS. 
 
 and as reverse functions thereof, there are obtained the trigo- 
 nometric functions. 
 
 x = sin 2, 1 
 
 Vl — x" = cos z. J 
 In the case of the plus sign, integral (16) gives 
 dx 
 
 . (18) 
 
 ■/; 
 
 -log{ Vl +x2 — a;j =arc sinh x, . (19) 
 
 Vl+x^ 
 and reverse functions thereof are the hyperbolic functions, 
 
 X = t: = sinh z ; 
 
 Vl+x^-- 
 
 S + '+i 
 
 -- cosh ,?. 
 
 (20) 
 
 The trigonometric functions are expressed by the series : 
 
 ■5;3 5^5 £7 
 
 sin. = 2-j3+j^-iy+ .. 
 
 cos 3 = 1— pj+TT- 
 
 • + 
 
 (21) 
 
 as seen in Chapter II, paragraph 58. 
 
 The hyperbolic functions, by substituting for e"*"^ and £~' 
 the series (13), can be expressed by the series: 
 
 ^3 ;S ;7 
 
 sinh 2 = + 7-r+7^+fz-.+ . 
 o p 7 
 
 (22) 
 
 cosh 3 = IH-TTY +i-j- +T77 + 
 
 179- In the next case, p = 3 or 4, 
 
 /V&o +5lX +?)2a;^ +53X3+^4X4 
 
 Co + cix + Cix'^ + . 
 
 dx, 
 
 (23) 
 
 already leads beyond the elementary functions, that is, (23) 
 cannot be integrated by rational, logarithmic or arc functions, 
 
APPENDIX A. 
 
 299 
 
 but^ gives a new class of functions, the elliptic integrals, and 
 their reverse functions, the elliptic functions, so called, because 
 they bear to the ellipse a relation similar to that, which the 
 trigonometric functions bear to the circle and the hyperbolic 
 functions to the equilateral hyperbola. 
 
 The integral (23) can be resolved into elementary functions, 
 and the three classes of elliptic integrals : 
 
 
 dx 
 
 \/x(l-a:)(l-c2x)' 
 
 xdx 
 Vx{l-x)(l-cH)' 
 dx 
 
 {x-b)Vx{l-x) (l-c^x) 
 
 (24) 
 
 (These three classes of integrals may be expressed in several 
 different forms.) 
 
 The reverse functions of the elliptic integrals are given by 
 the elliptic functions : 
 
 x = sin am(u, c); 
 
 Vl — X = cos am(u, c) ; ■ 
 
 (25) 
 
 \/l —c^x = Jam{u, c) ; 
 
 known, respectively, as sine-amplitude, cosine-amplitude, delta- 
 amplitude. 
 
 Elliptic functions are in some respects similar to trigo- 
 nometric functions, as is seen, but they are more general, 
 depending, as they do, not only on the variable x, but also on 
 the constant c. They have the interesting property of being 
 doubly periodic. The trigonometric functions are periodic, with 
 the periodicity 27t, that is, repeat the same values after every 
 change of the angle by 2^. The elliptic functions have two 
 periods pi and p2, that is. 
 
 sin am{u +npi +mp2, c) =sin am{u, c), etc.; 
 
 (26) 
 
 hence, increasing the variable u by any multiple of either 
 period pi and p2, repeats the same values. 
 
300 
 
 ENGINEERING MATHEMATICS. 
 
 The two periods are given by the equations, 
 
 dx 
 
 pi 
 
 
 Va;(l — x)(l — c^x)' 
 
 dx 
 
 (27) 
 
 2Vx(l-x)(l-c2x) 
 
 i8o. Elliptic functions can be expressed as ratios of two 
 infinite series, and these series, which form the numerator and 
 the denominator of the elliptic function, are called theta func- 
 tions and expressed by the symbol 6, thus 
 
 sin am{u, c) ■- 
 
 1 »■(£ 
 
 '6^ 
 
 (g)' 
 
 cos amiu 
 
 .'>-a/t? 
 
 ■te) 
 
 OA 
 
 Aam{u, c) = ^l — c^- 
 
 TIU 
 
 (28) 
 
 and the four d functions may be expressed by the series : 
 ^o(x) = 1 —2q cos 2x +254 cos 4x -2q^ cos 6x + -. . . : 
 
 25 
 
 ^i(x)=22i/*sinx-259/*sui3x+234 sin 5x- + .. . ; 
 
 25 
 
 ^2(2;) =2^1/* cos X +25"/* COS 3x +2g 4 cos 5x + ; 
 
 6i3(x) = 1+2(7 cos 2x +254 cos 4x +2^9 cos 6x + . • , 
 where 
 
 q= e." and a = jn— 
 
 Pi 
 
 , (29) 
 
 (30) 
 
 In the case of integral function (14), where p>4, similar 
 integrals and their reverse functions appear, more complex 
 
APPENDIX A. 301 
 
 than the eUiptic functions, and of a greater number of periodici- 
 ties. They are called hyperelliptic integrals and hyperelliptic 
 functions, and the latter are again expressed by means of auxil- 
 iarj' functions, the hyperelliptic 6 functions. 
 
 i8i. Many problems of physics and of engineering lead to 
 elliptic functions, and these functions thus are of considerable 
 importance. For instance, the motion of the pendulum is 
 expressed by elliptic functions of time, and its period thereby 
 is a function of the amplitude, increasing with increasing ampli- 
 tude: that is, in the so-called "second pendulum," the time of 
 one swing is not constant and equal to one second, but only 
 approximately so. This approximation is very close, as long 
 as the amplitude of the swing is very small and constant, but 
 if the amplitude of the swing of the pendulum varies and 
 reaches large values, the time of the swing, or the period ot 
 the pendulum, can no longer be assumed as constant and an 
 exact calculation of the motion of the pendulum by elliptic 
 functions becomes necessary. 
 
 In electrical engineering, one has frequently to deal with 
 oscillations similar to those of the pendulum, for instancp, 
 in the hunting or surging of synchronous machines. In 
 general, the frequency of oscillation is assumed as constant, 
 but where, as in cumulative hunting of synchronous machines, 
 the amplitude of the swing reaches large values, an appreciable 
 change of the period must be expected, and where the hunting 
 is a resonance effect with some other periodic motion, as the 
 engine rotation, the change of frequency with increase of 
 amplitude of the oscillation breaks the complete resonance and 
 thereby tends to limit the amplitude of the swing. 
 
 182. As example of the application of elliptic integrals, may 
 be considered the determination of the length of the arc of an 
 ellipse. 
 
