Illlll™ ™ ;i!l:l!lill'l!li:!li.ll|l mil iiiii ii Dt^ata, ^tm fattt BOUGHT WITH THE INCOME OF THE SAGE ENDOWMENT FUND THE GIFT OF HENRY W. SAGE 1891 Cornell University Library olin.anx 3 1924 031 245 1 Cornell University Library The original of tliis 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/cletails/cu31924031245156 ELEMENTARY MATHEMATICAL ANALYSIS MODERN MATHEMATICAL TEXTS EDITED BY Charles S. Slighter ELEMENTARY MATHEMATICAL ANALYSIS By CHABLsa S. Slichteb i97 pages, 5% 7ii,niiiatraled $2,60 MATHEMATICS FOR AGRICULTURAL STUDENTS By Henry C. Wolff 311 panes, 5 % 7H, lUustraled . $1.50 CALCULUS By Herman W. March and Henry C. Wolff 360 pages, 5 I 7H, Illustrated. . $2.00 PROJECTIVE GEOMETRY , By L. Watland Dowling 316 pages, a ii7H,IllustraUd $2.00 MODERN MATHEMATICAL TEXTS Edited bt Chaelbs S. Slichtee ELEMENTARY MATHEMATICAL ANALYSIS A TEXT BOOK FOR FIRST YEAR COLLEGE STUDENTS BY CHARLES S. SLICHTER, Sc. D. PROFESSOR OF APPLIED MATHEMATICS ■DNIVERSITY OP WISCONSIN Second Edition Revised and Entirely Reset McGRAW-HILL BOOK COMPANY, Inc. . 239 WEST 39TH STREET. NEW YORK LONDON: HILL PUBLISHING CO., Ltd. 6 & 8 BOUVBKIE ST., E. C- 1918 A- COPYEIGHT, 1914, 1918, BY THe] McGeaw-Hill Book Company, Inc. TMT- MAPtK PUESS roBK FA, PREFACE TO THE SECOND EDITION In rewriting the present book, simplification of the material has been the main end in view. Considerable matter has been omitted, and numerous worked exercises have been added. The second chapter is devoted to an introduction to rectangular coordinates and to the straight line. New sets of exercises and long lists of miscellaneous and review exercises have been inserted at appropriate places. Changes in order of material and in method of treatment have been made freely. Much greater use has been made of fine print than in the first edition. Sections of the text which can readily be omitted have a star attached to the section numbers. Some of these of a sub- ordinate illustrative character, or primarily intended for reference, are put in fine print. The review chapter on elementary algebra has been greatly enlarged. This material is placed in the last chapter or appendix, where the considerable amount of very elementary mathematics will not at once confront and perhaps discourage the well-prepared student. At the same time enough material is given so that students with but a single year of high school algebra can be gotten ready for the course. The elementary material is so classified that either a few days, or several weeks may be devoted to the review. The writer is greatly indebted to many persons for aid in the revision. Professors March and Wolff of the University of Wisconsin have contributed much, and Professor Wolff has read all of the galley proof. I am especially indebted to Professors Jordan and Lefschetz of the University of Kansas for many valuable suggestions and to Professor L. C. Plant of Lansing, Michigan. To all of these my especial thanks are due. Charles S. Slichter. FROM THE PREFACE TO THE FIRST EDITION This book is not intended to be a text on "Practical Mathe- matics" in the sense of making use of scientific material and of fundamental notions not already in the possession of the student, or in the sense of making the principles of mathematics secondary to its technique. On the contrary, it has been the aim to give the ftmdamental truths of elementary analysis as much promi- nence as seems possible in a working course for freshmen. The emphasis of the book is intended to be upon the notion of functionality. Illustrations from science are freely used to make this concept prominent. The student should learn early in his course that an important purpose of mathematics is to express and to interpret the laws of actual phenomena and not primarily to secure here and there certain computed results. Mathematics might well be defined as the science that takes the broadest view of all of the sciences — an epitome of quantitative knowledge. The introduction of the student to a broad view of mathematics can hardly begin too early. , The ideas explained above are developed in accordance with a two-fold plan, as follows : First, the plan is to group the material of elementary analysis about the consideration of the three fundamental functions: 1. The Power Function y = ax" (n any number) or the law "as X changes by a fixed multiple, y changes by a fixed multiple also." 2. The Simple Periodic Function y = asin mx, considered as fundamental to all periodic phenomena. 3. 'The Exponential Function, or the law "as x changes by a fixed increment, y changes by a fixed multiple." Second, the plan is to enlarge the elementary functions by the development of the fundamental transformations applicable to these and other functions. To avoid the appearance of abstruse- Viii PREFACE ness, these transformations are stated with respect to the graphs of the functions; that is, they are not called transformations, but "motions" of the loci. The facts are summarized in several simple "Theorems on Loci," which explain the translation, rota- tion, shear, and elongation or contraction of the graph of any function in the xy plane. Combinations of the fundamental functions as they actually occur in the expression of elementary natural laws are also dis- cussed and examples are given of a type that should help to ex- plain their usefulness. Emphasis is placed upoji the use of time as variable. This enriches the treatment of the elementary functions and brings many of the facts "analytic geometry" into close relation to their application in science. A chapter on waves is intended to give the student a broad view of the use of the trigonometric func- tions and an introduction to the application of analysis to periodic phenomena. It is difficult to understand why it is customary to introduce the trigonometric functions to students seventeen or eighteen years of age by means of the restricted definitions applicable only to the right triangle. Actual test shows that such rudimentary methods are wasteful of time and actually confirm the student in narrow- ness of view and in lack of scientific imagination. For that reason, the definitions, theorems and addition formulas of trigonometry are kept as general as practicable and the formulas are given general demonstrations. The possibiUties and responsibiUties of character building in the department of mathematics are kept constantly in mind. It is accepted as fundamental that a modern working course in mathe- matics should emphasize proper habits of work as well as proper methods of thought; that neatness, system, and orderly habits have a high value to all students of the sciences, and that a text- book should help the teacher in every known way to develop these in the student. ' The present work is a revision and rewriting of a preliminary form which has been in use for three years at the University of Wisconsin. During this time the writer has had frequent and valuable assistance from the instructional force of the department PREFACE IX of mathematics in the revision and betterment of the text. Ac- knowledgments are due especially to Professors Burgess, Dresden, Hart and Wolfif and to Instructors Fry, Nyberg and Taylor. Professor Burgess has tested the text in correspondence courses, and has kindly embraced that opportunity to aid very materially in the revision. He has been especially successful in shortening graphical methods and in adapting them to work on squared paper. Professor Wolff has read all of the final manuscript and made many suggestions based upon the use of the text in the class room. Mr. Taylor has read all of the proof and supphed the results to the exercises. Professor E. V. Huntington of Harvard University has read the galley proof and has contributed many important suggestions. The writer has avoided the introduction of new technical terms, or terms used in an unusual sense. He has taken the liberty, how- ever of naming the function ax", the "Power Function of x," as a short name for this important function seems to be an unfortu- nate lack — ^a lack, which is apparently confined solely to the Enghsh language. Chables S. Slichtbr. University or Wisconsin July 25, 1914 Note: The results to the exercises are issued aa a separate pamphlet. CONTENTS Faqb Preface . . v Intbodtjction .... . . xiii Mathematical Signs and Symbols. ... . . . xviii Chaptbr I. Vakiablbb and Functions op Vabiablbs ... . 1 II. Rectanqtjlak Cookdinatbs and the Straight Line ... ... .... 23 III. The Powbe Function . 48 MiSCELLANBOITS ExEBCISES. .... 92 IV. The Circle and the Circular Functions .... 97 V. The Ellipse and Htpebbola .152 VI. Single and Simultaneous Equations 174 VII. Permutations, Combinations; the Binomial Theorem . ... 198 VIII. Progressions . . ... 213 Questions and Exercises for Review, Chapters I to VIII 225 IX. The Logarithmic and Exponential Functions . 234 X. Tbigonombtbic Equations and the Solution of Tbiangles 304 XI. Simple Harmonic Motion and Waves . . . 339 XII. Complex Numbebs 357 XIII. Loci 387 XIV. The Conic Sections . . 399 XV. Appendix — A Review of Sbcondabt School Algbbba 451 Mathematical Tables 474 Index 491 INTRODUCTION Any course in mathematics requires the frequent use of geo- metrical constructions, and the carrying out of analytical and numerical computations. In order that this work may be per- formed neatly and accurately it is necessary that the student have a few simple instruments, and a supply of proper material for doing the work in a systematic and orderly manner. The indispensible instruments are as follows : I. Instruments. (1) Two 4:H hexagonal drawing pencils; one sharpened to a fine point for marking points upon paper or for sketch- ing free hand; the other sharpened to a chisel point for drawing straight lines. Some prefer to use a single pencil sharpened at both ends, one end round pointed, the other end chisel pointed. (2) A small drawing board' of soft wood — 10X12 inches is large enough. (3) A small T-square same length as the drawing board. (4) A 60° and a 45° transparent triangle. Five-inch triangles are large enough, although a larger 60° triangle will be found to be very convenient. (5) A protractor for laying off angles. (6) A triangular boxwood scale, decimally divided. (7) A pair of 6-inoh pencil compasses for drawing circles and arcs of circles, provided with medium hard lead, sharpened to a narrow chisel point. (8) A 10-inch sUde rule is required for Chapter IX, and may be used earlier at the discretion of the instructor. II. Materials. All mathematical work should be done on one side of standard size letter paper, 8^ X 11 inches. This is the smallest sheet that permits proper arrangement of mathematical work. A good equipment will include: (1) A notQ book cover to hold sheets of the above named size and ^Drawing boards of this size with T-square and two wood triangles are marketed by the Milton Bradly Co., Springfield, Mass. xiii XIV INTRODUCTION a supply of manUa paper "vertical file folders" for use in submit- ting work for the examination of the instructor. (2) A number of different forms of squared paper and computa- tion paper especially prepared for use with this book. These sheets will be described from time to time as needed in the work. Form M2 wiU be found convenient for problem work and for general cal- culation. M2 is a copy of a form used by a number of public utiUty and industrial corporations. Colleges usually have their own sources of supply of squared paper, satisfactory for use with this book. (3) Miscellaneous supphes such as thumb tacks, erasers, sandpaper- pencil-sharpeners, etc. in. General Directions. All drawings should be done in pencil, unless the student has had training in the use of the ruling pen, in which case he may, if he desires, "ink in" a few of the most important drawings. AU mathematical work, such as the solutions of problems and exercises, and work in computation should be done in ink. The student should acquire the habit of working problems with pen and ink. He will find that this habit will materially aid him in repressing carelessness and indifference and in acquiring neatness and system. TO THE STUDENT— SUGGESTIONS ON THE STUDY OF MATHEMATICS The following suggestions may assist the student to acquire habits of work essential to success in the study of mathematics and of the other exact sciences. Successful intellectual work depends very largely upon the power of concentration. Fortxmately this power can be acquired and culti- vated'. The student should study away from interruption and then must not permit his work to become interrupted by himseU or by others. By holding his attention upon his work and by keeping his mind from wandering to extraneous matters, the student will cultivate a fundamental habit that will tend to assure his success both in and out of college. In a course in mathematics a student (1) studies a textbook and (2) works exercises and problems. An assigrmient for a given day may therefore consist of the study of mathematical principles and theory (such as theorems, definitions, and explanations of processes), or it may consist of the working out of exercises and problems, or, as is usually the case, it may consist of the theory and principles of INTRODUCTION xv processes, together with an assignment of exercises illustrative of the theory. 1. THE STUDY OF THE TEXTBOOK Studying a mathematical textbook involves much more than the mere reading of the statements of principles and of the explanation of processes. The student must usually read the assigned paragraphs several times and must frequently turn back and re-read portions of the text included in previous lessons. In this manner the various points in the reasoning or explanations can be thought over, and the habit of asking self-put questions about the work can be acquired. First of all, in preparing a lesson, try to find out what 'it is about — what its purpose is. Try to decide how you yourself would go about the aocompUshment of the task and, if possible, make an independent attempt of your own. The more consideration you give to such an attempt, the greater scientific power you will gain. In particular: (A) The student should remember that the words in science have exact meanings and, of course, these meanings must be known to the student. In studying mathematics the student should acquire and use the language oi mathematics. For example, he should not say ' equation" when he means "expression." Indeed, he should go farther than this. He should make a conscious effort to use abso- lutely correct English, not only in written work but in oral work as well. (B) While studying the text, work out theorems or illustrative examples with pen and ink. Do not rely upon a mere reading — even repeated readings — of a new piece of reasoning or of the explana- tion of a new process. (C) Bead over all of the lesson assigned in the text a last time after working the assigned exercises. The text will probably have a new meaning after working out the special cases in the exercises. This habit will give a meaning to the words, "Learn by doing." (D) Finally, make a mental simimary of each lesson. (E) Review often. 2. THE WORKING OF EXERCISES (F) Read each exercise or problem carefully and plan a method of attack in advance in order to facilitate arrangements of equations and computations and the drawing of figures. (G) Look at your result and see if it is a reasonable one. XVI INTRODUCTION (H) Check result. (I) Indicate the results by a distinguishing mark, or summarize in logical qrder. (J) The figures and diagrams should have sufficient lettering, titles, etc., to make them self-explanatory. The units of measure used should, of course, be clearly indicated. (K) Do all work neatly the first time and (except drawings) invariably in ink. Try to have the first draft sufficiently neat in appearance and arrangement to hand in to your instructor. (L) After the first draft has been finished, read it over carefully to see where it may be unproved in method or arrangement and think about the processes you have used. If small changes only are needed to effect the desired improvement, make them by drawing lines through the portions to be changed and by making neat inser- tions. If considerable changes are necessary, do the work over. The study and improvement of the work will prove to be of fully as much importance to you as the doing of the work itself. (M) See to it that each piece of work or exercise is complete. On any piece of written work the nature of the problem should be clearly and briefly stated. The student should learn to think of each piece of work as a thing that is in itself worth while. Hence each detail should be attended to before the work is submitted to the instructor. See that sufficient explanation is given and that the numbers and magnitudes are adequately named and labelled. TO THE INSTRUCTOR The instructor cannot insist too emphatically upon the require- ment that all mathematical work done by the student — ^whether preliininary work, numerical scratch work, or any other kind (except drawings) — shall be carried out with pen and ink upon paper of suitable size. This should, of course, include all work done at home, irrespective of whether it is to be submitted to the instructor or not. The "psychological effect" of this requirement will be found to entrain much more than the acquirement of mere technique. If properly insisted upon, orderly and systematic habits of work will lead to orderly and systematic habits of thought. The final results will be very gratifying to those who sufficiently persist in this requirement. At institutions whose requirements for admisstion include more than one and one-half units of preparatory algebra, nearly all of Chapters VI, VII, and VIII may be omitted from the course. An asterisk attached to a section number indicates that the section INTRODUCTION xvii may he omitted. These sections will frequently be found useful in forming the basis of discussion by the instructor. The usual one and one-half year of secondary school Algebra, including the solution of quadratic equations and a knowledge of fractional and negative exponents, is required for the work of this course. In the appendix (Chapter XV) vrill be found material for a brief review of factoring, qitadratics, and exponents, upon which a week or ten days should be spent before beginning the regular work in this text. This review chapter is placed last because the amount of material in it is greater than need be taken in all cases and also because college students do not like to be confronted on the &st page of a scientific text-book with elementary work of high school grades. GREEK ALPHABET Capitals Lower case Names Capitals Lower case Names A a Alpha N V Nu B P Beta S i Xi r y Gamma o Omicron A s Delta II IT Pi E e Epsilon p 9 Rho Z f Zeta s a Sigma , H V Eta T T Tau e e Theta T V Upsilon I L Iota * Phi K K Kappa X X Chi A X Lambda * i' Psi M /< Mu Q U> Omega MATHEMATICAL SIGNS AND SYMBOLS read = read 5^ read = read =F read > read < read 1^ read (a,b) read |n read n! read limit r,, ,' read a; = oo read \a\ read log„a; read Iga; read In a; read and so on. is identical vnth. is not equal to. approaches. is approximately equal to. is greater than. is less than. is greater than or equal to. point whose coordinates are a and b. factorial n. factorial w or n admiration. limit of fix) as x approaches a. X becomes infinite, absolute value of a. logarithm of x to the base a. common logarithm of x. natural logarithm of x. Sw„ read summation from n = 1 io n = r of u„ ELEMENTARY MATHEMATICAL ANALYSIS CHAPTER I VARIABLES AND FUNCTIONS OF VARIABLES 1. Scales. Select a series of points along any curve and mark the points of division with the numbers of any sequence.' The result of such a construction is called a scale. Thus in Fig. 1 the points along the curve OA have been selected and marked in order with the numbers of the sequence: Oj 4j 2> 1> 25, 3, 5, 7, 8 A non-uniform scale. Thus primitive man might have made notches along a twig and then made use of it in making certain measurements of interest to him. If such a scale were to become generally used by others, it would be desirable to make many copies of the original scale. It would, therefore, be necessary to use a twig whose shape could be readily duplicated; such, for example, as a straight stick; and it would also be necessary to attach the same symbols invariably to the same divisions. Certain advantages are gained (often at the expense of others, however) if the distances between consecutive points of division are kept the same; that is, when the intervals are laid off by repe- tition of the same selected distance. When this is done, the scale is called a uniform scale. Primitive man might have selected for 1 A sequence of numbers here means a set of numbers arranged in order of magnitude. 1 2 ELEMENTARY MATHEMATICAL ANALYSIS [§1 such uniform distance the length of his foot, or sandal, the breadth of his hand, the distance from elbow to the end of the middle finger (the cubit), the length of a step in pacing (the yard), the amount he can stretch with both arms extended (the fathom), etc. Fig. 2. — An ammeter scale. We are familiar with many scales, such as those seen on a yardstick, the dial of a clock, a thermometer, a sun-dial, a steam- gage, an ammeter or voltmeter, the arm of a store-keeper's scales, etc. The scales on a clock, a yardstick, or a steel tape are uniform. Those on a sun-dial, on some ammeters or on a good thermometer, are not uniform. One of the most important advantages of a uniform scale is the fact that the place of beginning, or zero, maybe taken at any one of the points of divi- sion. This is not true of a non- uniform scale. If a sun-dial is not properly oriented, it is useless. If the needle of an ammeter be bent the instrument cannot be used. It is always necessary in using such an instrument to know that the zero is correct. If, however, a yarfistick or a steel tape be broken, it may stUl be used for measuring lengths. The student P.M A.M. Sun-dial scale. §2] VARIABLES AND FUNCTIONS OF VARIABLES 3 may think of many other advantages gajned in using a uniform scale. 2. Formal Definition of a Scale. If points be selected in order along any curve corresponding, one to one, to the numbers of any sequence, the curve, with its divisions, is called a scale. The notion of one to one correspondence, included in this definition, is frequently used in mathematics. Ii II I h I II h I I I I 111 il I M I I II II h I I I h I I I h I I I I II I I I 1.2 3 1 5 Fig. 4. — A uniform arithmetical scale. In mathematics we frequently speak of the arithmetical scale and of the algebraic scale. The arithmetical scale corresponds to the numbers of the sequence 0, 1, 2, 3, 4, 5, . . and such intermediate numbers as may be desired. It is usually represented by a uniform scale as in Fig. 4. The algebraic scale corresponds to the numbers of the sequence . . . -6, -5, -4, -3,-2,-1,0,+!, +2, +3, +4, +5, . . . and such intermediate numbers as may be desired. It is usually 1 I I I I I I I 1 I I I I I I I 11 I I I M I I I I I I I I I I I I I I I I I I I II I I II I I 1 -B -4 -S -2 -1 +1 +2 +3 +4 +5 Fig. 5. — A uniform algebraic scale. represented by a uniform scale as in Fig. 5. The arithmetical scale begins at and extends indefinitely in one direction. The algebraic scale has no point of beginning; the zero is placed at any desired point and the positive and negative numbers are then attached to the divisions to the right and the left, respectively, of the zero so selected. The algebraic scale extends indefinitely in both directions. Exercises 1. On a uniform algebraic scale, how far is the point marked 5 from the point marked 7? How far is the point marked 6 from the point marked 10.5? How far is th'e point marked —10.8 from the point marked 13.6? 4 ELEMENTARY MATHEMATICAL ANALYSIS [§3 2. Show that the distance between two points selected anywhere on the uniform algebraic scale is always found by subtraction. 3. What points of the uniform algebraic scale are distant 5 from the point 3 of that scale? What point of the uniform arithmetical scale is distant 5 from the point 3 of that scale? 4. -If two algebraic scales intersect at right angles, the common point being the zero of both scales, explain how to find the distance from any point of one scale to any point of the other scale. 3. Two Uniform Scales in Juxtaposition or Double Scales. The relation between two magnitudes or quantities, or between two numbers, may be shown conveniently by placing two scales side by side. Thus the relation between the number of centi- meters and the number of inches in any length may be shown by placing a centimeter scale and a foot-rule side by side with their zeros coinciding as in Fig. 6. From this figure it is seen that 1 inch corresponds to 2.6 centimeters; 3.3 inches correspond to 8.44 centimeters, 4.6 inches corresponds to 11.76 centimeters; that 5 centimeters correspond to 1.97 inches, 8.5 centimeters corresponds to 3.32 inches, etc. A thermometer is frequently seen bearing both Fahrenheit and the centigrade scales. See Fig. 7. It is obvious that the double scale of such a thermometer may be used (within the limits of its range) for converting any temperature reading Fahrenheit into the corresponding centigrade equivalent or vice versa. From Fig. 7 it is seen that 72°F. corresponds to 22.2°C., 212°F. to 100°C., 32°F. to zero degrees centigrade; that 21°C. corresponds to 69.8°F., 72''C. to 161.6°F. The construction of scales of the kind considered above may be made to depend upon the following problem in elementary geometry : To divide a given line into a given number of equal parts. Illustration. In Fig. 9 is given a double scale OA-OB showing the correspondence between speed expressed in miles per hour and speed expressed in feet per second. The student will reproduce neatly and accurately the drawing, on a larger scale, in accordance with the directions given below, A mUe contains 5280 feet, an hour contains 3600 seconds. Hence, one mile per hour equals |f-§-J or -f-| feet per second. Therefore, if two uniform scales. Fig. 9, one rejJresenting speed expressed in feet per second, and the other representing speed expressed in miles §3] VARIABLES AND FUNCTIONS OF VARIABLES -r-S _ — CO o -pas o « I a «3 6 M o o c3 &0 I - ag o V ■ g g =• »•§ •a ^ I S "» _ O I- _ 5? fe o o 00 f=( 6 ELEMENTARY MATHEMATICAL ANALYSIS [§3 20- B- 18- 11- i,;i2- 1^ 6- /o. Fia. 9. — Method of constraction of double scale showing relation between "miles per hour" and "feet per second." §3] VARIABLES AND FUNCTIONS OF VARIABLES 7 per hour, are constructed with their zeros coinciding and with the point marked 22 of the first coinciding with the point marked 15 of the second, the double scale may be used for converting speed expressed as miles per hour into speed expressed as feet per second, or vice versa. Lay off with a scale a line OA 11 inches long. Divide this line into 22 equal parts, and subdivide each division into 6 equal parts. Mark these divisions and subdivisions as indicated in Fig. 9. Draw the line OK, making the angle KOB about 30°. With a pair of bow dividers or with a scale lay off on OK 15 equal divisions, about f of an inch each. Let the last point of division be marked C. OC is then divided into 15 equal parts. Draw CA. With a pair of triangles draw lines through the points Ci, Ci, Cz, . Cn parallel to CA, intersecting the line OA in the points marked 15, 14, 13, 1, respectively. Why is OB a uniform scale divided into 15 equal parts? Mark the scales OA and OB in red ink with a new set of numbers so that the double scale may also be used for converting speeds if the readings fall between 15 and 30 feet per second instead of between and 15. From the double scale just constructed, find the speeds expressed as miles per hour corresponding to speeds of 2, 4, 5, 11, 14, 20, and 25 feet per second. The lengths selected to represent the various units in any dia- gram are, of course, arbitrary. As, however, the student is expected to prepare the various constructions and diagrams required for the exercises in this book on paper of standard letter size (that is, 8 J by 11 inches), the units selected should be such as to permit a convenient and practical construction upon sheets of that size. Exercises The student is expected to carry out the actual construction of only one of the double scales described in the following exercises. 1. Draw a double scale showing the relation between pressure expressed as inches of mercury and as feet of water, knowing that the density of mercury is 13.6 times that of water. These are two of the common ways of expressing pressure. Water pressure at water power plants, and often for city water service, is expressed in terms of head in feet. Barometric pressure, and the vacuum in the suction pipe of a pump and in the exhaust of a con- densing steam engine are expressed in inches of mercury. The approximate relations between these units, i.e., 1 atmosphere = 30 8 ELEMENTARY MATHEMATICAL ANALYSIS [§3 inches of mercury = 32 feet of water' = 15 pounds per square inch, are known to every student of elementary physics. To obtain, in terms of feet of water, the pressure equivalent of a; feet of mercury, multiply X by 13.6, the specific gravity of mercury. This product divided by 12, or 1.13a;, gives the number of feet of water corre- sponding to X inches of mercury. If we let the scale of inches of mercurj' range from to 10, then the scale of feet of water must range from to 11.3. Hence draw a line OA 10 inches long divided into inches and tenths to represent inches of mercury. Draw any line OC through and lay off on it 11.3 uni- form intervals (uich intervals will be satisfactory). Connect the end division on OA with the end division on OC by a line AC. Then from 1, 2, 3, inches on OC draw parallels to AC, thus forming adjacent to OA the scale of equivalent feet of water. Each of these intervals can then be subdivided into 10 equal parts corresponding to tenths of feet of water. 2. Draw a double scale showing pressure expressed as feet, of water, and as pounds per square inch, knowing that one cubic foot of water weighs 62.5 pounds. The weight of one cubic foot of water, 62.5 pounds, divided by 144, the number of square inches on one face of a cubic foot, gives 0.434 pounds per square inch as the equivalent of one foot of water pressure. One pound per square inch is equivalent, therefore, to 1/0.434 or 2.30 feet of water pressure. If we let the scale of pounds range from to 10, we may select 1 inch as the equivalent of 1 pound per square inch, and divide the scale OA into inches and tenths to repre- sent this magnitude. Draw OC through 0, and lay off 23 uniform intervals on OC, 1/2 inch being a convenient length for each of these parts. Connect the end division of OC with A and through all points of division of OC draw lines parallel to CA. The range may be extended to any amount desired by annexing ciphers to the numbers attached to the two scales. Extending the range by annexing ciphers to the attached numbers is obviously practicable so long as the various intervals or units are decimally subdivided. The- method is impracticable for scales that are not decimally subdivided, such as shilUngs and pence, degrees and minutes, feet and inches, etc. 3. Draw a double scale showing the relations between cubic feet, and gallons. One gallon equals 231 cubic inches, but use the approximate relation, 1 cubic foot equals 71 gallons. Divide the §4] VABIABLES AND FUNCTIONS OF VARIABLES 9 scale of cubic feet into tenths, the scale of gallons into fourths to correspond to quarts. It is obvious that it is always necessary first to select the range of the various scales, but it is quite as well in this case to show the equivalents for 1 cubic foot only, as numbers on the various scales can be multiplied by 10, 100, or 1000, etc., to show the equivalents for larger amounts. Select 10 inches = 1 cubic foot for the scale (OA) of cubic feet. Draw the line OC. On OC lay off 71 equal parts (say, 7^ inches). Connect the end division with A and draw the parallel lines exactly as with previous examples. The intervals of the scale of gallons can then be subdivided into the four equal parts to show quarts. 4. Draw a double scale showing the relation between cubic feet and liters. One cubic foot equals 28| liters. 5. If a double scale be drawn on a deformable body, as, for example, on a rubber band, would the double scale still represent true relations when the rubber band is stretched? What if the stretching were not uniform? 6. From Fig. 9, find the number of miles per hour corresponding to 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, and 15 feet per second. Place the results in tabular form, i.e., in the first of two adjacent vertical columns place the numbers 0, 1, 2, . . .15; opposite these numbers place in the second vertical column the corresponding numbers representing speed as mUes per hour. Give the first vertical column the heading "Speed-ft./sec," and the second column the heading "Speed-mi./hr." Aa the speed changes from 1 foot per second to 2 feet per second, the speed changes by what amount in miles per hour? As the speed changes from 3 feet per second to 4 feet per second, the speed in miles per hour changes by what amount? The change in speed as miles per hour is how many times the change in the speed as feet per second? 4. A Non-uniform Scale in Juxtaposition with a Uniform scale. Each scale of the double scales constructed in the preceding section were uniform scales. The construction of a double scale of this kind was possible because the change in the number of units of one magnitude represented was directly proportional to the corresponding change in the number of units of the other magnitude. It will, however, be sometimes desirable to construct double scales in which this proportionality does not exist. For example, if a double scale were to be constructed showing the 10 ELEMENTARY MATHEMATICAL ANALYSIS [§5 relation between the radius and the area of a circle, the preceding construction could not be used, since the change in area is pro- portional to the change in the square of the radius and not to the change in the radius. In this case both scales cannot be uniform. Figure 10 is a double scale representing the relation between the radius and the area of a circle. The area is repre- sented by the points on the uniform scale, the radius by the points on the non-uniform scale. The relation is A = irr^ where r is the radius in feet and A is the area in square feet. KadluB ol circle 4 5 6 iMiliMil[iiil||MliiiilMiJ|iMl|iii[iiiiliii-r lm ^il y i | li | l|ll|| - | l| i y l ^ il|li^^ Illlllllll llllllllllu Area of Circle Fig. 10. — Double scale showing the relation between the area of a circle and its radius. 5. Functions. The relation between two magnitudes expressed graphically by two scales drawn in juxtaposition, as above, may sometimes be expressed by means of an equation. Thus, F, the number representing the degrees Fahrenheit in a temperature reading, and C, the number representing the degrees centigrade of the same temperature, are connected by the equation F = iC + 32. (1) Again y, the number representing speed measured as miles per hour, and x, the number representing speed measured as feet per second, are connected by the equation y = iU. (2) Again u, the number representing pressure measured as feet of water, and v, the number representing the same pressure measured as pounds per square inch, are connected by the equation u = U^j^- (3) Again A, the number representing area of a circle measured as square feet, and r, representing the radius measured as feet, are connected by the relation A = irr^. (4) Note. The letters F, C, x, y, u, v, in the above equations stand for numbers; -to make this emphatic we sometimes speak of them as §5] VARIABLES AND FUNCTIONS OF VARIABLES 11 pure or abstract numbers. These numbers are thought of as arising from the measurement of a magnitude or quantity by the appUoation of a suitable unit of measure. Thus from the magnitude or quantity of water, 12 gallons, arises, by use of the unit of measure the gallon, the abstract number 12. Algebraic equations express the relation between numbers, and it is understood that the letters used in algebra stand for numbers and not for quantities or magnitudes. Quantity or Magnitude is an answer to the question: "How much?" Number is an answer to the question: "How many?" An interesting relation is given by the scales in Fig. 8. This diagram shows the fee charged for money orders of various amounts. The amount of the order may first be found on the upper scale and then the amount of the fee may be read from the lower scale. The relation here exhibited is quite different from those previously given. For example, note that as the amount of the order changes from $50.01 to $60 the fee does not change, but remains fixed at 20 cents. Then as the amount of the order changes from $60.00 to $60.01, the fee changes abruptly from 20 cents to 25 cents. For an order of any amount there is a cor- responding fee, but for each fee there corresponds not an order of a single value, but orders of a considerable range in value. This is quite different from the cases presented in Fig. 7. There for each reading Fahrenheit corresponds a certain reading centigrade, or vice versa, and for any change, however small, in one of the temperature readings a change, also small, takes place in the other reading. For this reason the latter number is said to be continuous. The relation between the temperature scales has been expressed by an algebraic equation. The relation between the value of a money order and the corresponding fee cannot be expressed by a similar equation. If we had given only a short piece of the centi- grade-Fahrenheit double scale, we could, nevertheless, produce it indefinitely in both directions, and hence find the corresponding readings for all desired temperatures. But by knowing the fees for a certain range of money orders we cannot determine the fees for other amounts. In both of these cases, however, we express the fact of dependence of one number upon another 12 ELEMENTARY MATHEMATICAL ANALYSIS [§6 number by sajnng that the first number is a function of the second number. 