 Let the ellipse of equation 
 
 'T»2 /i/2 
 
 ^24=1' (31) 
 
 be represented in Fig. 93, with the circumscribed circle, 
 
 2;2 + 2/2 = a2. ... . . (32) 
 
302 
 
 ENGINEERING MATHEMATICS. 
 
 To every point P = x, y of the ellipse then corresponds a 
 point Pi = x, 7/1 on the circle, which has the same abscissa x, 
 and an angle d = AOP\. 
 
 The arc of the ellipse, from A to P, then is given by the 
 integral, 
 
 ,.., f _41^5E=^, .... (33) 
 Jo 2^ - 
 where 
 
 Wz{l-z){l-c^z)' 
 
 2r = sin2 (9=(-l and c=- , 
 
 \a/ a 
 
 is the eccentricity of the ellipse. 
 
 ^ 
 
 A, 
 
 ^!«.yi 
 
 1 "^ 
 
 A/ 
 
 „ 
 
 I 
 
 b 
 
 y 
 
 (34) 
 
 Fig. 93. Rectification of Ellipse. 
 
 Thus the problem leads to an elliptic integral of the first 
 and of the second class. 
 
 For more complete discussion of the elliptic integrals and 
 the elliptic functions, reierence must be made to the text-books 
 of mathematics. 
 
 B. Special Functions. 
 
 i83- Numerous special functions have been derived by the 
 exigencies of mathematical problems, mainly of astronomy, but 
 in the latter decades also of physics and of engineering. Some 
 of them have already been discussed as special cases of the 
 general Abelian integral and its reverse function, as the expo- 
 nential, trigonometric, hyperbolic, etc., functions. 
 
APPENDIX A. 303 
 
 Functions may be represented by an infinite series of terms; 
 that is, as a sum of an infinite number of terms, which pro- 
 gressively decrease, that is, approach zero. The denotation of 
 the terms is commonly represented by the summation sign 2. 
 
 Thus the exponential functions may be written, when 
 defining, 
 
 [0 = 1; 'ln = lX2x3X4X. . .Xn, 
 as 
 
 ex = i+,+_4-- + .....S.- .... (35) 
 
 which means, that terms j — are to be added for all values of n 
 
 n 
 
 from n = to n=oc . 
 
 The trigonometric and hyperbolic functions may be written 
 in the form : 
 
 X^ X^ X^ ^ , .s X^"*! ,„„. 
 
 sinx = x-T^+fv-^+ .. = 2"(-l)"^5-— r; . (36) 
 
 ]3 |5 |7 ' ■ ■ ' ' | 2n + l 
 
 j'2 '^4 ^6 00 X^" 
 
 cosx = l-i2+|T-|6+ •• = ?"(-l)"g ■ ■ (37) 
 
 » X- 
 
 ■2n+l 
 
 sinhx-X+T7r+TV+7=-+... = 2n— — —; . . . (38) 
 
 13 15 ' 17 2n + l 
 
 X2 X* x" » X2" 
 
 coshx = l+i2+|^+ig+... = Snj^ (39) 
 
 Functions also may be expressed by a series of factors; 
 that is, as a product of an infinite series of factors, which pro- 
 gressively approach unity. The product series is commonly 
 represented by the symbol JJ. 
 
 Thus, for instance, the sine function can be expressed in the 
 form, 
 
 sm X = X 
 
 (i-S)(i-&)('-S--=^w(i-i)- «« 
 
 184. Integration of known functions frequently leads to new 
 functions. Thus from the general algebraic functions were 
 
304 ENGINEERING MATHEMATICS. 
 
 derived the Abelian functions. In physics and in engineering, 
 integration of special functions in this manner frequently leads 
 to new special functions. 
 
 For instance, in the study of the propagation through space, 
 of the magnetic field of a conductor, in wireless telegraphy, 
 lightning protection, etc., we get new functions. If i=f (J) 
 is the current in the conductor, as function of the time t, at a 
 distance x from the conductor the magnetic field lags by the 
 
 X 
 
 time h=—, where S is the speed of propagation (velocity of 
 o 
 
 hght). Since the field intensity decreases inversely propor- 
 tional to the distance x, it thus is proportional to 
 
 X 
 
 y-^^>- ■ ■ ■ (41) 
 
 and the total magnetic flux then is 
 z= j ydx 
 
 =J J— rf^ (42) 
 
 If the current is an alternating current, that is, f (t) a 
 trigonometric function of time, equation (42) leads to the 
 functions. 
 
 Tsin X , 
 M= I ax; 
 
 -J 
 
 cos X , 
 -ax. 
 
 (43) 
 
 If the current is a direct current, rising as exponential 
 function of the time, equation (42) leads to the function, 
 
 w 
 
 /e^x 
 
 (44) 
 
APPENDIX A. 
 
 305 
 
 Substituting in (43) and (44), for sin x, cos x, e^ their 
 infinite series (21) and (13), and then integrating, gives the 
 following : 
 
 
 sm X 
 
 — d2; = X-g|g+^-y=+■ 
 
 C0S X , , X^ X* x^ 
 
 / 
 
 t-dx = hgx+x+^^+^ + . 
 
 . (45) 
 
 For further discussion and tables of these functions see 
 '■' Theory and Calculation of Transient Electric Phenomena and 
 Oscillations," Section III, Chapter VIII, and Appendix. 
 
 i8S. If y=f{x) is a function of x, and z= j f (x)dx = ^(x) 
 
 n 
 
 its integral, the definite integral, Z= \ f{x)dx, is no longer 
 
 mJa 
 
 a function of x but a constant, 
 
 Z = <^(&)-</,(a). 
 For instance, if y=^c(x—n)^, then 
 
 c{x—ny 
 
 -/ 
 
 c(x—nydx=- 
 
 and the definite integral is 
 
 Z= I c(z-n)2dx = |((E>-n)3-(a-n)3}. 
 
 This definite integral does not contain x, but it contains 
 all the constants of the function / (x), thus is a function of 
 these constants c and n, as it varies with a variation of these 
 constants. 
 
 In this manner new functions may be derived by definite 
 integrals. 
 
 Thus, if 
 
 y=f{x,u,v...) (46) 
 
 is a function of x, containing the constants u, v . . . 
 
306 ENGINEERING MATHEMATICS. 
 
 The definite integral, 
 
 Z= I f{:£,u,v.. . )dx, (47) 
 
 is not a function of x, but still is a function of w, v . . . , and 
 may be a new function. 
 i86. For instance, let 
 
 J/=£-:r2;«-l; (48) 
 
 then the integral, 
 
 (49) 
 
 /(w)= rr--x«-iofx, 
 
 is a new function of u, called the gamma function. 
 