6. Definition. Any number, u, is said to be a function of another number, t, if, when the value of t is given, the value of u is determined. The number t is called the argument of the function u. Illustrations. The length of a rod is a function of its tempera- ture. The area of a square is a function of the length of a side. The area of a circle is a function of its radius. The square root of a number is a function of the number. The strength of an iron rod is a function of its diameter. The pressure in the ocean is a function of the depth below the surface. The price of a railroad ticket is a function of the distance to be travelled. The temperature Fahrenheit is a function of the temperature centigrade. It is obvious that any mathematical expression is, by the above definition, a function of the letter or letters that occur in it. Thus, in the equations u = t^ + it + l _ t - 1 " ~ 2( + 2 u = Vt + 4: + P -\ u is in each case a function of t. Goods sent by freight are classified into first, second, third, fourth, and fifth classes. The amount of freight on a package is a function of its class. It is also a function of its weight. It is also a function of the distance carried. Only the second of these functional relations just named can readily be expressed by an algebraic equation. It is possible, however, to express all three graphically by means of parallel scales. The definition of the function is given (for any particular railroad) by the complete freight tariff book of the railroad. The fee charged for a money order is a function of the amount of the order. The functional relation has been expressed graphically in Fig. 8. Note that for orders of certain amounts, namely, $2i $5, $10, $20, $30, $40, $50, $60, $75, the function is not de- fined. The graph alone cannot define the function at these values, §6] VARIABLES AND FUNCTIONS OF VARIABLES 13 as one cannot know whether the higher, the lower, or an inter- mediate fee should be demanded. One can, however, define the function for these values by the supplementary statement (for example): "For the critical amounts, always charge the higher fee." As a matter of fact, however, the lower fee is always charged. A function having sudden jumps like the one just considered, is said to be discontinuous. Illustration 1. One side of a rectangle is 2 centimeters. The other side is (x + 2) centimeters. Express the area A of the rectangle as a function of x. The area is the product of the breadth by the length of the rectangle. Hence A =2(x + 2) =2x + 4, (l; which is the function of x sought. Illustration 2. The hypotenuse of a right triangle is 10 inches. One side is x inches. Express A, the area of the triangle, as a function of x. Since the hypotenuse squared equals the sum of the squares of the two legs, we may write 102 = a;2 -f yi^ (1) where y stands for the length in inches of the second leg of the triangle. But we know that A = kxy. (2) From (1) y = Vl02 - xS ' (3) Substituting in (2), we have A = kxy/im -x\ (4) which is the function of x desired. , Illustration 3. Express the amount A of $1 at simple interest at 6 per cent, for n years as a function of n. The interest on $1 for n years equals Sy^Tfre. Hence the amount (which is the principal plus the interest) is' expressed by ,'' A = 1 + TBTO. Exercises In the following exercises the function described can be represented by a mathematical expression. The problem is to set up the expres- sion in each case. 14 ELEMENTARY MATHEMATICAL ANALYSIS [§7 1. One side of a rectangle is 10 feet. Express the area il as a func- tion of the other side x. 2. One leg of a right triangle is 15 feet. Express the area .A as a function of the other leg x. 3. The base of a triangle is 12 feet. Express the area as a function of the altitude I. 4. Express the circumference of a circle as a function (1) of its radius r; (2) of its diameter d. 6. Express the diagonal doia, square as a function of one side x, 6. One leg of a right triangle is 10. Express the hypotenuse h as a function of the other leg x. 7. A Ship B sails on a course AB perpendicular to OA. If OA = 30 mUes, express the distance of the ship from as a function of AB. 8. A circle has a radius 10 units. Express the length of a chord as a function of its distance from the center. 9. An isosceles triangle has two sides each equal to 15 centimeters, and the third side equal to x centimeters. Express the area of the triangle as a function of x. 10. A right cone is inscribed in a sphere of radius 12 inches. Ex- press the volume of the cone as a function of its altitude I. Hint: The distance from the center of the sphere to the base of th e cone is (t— 1 2), if I >12. The radius of the base of the cone is Vl2'-(.l-12)' or V24J-Z2. What if 2 < 12? 11. A right cone is inscribed in a sphere of radius a. Express the volume of the cone as a function of its altitude I. 12. One dollar is at compound interest for 20 years at r per cent. Express the amount A as a function of r. 7. Functional Notation. The following notation is used to ex- press that one number is a function of another; thus, if u is a function of t we write Likewise y = /W means that y is a function of x. Other symbols commonly used to express functions of x are : Hx), Xix), f'(x), F(x), etc. These may be read the "(^-function of x," the "Z-function of x," etc., or more briefly, "the of x," "the X of x," etc. Expressing the fact that temperature reading Fahrenheit (.F) is §8] VARIABLES AND FUNCTIONS OF VARIABLES 15 a function of temperature reading centigrade (C), we may write: F=f(.C). This is made specific by writing F = iC + 32. Likewise the fact that the charge for freight is a function of class, weight, and distance, may be written r = fie, w, d). To make this functional symbol explicit, might require that we be furnished with the complete schedule as printed in the freight tariff book of the railroad. The dependence of the tariff upon class and weight can usually be readily expressed, but the dependence upon distance often contains arbitrary elements that cause it to vary irregularly, even on different branches of the same railroad. A complete specification of the functional symbol / would be con- sidered given in this case when the tariff book of the railroad was in our hands. 8. Variables and Constants. In elementary algebra, a letter is always used to stand for a number that preserves the same value in the same problem or discussion. Such numbers are called constants. In the discussion above we have used letters to stand for numbers that are assumed not to preserve the same value but to change in value; such numbers (and the quantities or magnitudes which they measure) are called variables. If r stands for the distance of the center of mass of the earth from the center of mass of the sun, r is a variable. In the equation s = igt' (the law of falling bodies), if i be the elapsed time, s the distance traversed from rest by the falling body, and g the acceleration due to gravity, then s and t are variables and g is the constant 32.2 feet per second per second. The following are constants: Ratio of the diameter to the circum- ference in any circle; the electrical resistance of pure copper at 60° F. ; the combining weight of oxygen; the density of pure iron; the velocity of light in empty space. The following are variables: the pressure of steam in the cyhnder of an engine; the price of wheat; the electromotive force in an alter- nating current; the elevation of groundwater at a given place; the discharge of a river at a given station. When any of these magnitudes 16 ELEMENTARY MATHEMATICAL ANALYSIS [§9 are assumed to be measured, the numbers resulting are also variables. The volume of the mercury in a common thermometer is a variable; the mass of mercury in the thermometer is a constant. 9.* Graphical Computation. The ordinary operations of arith- metic, such as multiplication, division, involution and evolution, can be performed graphically as explained below. The graphical construction of products and quotients is useful in many problems of science. The fundamental theorem in all graphical computa- tion is : The homologous sides of similar triangles are in proportion. Its application is very simple, as wiU appear from the following work. Fboblem 1 : To compute graphically the product of two numbers. Let the two numbers whose product is required be a and b. On any line lay off the unit y jj of measurement, 01, Fig. i" 11. On the same line, and, of course, to the same , ,f / / / / 1 1 U) Ai r=c A- B / / (B) \l = AC OA / / B / \^ O 1 Fig. 11. — Graphical multiplication by proper- ties of similar triangles. 5 6 7 8 9 10 Fig. 12.— Method of graphical mul- tiplication and division carried out on squared paper. The figure shows 1 . 9 X4.4 = 8.4. scale, lay off OA equal to one of the factors a. On any other line passing through 1 lay off a Une IB equal to the other factor 6. Join OB and produce it to meet AC drawn parallel to IB. Then AC is the required product. For, from similar triangles. or AC:\B = 0A: 01, AC = OA X IB. (1) (A) §9] VARIABLES AND FUNCTIONS OF VARIABLES 17 AC is to be measured with the same scale used in laying off 01, OA, and IB. The number of unit's in AC is then the product of a by 6. It is obvious that the angle OiB may be of any magnitude. Hence it may conveniently be taken a right angle, in which case the work may readily be carried out on ordinary squared paper. Many prefer, however, to do the work on plain paper, la3dng off the required distances by means of a boxwood triangular scale. If squared paper is preferred draw the two lines OX and OF at right angles and the unit line If/, as shown in Fig. 12.'! In the exercises that follow the dimensions are given in inches. If the centimeter scale or squared paper Form M\ be used, use 2 cer\timeters everywhere in place of 1 inch. Exercises 1. Find graphically the product of 1.63 by 2.78. Hird: Choose 2 inches to represent one unit. Draw a horizontal line OA 5.56 inches long. Lay off the distance 01 2 inches in length. Draw IB perpendicular or nearly perpendicular to OA and lay off IB equal to 3.26 inches in length. Draw OB. Draw AC parallel to IB. Measure AC. One-half of the length of AC in inches win be the desired product. It will be noticed that the smaller factor is laid off on IB. 2. Find graphically the product of 3.15 by 6.27. Let 1 inch represent one unit. 3. Fmd graphically the product of 36.7 by 5.82. Hivi: Find the product of 3.67 by 5.82 and then move the decimal point one place to the right. 4. Find graphically the product of 936 by 3.17. HiTii: VmA the product of 0.936 by 3.17 and move the decimal point three places to the right in the result obtained. Let 2 inches represent one unit. 5. Fiud graphically the product of 9.36 by 7.23. Hint: Ymd the product of 0.936 by 0.723 and move the decimal point three places to the right in the result obtained. Let 5 inches represent one unit. Problem 2 : To compute graphically the quotient of two numbers a and b. Formula (A) above can be written 18 ELEMENTARY MATHEMATICAL ANALYSIS [§9 From this it is seen that the quotient of two numbers a and 6 can readily be computed graphically by use of Figs. 11 or 12. Exercises 1. Compute graphically the quotient of 1.33 divided by 1.72. Hint: Let 5 inches represent one unit. Lay off OA equal to 8.6 inches. Draw AC perpendicular or nearly perpendicular to OA. Lay off AC equal to 6.65 inches. Draw OC. Draw IB parallel to AC. One-fifth the number of inches in the length of IB is the required quotient. 2. Compute graphically the quotient of 7.32 divided by 1.26. Hint: Find the quotient of 0.732 by 1.26, using 5 inches to represent one unit. 3. Compute graphically 137 divided by 732. Hint: Calculate 1.37 divided by 0.732 and move the decimal point one place to the left in the result obtained. Use 5 inches to represent one unit. Pboblem 3 : To compute graphically the square of any number N. This is a special case of Problem 1, when the two factors are equal. Exercises 1. Find graphically the square of (o) 5; (6) 3; (c) 2, Hint: In finding the square of 5, first find square of^O.S. Let 10 inches represent one unit. 2. Find graphically the square of 93.6. Hint: Find the square of 0.936. 3. Find graphically the square of 0.0672. Hint: Find the square of 0.672. 4. Find graphically the square of 112. Hint: Find the square of 1.12. Phoblbm 4 : To compute graphically the reciprocal of any numherN. This is a special case of Problem 2, when the dividend is 1 and the divisor is N. Exercises Find graphically the reciprocals of the following: (o) 2; (b) 3.5; (c) 12.3; (d) 0.817. , Peoblem 5: To compute graphically the square root of any VARIABLES AND FUNCTIONS OF VARIABLES 19 number N. On OX, Fig. 13, lay off 01 = 1 and lA = N. Upon OA as diameter describe a semicircle OCA. At 1 erect a per- pendicular, IC, to OA. Then IC is the square root of lA. Another construction is to place a celluloid triangle in the position shown in Fig. 13, so that the two edges pass through and A and the vertex of the right angle Ues on the line 1 U. Fig. 13 shows the construction for \/7. in' — u q g ■^ -^ ^ s //' ■^- \ B ^~ A X "< , i / . > ( "^ w b / r "■- , '- -._ / 1 ->. " ~- / f / — — — Fig. 13. — Graphical method of the extraction of square roots. The figure shows Vt = 2 . Q5. Exercises Find graphically the square roots of the following: (o) 2; (b) 3; (c) 5; (d) 10; (e) 932. Hivi: In part (e) find the square root of 9.32 and move the decimal point one place to the right in the result obtained. Problem 6: To compute graphically the integral powers of any number N. This problem is solved by the successive applica- tion of Problem 1 to construct N'^, N^, N*, etc., and of Problem 2 to construct iV"', N~^, N'^, etc. This construction is shown for the powers of 1.5 in Fig. 14. Exercises 1. Compute graphicaUy (a) (1.2)^; (6) (0.85)»; (c) (1.72)-2. Hint: Let 5 inches represent one unit. 20 ELEMENTARY MATHEMATICAL ANALYSIS 2. Show that (1.05)^^ = 2.08, so that money at 5 percent compoimd interest more than doubles itself in fifteen years. Note: The work is less if (1.05)* is firstfound and then this result cubed. 3. From the following outline the student is to produce a complete method, including proof, of constructing successive powers of any number. R 4 / / / / / / // 7 3 1 / / A 2 / // / 1 // // / 1 4 .1 1 ^ -3 -4 1 N 2 3 i Fig. 14. — Graphical computation of (1.5)" for n = —4, 0, 1, 2, 3, 4, 6. -3, -2, -1, Let OA (Fig. 15) be a radius of a circle whose center is 0. Let OB be any other radius making an acute angle with OA. From B drop a perpendicular upon OA, meeting the latter at Ai. From Ai drop a perpendicular upon OB meeting OB at Ai. From .Aj drop a perpendicular upon OA meeting OA at A3, and so on indefinitely. Then, if OA be unity, OAi is less than unity, and OAi, OAs, OAt . . . are, respectively, the square, cube, fourth power, etc.. of OAi. §10] VARIABLES AND FUNCTIONS Of VARIABLES 21 Instead of the above construction, erect a perpendicular to OB meeting OA produced at ai. At Oi erect a perpendicular meeting OB produced at 02, and so on indefinitely. Then if OA be unity, ai is greater than unity and az, 03, 04, . are, respectively, the square. Fig. 15.= — Graphical computation of powers of a number. cube, etc., of oi. As an exercise, construct powers of 4/5 and of 2.5. 4. Show that the successive "treads and risers" of the steps of the "stairways" of Fig. 16a and 166 are proportional to the powers of r. The figures are from Milaukovitch, Zeitschrift fiir Math, und Nat. Unterricht, Vol. 40, p. 329. Fig. 16. — Computation of ar, or', ar', . . . for r < 1 and for r > 1. 10.* Double Scales for Several Simple Algebraic Functions. We may make use of the graphical method of computation explained above to construct graphically double scales representing simple 22 ELEMENTARY MATHEMATICAL ANALYSIS [§10 algebraic relations. For example, we may construct a double scale for determining the square of any desired number. Call OA (see Fig. 17) the scale on which we desire to read the number; call OB the scale on which we read the square. Let us agree to lay off OA as a uniform scale, using 01 as the unit of measure. Since we desire to read opposite 0, 1, 2, 3, . . . of the uniform scale, the squares of these numbers, the lengths along the scale OB must be laid off proportional to the sqvare roots of the numbers Fig. -1 17. 01234567 -Method of constructing a double scale of squares or of square roots. 0, 1, 2, 3, . . . that is, the square root of any length, when laid off on OB, and marked with the symbol of the original length, will he opposite the square root of that number on OA. No difficulty need be experienced in carrying out the actual con- struction of double scales representing algebraic relations, either by use of a table of numerical values of the function or by means of graphical construction. As a less laborious method of graphically expressing functional relations will be explained in the next chapter, the matter of double scales will not be discussed further at this place . CHAPTER II RECTANGULAR COORDINATES AND THE STRAIGHT LINE 11. Statistical Graphs. Prom work in elementary algebra the student is familiar with the construction of statistical graphs simUar to Figs. 18 and 19. The student should carefully study the construction of these two graphs. In Fig. 18, the point at the center of any small circle represents the maximum temperature (or the minimum temperature) on a particular day. This circle is joined by a straight hne to the circle representing the maximum (or minimum) temperature on the next day, and so on. The lines joining the circles enable the eye to foUow at a glance the changes of temperature for the entire month. However, a point on a line between two circles has no meaning, because a point on the horizontal scale between two consecutive points has no meaning, for of course there is but one daily maximum for each day. The student should especially note that the ratio of the distance on the horizontal scale representing days to the distance on the vertical scale representing a chajige of one degree in temperature is so chosen as to make the fluctuations in the temperatures stand out prominently. In constructing statistical graphs, the student should always choose this ratio so that the graph will clearly convey its intended meaning. Smooth curves are drawn through the plotted points of Fig. 19 (not straight lines as in Fig. 18) because in this case intermediate points have meaning; they represent temperatures at various times of the day. Fig. 20 is a barograph, or autographic record of the atmospheric pressure recorded November 24, 1907, during a balloon journey from Frankfort to Marienburg in West Prussia. The zero of the scale of pressure does not appear in the diagram. Note also - that the scale of pressure is an inverted scale, increasing downward. 23 24 ELEMENTARY MATHEMATICAL ANALYSIS t§ll ::: ^$; :: :^4::::::: :::": -~-t,,_|V =5^ -,- ^, ...bl,(^--- ,.^:!j ,,-"-" ,.^:I "x. / \ ___3,_.^r- 1 3.^^^^ -t- [ :v, \ J 1 .::l 1 ' ^r 1 Mi___g====:::.._.e^_ ^ '^6,1 g 1 P P5 - - --X T 1 . ^> - II -- i -- J -- "is ..^Z____Z 1 i-"" _>. V^ SH^g ) S-^^M "' ..^- / ^ is 1 v___. _,__.,.§ %° :^ l_ < ^ /l.f-'-'. 1— g -- ---.^ U —-J » ::::::::^;:::::;:::=;;:::::t:::::i^:::": v___.e4--ll-- 1 asss8§ggs» -§ 7|2 Si* si -- B --5J 3g ■d ^isqusiqcj saajSaQ — siTHEjaamaj; §11] RECTANGULAR COORDINATES 25 The scale of time is an algebraic scale,, the zero of which may be arbitrarily selected at any convenient point. The scale of pres- sure is an arithmetical scale. The zero of the barometric scale 'i70 — — — — — — ~ — — — — ~~ ' »^ ■%eo *-=k^ ■g ' •S^" & — — — — — •^ b- — — — h^ n $ 40 — — ~ • =u= ^- Zt ^ — — — — _ G) , i>-j 1 -^ 1 *? 30 ~ " -_ r:^ rt: — ~~ ~ — ~ ~ — ~ § Houriy_Air Temperatures at Madison Wisconsin May,14.i910 and Oct.. 10. 1910 0) . |io 2 4 3 8 10 12_ 2 4 _ i 8 10 1? Til ne oil iaV Hours 1/ Fig. 19. — Hourly air temperatures. corresponds to a perfect vacuum — no less pressure and hence, in this case, no negative value exists. Fig. 21 is a graphical time-table of certain passenger trains be- tween Chicago and Minneapolis. The curves are not continuous. s g -3 ^ s =1 Fig. 20. — Barograph taken during a balloon journey. The vertical scale is atmospheric pressure in miUimeters of mercury. as in the case of the barograph, but contain certain sudden jumps. What is the meaning of these? What indicates the speed of the trains? Where is the fastest track on this railroad? What shows the meeting jxrint of trains? 26 ELEMENTARY MATHEMATICAL ANALYSIS [§11 If the diagram, Fig. 21, he wrapped around a vertical cylinder of such size that the two midnight lines coincide, then each train line may be traced through continuously from terminus to terminus. Functions having this remarkable property are said to be peri- odic. In the present case the trains run at the same time every day, that is, periodically. In mathematical language, the po- sition of the trains is said to be a periodic function of the time. Chicago £au Olaire Menomoaie Hudson St Paal MinneapolieJf 10 12 2 A.M. Noon P.M. Fig. 21. — Graphical time table of certain passenger, trains between Chicago and Minneapolis. Fig. 22 represents the fluctuation of the elevation of the ground- water at a certain point near the sea coast on Long Island. The fluctuations are primarily due to the tidal wave in the near-by ocean. The curve is continuous. Is the curve periodic? What indicates the rate of change in the elevation of the ground- water? When is the elevation changing most rapidly? When is it changing most slowly? Fig. 23 represents the functional relation between the amount of a domestic money order and the fee. This is an excellent illustra- §12] RECTANGULAR COORDINATES 27 tion of a discontinuous function. On account of the sudden jumps in the values of the fee, the fee, as explained in the preceding chapter, is said to be a discontinuous function of the amount of the order. Fig. 22. — Upper curve, elevation of water in a well on Long Island. Lower curve, elevation of water in the nearby ocean. 12. Suggestions on the Construction of Graphs. Two kinds of rectangular coordinate paper have been prepared for use with this book. Form Ml is ruled in centimeters and fifths. Form M2 is ruled without major divisions in uniform 1/5-inch intervals. It is a mistake to assume that more accurate work can be done on finely ruled than on more coarsely ruled squared paper. Quite the 28 ELEMENTARY MATHEMATICAL ANALYSIS [§12 contrary is the case. Paper ruled to 1/20-inoh intervals does not per- mit interpolation within the small intervals while paper ruled to 1/10 or 1/5-inch intervals permits accurate interpolation to one-tenth of the smallest interval. Form Ml is ruled to 2-mm. intervals, and is fine enough for any work. The centimeter unit has the very considerable advantage of permitting twenty of the units within the width of an ordinary; sheet of letter paper (SJ X 11 inches) while seven is the largest number of inch units available on such paper. In order to secure satisfactory results, the student must recog- nize that there are several varieties of statistical graphs, and that each sort requires appropriate treatment. -50 MO 1 -30 r-20 HlO 1. 1. .Ill j_ _L J. _!_ I _L J_ 10 90 20 80 40 50 60 70 * 80 Amount of the Money Order in Dollars Fig. 23. — The graph of a discontinuous function. 1. It is possible to make a useful graph when only one variable is given. Thus Table I gives the ultimate tensile strength of various materials. A graph showing these results is given in Fig. 24. There are two practical ways of showing the numerical values pertaining to each material, both of which are indicated in the diagram; either rectangles of appropriate height may be erected opposite the name of each material, or points marked by circles, dots or crosses may be located at the appropriate height. It is obvious in this case that a smooth curve should not be drawn through these points — such a curve would be quite meaningless. In this case there §12] RECTANGULAR COORDINATES 29 Table I. — Ultimate Tensile Strength of Various Materials Material Tensile strength, tons per square inch Hard steel 50.0 30.0 25.0 21.5 16.0 12.0 11.0 10.0 5.0 Wrought iron Drawn brass Cast brass Timber, with grain are not two scales, but merely the single vertical scale. The hori- zontal axis bears merely the names of the different materials and has no numerical or quantitative signifioance. The result is obviously not the graph of a function, for there are not two variables, but only one. The graph is merely a convenient ex- pression for certain discrete and independent results arranged in order of descending magnitude. 2. It is possible to have a graph involving two variables in which it is either impossible or undesirable to represent the graph by a continuous curve or line. For example. Fig. 18 is a graph representing the maximum temperature on each day of a certain month. Because there is only one maximum temperature on each day, the value corresponding to this should be shown by an appropriate rectangle, or by marking a point by a circle, or by a dot or cross, as in the preceding case. A continuous curve through these points has no meaning. The horizontal scale may be marked by the names of the days of the week or by numbers, but in either case the horizontal line is a true scale, as it cor- responds to the lapse of the variable time. Sometimes, as in Fig. 18, graphs of this kind are represented by marking the appropriate points by dots or circles and then connecting the successive points by straight lines. These lines have no special meaning in such a case, but they aid the eye in following the succession of separate points. 30 ELEMENTARY MATHEMATICAL ANALYSIS [§12 If a graph be made of the noonday temperatures of each day of the same month referred to in Fig. 18, one of the same methods indicated above would be used to represent the results; that is, either rectangles, marked points, or marked points joined by lines. Although a smooth curve drawn through the known points would have a meaning (if correct), it is obvious that the noonday temperatures alone are not sufficient for determining its form. In all such cases a smooth curve should not be drawn. 3. If the data are reasonably sufficient, a smooth curve may, and often should, be drawn through the known points. Thus if the temperature be observed every hour of the day and the re- sults be plotted, a smooth curve drawn carefully through the plotted points will probably very accurately represent the un- known temperatures at interme- diate times. The same may safely be done in exercises (3) and (4) below. In scientific work it is desirable to mark by circles or dots the values that are actually given to distinguish them from the intermediate values "guessed" and repre- sented by the smooth curve. In addition to the above suggestions, the student should adhere to the following instruc- tions : 4. Every graph should be marked with suitable numerals along both numerical scales. 5. Each scale of a statistical graph should bear in words a description of the magnitude represented and the name of the unit of measure used. These words should be printed in drafting letters and not written in script. •9 2 H 1 JS!a(\ Id- 1 ', \ i2 (- 2.- 1) -? I] I IV -3 Y Pi (2,-S ) ^ Fig. 25. — Rectangular coordinates. suitable unit of measiu'e (as centimeter, inch, etc.), the distance of -the point to the right or left of the vertical scale and its distance above or below the horizontal scale. Thus the point Pi, in Fig. 25, is 2j units to the right and 3j units above the standard scales. P2 is 3 units to the left and 2 units above the standard scales, etc. Of course these directions are to be given in mathe- matics by the use of the signs "-)-" and " — " of the algebraic scales, and not by the use of the words "right" or "left," "up" or "down." The above scheme corresponds to the location of a place on the earth's surface by giving its angular distance in 3 34 ELEMENTARY MATHEMATICAL ANALYSIS [§13 degrees of longitude east or west of the standard meridian, and also by giving its angular distance in degrees of latitude north or south of the equator. The sort of latitude and longitude that is set up in the manner described above is known in mathematics as a system of rectangu- lar coordinates. It has become customary to letter one of the scales XX', called the X-axis, and to letter the other YY', called the Y-axis. In the standard case these are drawn to the right and left, and up and down, respectively, as shown in Fig. 25. The distance of any point from the F-axis, measured parallel to the X-axis, is called the abscissa of the point. The distance of any point from the X-axis, measured parallel to the F-axis, is called the ordinate of the point. Collectively, the abscissa and ordinate are spoken of as the coordinates of the point. Abscissa corresponds to the longitude and ordinate corresponds to the latitude of the point, referred to the X-axis as equator, and to the F-axis as standard meridian. In the standard case, abscissas measured to the right of YY' are reckoned positive, those to the left, negative. Ordinates measured up are reckoned positive, those measured down, negative. Rectangular coordinates are frequently called Cartesian co- ordinates, because they were first introduced into mathematics by Ren6 Descartes (1596-1650). The point of intersection of the axes is lettered and is called the origin. The four quadrants, XOY, YOX', X'OY', Y'OX, are called the first, second, third, and fourth quadrants, respectively. A point is designated by writing its abscissa and ordinate in a parenthesis and in this order: Thus, (3, 4) means the point whose abscissa is 3 and whose ordinate is 4. Likewise (—3, 4) means the point whose abscissa is (—3) and whose ordinate is (+4). Abscissas are usually represented by the letter x and ordinates are usually represented by the letter y. Thus the point whose abscissa is 3 and whose ordinate is 4, may be described as the point (3, 4), or equally well as the point x = 3, y — 4. Unless the contrary is expUcitly stated, the scales of the eo- ■ ordinate axes are assumed to be straight and uniform and to inter- sect at right angles. Exceptions to this are not uncommon. §14] RECTANGULAR COORDINATES 35 Exercises On suitable squared paper, select and mark a horizontal line as the X-axis (or axis of abscissas) and select and mark a vertical line as the F-axis (or axis of ordinates). Select and mark a suitable unit of measure on each axis, for example 1 centimeter or 1/2 inch. Then locate the points whose coordinates are given in the following exercises. 1. Draw the coordinate axes on squared paper and locate the points (3, 3), (2, 2), (1, 1), (0, 0), (-1, -1), (-2, -2), (-3, -3). 2. Draw the axes and locate the points (2, 3), (—2, 3), ( — 2, —3), (2, -3). 3. Draw the coordinate axes and locate the points (5, 0), (4, 3), (3, 4), (0, 5), (-3, 4), (-4, 3), (-5, 0), (-4, -3), (-3, -4), (0, -5), (3, -4), (4, -3). 4. Draw suitable axes and locate the points ( — 3, —5), (—2, —3), (-1, -1), (0, 1) (1, 3), (2, 5), (3, 7), (4, 9). A brief way of describing a ^et of points is to place the abscissas and ordinates in tabular form, indicating abscissas by the letter x and indicating ordinates by the letter y, as follows : a; I -3' -2 -10 1 2 3 4 y \ -5-3-113579 14. Mathematical, or Non-statistical Graphs. — Instead of the expressions "abscissa of a ■point," or "ordinate of a point," it has be- come usual to speak merely of the "x of a point," or of the "y of a point," since these distances are conventionally represented by the letters x and y, respectively. If we impose certain conditions upon X and y, then it will be found that we have, by that very fact, restricted the possible points of the plane located by them to a certain array, or set of points, and that all other points of the plane fail to satisfy the conditions or restrictions imposed. It is obvious that the command, "Find the place whose latitude equals its longitude," does not restrict or confine a person to a par- ticular place or point. The places satifying this condition are unlimited in number. We indicate all such points by drawing a line bisecting the angles of the first and third quadrants; at all points on this line latitude equals longitude. We speak of this line as the locus of the point satisfying the conditions. We might describe the same locus by saying "the y of each point of the 36 ELEMENTARY MATHEMATICAL ANALYSIS [§14 locus equals the x" or, with the maximum brevity, simply, write the equation "y =; x." The equation "y = x" is called the equation of the locus, and the line is called the locus of the equation. It is of the utmost importance to be able readily to interpret any condition imposed upon, or, what is the same thing, any relation between variables, when these are given in words. It will greatly aid the beginner in mastering the concept of what is meant by the term fimction if he will try to think of the meaning in words of the relations commonly given by equations, and vice versa. The very elegance and brevity of the mathematical expression of rela- tions by means of equations, tends to make work with them formal y "h i 2 ^ / / 1 /■ X' / X 1 D / ^' Fig. 26. — The straight line y = x. Fig. 27.— The straight line y = 2x. and mechanical unless care is taken by the beginner to express in words the ideas and relations so briefly expressed by the equa- tions. Unless expressed in words, the ideas are liable not to be expressed at all. The equation of a curve is an equation satisfied by the co- ordinates of every point of the curve and by the coordinates of no other point. The graph of an equation is the locus of a point whose coordi- nates satisfy the equation. Illustration I. Find the equation of the Une of Fig. 26. The Une §14] RECTANGULAR COORDINATES 37 of this figure states that the y of any point of the line equals the x of that point. Hence the equation of the line isy = x. Illustration 2. The line of Fig. 27 is drawn through the origin and the point (1, 2). Find the equation of the line. Let OB and DP be the abscissa and ordinate, respectively, of any point on the line. Then from similar triangles OPD and OPil, DP:OD = 2 : 1, or y : a; = 2: 1, or y = 2x, which is the equation of the line. Exercises What is its What is What is 1. Draw a hne through the origin and the point (1, 3). equation? 2. Draw a line through the origin and the point (1, -f)- its equation? 3. Draw a line through the origin and the point (1, —1). its equation? Draw a line through the origin and the point (1, —2). What is its equation? 4. Draw loci for the following and show that each locus is a straight line passing through the origin: (a) The ordinate of any point of a certain locus is twice its abscissa; (b) the x of every point of a cer- tain locus is half its y; (c) the yoia. point is 1/3 of its x; (d) a point moves in such a way that its latitude is always treble its longitude; (e) the sum of the latitude and longitude of a point is zero; (/) a point moves so that the difference in its latitude and longitude is always zero. Hint: In part (a) let Pi (Fig. 28) be any point on the locus and let Pi be any second point on the locus. Draw OPi and OPi] draw PiDi and P2D2 perpendicular to OX. By the conditions of the problem PiDi = 20Di and P2D2 = 20Di. Hence PiDi _ P,D2 ODi OD2' and the triangles OPiDi and OP2D2 are similar. Then the angles PiODi and P2OD2 are equal. Hence OPi and OP2 coincide in direction and 0, Pi, and P2, are upon a straight line. 5. Draw the locus: Beginning at the point (1, 2), a point moves so Fig. 28. — Diagram for exercise 4 (o) §14 and , exercise 3 §15. 38 ELEMENTARY MATHEMATICAL ANALYSIS [§15 that its gain ia latitude is always twice as great as its gain in longitude. Show that the locus is a straight line. 6. A point moves so that its latitude is always greater by 2 units than three times its longitude. Write the equation of the locus and construct- Show that the locus is a straight line. 15. Slope. The slope of a straight line is defined to be the change in y for an increase in x equal to 1. It will be represented in this book by the letter to. In Fig. 26 the line OP has slope 1 and in Fig. 27 the line OP has slope 2. Also in Fig. 29 the line A has the slope to = 1.5, for it is seen that at any point of the linb the ordinate y gains 1.5 units for an increase of 1 in X. The line B, parallel to the line A, is also seen to have the slope equal to 1.5. The equation of the line A is obviously y = 1.5a;. In the same figure the slope of the line C is ( — 2), for at any point of this line the ordinate 2/ decreases 2 units for an increase in X equal to 1. The equation of the line C is obviously y = —2x. Line D, parallel to line C, also has slope ( — 2). If h be the change in y for an increase of x equal to k, then the slope TO is the ratio h/k. Hence the practical method of determin- ing the slope of a line drawn upon squared paper is: Select two convenient points on the line rather far apart, and divide the change in y by the increase in x. The technical word slope differs from the word slope or slant in common language only in the fact that slope, in its technical use, is always expressed as the ratio of two algebraic numbers. In Y U C 1 s B A \ \ / / \ I \ m .