 Some properties of this function may be derived by partial 
 integration, thus : 
 
 r{u + l)=ur{u); (50) 
 
 if n is an integer number, 
 
 r{u) = {u-l){u-2)...{u-n)r{u-n), . . (51) 
 and since 
 
 Al) = l, (52) 
 
 if u is an integer number, then, 
 
 r{u) = \u-l. (53) 
 
 C. Exponential, Trigonometric and Hyperbolic Functions. 
 
 (a) Functions of Real Variables. 
 
 187. The exponential, trigonometric, and h^'perbolic func- 
 tions are defined as the reverse functions of the integrals, 
 
 Cdx . 
 a- w = J— = log2;, (54) 
 
 and: x=£" (55) 
 
 f dx 
 
APPENDIX A. (307/ 
 
 and: x=sm.u, (57) 
 
 Vl-x2 = C03 M, (58) 
 
 /fix 
 
 c- ■"= I 77T=5=-log{Vl+a;2-x}; . . . . (59) 
 
 and x= 2 — =sinhw; .... (60) 
 
 £" -\- e~" 
 Vl+x2 = — ■=coshM (61) 
 
 From (57) and (58) it follows that 
 
 sin^ m+cos2m = 1 (62) 
 
 From (60) and (61) it follows that 
 
 cos^ /iM— sin 2/iM=:l. (63) 
 
 Substituting (— x) for x in (56), gives (— m) instead of u, 
 and therefrom, 
 
 sin ( — w) = — sin M (64) 
 
 Substituting (— m) for u in (60), reverses the sign of x, 
 that is, 
 
 sinh (— m) = — sinh w. . . . (65) 
 
 Substituting (— x) for x in (58) and (61), does not change 
 the value of the square root, that is, 
 
 cos (— m)=cos M, (66) 
 
 cosh (—w)= cosh M, (67) 
 
 Which signifies that cos u and cosh u are even functions, while 
 sin u and sinh u are odd functions. 
 
 Adding and subtracting (60) and (61), gives 
 
 £=■=" = cosh w± sinh M (68) 
 
/ 308 ENGINEERING MATHEMATICS. 
 
 / (5) Functions of Imaginary Variables. 
 
 
 i88. Substituting, in (56) and (59), x= —jy, thus y = jx, gives 
 
 dx 
 
 ■U c ~U 
 
 a;=sinu; a; = sinh« 
 
 V1+x2 = cosm; vT+x2 = coshM = 
 
 2 ' 
 
 hence, ju ■■ 
 
 /—. , hence, m= i 
 
 VT+f' " J VI^' 
 
 i/ = sinhjM = ;^ ; y=smiu; . . . (69) 
 
 £)" -)_ £-;u 
 
 Vl +2/2 = cosh /m= ^^ ; Vl — j/2 = cos ju; . . . (70) 
 
 Resubstituting a; in both 
 
 sinh fw £3"— £-'■" . £>._£-« sin m _^^ 
 
 a;=sinM = r^ = — ^-^ : x = sinhii= — ^ — = — : — ; (71) 
 
 .« 4_ p—u 
 
 £"+£ 
 
 2 
 
 £)■« + 5-1 
 
 Vl — x^ = COS ti = cosh /m vT+x^ = cosh u = 
 
 = COS ju. . (72) 
 
 Adding and subtracting, 
 
 £±'" = cos M±/.sin M = cosh jM±siuh ju 
 and £±" = cosh ii±sinhu = cos j'mTJ sin jM. . . (73) 
 
 (c) Functions of Complex Variables 
 
 189. It is: 
 
 e"±i^=s:"e^'''=£"{cosv±ismv); . . . (74) 
 
APPENDIX A. 309 
 
 sin {u±jv) =Bin u cos jv±cos u sin jv 
 
 = Sin u cosh v±] cos w smh v = — ^^ — sin u ± ] — ^z — cos « ; 
 
 cos (u ± ]v) = COS w COS ]v T sin u sin p 
 
 £" + £-" .£"-£-" . 
 
 = COS u cosh u T J sin K smh ?; = — -x — cos u =F J — ^ — sin w ; 
 
 sinh(w±p)= :^ = — ^ — cosvij — ^ — sinr 
 
 = sinh u cos v±j cosh m sin v; 
 
 cosh(M±p)= 2 = — 2 — ^osv±] — ^ — sinu 
 
 = cosh u cos v±j sinh w sin v ; 
 etc. 
 
 (75) 
 
 (76) 
 
 (77) 
 
 (78) 
 
 (d) Relations. 
 
 190. From the preceding equations it thus follows that the 
 three functions, exponential, trigonometric, and hyperbolic, 
 are so related to each other, that any one of them can be 
 expressed by any other one, so- that when allowing imaginary 
 and complex imaginary variables, one function is sufficient. 
 As such, naturally, the exponential function would generally 
 be chosen. 
 
 Furthermore, it follows, that all functions with imaginary 
 and complex imaginary variables can be reduced to functions 
 of real variables by the use of only two of the three classes 
 of functions. In this case, the exponential and the trigono- 
 metric functions would usually be chosen. 
 
 Since functions with complex imaginary variables can be 
 resolved into functions with real variables, for their calculation 
 tables of the functions of real variables are sufficient. 
 
 The relations, by which any function can be expressed by 
 any other, are calculated from the preceding paragraph, by 
 the following equations : 
 
310 
 
 ENGINEERING MATHEMATICS. 
 
 «=•=" = cosh M ±sinh u = cos ju ^ j sin ju; 
 £ ± '" = cos V ± y sin t; = cosh jv ± j sinh jv ; 
 £"*'"=£" (cos rijsin v), 
 
 sinh jM £'"— £"'" 
 
 sin M = 
 
 J 
 
 sin jv = j sinh v = j 
 
 2] ' 
 
 £"-£-" 
 
 sin (u±jv) =sin w cosh 2> ± / cos u sinh r 
 
 £" + £~" . £* — e~'" 
 
 = — ^ — sin u±j — cosw; 
 
 cos w = cosh ju = -^^ — ; 
 
 cos J j; = cosh V = 
 
 £'" + £-'" 
 
 COS {u±jv) = cos w cosh vT J sin u sinh j; 
 
 sinh w = - 
 
 -^ — cos wT; — p — sin u; 
 — £~" sin ju 
 
 2 
 
 sinh jv = j sin v -- 
 
 1 ' 
 
 ^v_ g-jv 
 
 sinh {u ± jv) = sinh ucosv±j cosh u sin v 
 
 £«_£-« £"+£-" 
 
 = — 2 ^°^ ^±^ — 2 ^^'^ ^' 
 
 cosh M = 
 
 ■U _J_ c~ « 
 
 £"+£ 
 
 2 = cos ]u; 
 
 cosh p = cos v = 
 cosh (w ± jv) = cosh M cos V ± / sinh w sin t; 
 
 "2 — cosr±7 — 2" 
 
 • sm V. 
 
 (a) 
 
 (&) 
 
 (c) 
 
 (d) 
 
 (e) 
 
APPENDIX A. 311 
 
 And from (b) and (d), respectively (c) and (e), it follows that 
 
 sinh (u ± jv) = J sin ( ± i' — j'u) = ± j sin (?; ± ju) ; 
 cosh (u ± jv) = cos (i' T ju) . 
 
 if) 
 
 Tables of the exponential functions and their logarithms, 
 and of the hyperboHc functions with real variables, are given 
 in the following Appendix B. 
 
APPENDIX B. 
 
 TWO TABLES OF EXPONENTIAL AND HYPERBOLIC 
 
 FUNCTIONS. 
 
 Table I. 
 
 £ = 2,7183, 
 
 log £ = 0.4343. 
 
 X 
 
 X10-' 
 
 XlO-2 
 
 xio-i 
 
 XI 
 
 1.0 
 
 0.999 
 
 0.990 
 
 0.905 
 
 0.368 
 
 1.2 
 
 
 
 0.988 
 
 0.887 
 
 0.301 
 
 1.4 
 
 
 0.986 
 
 0.869 
 
 0.247 
 
 1.6 
 
 
 
 0.984 
 
 0.852 
 
 0.202 
 
 1.8 
 
 
 0.982 
 
 0.835 
 
 0.165 
 
 2.0 
 
 0.998 
 
 0.980 
 
 0.819 
 
 0.135 
 
 2.5 
 
 
 0.975 
 
 0.779 
 
 0.082 
 
 3.0 
 
 0.997 
 
 0.970 
 
 0.741 
 
 0.050 
 
 3.5 
 
 
 0.966 
 
 0.705 
 
 0.030 
 
 4.0 
 
 0.996 
 
 0.961 
 
 0.670 
 
 0.018 
 
 4.5 
 
 
 0.956 
 
 0.638 
 
 0.011 
 
 5.0 
 
 0.995 
 
 0.951 
 
 0.607 
 
 0.007 
 
 6 
 
 0.994 
 
 0.942 
 
 0.549 
 
 0.002 
 
 7 
 
 0.993 
 
 0.932 
 
 0.497 
 
 0.001 
 
 8 
 
 0.992 
 
 0.923 
 
 0.449 
 
 0.000 
 
 9 
 
 0.991 
 
 0.914 
 
 0.407 
 
 
 10 
 
 0.990 
 
 0.905 
 
 0.368 
 
 
 312 
 
APPENDIX B. 
 
 313 
 
 Table II. 
 
 EXPONENTIAL AND HYPERBOLIC FUNCTIONS. 
 
 £ = 2.718282 — 2.7183, log e = 0.4342945~0.4343. 
 
 p.p. 
 
 
 434 
 
 435 
 
 
 
 
 
 
 
 0.1 
 
 43 
 
 43 
 
 0.2 
 
 87 
 
 87 
 
 0.3 
 
 130 
 
 130 
 
 0.4 
 
 174 
 
 174 
 
 0.5 
 
 217 
 
 217 
 
 0.6 
 
 261 
 
 261 
 
 0.7 
 
 304 
 
 304 
 
 0.8 
 
 347 
 
 348 
 
 0.9 
 
 391 
 
 391 
 
 1.0 
 
 434 
 
 436 
 
 
 cosh X = 
 
 ii£ + ^+£-^i, 
 
 sinhx = i|£ 
 
 fX_j-2 
 
 !• 
 
 
 
 
 Jlog 
 
 
 
 
 
 
 
 X 
 
 loge + ^ 
 
 £±I 
 
 log £-^ 
 
 ^ + x 
 
 f-x 
 
 cosh X 
 
 sinh X 
 
 X 
 
 
 
 
 
 434 
 435 
 434 
 434 
 
 
 
 1 
 
 1 
 
 1 
 
 
 
 
 
 0.001 
 
 0.000434 
 
 9.999666 
 
 1.00100 
 
 0.99900 
 
 1.00000 
 
 0.00100 
 
 0,001 
 
 0.002 
 
 0.000869 
 
 9.999131 
 
 1.00200 
 
 0.99800 
 
 1.00000 
 
 0.00200 
 
 0,002 
 
 0.003 
 
 0.001303 
 
 9.998697 
 
 1.00301 
 
 0.99700 
 
 1.00000 
 
 0.00300 
 
 0.003 
 
 0.004 
 
 0.001737 
 
 9.998263 
 
 1.00401 
 
 0.99601 
 
 1.00001 
 
 0.00400 
 
 0.004 
 
 
 
 434 
 
 
 
 
 
 
 
 0.005 
 
 0.002171 
 
 9.997829 
 
 1.00601 
 
 0.99601 
 
 1.00001 
 
 0.00600 
 
 0.005 
 
 
 
 435 
 
 434 
 434 
 435 
 
 434 
 
 
 
 
 
 
 