- 2 4n = 1 ■y \ \ / / t m\ B- 2\ (^ / 1 / \ \, / \ \ -> / ^m = + 1.. \, \ / / 1 \ \ / / \, / \; ^m -1 .5 X / ^ / X -5 -4 -3 -2 / i^-l \ / \ 1 2 3 4 6 / \ -1 \ m -- 2 / / V \ ' / \ \ / t \ s / / -9 \ \ / / V / f -\ \ / / \ \ / / -5 Y s ..S Fig. 29.- -Lines of slope (1 . 5) and of slope (-2). §16] RECTANGULAR COORDINATES 39 common language we speak of a "slope of 1 in 10," or a "grade of 50 feet per mile," etc. In mathematics the equivalents are "slope = 1/10," "slope = 50/5280," etc. As already indicated, the definition of slope requires us to speak in mathematics of positive slope and negative slope. A line of posi- tive slope extends upward with respect to the standard direction OX and a line of negative slope extends downward with reference toOZ. In a similar way we may speak of the slope of any curve at a given point on the curve, meaning thereby the slope of the tangent line drawn to the curve at that point. Exercises 1. Give the slopes of the lines in exercises 1 to 6 of the preceding set of exercises. 2. Draw y = x; y = 2x; y = Zx; y = -^-t y = ^, y = ^, y = — 2x; y = — 3z; y = Ox. 3. Prove that y = mx always represents a straight line, no matter what value m may have. Hint: Make use of Fig. 28. 16. Equation of a Straight Line. Intercepts. — In Fig. 30, the line MN expresses that the ordinate y is, for all points on the line, always 3 times the abscissa x, or it says that y = 3a;. The line HK is 2 units higher than MN, so that it states that "2/ is 2 more than 3a;." Thus the line HK has the equation y = Zx + 2. In Fig. 29 the line 5 is 2 units higher than the line y = 1.5a;, hence its equation is y = 1.5x -{- 2. The line D is 2 units lower than the line C, whose equation is y = —2x, hence the equation of Disy = -2x - 2. In general, since y = mx is always a straight line,^ then y = ma; + 6 is a straight line, for the y of this locus is merely, in each case, the y of the former increased by the constant amount 6 (which may, of course, be positive or negative). Therefore, y — mx + 6 is a line parallel to y = mx. The line y = mx + 6 is 6 units higher than, or above, the line y = mx ii b stands for a positive number and the line y = mx + b is b units lower than, or below the line y = mx if 6 stands for a negative number. The distance 1 See exercise 3, §15, above. 40 ELEMENTARY MATHEMATICAL ANALYSIS l§16 OB (Fig. 30) is equal to b, and is called the Y-intercept of the graph. The distance OA is equal to — b/m, for it is the value of X obtained from the equation when y is given the value zero. It is called the X-intercept of the locus. The equation 7 = mz -|- b is called the common equation of the straight line. X' 1 K a 1 2 r s / 1 B 1 / T—r A / -1 : 3 1 1 ' / 2 / -9. L / -S 1 J y' -4 X Fig. 30. — Intercepts. MN is the line j/ = 3x; UK is the line y = 3s + 2 ; OB, the F-intercept of HK, is equal to 2 ; OA, the Z-intercept, is equal to —2/3. IllustraMon 1. Sketch the line y = 2s + 3. This line is 3 units higher than the line y = 2x. Hence through the point (0, 3) on the 7-axis, draw a line of slope 2, which is the required line. I Illustration 2. Sketch the line ?/ = — 2.r — 1. §16] RECTANGULAR COORDINATES 41 The line is 1 unit lower than the line y = —2x. Hence through the point (0, —1) on the F-axis, draw a line of slope (—2). The line lies halfway between lines C and D (Fig. 29). Illustration 3. Draw the line whose equation is 4a! — 2?/ — 3 = 0. Solve the equation for y by transposing the terms 4a; and (—3) to the right member and dividing both members by ( — 2), then y = 2x-i. Hence through the point (0, — f ) on the y-axis draw a line of slope 2. Exercises 1. Sketch, from inspection of the equations, the lines given by: (a) y ^ X. (d) y = X + 3. (b) y = X + 1. (e) y = X - 1. (c) y =x + 2. if) y =x -2. 2. Sketch, from inspection of the equations, the lines given by: (a) y = ix. (/) y = -\x. (6) y = ix. (g) y = -x. (c) y = X. (h) 2/ = -2a;. {d) y = 2x. (i) y = -3x. - (e) 2/ = 3a;. (j) y = y/2 x. 3. Sketch the lines given by the equations: (a) a; = 3. {d) y = 1. (s) 2/ = 0. (6) a; = 6. (e) y = 5. (A) x = 0. (c) X 2. t/) y = -3. (i)a;2 = 4. 4. Sketch and name the slope and F-intercept in each of the following: (o) 2/ = a; + 1. (/) 2/ = 3x + 4. (6) 2/ = ia; + 1. (?) 2/ = a; - 6. (c) 2/ = -2a; + 4. Qi) 2/ = fx + 8. {d) 2/ = 6x + 3. (i) 2/ = -3x + 4. (e) y = —Sx — 2. 0') 2/ = — ia; — 3. 5. Give the slope and F-intercept for each of the following : (a) y =2x + Z. (/) 3y - 6x = 12. (6) y = 3x -2. (9) y +x = 1. (c) 2/ = -3x - 1. I^h) 3y'+ 2x = 7. (d) 2/ = 5x— 6. (i) X — ^ = 6. (e) 22/ = X + 4. (j) X - 22/ = I. 42 ELEMENTARY MATHEMATICAL ANALYSIS [§16 6. Find the X-intercept and the }'-intercept for each of the following : (.a) Sx - 2y = 5. (e) y - 2x ~ 6 ,= 0. (6) 2x + y =Q. if) 2y + 3x + 5 = 0. (c) X -y = 7. (g) X + y + I = 0. (d) y -3x = 5. (h) 5y - 3x + 10 = 0. 7. Name the slope and the F-intercept in each of the following; (a) 2y =x + 4:. {f) ix = 3y - 6. (6) y -2x-3 =0. (ff) Wx-y = 7. (c) y + fa; + J = 0. (h) ax -\=y. (d) 2y -\-3x = 4. (i) ax + by = c. (e) 2x -3y = 6. 0') x/a + y/b = 1. 8. What is the equation of the X-axis? Of the K-axis? 9. What ia the equation of a line parallel to the X-axis 4 units above? 3 units above? 10 units below? 60 units below? 10. What is the equation of a line parallel to the F-axis 3 units to the right? 20 units to the right? 7 units to the left? 100 units to the left? 11. Plot the following pairs of points on squared paper, and draw the line determined by each pair: (a) (-1, 3) and (5, -6) (6) (-2, -5) and (3, 4) (c) (1, 1) and (7, -8). Find the slope and, by means of similar triangles, find the F-intercept of each line. Write the equation of each line by replacing m and 6 in y = mx -|- 6 by the values found for slope and intercept. Test the correctness of the equations by substituting for x and y the co- ordinates of the given points. 12. A head of 100 feet of water causes a pressure at the bottom of 43.4 pounds per square inch. Draw a graph showing the relation between head and pressure, for all heads of water from to 200 feet. StTGGESTiON: There are several ways of proceeding. Let pounds per square inch be represented by abscissas or x, and feet of water be represented by ordinates or y. Since negative numbers are not in- volved ia this exercise, the origin may be taken at or near the lower left corner of the squared paper. Draw a line through the points (0, 0) and (86.8, 200). This will be the required graph. Otherwise 100 produce the equation y = 73-72; from the proportion x:y= 43.4 : 100 §16] RECTANGULAR COORDINATES 43 and then draw the graph from the fact that the latitude is always TK~r of the longitude. In drawing this graph let 2 centimeters on the X-axis represent 10 units, and 1 centimeter on the F-axis represent 10 units. Be sure that the scales are numbered and labelled in accordance with suggestions (4), (5), and (6) of §12. On the X-axis mafk only the points corresponding to hundreds of pounds, and on' the y-axis mark only the points corresponding to tens of feet. 13. From the straight line drawn in exercise 12, find pressure meas- ured in pounds corresponding to 13.1, 112.6, 93.7, and 187.5 feet of water. 14. From the straight line drawn in exercise 12, find the head in feet of water corresponding to 1123, 178, and 89 pounds per square inch. 16. A pressure of 1 pound per square inch is equivalent to a column of 2:04 inches of mercury, or to one of 2.30 feet of water. Draw a graph showing the relation between pressure expressed in feet of water and pressure expressed in inches of mercury. StrGGBSTiON: Let x = inches of mercury and y = feet of water. First properly number and label the X-axis to express inches of mer- cury and number and label the K-axis to express feet of water. Since negative numbers are not involved in this exercise, the origin may be taken at the lower left-hand corner of the squared paper. First locate the point x = 2.04; y = 2.30 (which are the corresponding values given by the problem) and draw a line through it and the origin. This is the required locus since at all points we must have the proportion x:y:: 2.04 : 2.30, which says that the ordinate of every point of the locus is 2.30/2.04 times the abscissa of that point. 16. A certain mixture of concrete (in fact, the mixture 1:2:5) con- tains 1.4 barrels of cement in a cubic yard of concrete. Draw a graph showing the cost of cement per cubic yard of concrete for a range of prices of cement from $0.80 to $2.00 per barrel. Suggestion: Let x be the price per barrel of cement and y be the cost of the cement in 1 cubic yard of concrete. Let 2 centimeters on both vertical and horizontal scales represent 10 cents. Number only the points representing multiples of 10 cents. Since that portion of the graph near the origin, namely to the left of 0,.80 and below 1.12 will not be used, place the scales on the horizontal and vertical lines passing through the point (0.80, 1.00) and place this point at or near the lower left corner of the paper. The X- and F-axes will not appear on the drawing. 17. Draw a graph showing the cost per cubic yard of concrete for 44 ELEMENTARY MATHEMATICAL ANALYSIS [§17 various prices of cement, provided $2.10 per yard must be added to the results of example 16 to cover cost of sand and crushed stone. 18. Cast iron pipe, class A (manufactured for heads under 100 feet), weighs, per foot of length: 4-inch, 20.0 pounds; 6-inch, 30.8 pounds; 8-inch, 42.9 pounds. Upon a single sheet of squared paper, construct a graph for each size of pipe, showing the cost per foot for all variations in market price between $20 and $40 per ton. Suggestion: If the horizontal scale be selected to represent •price per ton, the scale may begin at 20 and end at 40, as this covers the range required by the problem. Therefore let 1 centimeter represent $1.00. Since the range of prices is from 1 cent to 2 cents per pound, the cost per foot will range from 20 cents to 40 cents for 4-inch pipe and from 42.9 cents to 85.8 cents for 8-inch pipe. Hence for the vertical scale 10 cents may be represented by 2 centimeters. In this case the vertical scale may quite as well begin at cents instead of at 20 cents, as there is plenty of room on' the paper. 19. Show that the shortest distance between y — mx and y = mx + 6 is not 6, but — , 20. Pick out two pairs of parallel Unes in exercise 5, above . Pick out a pair of parallel lines in exercise 4, above. 17. Line with Slope and One Point, or with Two Points Given. — The equation of any line parallel to the F-axis is of the form x = a, which is an equation in which the variable y does not appear. The equation of all other lines may be written in the form y = mx + h, j (1) in which m is the slope of the line and h is the F-intercept. Two important special cases are explained below. Illustration 1. Find the equation of the line of slope 4 which passes through the point (2, 3). Since to = 4, equation (1) becomes y =4x + b. (2) Replacing a; by 2 and y by 3, we get 3 = 8 -F 6, or 6 = -6. Hence the equation of the line of slope 4 passing through (2, 3) is y = 4x - 5, (3) : §17] RECTANGULAR COORDINATES 45 Illustration 2. Find the equation of the line passing through the pomts (2, 3) and (4, 1). Substituting the given values of x and y in equation (1) we have 3 = 2m + 6 1 = 4w + 6. Solving these equations for the two unknown numbers, m and 6, we find 6 = 5 m = —1, so that the equation of the line passing through the given points is 2/ = -X + 5. In like manner the equation of a line passing through any two given points may be found. In geometry we learned that two points de- termine a straight line, and in the present problem the coordinates of two given points are necessary and sufficient for the determination of the equation of the line. Exercises 1. Find the equation of the line determined by each of the following conditions: (a) Passes through (2, 5) and has slope 3. (6) Passes through ( — 2, 6) and has slope — 2^. (c) Passes through (4, —1) and has slope 7.41. 2. Find the equation of the line determined by each of the following pairs of points: (o) (3, 2) and (1, 5) (c) (4, 6) and (3, -2) (6) (1, 2) and (- 2, 6) (d) (0, 0) and (-2, -3) 3. Show that the equation of the line passing through (a, 0) and (0, 6) niay be written in the form -+! = !■ a 4. Find the equations of the three sides of the triangle whose ver- tices are the points (0, 3), (2, 4), and (5, 9). In each of the following exercises certain observed data are tabu- lated which will be found in each case to give points lying on a straight line. The law connecting y and x must then be of the form y = mx + b. 46 ELEMENTARY MATHEMATICAL ANALYSIS [§17 6. Find the law connecting y and x when the following correspond- ing values are given : X I 10 25 54 72 17 47 105 141 Hint: In plotting, take the origin at the lower left corner of the squared paper. Let 2 centimeters represent 10 units on the X-axis and 1 centimeter represent 10 units on the K-axis. To find the slope divide the change in ?/ or (141 — 17) by the increase in a; or (72 — 10) which gives 2. Find F-intercept by method of Illustration 1. 6. Find the law connecting x and y from the following data : X I 12.0 15.3 17.8 19.0 y \ 24.2 29.0 32.6 34.2 Hint: Take origin near left lower corner of the squared paper and let 4 centimeters equal 10 units on each axis. 7. L is the latent heat of steam at a temperature i° C Find a simple formula giving L in terms of t. « I 75 90 100 115 125 L I 554 544 536 526 519 Hint: Call the lower left corner of the squared paper the point t = 75, L = 500. Let 1 centimeter = 5 units on each axis. 8. V is the volume in cubic centimeters of a certain weight of gas at temperature t° C, the pressure being constant. Find the law connecting V and t. t I 27.0 33.0 40.0 55.0 68.0 V I 109.9 112.0 114.7 120.1 125 Hint: Call the lower left corner of the squared paper the point t = 25, V = 100. Let 2 centimeters equal 5 units on the t scale and 1 centimeter equal 1 unit on the V scale. 9. I feet is the length of an iron bar under a pulling stress of W tons. Find the law connecting I and W. W I 1 1^8 3^2 4^2 6.0 I I 10 10.005 10.010 10.0175 10.0225 10.0325 Hint: Call the lower left comer of the squared paper the point TF = 0, Z'= 10. Let 2 centimeters = 1 unit on the W scale and 1 qentimpter = 0.005 unit? on the I scale. §17] RECTANGULAR COORDINATES 47 10. The following table gives the draw-bar pull in pounds (P) of an electric locomotive in terms of the current consumed (A). Find an approximately correct algebraic formula giving A for any value of P. Find the current required for a pull of 2000 pounds. P I 400 800 1370 1600 2080 2400 , A I 65 86 106 116 137 150 Hint: Call the lower left corner of the squared paper A = 50, P = 400. Let 2 centimeters equal 10 units on the A scale and 1 centimeter equal 100 units on the P scale. / Exercises 5-10 above are taken from Saxelby's "Practical Mathe- matics," Longmans, Green and Co., New York, 1905. CHAPTER III THE POWER FUNCTION 18. Definition of the Power Function. The algebraic function consisting of a single power of the variable, such for example as the functions x'^, x', 1/x, 1/x'', a;^''', etc., stand next to the linear function of a single variable, mx + b, in fundamental impor- tance. The function a;" is known as the power function of x. 19. The Graph of x^. The variable part of many functions of practical importance is the square of a given variable. Thus the area of a circle depends upon the square of the radius; the distance traversed by a falling body depends upon the square of the elapsed time; the pressure upon a flat surface exposed directly to the wind depends upon the square of the velocity of the wind; the heat generated in an electric current in a given time depends upon the square of the number of amperes of current, etc. Each of these relations is expressed by an equation of the form y = ax^, in which x stands for the number of units in one of the variable quantities (radius of the circle, time of fall, velocity of the wind, amperes of current, respectively, in the above named cases) and in which y stands for the other variable dependent upon these. The number a is a constant which has a value suitable to each particular problem, but in general is not the same constant in different problems. Thus, if y be taken as the area of a circle, y = irx^, in which x is the radius measured in feet or inches, etc., and y is measured in square feet or square inches, etc. ; or if s is the distance in feet traversed by a falling body, thens = 16. li'', where i stands for the elapsed time in seconds. In one case the value of the constant a is 3.1416 and in the other its value is 16.1. Let us first graph the abstract law or equation y = x^, in which a concrete meaning is not assumed for the variables x and y but 48 §20] THE POWER FUNCTION 49 in which both are thought of as abstract variables, suitable table of values for x and x^ as follows: First form a X a;2 or y X x^ or y - 3.0 9.00 0.2 0.04 - 2.5 6.25 0.4 0.16 - 2.0 4.00 0.6 0.36 - 1.8 3.24 0.8 0.64 - 1.6 2.56 1.0 1.00 - 1.4 1.96 1.2 1.44 - 1.2 1.44 1.4 1.96 - 1.0 1.00 1.6 2.56 - 0.8 0.64 1.8 3.24 - 0.6 0.36 2.0 4.00 - 0.4 0.16 2.5 6.26 - 0.2 0.04 3.0 9.00 0.0 0.00 Here we have a series of pairs of values of x and y which are asso- ciated by the relation y — x'^. Using the x of each pair of values as abscissa with its corresponding y there can be located as many- points as there are pairs of values in the table, and the array of points thus marked may be connected by a freely drawn curve. To ,draw the curve upon coordinate paper, form Ml, the origin may be. taken near the lower mid-point of the sheet, and 2 centimeters used as the unit of measure for x and y. If the points given by the pairs of values are not located fairly close together, it is obvious that a smooth curve cannot be satisfactorily sketched between the points until intermediate points are located by using intermediate values of x in forming the table of values. The student should think of the curve as extending indefinitely beyond the limits of the sheet of paper used; the entire locus consists of the part actually drawn and of the endless portions that must be followed in imagination beyond the range of the paper. If the graph of i/ = a;^ be folded about the F-axis, OY, it will be noted at once that the left and right portions of the curve will exactly coincide. The student will explain the reason for this fact. 20. Parabolic Curves. The equations y = x, y — x', y = xi, y = x^ should be graphed by the student on a single sheet'of coor- 60 ELEMENTARY MATHEMATICAL ANALYSIS [§20 dinate paper, using 2 centimeters as the unit of measure in each case.