 0.006 
 
 0.002606 
 
 9.997394 
 
 1.00602 
 
 0.99402 
 
 1.00002 
 
 0.00600 
 
 0.006 
 
 0.007 
 
 0.003040 
 
 9.996960 
 
 1.00702 
 
 0.99302 
 
 1.00002 
 
 0.00700 
 
 0.007 
 
 0.008 
 
 0.003474 
 
 9.996626 
 
 1.00803 
 
 0.99203 
 
 1,00003 
 
 0.00800 
 
 0.008 
 
 0.009 
 
 0.003909 
 
 9.996091 
 
 1.00904 
 
 0.99104 
 
 1.00004 
 
 O.OO90O 
 
 0.009 
 
 0.010 
 
 0.004343 
 
 9.995667 
 
 1.01005 
 
 0.99005 
 
 1.00005 
 
 0.01000 
 
 0.010 
 
 0.012 
 
 0.006212 
 
 
 9.994788 
 
 1.01207 
 
 0.98807 
 
 1.00007 
 
 0.01200 
 
 0.012 
 
 0.014 
 
 0.006080 
 
 
 9.993920 
 
 1.01410 
 
 0.98610 
 
 1.00010 
 
 0.01400 
 
 0.014 
 
 0.016 
 
 0.006949 
 
 
 9.993051 
 
 1.01613 
 
 0.98413 
 
 1.00013 
 
 0.01600 
 
 0.016 
 
 0.018 
 
 0.007817 
 
 
 9.992183 
 
 1.01816 
 
 0.98216 
 
 1.00016 
 
 0.01800 
 
 0.018 
 
 0.020 
 
 . 008686 
 
 
 9.991314 
 
 1.02020 
 
 0.98020 
 
 1.00020 
 
 0.02000 
 
 0.020 
 
 0.025 
 
 0.010857 
 
 
 9.989143 
 
 1.02531 
 
 0.97531 
 
 1.00031 
 
 0.02500 
 
 0.025 
 
 0.030 
 
 0.013029 
 
 
 9.986971 
 
 1.03046 
 
 0.97046 
 
 1.00046 
 
 0.03000 
 
 0.030 
 
 0.035 
 
 0.015200 
 
 
 9.984800 
 
 1.03562 
 
 0.96661 
 
 1.00062 
 
 0.03500 
 
 0.036 
 
 0.040 
 
 0.017372 
 
 
 9.982628 
 
 1.04081 
 
 0.96079 
 
 1.00080 
 
 0.04001 
 
 0.040 
 
 0.045 
 0.050 
 
 0.019543 
 
 
 9 . 980467 
 
 1.04603 
 
 0.95600 
 
 1.00102 
 
 0.04502 
 
 0.045 
 
 0.021715 
 
 
 9.978285 
 
 1.05127 
 
 0.96123 
 
 1.00125 
 
 0.05003 
 
 0.050 
 
 0.06 
 
 0.026068 
 
 
 9.973942 
 
 1.06184 
 
 0.94176 
 
 1.00180 
 
 0.06004 
 
 0.06 
 
 0.07 
 
 0.030401 
 
 
 9.969599 
 
 1.07251 
 
 0.93239 
 
 1.00245 
 
 0.07006 
 
 0.07 
 
 0.08 
 
 0.034744 
 
 
 9.966266 
 
 1.08329 
 
 0.92312 
 
 1.00321 
 
 0.08008 
 
 0.08 
 
 0.09 
 
 0.039086 
 
 
 9.960914 
 
 1.09417 
 
 0.91393 
 
 1.00405 
 
 0.09011 
 
 0.09 
 
 0.10 
 
 0.043429 
 
 
 9.966571 
 
 1.10616 
 
 0.90484 
 
 1.00500 
 
 0.10016 
 
 0.10 
 
 0,12 
 
 0.062115 
 
 
 9.947885 
 
 1.12760 
 
 0.88692 
 
 1,00721 
 
 0.12028 
 
 0.12 
 
 0.14 
 
 0.060801 
 
 
 9.939199 
 
 1.15027 
 
 0.86936 
 
 1.00982 
 
 0.14046 
 
 0.14 
 
 0.16 
 
 0.069487 
 
 
 9.930513 
 
 1.17361 
 
 0.85214 
 
 1.01283 
 
 0.16069 
 
 0.16 
 
 0.18 
 
 0.078173 
 
 
 9.921827 
 
 1.19721 
 
 0.83527 
 
 1.01624 
 
 0.18097 
 
 0.18 
 
 0.20 
 
 0.086869 
 
 
 9.913141 
 
 1.22140 
 
 0.81873 
 
 1.02006 
 
 0.20134 
 
 0.20 
 
 j + 0,001 = 1.001000494, £-o,ooi ^ 0,99900049. 
 
314 ENGINEERING MATHEMATICS. 
 
 Table II — Continued. 
 EXPONENTIAL AND HYPERBOLIC FUNCTIONS. 
 