^ Table II may be used to save numerical computation in the construction of the graphs of these power functions. As in the case oi y = x^, a smooth curve should be sketched free-hand t L 1 1 1 1 1 I 5 Y I 1 1 1 1 1 1/' -\ |--- 1 1 1 1 4 ^ /.. Vf 1 ^ --f-- --\-'f 4-- 1 2 --f- -A--- I\ 1 1 --J-- / j X' — -|-- \L.. ---]/- — I X 2 1 1 1 1 Y' j 1 ! 2 Fig. 31. — The parabola y = x'. through the points located by means of the table of values, and intermediate values of x and y should be computed when doubt exists in the mind of the student concerning the course of the curve between any two points. 1 Wlieii.aquBred paper ruled in inches is used instead of form M 1, one inch should b« taken as th« unit of measure. §20] THE POWER FUNCTION Table II 51 X x^' X' Vi 0 (parabolic curves) in the first quadrant. Exercises 1. On coordinate paper draw the curves y — x', y = x^, y = x^, y = x^, using 2 centimeters as the unit of measure. On the same sheet draw the Unes x = +1, y = ±1, y — +x. 2. On coordinate paper sketch the curves x = y^, x = y^, x = yi, X = y^. Compare with the curves of Exercise 1. 56 ELEMENTARY MATHEMATICAL ANALYSIS [§23 3. Sketch and discuss the curves y = xi, y = xi, y = xi. Can any of these curves be drawn from patterns made from the curves of Exercise 1? Why? 4. Draw the curve y^ = x*. Compare with the curve y = x^. 6. Name in each case the quadrants in which the curves of Exer- cises 1-4 lie, and state the reasons why each curve exists in certain quadrants and not in the other quadrants. 23. Discussion of the Parabolic Curves. — Draw the straight ines X = 1, X = —l,y=l,y= —1 upon the same sheet upon which a number of para- ■,/J;^. :^ I B bolic curves have been drawn. These lines to- gether with the coordi- nate axes divide the plane into a number of rect- angular spaces. In Fig. 35 these spaces are shown divided into two sets, those represented by the cross-hatching, and those shown plain. The cross- hatched rectangular spaces 'iontain the lines y = x and y = —x and also all curves of the parabolic type. No parabolic curve enters the rectangular strips shown plain in Fig. 35. The line y = x divides the spaces occupied by the parabolic curves into equal portions. Why does the curve y = x' (in the first quadrant) lie below this line in the interval a; = to a; = 1, but above it in the interval to the right of x = 1 ? On the other hand, why does the curve y = xi, or y^ = x (in the first quad- rant), lie above the line 2/ = a; in the interval a; = to a; = 1 and below y = xin the interval to the right of x = 1? One part of the parabolic curve y = x" always lies in the first quadrant. If n be an even integer, another part of the curve lies in which quadrant? If n be an odd integer, the curve lies in which quadrants? Fig. 35. — The regions of the parabohc and the hyperboUc curves. All parabolic curves he within the cross-hatched region. All hj^jerboUc curves he within the region shown plain. §23] THE POWER FUNCTION 57 If the exponent n of any power function be a positive fraction, say m/r, the equation of the curve may be written y = x". If in this case both m and r be odd, the curve lies in which quadrants? If m be even and r be odd, the curve lies in which quadrants? If m be odd and r be even, the curve lies in which quadrants? If both m and r be even the curve lies in which quadrants? A curve which is symmetrical to another curve with respect to a line may figuratively be spoken of as the reflection or image of the second curve in a mirror represented by the given line. Exercises Exercises 1-5 refer to curves in the first quadrant only. 3 1. The expressions x', x^, x', x* are numerically less than x for values of x between and 1. How is this fact shown in Fig. 34? 2. The expressions x^, x^, a;', s* are numerically greater than x for all values of x numerically greater than unity. How is this fact pictured in Fig. 34? 3. For values of x between and 1, x* 1, X* > x^ > x^ > ^ > X. Explain how each of these facts is expressed by the curves of Fig. 34. 2 i i 1 4. Show that the graphs y = x^, y = a;', y = x'l y = x^ are the reflections of i/ = x^, y = x', y = x\ y = x\ in the line y = x. 6. Sketch on a single sheet of squared paper without tabulating the numerical values, the following loci: y = x^", y = a;"', y = a;'"", y = a;0.01 The following are to be discussed for all quadrants. 6. Sketch, without tabulating numerical values, the following loci y' = x\ y* = x', y^ = x^, y' = x^, y^ = x^. 7. Show that y = —x' is the reflection oi y = x' in the X-axis. 8. Sketch the following; y = —x, y = —x^, y = —x', y' = — x^, y^ — — x^. 9. A ball roUs down a smooth inclined plane making an angle of 45° with the horizontal. The distance s measured in feet along the incline is given by the formula s = 11.4*2 where t is time in seconds. Draw a, graph for this equation. Let time t be represented by distances along the axis of abscissas and dis- 58 ELEMENTARY MATHEMATICAL ANALYSIS [§24 tance s along the axis of ordinates. Let 2 centimeters represent one second on one axis and 10 feet on the other. 24. Hyperbolic Curves. Loci of equations of the form yx" = 1, OT y = l/x", where n is positive, are called hyperbolic curves. The fundamental curve xy = 1, or y = l/x is called the rec- tangular hyperbola. Its graph is given in Fig. 36, but the curve Fig. 36. — The hyperbolas y = x" ioi n = — 1, —2, and —3. should be drawn independently by the student, using 2 centi- meters as the unit of measure. Its relation to the X- and y-axes is most characteristic. For a very small positive value of X, the value of y is very large, and as x approaches 0, y increases indefinitely. But the function is not defined for x = 0, for the product xy cannot equal 1 if x be zero. For numerically small but THE POWER FUNCTION 59 negative values of x, y is negative and numerically very large, and becomes numerically larger as x approaches 0. Instead of saying that "y increases in value without limit," it is just as common to say "y becomes infinite;" in fact, "infinite" is merely the Latin equivalent of "no limit." It is often written 2/ = 00 . This is a mere abbreviation for the longer expressions, "y becomes infinite" or "y increases in value without limit." The student must be cautioned that the symbol <» does not stand Fig. 37. — The hyperbolas y = x" for n -i, —i, —21 and —J for a number, and that "y = oo" must not be interpreted in the same way that "y = 5" is interpreted. As X increases from numerically large negative values to 0, y continually decreases and becomes negatively infinite (abbre- viated 2/ = — 00 ). As a; decreases from numerically large positive values to 0, y continually increases and becomes infinite. Thus, in the neighborhood oi x = 0, y is discontinuous, and, in this case, the discontinuity is called an infinite discontinuity. 60 ELEMENTARY MATHEMATICAL ANALYSIS [§24 0.1 0.2 0.3 0.4 O.S 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.1 1.6 1.6 1.7 1.8 1.9 2.0 Fig. 38. — Hsrperbolic curves in the first quadrant. y = l/s^-^o'is the adiabatic curve for air. On account of the symmetry in xy = l,]i we look upon x as a function o^ y, all of the above statements may be repeated, merely interchanging x and y wherever they occur. Thus, there is an in- finite discontinuity in x, as y passes through the value 0. The lines XX' and YY' which these curves approach as near as' we please, but never meet, are called the asymptotes of the hyperbola. All other curves of the hyper- bolic family, such as yx"^ = 1, ^2/" = 1, 2/'a;' = 1, y^x* = 1 and , the like, approach the X- and y-axes as asymptotes. The rates at which they approach the ^\ ^=^^ ^^m ^^ 1 wMmm ~\ ^ =^= — 1 Fig. 39. — A hyperbola formed by capillary action of two converging plane plates. §25] THE POWER FUNCTION 61 axes depends upon the relative magnitudes of the exponents of the powers of x and y; the quadrants in which the branches lie depend upon the oddness or evenness of these exponents. Exercises 1. Draw accurately upon squared paper the loci, xy = 1, xy' = 1, x'y = 1. Use 2 centimeters as unit and make a pattern for xy = 1. 2. Show that the curves of the hyperboUc type lie in the rectangu- lar regions shown plain, or not cross-hatched, in Fig. 35. 3. In what quadrants do the branches of x^^y'' = 1 he? 4. How does the locus of x^y^ — 1 differ from that oixy = 1 ? 6. Sketch, showing the essential character of each locus, the curves x^y = 1, x^y = 1, s"""?/ = 1. 6. Show that xy = d passes through the point (\/a, -y/a) ', that xy = a' passes through (a, a) and can be made from xy = Ihy "stretch- ing" (if a > 1) both abscissas and ordinates of xy = 1 in the ratio l:a.i 25. Symmetry. Some of the facts of symmetry respecting two portions of the same parabola or hyperbola may be readily ex- tended by the student to other curves. First answer the following questions : How are the points (a, 6) and ( — a, b) related to the F-axis? How are the points (a, 6) and (a, —6) related to the X-axis? How are the points (a, b) and (6, a) related to the line y = xl Prove the result by plane geometry. The following may then be readily proved by the student: Theorems on Loci I. 7/ X he replaced by (—x) in any eqvution containing x and y, the new graph is the reflection of the former in the axis YY'. II. If y be replaced by (—y) in any equation containing x and y, the new graph is the reflection of the former in the axis XX'. III. If x and y be interchanged in any equation containing x and y, the new graph is the reflection of the former in the line y = x. IV. If an equation remains unchanged when x is replaced by {—x), its graph is symmetrical with respect to the Y-axis. 1 To "elongate" or "stretch" in the .ratio 2 :3 naeans to change tfie length of a line segment so that (original length) : (new or stretched length) = 2:3. 62 ELEMENTARY MATHEMATICAL ANALYSIS [§26 v. If an equation remains unchanged when y is replaced by {—y), its graph is symmetrical with respect to the X-axis. VI. If an equation remains unchanged when x and y are inter- changed, its graph is symmetrical with respect to the line y = x. VII. If an equation remains unchanged when x is replaced by i—x) and y by {—y), its graph is symmetrical with respect to the origin. VIII. If an equation remains unchanged when x is replaced by (—J/) and y is replaced by (.—x), its graph is symmetrical with re- spect to the line y = —x. 26. The Variation of the Power Function. The symmetry of the graphs of the power function with respect to certain lines and points, while of interest geometrically, nevertheless does not con- stitute the most important fact in connection with these functions. Of more importance is the law of change of value or the law by which the function varies. Thus returning to a table of values for the power function x^ for the first quadrant. X xi d 1 1 1 4 ? 4, a 1 \ 4 5. 3 g 4 1 % 7 2 ¥- 5 ¥ I 3 ¥ i -V- 5 . 4 ¥- i we note that as x changes from to ^ the function grows by the small amount \. As x changes from \ by another increment of \ to the value 1, the function increases by f to the value 1. As X grows by successive steps or increments of'| unit each, it is seen that x"^ grows by increasingly greater and greater steps, until finally the change in x"^ produced by a small change in x becomes very large. Thus the step by step increase in the func- tion is a rapidly augmenting one, as is shown by the column of §27] THE POWER FUNCTION 63 differences headed "d" in the table. Even more rapidly does the function x^ gain in value as x grows in value. On the contrary, for positive values of x the power functions 1/x, 1/x^, 1/x^, etc., decrease in value as x grows in value. Referring to the definition of the slope of a curve given in §15, we see that the parabolic curves have a positive slope in the first quadrant, while the hyper- bolic curves have always a negative slope in the first quadrant. The law of the power function is stated in more definite terms in §34. That section may be read at once, and then studied again in connection with the practical work which precedes it. 27. Increasing and Decreasing Functions. — As a point passes from left to right along the X-axis, x increases algebraically. As a point moves up on the F-axis, y increases algebraically and as it moves down on the F-axis, y decreases algebraically. An increasing function of x is one such that as x increases algebraic- ally, y, or the function, also increases algebraically. By a decreasing function of x is meant one such that as x increases algebraically, y decreases algebraically. Graphically, an increas- ing function is indicated by a rising curve as a point moves along it from left to right. The power function y = s^ {n positive) is an increasing function of x in the first quadrant and y = x~^ (— n negative) is a decreasing function of x in the first quadrant. The power function y = x^ \& an increasing function for all positive atid for all negative values of x, while y = x'^\&& decreasing function in the second quadrant but an increasing function in the first quadrant. In a case like y = +xi, where y has two values for each positive value of x, it is seen that one of these values increases with x while the other decreases with x. Exercises 1. Consider the function y = +x^. As x grows by successive steps of one unit each, does the function grow by increasingly greater and greater steps or not? Is the slope of the curve an increasing or a decreasing function of x? 2. Does the algebraic value of the slope oi xy = 1 increase with x in the first quadrant? 3. As £ changes from —5 to +5 does the slope oi y = x^ always increase algebraically? 64 ELEMENTARY MATHEMATICAL ANALYSIS [§28 4. Express in the language of mathematics the fact that the curves y = I", when n is a rational number greater than unity, are concave upward. Answer: "When n is greater than unity, the slope of the curve increases as x increases." Express in a similar way the fact that the curves y = x^/" are concave downward. 28. The Graph of the Power Function when x» has a Coeffi- cient. If numerical tables be prepared for the equations y = x^ y' = 2x' and 0\ 1 X> 2 -4 -3 -2 -1 O (a) (b) Fig. 40. — (o) The curves y = x' and y' = 2x'. (b) The curves y = x^ and 2/ = (I) • then for like values of x each ordinate of the second curve will be two times the corresponding ordinate of the first curve. These curves are shown in Fig. 40a. For each position of P on the curve y' = 2x\ DP = 2DQ. It is obvious that the curve J i-u V = X" (1) and the curve y, ^ a^„ ^2) are similarly related; the ordinate of any point of the second graph can be made from the corresponding ordinate (i.e., the ordinate having the same abscissa) of the first graph by multiplying the former by a. If o be positive and greater than unity, this corre- §28] THE POWER FUNCTION 65 sponds to stretching or elongating all ordinates of (1) in the ratio 1 : o; if a be positive and less than unity, it corresponds to con- tracting or shortening all ordinates of (1) in the ratio 1 : a. For example, the graph of y' = ax' can be made from the graph of y = x"ii the latter be first drawn upon sheet rubber, and if then the sheet be uniformly stretched in the y direction in the ratio 1 : a. If the curve be drawn upon sheet rubber which is already under tension in the y direction and if the rubber be allowed to contract in the y direction, the resulting curve has the equation y = ax" where a is a proper fraction or a positive number less than unity. The above results are best kept in mind when expressed in a slightly different form. The equation y' = a-x" can, of course, be written in the form (y'/a) = a;". Comparing this with the equa- tion y = x", we note that (y'/a) = y or y' = ay, therefore we may conclude generally that substituting (y'/a) for y in the equation of any curve multiplies all of the ordinates of the curve by a. For example, after substituting (y'/2) for y in any equation, the new ordinate y' must be twice as large as the old ordinate y, in order that the equation remain true for the same value of x. (x'\ ** — I , that is, substituting ( — ) for x in any equation multiplies all of the abscissas of the curve by a. See Fig. 406. Multiplying all of the abscissas of a curve by a elongates or stretches all of the abscissas in the ratio ^ 1 : a if a > 1, but contracts or shortens all of the abscissas if o < 1. As the above reasoning is true for the equation of any locus, we may state the results more generally as follows: Theorems on Loci IX. Substituting ( -) for z in the equation of any locus multi- plies all of the abscissas of the curve by a. X. Substituting I - ) for y in the equation of any locus multiplies all of the ordinates of the curve by a. 1 See footnote, p. 61. 66 ELEMENTARY MATHEMATICAL ANALYSIS [§29 Note: It is not necessary to retain the symbols x' and y' to indicate new variables, if the change in the variable be otherwise understood. Exercises 1. Without actual construction, compare the graphs y, = a;^ and ^2 1 2 y = 5x2; J/ = a;2 and ^ = "2 ; 2/ = - and J/ = -; y = *» and y = 2x^; s y = x^ and 2/ = -g ' 2. Without actual construction, compare the graphs y = x' and 2/ = f|V; 2/ = s' and ^= x^;y = x^a.ndy = {^j; y = x^ and | = x\ 3. Compare y^ = a;= and i/^ = \k] ; j/^ = i' and (gj = x\- y' =x' 4. Compare the curves t/ = 2x^ and 2/ = 2 ( s j ; 3?/'' = a;' and 32/== (l) '; 2/^ = X' and (2)/)^' = {Zxy. 5. Compare ?/ = a; + 3 and y = 2 {x + 3); y = 2x — 1 and I =2a; - 1; 2/ = 2a; - 1 and 22/ = 2a; - 1. The following exercise involves a different principle from that used above, which the student should reason out for himself. 6. Without actually constructing the curves, compare the curves, for 2/ = 2a; + 3 and y = 2x + 5; y = x^ and y = x^ + l; y == x' and 2/ = a;' + 2; 2/ = a;' and y = x' — 1; y = x' and y = x' + i; y =~ and y = — \- 2;y = x^ and 2/ = (x — 1)^. 29. Change of Unit. To produce the graph of 2/ = lOa;^ from that of 2/ =^ a;^, the stretching of the ordinates in the ratio 1 : 10 need not actually be performed. If the unit of the vertical scale of 2/ = a;^ be taken 1/10 of that of the horizontal scale, and the proper numerical values be placed upon the divisions of the scales, then obviously the graph ol y = x^ may be used for the graph of 2/ = lOx''. Suitable change in the unit of measure on one or both of the scales of 2/ = a?" is often a very desirable method of representing the more general curve y — ax^. An interesting example is given in Fig. 41. The period of vi- THE POWER FUNCTION 67 1.6 1.4 ■1.2 ^ "^ -1.0 |0.8 |o.6 0.4 2 ^ ^ y y / / / / 20 40 60 80 100 120 140 160 180 200 liength in Ooi. Fig. 41.