 T 
 
 log e + ^ 
 
 log £-^ 
 
 £ + ^ 
 
 f-^ 
 
 cosh X 
 
 sinh X 
 
 X 
 
 0.20 
 
 0.086859 
 
 9.913141 
 
 1.22140 
 
 0.81873 
 
 1.02006 
 
 0.20134 
 
 0.20 
 
 0.25 
 0.30 
 0.35 
 0.40 
 0.45 
 
 0.108574 
 0.130288 
 0.152003 
 0.173718 
 0.195433 
 
 9.891426 
 9.869712 
 9.847997 
 9 . 826282 
 9.804567 
 
 1.28403 
 1 . 34986 
 1.41907 
 1.49183 
 1.56831 
 
 0.77880 
 0.74082 
 0.70469 
 0.67032 
 0.63763 
 
 1.03142 
 1 . 04634 
 1.06188 
 1.08108 
 1 . 10297 
 
 0.25261 
 0.30457 
 0.35719 
 0.41076 
 0.46634 
 
 0.25 
 0.30 
 0.35 
 0.40 
 0.45 
 
 0.50 
 
 0.217147 
 
 9.782853 
 
 1.64870 
 
 0.60653 
 
 1.12761 
 
 0.62108 
 
 0.50 
 
 0.6 
 0.7 
 0.8 
 0.9 
 
 0.260577 
 . 304006 
 0.347436 
 0.390865 
 
 9.739423 
 9.695994 
 9.652664 
 9.609135 
 
 1.82212 
 2.01375 
 2 . 22554 
 2.45960 
 
 0.54881 
 0.49659 
 0.44933 
 0.40657 
 
 1.19646 
 1.26617 
 1.33744 
 1.43309 
 
 0.63666 
 0.75858 
 0.88811 
 1.02667 
 
 0.6 
 0.7 
 0.8 
 0.9 
 
 1.0 
 
 0.434294 
 
 9.565706 
 
 2.71828 
 
 0.36788 
 
 1.54308 
 
 1 . 17620 
 
 1.0 
 
 1.2 
 1.4 
 1.6 
 1.8 
 
 0.521153 
 0.608012 
 0.694871 
 0.781730 
 
 9.478847 
 9.391988 
 9.305129 
 9.218270 
 
 3.32011 
 4 . 05520 
 4.95304 
 6.04965 
 
 0.30119 
 . 24660 
 0.20190 
 . 16530 
 
 1.81065 
 2.15090 
 2.57746 
 3.10746 
 
 1.60946 
 1.90430 
 2.37557 
 3.44218 
 
 1,2 
 1.4 
 1.6 
 1.8 
 
 2.0 
 
 0.868589 
 
 9.131411 
 
 7.38906 
 
 0.13534 
 
 3.76220 
 
 3.62686 
 
 2.0 
 
 2.5 
 3.0 
 3.5 
 4.0 
 4.5 
 
 1.085736 
 1.302883 
 1.520030 
 1.737178 
 1.954325 
 
 8.914264 
 8.694117 
 8.479970 
 8.262822 
 8.045675 
 
 12.1825 
 20 . 0855 
 33.1154 
 54.5983 
 90.0170 
 
 0.082085 
 0.049797 
 0.030197 
 0.018316 
 0.011109 
 
 6.1323 
 10.0677 
 16.5728 
 27.3083 
 45.0141 
 
 6.0002 
 10.0178 
 16.5426 
 27.2900 
 45 . 0030 
 
 2.5 
 3.0 
 3.5 
 4.0 
 4.5 
 
 5.0 
 
 2.171472 
 
 7.828528 
 
 148.413 
 
 0.006738 
 
 74.210 
 
 74.203 
 
 5.0 
 
 6 
 7 
 8 
 9 
 
 2 . 605767 
 3.040061 
 3.474356 
 3.908650 
 
 7.394233 
 6.959939 
 6.525644 
 6.091350 
 
 403.428 
 1096.63 
 2980.96 
 8103.08 
 
 0.002479 
 0.000912 
 0.000335 
 0.000123 
 
 201.715 
 
 201.713 
 
 6 
 
 7 
 8 
 9 
 
 10 
 
 for x> 6 
 
 10 
 
 4.342945 
 
 5.657055 
 
 22026 . 5 
 
 0.0000454 
 
 
 12 
 14 
 16 
 18 
 
 5.211534 
 6.080123 
 6.948712 
 7.817301 
 
 4.788466 
 3.919877 
 3.051288 
 2.182699 
 
 162755 
 
 1202610 
 
 888612(1 
 
 65660000 
 
 0.0000061 
 0.00000083 
 0.00000011 
 0.00000002 
 
 ■ 
 
 12 
 14 
 16 
 18 
 
 20 
 
 20 
 
 8.685890 
 
 1.314110 
 
 485166000 
 
 0.00000000 
 
 
INDEX 
 
 B 
 
 Abelian integrals and functions, 305 
 Absolute number, 4 
 value of fractional expression, 49 
 of general number, 30 
 Accuracy, loss of, 281 
 
 of approximation estimated, 200 
 of calculation, 279 
 of curve equation, 210 
 of transmission line equations, 208 
 Addition, 1 
 
 of general number, 28 
 and subtraction of trigonometric 
 functions, 102 
 Algebra of general number or com- 
 plex quantity, 25 
 Algebraic expression, 294 
 
 function, 75 
 Alternating current and voltage vec- 
 tor, 41 
 functions, 117, 125 
 waves, 117, 125 
 Alternations, 117 
 
 Alternator short circuit current, ap- 
 proximated, 195 
 Analytical calculation of extrema, 
 152 
 function, 294 
 Angle, see also Phase angle. 
 Approximation calculation, 280 
 
 by chain fraction, 208c 
 Approximations giving (1 + s) and 
 (1 - s), 201 
 of infinite series, 53 
 methods of, 187 
 Arbitrary constants of series, 69, 79 
 Area of triangle, 106 
 Arrangement of numerical calcula- 
 tions, 275 
 Attack, method of, 275 
 
 Base of logarithm, 21 
 
 Binomial series with small quanti- 
 ties, 193 
 theorem, infinite series, 59 
 of trigonometric function, 104 
 
 Biquadratic parabola, 219 
 
 Calculation, accuracy, 279 
 checking of, 291 
 numerical, 258 
 reliability, 271 
 
 Capacity, 65 
 
 Chain fraction, 208 
 
 Change of curve law, 211, 234 
 
 Characteristics of exponential curves, 
 228 
 of parabolic and hyperbolic curves, 
 223 
 
 Charging current maximum of con- 
 denser, 176 
 
 Checking calculations, 293a 
 
 Ciphers, number of, in calculations, 
 282 
 
 Circle defining trigonometric func- 
 tions, 94 
 
 Coefficients, unknown, of infinite 
 series, 60 
 
 Combination of exponential func- 
 tions, 231 
 of general numbers, 28 
 of vectors, 29 
 
 Comparison of exponential and hy- 
 perbolic curves, 229 
 
 Complementary angles in trigono- 
 metric functions, 99 
 
 Complex imaginary quantities, see 
 General number. 
 
 315 
 
316 
 
 INDEX 
 
 Complex, quantity, 17 
 algebra, 27 
 see General number. 
 Conjugate numbers, 31 
 Constant, arbitrary of series, 69, 79 
 errors, 186 
 
 factor with parabolic and hyper- 
 bolic curves, 223 
 phenomena, 106 
 terms of curve equation, 211 
 of empirical curves, 234 
 in exponential curves, 230 
 with exponential curves, 229 
 in parabolic and hyperbolic 
 curves, 225 
 Convergency determinations of 
 potential series, 215 
 of series, 57 
 Convergent series, 56 
 Coreless by potential series, 213 
 
 curve evaluation, 244 
 Cosecant function, 98 
 Cosh function, 305 
 Cosine-amplitude, 299 
 
 components of wave, 121, 125 
 function, 94 
 series, 82 
 
 versed function, 98 
 Cotangent function, 94 
 Counting, 1 
 
 Current change curve evaluation, 
 241 
 of distorted voltage wave, 169 
 input of induction motor, ap- 
 proximated, 191 
 maximum of alternating trans- 
 mission circuit, 159 
 Curves, checking calculations, 2936 
 empirical, 209 
 law, change, 234 
 rational equation, 210 
 use of, 284 
 
 D 
 
 Data on calculations and curves, 271 
 
 derived from curve, 285 
 Decimal error, 2936 
 
 Decimals, number of, in calculations, 
 282 
 
 in logarithmic tables, 281 
 Definite integrals of trigonometric 
 
 functions, 103 
 Degrees of accuracy, 279 
 Delta-amplitude, 299 
 Differential equations, 64 
 
 of electrical engineering, 65, 78, 86 
 
 of second order, 78 
 Differentiation of trigonometric 
 
 functions, 103 
 Diophantic equations, 186 
 Distorted electric waves, 108 
 Distortion of wave, 139 
 Divergent series, 56 
 Division, 6 
 
 of general number, 42 
 
 with small quantities, 190 
 Double angles in trigonometric 
 functions, 103 
 
 peaked wave, 255, 260, 266 
 
 periodicity of elliptic functions, 
 299 
 
 scale, 289 
 
 E 
 
 Efl&ciency maximum of alternator, 
 162 
 of impulse turbine, 154 
 of induction generator, 177 
 of transformer, 1 55, 174 
 Electrical engineering, differential 
 
 equations, 65, 78, 86 
 Ellipse, length of arc, 301 
 Elliptic integrals and functions, 299 
 Empirical curves, 209 
 , evaluation, 233 
 equation of curve, 210 
 Engineering differential equations, 
 65, 78, 86 
 reports, 290 
 Equilateral hyperbola, 217 
 Errors, constant, 186 
 numerical, 2936 
 of observation, 180 
 
INDEX 
 
 317 
 
 Estimate of accuracy of approxima- 
 tion, 200 
 Evaluation of empirical curves, 233 
 Even functions, 81, 98, 305 
 periodic, 122 
 harmonics, 117, 266 
 
 separation, 120, 125, 134 
 Evolution, 9 
 
 of general number, 44 
 of series, 70 
 Exact calculation, 281 
 Exciting current of transformer, 
 
 resolution, 137 
 Explicit analytic function, 294 
 Exponent, 9 
 Exponential curves, 227 
 forms of general number, 50 
 functions, 52, 297, 304 
 
 with small quantities, 196 
 series, 71 
 
 tables, 312, 313, 314 
 and trigonometric functions, rela- 
 tion, 83 
 Extrapolation on curve, limitation, 
 
 210 
 Extrema, 147 
 analytic determination, 152 
 graphical construction of differen- 
 tial function, 170 
 graphical determination, 147, 150, 
 
 168 
 ■with intermediate variables, 155 
 with several variables, 163 
 simplification of function, 157 
 
 Factor, constant, with parabolic 
 
 and hyperbolic curves, 223 
 Fan motor torque by potential ser- 
 ies, 215 
 Fifth harmonic, 261, 264 
 Flat top wave, 255, 260, 265, 268 
 zero waves, 255, 258, 261, 265 
 Fourier series, see Trigonomelric 
 
 series. 
 Fraction, 8 
 as series, 52 
 chain-, 208 
 
 Fractional exponents, 11, 44 
 
 expressions of general number, 49 
 Full scale, 289 
 Functions, theory of, 294 
 
 G 
 
 Gamma function, 304 
 General number, 1, 16 
 algebra, 25 
 
 engineering reports, 291 
 exponential forms, 50 
 reduction, 48 
 
 reports on engineering matters, 
 292 
 Geometric scale of curve plotting, 
 
 288 
 Graphical determination of extrema, 
 147, 150, 168 
 
 H 
 
 Half angles in trigonometric func- 
 tions, 103 
 Half waves, 117 
 Half scale, 289 
 Harmonics, even, 117 
 odd, 117 
 
 of trigonometric series, 114 
 two, in wave, 255 
 High harmonics in wave shape, 255, 
 
 269 
 Hunting of synchronous machines, 
 
 257 
 Hyperbola, arc of, 61 
 
 equilateral, 217 
 Hyperbolic curves, 216 
 functions, 294 
 
 curve, shape, 232 
 integrals and functions, 298 
 tables, 313, 314 
 Hyperelliptic integrals and func- 
 tions, 301 
 Hysteresis curve of silicon steel, in- 
 vestigation of, 248 
 
 Imaginary number, 26 
 quantity, see Quadrature number. 
 
318 
 
 INDEX 
 
 Incommensurable waves, 257 
 Indeterminate coefficients, method, 
 
 71 
 Indeterminate coefficients of infi- 
 nite series, 60 
 Individuals, 8 
 Inductance, 65 
 Infinite series, 52 
 
 values of curves, 211 
 of empirical curves, 233 
 Inflection points of curves, 153 
 Impedance vector, 41 
 Implicit analytic function, 294 
 Integral function, 295 
 Integration constant of series, 69, 79 
 
 of differential equation, 65 
 
 by infinite series, 60 
 
 of trigonometric functions, 103 
 Intelligibility of calculations, 283 
 Intercepts, defining tangent and co- 
 tangent functions, 94 
 Involution, 9 
 
 of general numbers, 44 
 Irrational numbers, 11 
 Irrationality of representation by 
 potential series, 213 
 
 J, 14 
 
 Least squares, method of, 179, 186 
 Limitation of mathematical repre- 
 sentation, 40 
 
 of method of least squares, 186 
 
 of potential series, 216 
 Limiting value of infinite series, 54 
 Linear number, 33 
 
 see Positive and Negative number. 
 Line calculation, 276 
 
 equations, approximated, 204 
 Logarithm of exponential curve, 229 
 
 as infinite series, 63 
 
 of parabolic and hyperbolic curves, 
 225 
 
 with small quantities, 197 
 
 Logarithmation, 20 
 
 of general numbers, 51 
 Logarithmic curves, 227 
 
 functions, 297 
 
 paper, 233, 287 
 
 scale, 288 
 
 tables, number of decimals in, 281 
 Loss of curve induction motor, 183 
 
 M 
 
 Magnetic characteristic on semi- 
 logarithmic paper, 288 
 Magnetite arc, volt-ampere charac- 
 teristic, 239 
 
 characteristic, evaluation, 246 
 Magnitude of effect, determination, 
 
 280 
 Maximum, see Extremum. 
 Maxima, 147 
 
 McLaurin's series with small quan- 
 tities, 198 
 Mechanism of calculating empirical 
 
 curves, 237 
 Methods of calculation, 275 
 
 of intermediate coefficients, 71 
 
 of least squares, 179, 186 
 
 of attack, 275 
 Minima, 147 
 
 Minimum, see Extremum. 
 Multiple frequencies of waves, 274 
 Multiplicand, 39 
 Multiplication, 6 
 
 of general numbers, 39 
 
 with small quantities, 188 
 
 of trigonometric functions, 102 
 Multiplier, 39 
 
 N 
 
 Negative angles in trigonometric 
 functions, 98 
 
 exponents, 11 
 
 number, 4 
 Nodes in wave shape, 256, 270 
 Non-periodic curves, 212 
 Nozzle efficiency, maximum, 150 
 Number, general, 1 
 
INDEX 
 
 319 
 
 Numerical calculations, 275 
 values of trigonometric functions, 
 101 
 
 
 
 Observation, errors, 180 
 Octave as logarithmic scale, 288 
 Odd funcfons, 81, 98, 305 
 
 period c, 122 
 harmonics in symmetrical wave, 
 117 
 
 separation, 120, 125, 134 
 Omissions in calculations, 2936 
 Operator, 40 
 
 Order of small quantity, 188 
 Oscillating functions, 92 
 Output, see Power. 
 
 IT and 2 added and subtracted in 
 
 trigonometric function, 100 
 approximated by chain fraction, 
 208c 
 Pairs of high harmonics, 270 
 Parabola, common, 218 
 Parabolic curves, 216 
 Parallelogram law of general num- 
 bers, 28 
 of vectors, 29 
 Peaked wave, 255, 258, 261, 264 
 Pendulum motion, 301 
 Percentage change of parabolic and 
 
 hyperbolic curves, 223 
 Periodic curves, 254 
 decimal fraction, 12 
 phenomena, 106 
 Periodicity, double, of elliptic func- 
 tions, 299 
 of trigonometric functions, 96 
 Permeability maximum, 148, 170 
 Phase angle of fractional expression, 
 49 
 of general number, 28 
 Plain number, 32 
 
 see General number. 
 