^ — Relation of length of a, simple pendulum to period of vibration. bration of a simple pendulum is given by the formula T = Tr-\/l/g. When g' = 981 cm. per second per second (abbreviated cm. /sec. 2) this gives T = O.W03-\/l, which for many purposes is sufficiently accurate when written T = 0.10\/l. In this equation T must be in seconds and I in centimeters. Thus when I = 100 cm., T = 1 sec, so that the graph may be made by drawing the parabola y = ^/x from the. pattern previously made and then attaching the proper num- bers to the scales, as shown in Fig. 41. 30. Variation. The re- lation between y and a; ex- pressed by the equation y = ax", where n is any positive number, is often expressed by the state- ment "y varies as the nth power of X," or by the statement "y is proportional to X"." Likewise, the relation y = a/x", where n is positive, is expressed by the statement "y varies inversely as the nth power of x." The statement "the elongation of a coil spring is proportional to the weight of the suspended mass" tells us y = mx (1) where y is the elongation (or increase in length from the natural or unloaded length) of the spring, and x is the weight suspended by the spring, hut it does not give us the value of m. The value of m may readily be determined if the elongation corresponding to a given weight be given. Thus if a weight of 10 pounds when sus- pended from the spring produces an elongation of 2 inches in the length of the coil, then, substituting x = 10 and y = 2 m (1), and hence 2 = mlO m = . If this spring be used in the construction of a spring balance, the length of a division of the uniform scale corresponding to 1 pound will be 1/5 inch. 68 ELEMENTARY MATHEMATICAL ANALYSIS [§30 A special symbol, « , is often used to express variation. Thus states that y varies inversely as d^. It is equally well expressed by k where A; is a constant called the proportionality factor. The statements "y varies jointly as u and v," and "y varies directly as u and inversely as v,'' mean, respectively, y = kuv hu Thus the area of a rectangle varies jointly as its length and breadth, or A = kLB. If the length and breadth are measured in feet and A in square feet, k is unity. But, if L and B are measured in feet and A in acres, then k = 1/43560. If L and B are measured in rods and A in acres, then k = 1/160. From Ohm's law, we say that the electric current in a circuit varies directly as the electromotive force and inversely as the resistance, or C a -51 or C = A; ^• K K The constant multiplier is unity if C be measured in amperes, E in volts, and R in ohms, so that for these units ^ - R Exercises 1. The original length of a spring is 10 inches. The force, F, necessary to stretch the spring is directly proportional to its elongation, s. (o) Find the proportionality factor if a force of 200 pounds will hold the spring at a length of 12 inches, (b) What force will be required to hold the spring at a length of 13 inches? (c) What force §31] THE POWER FUNCTION 69 will be required to elongate the spring 1 inch? Note that the elon- gation is the extension of the length beyond the original length and not the total length after elongation. StfGGBSTioN: Since the force F is directly proportional to the elon- gations, we may write F = ks, where k is the proportionality factor. We have given that F is 200 pounds when s is 2 inches. 2. Hooke's Law states that the elongation of a steel bar is propor- tional to the force applied. A bar 500 inches long is stretched to a length of 500.5 inches when a force of 1000 pounds is applied. Find the proportionality factor. 3. Boyle's Law states that the volume of a perfect gas at constant temperature varies inversely as the pressure. If volume is measured as cubic feet and pressure as pounds per square inch, find the pro- portionality factor if the volume is 13 cubic feet when the pressure is 60 pounds per square inch. What will be the volume of the same gas, according to Boyle's Law, if the pressure becomes only 15 pounds per square inch? 31. Illustrations from Science. Some of the most important laws of natural science are expressed by means of the power func- tion' or graphically by means of loci of the parabolic or hyperbolic type. The linear equation y = mx is, of course, the simplest case of the power function and its graph, the straight line, may be regarded as the simplest of the curves of the parabolic type. The following illustrations will make clear the importance of the power function in expressing numerous laws of natural phenomena. Later the student will learn of two additional types of fundamental laws of science expressible by two functions entirely different from the power function now being discussed. The instructor will ask oral questions concerning each of the following illustrations. The student should have in mind the general form of the graph in each case, but should remember that the law of variation, or the law of change of value which the func- tional relation expresses, is the matter of fundamental importance. The graph is useful primarily because it aids to form a mental pic- ture of the law of variation of the function. The practical graph- 1 For brevity ax" as well as a:" will frequently be called a power function of x. 70 ELEMENTARY MATHEMATICAL ANALYSIS [§31 ing of the concrete illustrations given below will not be done at present, but will be taken up later in §33. (a) The pressure of a fluid in a vessel may be expressed in either pounds per square inch or in terms of the height of a column of mer- cury possessing the same static pressure. Thus we may write p = 0.492/!, (1) in which p is pressure in pounds per square inch and h is the height of the column of mercury in inches. The graph' is the straight line through the origin of slope 492/1000. The constant 0.492 can be computed from the data that the weight of mercury is 13.6 times that of an equal volume of water and that 1 cubic foot of water weighs 62.5 pounds. In this and the following equations, it must be remembered that each letter represents a number, and that no equation can be used until all the magnitudes involved are expressed in terms of the particular units which are specified in connection with that equation. (6) The velocity of a falling body which has fallen from a state of rest during the time t, is given by V = 32.2/, (2) in which t is the time in seconds and v is the velocity in feet per second. If t is measured in seconds and v is in centimeters per second, the equation becomes' v = 98K. In either case the graph is a straight line, but the lines have different slopes. ^ A full discuBsion of the process of changing formulas like the ones in the present section into a new set of units should be sought in text-books on physics and mechan- ics. The following method is sufficient for elementary purposes. First, write (for the present example) the formula v = 32. 2£ where v is in ft./sec. and t is in seconds. For any units of measure that may be used, there holds a general relation u = ct, where c is a constant. To determine what we may call the dimensions of c, sub- stitute for all letters in the formula the names of the units in which they are ex- pressed, treating the names as though they were algebraic numbers. From v = ct write, ft./sec, = csec. Hence (solving for dimensions of c), c has dimensions ft./sec.^ Therefore in the given case, we know c = 32.2 ft. /sec. 2. To change to any other units simply substitute equals for equals. Thus 1 ft. = 30.5 cm., hence c = 32.2 X 30.5 cm./sec.2 = 981 cm./sec.^ To change velocity from mi./hr. to ft./sec. in formula (19) below, we have R = 0.00372 where R is in Ib./sq. ft. and V is in mi./hr. Write the general formula R = cY^. The dimensions of c are (lb./ft.2) -7- (mi.Vhr.2) or (lb./ft.2) X (hr.^/mi.sy. In the given case we have the value of c = 0.003 (lb. /ft. 2) X (hr.2/mi.2). To change V to ft./sec, substitute equals for equals, namely 1 hr. = 3600 sec, 1 mi. = 5280 ft., which gives the formula R = 0.0013972^ where V is expressed as ft./sec and R ig expressed as lb./ft.2. Note that 1 mi./hr. = f ft./sec. approximately. §31] THE POWER FUNCTION 71 (c) The space traversed by a falling body is given by s = igt\ (3) or in English itaits (s in feet and t in seconds) s = 16.1(2. ' (4) (d) The velocity of the f aUing body, from the height h, is V = ■\/2gh = V&iAh. (5) The resistance of the air is not taken into account in formulas (2) to (5). The formula equivalent to (5) jTOD^ = mgh, (6) where to is the mass of the body, expresses the equivalence of ^mv^, the kinetic energy of the body, and mgh, the work done by the force of gravity mg, working through the distance h. (e) The intensity of the attraction exerted on a unit mass by the sun or by any planet varies inversely as the square of the distance from the center of mass of the attracting body. If r stand for that distance and if / be the force exerted on unit mass of the attracted body, then / = ^- (7) The constant m is the value of the force when r is unity. (f) The formula for the horse power transmissible by cold-roUed shafting is where H is the horse power transmitted, d the diameter of the shaft in inches, and N the number of revolutions per minute. The rapid increase of this function (as the cube of the diameter) accounts for some interesting facts. Thus doubling the size of the shaft operating at a given speed increases 8-fold the amount of power that can be transmitted, while the weight of the shaft is increased but 4-fold. If H be constant, N varies inversely as d^ Thus an old-fashioned 5.0-h.p. overshot water-wheel making three revolutions per minute requires about a 9-inch shaft, while a DeLaval 50-h.p. steam turbine making 16,000 revolutions per minute requires a turbine shaft but little over J^ inch in diameter. 72 ELEMENTARY MATHEMATICAL ANALYSIS [§31 (g) The period of the simple pendulum is T = irVrTg, (9) where T is the time of one swing in seconds, I the length of the pendu- lum in feet, and g = 32.2 ft. /sec.', approximately. (h) The centripetal force on a particle of weight W pounds, rotat- ing in a circle of radius R feet, at the rate of JV revolutions per second is F = ^ ^"^ , (10) 9 or if ff = 32.16 ft. /sec.?, F = 1.227GWRN' (11) where F is measured in pounds. If N be the number of revolutions per minute, then ^ - 36009 ^^^^ = 0.000341 TFiJi\r2_ (13) (i) An approximation formula for the indicated horse power required/ for a steamboat is I.H.P. = ^, (14) where S is speed in knots, D is displacement in tons, and C is a con- stant appropriate to the size and model of the ship to which it is appUed. The constant ranges in value from about 240, for finely shaped boats, to 200, for fairly shaped boats. (j) Boyle's law for the expansion of a gas maintained at constant temperature is pv = C, (15) where p is the pressure and v the volume of the gas, and C is a constant. Since the density of a gas is inversely proportional to its volume, the above equation may be written in the form P = cp, (16) in which p is the density of the gas. (fc) The flow of water over a trapezoidal weir is given by q= Z.S7Lh^, (17) where q is the quantity in cubic feet per second, L is the length of the weir' in feet, and h is the head of water on the weir, in feet. I The instructor is expected to explain the meaning of the terms here used. §31] THE POWEE FUNCTION 73 {I) The physical law holding tor the adiabatic expanBion of air, that is, the law of expansion holding when the change of volume is not accompanied by a gain or loss of heat/ is expressed by p = cp'-^'' (18) This is a good illustration of a power function with fractional expo- nent. The graph is not greatly different from the semi-cubical parabola y = ci' (to) The pressure or resistance of the air upon a flat surface per- pendicular to the current is given by the formula R = 0.003F^ (19) in which V is the velocity of the air in miles per hour and B is the resulting pressure upon the surface in pounds per square foot. Ac- cording to this law, a 20-mile wind would cause a pressure of about 1.2 pounds per square foot upon the flat surface of. a building. One foot per second is equivalent to about 2/3 mile per hour, so that the formula when the velocity is given in feet per second becomes : R = 0.0013F2. (20) (n) The power used to drive an aeroplane may be, divided into two portions. One portion is utilized in overcoming the resistance of the air to the onward motion. The other part is used to sustain the aeroplane against the force of gravity. The first portion does "use- less" work — ^work that should be made as small as possible by the shapes and sizes of the various parts of the machine. The second part of the power is used to form continuously anew the wave of compressed air upon which the aeroplane rides. Calling the total power'' P, the power required to overcome the resistance Pr, and that used to sus- tain the aeroplane P«, we have P =Pr+P, (21) We learn from the theory of the aeroplane that P, varies as the cube of the velocity, while P, varies inversely as V, so that Pr = cV^ (22) ^ Note that when a vessel containing a gas is insulated by a non-conductor of heat, so that no heat can enter or escape from the vessel, that the temperature of the gaa will rise when the gas is compressed, and fall when it is expanded. Adiabatic expansion may be thought of, therefore, as taking place in an inaulated vessel. 2 Power (= work done per unit time) is measured by the unit horse power, which ia 550 foot-pounds per second. 74 ELEMENTARY MATHEMATICAL ANALYSIS [§31 and P, = k (23) Thus at high velocity less and less power is requireii to sustain the aeroplane but more and more is required to overcome the frictional resistance of the medium. The law expressed by (23) that less and less power is required to sustain the aeroplane as the speed is increased is known as Langley's Law. From this law Langley was convinced 25 24 .23 S22 S21 S20 §19 h18 |17 gl6 3 15 014 1 13 ^112 Sll Sio 39 g 8 « 6 ■g 5- fl 4 a Hi 2 1 in 1 T iL_r rr r /^ M- t '^t t T _/J^z 3L txi t^^ 4 J Z^^/ W^Tf/ '! /b/ /-7/ / ^l^LJ ti Ij^Xj^ ^^ 4ZJ-t tJ L/t/y^V 4 IJjty r /.ryY/^ J ^1^4 Tl_/^Tl/Zy& ti^ tt-i/tj/L^ui^TO ttl--4J^'tij.^\ftiy^4^ ttttj^-/^tKttZZC^ '^trr'^ttZt^Z-^t^ ^ XlTH-JltZC^"^^^^ H 37^^555277^^53?^ Iir77Z55/:522^;^2^ tinmlt2z4>^t^ [I/QZ6Z232|g?^ IDAZgggglpJ^ ffizzpppp^ Wl /Aw^, ^^' m™ ^^^^^ J aj^w L t )Tl*lO«DC-000>o,-IOaCQ-3'»rt(ot-d60»^ Gals.for One Foot Depth Fig. 42. — Capacity of rectangular and circular tanks per foot of depth. that artificial flight was possible, for the whole matter seemed to depend primarily upon getting up sufficient speed. It is really this law that makes the aeroplane possible. An analogous case is the well-known fact that the faster a person skates, the thinner the ice necessary to sustain the skater. In this case part of the energy of the skater is continually forming anew on the thin ice the wave of depression which sustains the skater, while the other part overcomes the frictional resistance of the skates on the ice and the resistance of the air. §32] THE POWER FUNCTION 75 (o) The capacity of cast-iron pipe to transmit water is often given by the formula 9'-88 = 1.68W-" (24) in which q is the quantity of water discharged in cubic feet per second, d is the diameter of the pipe in feet, and h is the loss of head measured in feet of water per 1000 linear feet of pipe. This is a good illustra- tion of the equation of a parabolic curve with complicated fractional exponents. The curve very roughly approximates the locus of the equation y = cVhxi. (25) (p) The contents in gallons of a rectangular tank per foot of depth, 6 feet wide and I feet long, is q = 7.5W. (26) The contents in gallons per foot of depth of "a cylindrical tank d feet in diameter is q = 7.5^^74. (27) Fig. 42 shows the graph of (26) for various values of 6 and also shows to the same scale the graph of (27). 32. Rational and Empirical Equations. — A number of the formulas given above are capable of demonstration by means of theoretical considerations only. Such for example are equations (1), (2), (3), (4), (5), (7), (8), (9), (10), etc., although the constant coefficients in many of these cases were experimentally deter- mined. Formulas of this kind are known in mathematics as rational equations. On the other hand certain of the above for- mulas, especially equations (14), (17), (19), (22), (23), (24), including not only the constant coefficients but also the law of variation of the function itself, are known to be true only as the result of experiment. Such equations are called empirical equations. Such formulas arise in the attempt to express by an equation the results of a series of laboratory measurements. For example, the density of water (that is, the mass per cubic centimeter or the weight per cubic foot) varies with the tem- perature of the water. A large number of experimentors have prepared accurate tables of the density of water for wide ranges of temperature centigrade, and a number of very accurate empir- ical formulas have been ingeniously devised to express the results, of which the following four equations are samples : 76 ELEMENTARY MATHEMATICAL ANALYSIS [§33 Empirical fonnulas jor the density, d, of water in terms of tem- perature centigrade, B. 96(9 - 4)2 (a) d = 1 - 10' (K^ ^ 1 93(0 - 4)i»«^ (6) d = 1 jq^^ , , J , 6fl2 - 369 + 47 (c) "^ = 1 io« , „ J , , 0.4859» - 81.39^ + 6029 - 1118 (d; d = 1 H jq^ Exercises 1. Among the power functions named in the above illustrations, pick out examples of increasing functions and of decreasing functions. 2. Under the same difference of head or pressure, show by formula (24) that an 8-inch pipe will transmit much more than double the quantity of water per second that can be transmitted by a 4-inch pipe. 3. Wind velocities during exceptionally heavy hurricanes on the Atlantic coast are sometimes over 140 miles per hour. Show that the wind pressure on a flat surface during such a storm is about fifty times the amount experienced during a 20-mile wind. 4. Show that for wind velocities of 10, 20, 40, 80, 160 miles per hour (varying in geometrical progression with ratio 2), the pressure exerted on a flat surface is 0.3, 1.2, 4.8, 19.2, 76.8 pounds per square foot respectively (varying in geometrical progression with ratio 4) . 6. A 300-h.p. DeLaval turbine makes 10,000 revolutions per min- ute. Find the necessary diameter of the propeller shaft. 6. A railroad switch target bent over by the wind during a tornado in Minnesota indicated an air pressure due to a wind of 600 miles per hour. Show that the equivalent pressure on a flat surface would be 7.5 pounds per square inch. 7. Show that a parachute 50 feet in diameter and weighing 50 pounds will sustain a man weighing 205 pounds when falling at the rate of 10 feet per second. Suggestion: Use approximate value ir = 22/7 in finding area of parachute from formula for circle, nr^, and use formula (20) above. 