 Plotting of curves, 212 
 
 proper and improper, 286 
 of empirical curve, 234 
 Polar co-ordinates of general num- 
 ber, 25, 27 
 expression of general number, 25, 
 27, 38, 43, 44, 48 
 Polyphase relation, reducing trigo- 
 nometric series, 134 
 of trigonometric functions, 104 
 system of points or vectors, 46 
 Positive number, 4 
 Potential series, 52, 212 
 Power factor maximum of induction 
 motor, 149 
 maximum of alternating trans- 
 mission circuit, 158 
 of generator, 161 
 of shunted resistance, 155 
 of storage battery, 172 
 of transformer, 173 
 of transmission line, 165 
 not vector product, 42 
 of shunt motor, approximated, 189 
 with small quantities, 194 
 Probability calculation, 181 
 Product series, 303 
 
 of trigonometric functions, 102 
 Projection, defining cosine function, 
 
 94 
 Projector, defining sine function, 94 
 
 Q 
 
 Quadrants, sign of trigonometric 
 
 functions, 96 
 Quadrature numbers, 13 
 Quarter' scale, 289 
 Quaternions, 22 
 
 R 
 
 Radius vector of general number, 28 
 Range of convergency of series, 56 
 Ratio of variation, 226 
 Rational equation of curve, 210 
 
 function, 295 
 Rationality of potential series, 214 
 
320 
 
 INDEX 
 
 Real number, 26 
 
 Rectangular co-ordinates of general 
 
 number, 25 
 Reduction to absolute values, 48 
 Relations of hyperbolic trigono- 
 metric and exponential func- 
 tions, 309 
 Relativeness of small quantities, 188 
 Reliability of numerical calculations, 
 
 293 
 Reports, engineering, 290 
 Resistance, 65 
 Resolution of vectors, 29 
 Reversal by negative unit, 14 
 
 double, at zero of wave, 258, 261 
 Reverse function, 294 
 Right triangle defining trigonomet- 
 ric functions, 94 
 Ripples in wave, 45 
 
 by high harmonics, 270 
 Roots of general numbers, 45 
 
 expressed by periodic chain frac- 
 tion, 208e 
 with small quantities, 194 
 of unit, 18, 19, 46 
 Rotation by negative unit, 14 
 by quadrature unit, 14 
 
 Saddle point, 165 
 
 Saw-tooth wave, 246, 255, 258, 260, 
 
 265 
 Scalar, 26, 28, 30 
 
 Scale in curve plotting, proper and 
 improper, 212, 286 
 
 full, double, halt, etc., 287 
 Scientific engineering records, 291 
 Secant function, 98 
 Second harmonic, effect of, 266 
 Secondary effects, 210 
 
 phenomena, 234 
 Semi-logarithmic paper, 287 
 Series, exponential, 71 
 
 infinite, 52 
 
 trigonometric, 106 
 Seventh harmonic, 262 
 
 Shape of curves, 212 
 
 proper in plotting, 286 
 
 of exponential curve, 227, 230 
 
 of function, by curve, 284 
 
 of hyperbolic functions, 232 
 
 of parabolic and hyperbolic curves, 
 217 
 Sharp zero wave, 255, 260, 265 
 Short circuit current of alternator, 
 
 approximated, 195 
 Sign error, 293c 
 
 of trigonometric functions, 95 
 Silicon steel, investigation of hystere- 
 sis curve, 248 
 Simplification by approximation, 187 
 Sine-amplitude, 199 
 
 component of wave, 121, 125 
 
 function, 94 
 
 series, 82 
 
 versus function, 98 
 Sine function, 305 
 Slide rule accuracy, 281 
 Small quantities, approximation, 187 
 Special functions, 302 
 Squares, least, method of, 179, 186 
 Steam path of turbine, 33 
 Subtraction, 1 
 
 of general number, 28 
 
 of trigonometric functions, 102 
 Summation series, 303 
 Superposition of high harmonics, 273 
 Supplementary angles in trigono- 
 metric functions, 99 
 Surging of synchronous machines, 
 
 301 
 Symmetrical curve maximum, 150 
 
 periodic function, 117 
 
 wave, 117 
 
 Tabular form of calculation, 275 
 
 Tangent function, 94 
 
 Taylor's series with small quantities, 
 
 199 
 Temperature wave, 131 
 Temporary use of potential series, 
 
 216 
 
INDEX 
 
 321 
 
 Termmal conditions of problem, 69 
 Terms, constant, of empirical curves, 
 234 
 in exponential curve, 229 
 with exponential curve, 229 
 in parabolic and hyperbolic 
 curves, 225 
 of infinite series, 53 
 Theorem, binomial, infinite series, 
 
 59 
 Thermomotive force wave, 133 
 Theta functions, 300 
 Third harmonic, 136, 255 
 Top, peaked or flat, of wave, 255 
 Torque of fan motor by potential 
 
 series, 215 
 Transient current cnrve, evaluation, 
 241 
 phenomena, 106 
 Transmission equations, approxi- 
 mated, 204 
 line calculation, 275 
 Treble peak of wave, 262 
 Triangle, defining trigonometric 
 functions, 94 
 trigonometric relations, 106 
 Trigonometrical and exponential 
 functions, relations, 83 
 functions, 94, 304 
 series, 82 
 
 with small quantity, 198 
 integrals and functions, 298 
 series, 106 
 
 calculation, 114, 116, 139 
 Triple harmonic, separation, 136 
 peaked wave, 255 
 scale, 289 
 
 Tungsten filament, volt-ampere 
 
 characteristic, 235 
 Turbine, steam path, 33 
 
 U 
 
 Unequal height and length of half 
 
 waves, 268 
 Univalent functions, 106 
 Unsymmetric curve maximum, 151 
 wave, 138 
 
 Values of trigonometric functions, 
 
 101 
 Variation, ratio of, 226 
 Vector analysis, 32 
 multiplication, 39 
 quantity, 32 
 
 see General number. 
 representation by general number, 
 29 
 Velocity diagram of turbine steam 
 path, 34 
 functions of electric field, 304 
 Versed sine and cosine functions, 
 
 98 
 Volt-ampere characteristic of mag- 
 netite arc, 239 
 of tungsten filament, 235 
 
 Z 
 
 Zero values of curve, 211 
 of empirical curves, 233 
 of waves, 255