8. Show that empirical formulas (a) and (6) for the density of water reduce to a power function if the origin be taken at 9 = 4, d = 1. 33. Practical Graphs of Power Functions. The graphs of the power function 2/ = a;^ y = x^, y = -> y = x\, etc., (1) §33] THE POWER FUNCTION 77 can, of course, be made the basis of the laws concretely expressed by equations (1) to (27) of §31. If, however, the graph of a scientific formula is to serve as a numerical table of the function for actual use in practical work, then there is much more labor in the proper construction of the graph than the mere plotting of the abstract mathematical function. The size of the unit to be selected, the range over which the graph should extend, the permissible course of the curve, become matters of practical importance. If the apparent slope^ of + 1 or —1, it is desirable to make an abrupt change of unit in the vertical or the horizontal scale, so as to bring the curve back to a desirable course, for it is obvious that numeri- cal readings can best be taken from a curve when it crosses the rulings of the coordinate paper at ap- parent slopes differing but little from + 1. The above suggestions in practical graphing are the follow- a graph departs too widely from E 350 ■ i ' 1 \ / ' / / ( ' > 1 / ' > / / / ya -A 7 ir J 2 3 4 5 6 7 8 Diameter of Shaft in Inches 9 10 Pig. 43.— Capacitjr at 100 R.P.M. of cold-rolled shafting to transmit power. illustrated by ing example : Graph the formula (equation (8), §31), for the horse power transmissible by cold- rolled shafting. in which d is the diameter in inches and N is the number of revolutions per minute. The formula is of interest only for the range of d between and 24 inches, as the dimensions of ordinary 1 Of course the real slope of a curve is independent of the scales used. By apparent slope =3 1 is meant that the graph appears to cut the ruling of the squared paper at about 45°. 78 ELEMENTARY MATHEMATICAL ANALYSIS [§33 shafting lie well within these limits. Likewise one would not ordinarily be interested in values of N except those lying between 10 and 3000 revolutions per minute. Fig. 43 shows a suitable graph of this formula for the range 1 < d < 10 for the fixed value oiN = 100. In order properly to graph this function, three different scales have been used for the ordinate H, so that the slope of the curve may not depart too widely from unity. If similar graphs be drawn for N = 200, N = 300, N = 400, etc., a set of parabolas is obtained from which the horse power of shafting for various speeds of rotation as well as for various diameters may be obtained at once. A set of curves systematic- ally constructed in a manner similar to that just described, is often called a family of curves. Fig. 42 shows a family of straight lines expressing the capacity of rectangular tanks corresponding to the various widths of the tanks. Inasmuch as many of the fqrmulas of science are used only for positive values of the variables, it is only necessary in these cases to graph the function in the first quadrant. For such problems the origin may be taken at the lower left corner of the coordinate paper so that the entire sheet becomes available for the curve in the first quadrant. The illustrations of §31 are sufficient to make clear the impor- tance in science of the functions now being discussed. The follow- ing exercises give further practice in the useful application of the properties of the functions. Exercises The graphs for the following problems are to be constructed upon rectangular coordinate paper. The instructions are for centimeter paper (form Ml) ruled into 20 X 25 cm. squares. On other paper use J inch in place of 1 centimeter. In each case the units for abscissa and for ordinates are to be so selected as best to exhibit the functions, considering both the workable range of values of the variables and ' the suitable slope of the curves. The student should read §12 a second time before proceeding with the following exercises, giving especial care to instructions (4), (5), and (6) of that section. 1. Classify the graphs of formulas (1) to (27), §31, as to parabolic or hyperbolic type. §33] THE POWER FUNCTION 79 2. Graph the formula v^ = 2gh, or v = y/lgh = 8.02hi, if h range between. 1 and 100, the second and foot being the units of measure. See formula (5), §31. The following table of values is readily obtained : h\ 1 5 10 20 30 40 50 60 70 80 90 100 v\ 8.02 17.9 25.3 35.8 43.9 50.7 56.7 62.1 67.1 71.7 76.0 80.2 Use 2 cm. = 10 feet as the horizontal unit for h, and 2 cm. = 10 feet per second as the vertipal unit for v. The graph is then readily con- structed without change of unit or other special expedient.^ 3. Graph the formula q = 3.37Lhi fori = 1, and for h = 0, 0.1, 0.2, 0.3, 0.4, 0.5. See formula (17), §31. Use 4 cm. = 0.1 for horizontal unit for h and 2 cm. =0.1 for vertical unit for q. 4. Draw a curve showing the indicated horse power of a ship I.H.P. = S'Di/C for C = 200 if the displacement D = 8000 tons, and for the range of speeds iS = 10 to S = 20 knots. See formula (14), §31. For the vertical unit use 1 cm. — 1000 h.p. and for the horizontal unit use 2 cm. = 1 knot. Call the lower left-hand corner of the paper the point (S = 10, I.H.P. = 0). 5. From the formula expressing the centripetal force in pounds of a rotating body, F = 0.000341 ITiJiV^ draw a curve showing the total centripetal force sustained by a 36-inch automobile tire weighing 25 pounds, for all speeds from 10 to 40 miles per hour. See formula (13), §31. Miles per hour must first be converted into revolutions per minute by .dividing 5280 by the circumference of the tire and then dividing the result by 60. This gives 1 mile per hour = 9J revolutions per minute If V be the speed in miles per hour the formula for F becomes F = 0.000341(1.5)25(9J)2F2 = l.llF^ For horizontal scale let 4 cm. = 10 miles per hour and for the vertical scale let 1 cm: = 100 pounds. 6. Draw a curve from the formula / = m/r'^ showing the accelera- tion of gravity due to the earth at all points between the surface of the earth and a point 240,000 miles (the distance to the moon) from the center, if / = 32.2 when radius of the earth = 4000 miles. It is convenient in constructing this graph to take the radius of the earth as unity, so that the graph will then bo required of / = 32.2/r^ 80 ELEMENTARY MATHEMATICAL ANALYSIS [§34 from r ■« 1 to r = 60. In order to construct a suitable curve several changes of units are desirable. See Kg. 44. One centimeter repre- sents one radius (4000 miles) from r = to r = 10, after which the scale is reduced so that 1 mm. represents one radius. In the vertical direction the scale is 4 cm. = 10 feet per second for < r < 5, 4 cm. = 1 foot a second for 5 < r < 10, and 4 cm. =0.1 foot a second for 10 < r < 60. Even with these four changes of units just used the first and third curves are somewhat steep. The student can readily improve on the scheme of Fig. 44 by a better selection of units. 40 80 20 90 I I \ 1 I \ N> \ \ S ' V ^ \, \ ^, --. \ s "^ ^ — g 43 o I 10 1 2 3 4 6 6 7 8 9 10 20 SO 40 EO 60 Distance from Earth's. Oeuter, Earth's Eadiuscl Fig. 44. — Gravitational acceleration at various distances from the earth's center. The moon is distant approximately 60 earth's radii from the center of the earth. 34. The Law of the Power Functions. Sufficient illustrations have been given to show the fundamental character of the power function as an expression of numerous laws of natural phenomena. How may a functional dependence of this sort be expressed in words? If a series of measurements are made in the laboratory, so as to produce a numerical table of data covering certain phe- nomena, how can it be determined whether or not a power func- tion can be written down which will express.the law (that is, the function) defined by the numerical table of laboratory results? §34] THE POWER FUNCTION 81 The answers to these questions are readily given. Consider first the law of the falling body s = 16.1i^ (1) Make a table of values for values oi t = 1, 2, 4, 8, 16 seconds, as follows : t 1 2 4 8 16 s 16.1 64.4 257.6 1030.4 4121.6 The values of t have been so selected that t increases by a fixed multiple, that is, each value of t in the sequence is twice the pre- ceding value. From the corresponding values of s it is observed that s also increases by a fixed multiple, namely 4. Similar conclusions obviously hold for any power function. Take the general case y = ax", (2) where n is any exponent, positive, negative, integral or fractional. Let X change from any value xi to a multiple value mxi and call the corresponding values of y, yi and 2/2. Then we have 2/1 = axi", (3) and 2/2 = o(wia;i)" = aiwxi". (4) Divide the members of (4) by the members of (3) and we have ^ = m». (5) 2/1 That is, if a; in any power function change by the fixed multiple m, then the value of y will change by a fixed multiple w. Thus the law of the power function may be stated in words in either of the two following forms : In any power function of x, if x change by a fixed multiple, y will change by a fixed multiple also. In any power function of x, if x increase by a fixed percent, the function will increase or decrease by a fixed percent also. This test may readily be applied to laboratory data to determine whether or not a power, function can be set up to represent as a formula the data in hand. To apply this test, select at several places in one column of the laboratory data, pairs of numbers which change by a selected fixed percent, say 10 per cent, or 20 82 ELEMENTARY MATHEMATICAL ANALYSIS 1§35 percent, or any convenient percent. Then the corresponding pairs of numbers in the other column of the table must also be related by a fixed percent (of course, not in general the same as the first- named percent), provided the functional relation is expressible by means of a power function. If this test does not succeed, then the function in hand is not a power function. Since the fixed percent for the function is ot" if the fixed percent for the variable be m, the possibility of determining n exists, since the table of laboratory data must yield the numerical values of both m and to" 36. Simple Modifications of the Parabolic and of the H]rperbolic Types of Curves. In the study of the motion of objects it is convenient to divide bodies into two classes: first, bodies which retain their size and shape unaltered during the motion; second, bodies which suffer change of size or shape or both during the motion. The first class of bodies are called rigid bodies ; a mov- ing stone, the reciprocating or rotating parts of a machine, are illustrations. The second class of bodies are called elastic bodies ; a piece of rubber during stretching, a spring during elongation or contraction, a rope or wire while being coiled, the water flowing in a set of pipes, are all illustrations of this class of bodies. When a body changes size or shape the motion is called a strain. Bodies that preserve their size and shape unchanged may possess motion of two simple types: (1) Rotation, in which all particles of the body move in circles whose centers lie in a straight fine called the axis of rotation, which line is perpendicular to the plane of the circles, and (2) translation, in which every straight line of the body remains fixed in direction. We have already noted that the curve. (1) 2/1 = ax' or a s". can be made from the curve y = x" (2) by multiplying all the ordinates of (2) by a. The effect is either §36] THE POWER FUNCTION 83 to elongate or to contract all of the ordinates, depending upon whether a > 1 or a < 1, respectively. The substitution of (j/i/a) for y has therefore produced a motion or strain in the curve y = x". Likewise »=(?)■ (« can be made from 2/ = X" (4) by multiplying all of the abscissas of (4) by a. The effect is either to stretch or to contract all of the abscissas, depending upon whether a > 1, or a < 1, respectively. In general, if a curve has the equation V = fix), (5) then (6) »=/(?) is nlade from curve (5) by ' lengthening or stretching the XY- plane uniformly in the x direction in the ratio 1 : a. The statement just given is made on- the assumption that a>l. If a; (4) y =2x + 7. 6. The loci named in exercise 4 have their ordinates shortened in the ratio 2:1; write their equations. 6. Show that y = — -r-r and y = r are hyperbolas. X ~i~ X o 7. Show that y = — -j-r^ is a hyperbola. Note : Divide the numerator by the denominator, obtaining the b b equation y = 1 ^—r> oi y — 1 = — 8. Show that y =^ or x+b'"'" ' - x+b X + a xH>' a — b y = i+: x+b is a hyperbola, namely the curve xy = a — b translated to a new position. « oi i 1 , N ^ ,i_s ^ + 3 , ^ 3x + 2 , 9. Sketch: (a) y = ^-^7^; (b) y = ^qrj; (c) V = "Jipj"; ^"" id) y = — 3-0"' Sketch a curve from which each curve is obtained by translation. 10. Show how the graph for t/ = x^ + 4:X + 5 may be obtained from the graph for y = x". Hint: 2/ = x^ + 4x + 5 = x^ + 4x + 4 + 1 = (x + 2)^ + 1, or 2/ — 1 = (x + 2)2. Thus, the graph for y = x' + 4:X + 5 may be obtained by translating the graph for y = x^ one unit up and two units to the left. 11. Sketch the curves for: (a) 2/ = x2 + 4x + 4; (b) y = x^ + 6x + 10; (c) 2/ = x2 + 2x - 3; (d) 2/ = 4x2 4. 4^; + j. (e) y = 4x' + 2x - 1; (f) y = ^x - x"; (s) 2/ = 6x - x"; (h) 2/ = x^ + 3x - 1; (i) y ^2x^ +Zx; (j) j/ = 3x - 2x2; (fc) 2/^ = X + 1. 12. Which of the curves of exercise 11 pass through the origin? 13. Sketch: (a) x2 +2/2 = 1; (6) x^ + j/^ = 4; (c) x2 + (2/ - i;2 = 1; (d) (x - 1)2 + 2/2 = 4; (e) (x + 1)2 + (y - 2)2 = 5; (/) x2 + 2x + 2/* = 3. §37] THE POWER FUNCTION 87 37. Shearing Motion. Let the dotted curve Pi'OPi, Fig. 46 be the graph of the semicubical parabola y = xi and OP2 the graph of y "= X. The graph P'OP is constructed by taking its ordinate, for any value of the abscissa, equal to the (algebraic) sum of the ordinates of the two given curves. Thus, DP = DPi + DPi and DP' = DPi' + DP,. The equation of P'OP is y = a;t + X, since DPi = xi and DP2 = X. Exercises 1. From the curves for y = x^ and y = ^x, sketch y = x^ -\- |.-c. 2. From the curves for y = x'^ and 2/ = — |x, sketch y = x'' — ^x. 3. From the curves for y = — x^ and y = x, sketch y = x — x^. 4. From the curves for y = - X and y = X, sketch y = — \- x- 5. From the curves for y = - X ■ and y = X, sketch y = x 38. General Case. Consider the production of the curve y = fix) + mx (1)' from the curve Fig — The shear of y the line y = x. and the straight line y' = m (2) (3) Graphically, the curve (1) is seen to be formed by the addition of the ordinates of the straight line y" = mx to the corresponding ordinates of y' = f(x) . Thus, in Fig. 47, the graph of the func- tion a;^ + a; is made by adding the corresponding ordinates of 88 ELEMENTARY MATHEMATICAL ANALYSIS [§38 y = x^ and y = x. Mechanically, this might be done by draw- ing the curve on the edge of a pack of cards (see Fig. 48), and then slipping the cards over each other uniform amounts. The change of the shape of a body, or the strain of a body, here illustrated, is called lamellar motion or shearing motion. It is a form of motion of very great importance. 4 m \ / / / " / / /a / \ Jl A -3 -2 I Vx X A ! 3 4 -•>. \ / / -3 \ ■4 Fig. 47. — The shear of the cubical parabola 2/ = a' in the line y = X, and also in the fine y = — x. We shall speak of the locus y = f{x) + mx as the shear of the curve y = f(x) in the line y = mx. Theorems on Loci XIII. The addition of the term mx to the right side of y = f{x) shears the locus y = f(x) in the line y = mx. The locus y — ax^ + TOcc + 6 is made from y = a;' by a combination of first, a uniform elongation THE POWER FUNCTION 89 [a], second,, a shearing motion [m], and third, a translation [6]. Either motion may be changed in sense by changing the sign of a, m, or 6, respectively. The student may easily show that the effect of a shearing motion upon the straight line y = mx + b is merely a rotation about the fixed point (0, b). The line is really stretched in the direction M H K 8 7 5 4 3 2 1 - o o o o 2 o o o I o )" 1 2 3 4 5 6 7 3 - 1 2 Fig. 48. — Shearing motion illustrated by the slipping of the members of pack of cards. of its own length, but this does not change the shape of the line nor does it change the line geometrically. A line segment (that is, a hne of finite length) would be modified, however. The parabola y = x^ is transformed under a shearing motion in a most interesting way. For, after shear, y = x' becomes y = x^ + 2mx, (4) where, for convenience, the amount of the shearing motion is 90 ELEMENTARY MATHEMATICAL ANALYSIS [§38 represented by 2w instead of by m. Writing. this in the form y = x^ + 2mx + m^ — m^, or 2/ = (x + my — m^, y + m'' = (x + mY, (5) we see that (4) can be made from the parabola y = x'^ hy trans- lating the curve to the left the amount m and down the amount m^. (See Fig. 49.) \ \ 4 S "11 <='7 / "V / \ \ •>. \\ -^ fe*/' ■3 -2 T -1 . I 3 y^ -9. .9. -4 Fig. 49. — The shear oi y = x^ in the line y = . 6x. Shearing motion, therefore, rotates the straight line and trans- lates the parabola. The effect on other curves is much more complicated, as is seen from Figs. 46 to 48. The parabola y = x^ after shear is identical in size and shape with y = x^ -\- mx + b. Likewise, y = ax' -\- bx + c is a para- bola differing only in position from y = ax'. Exercises 1. Explain how the curve y = x^ -\- 2x may be made from the curve y = x^. How can the curve y = 2x' + 3x be made from the curve y = 2x'? §39] THE POWER FUNCTION 91 2. Find the coordinates of the lowest point oi y — x^ — ix, that is, put this equation in the form y — b = {x — a)^. 3. Compare the curves y = x' -\- 2x and y = x^ — 2x. (Do not draw the curves.) 4. Explain how the curve y = 1/x + 2x may be formed from the curve y = 1/x and oi y = 2x. • , 39. Rotation of a Locus. The only simple type of displace- ment of a locus not yet considered is the rotation of the locus about the origin 0. This will be taken up in the next chapter. 40. Roots of Functions. The roots, or zeros, of a function are the values of the argument for which the corresponding value of the function is zero. Thus, 2 and 3 are rgots of the function x^ — 5x + 6, for substituting either number for x causes the function to become zero. The roots of a;^ — a; — 6 are + 3 and - 2. The roots of x^ - 6x^ + llx - 6 are 1, 2, 3. The word root, used in this sense, has, of course, an entirely different significance from the same word in "square root," "cube root,'' etc. But the roots of the function x'' — 5x — 6 are also the roots of the equation x' — 5x — Q = 0. In the graph of the cubic function y = x' — x in Fig. 47, the curve crosses the X-axis at a; = — 1, x = 0, and x = 1. These are the values of x that make the function x^ — x zero, and are, of course, the roots of the function a;' — x. No matter what the function may be, it is obvious that the intercepts on the X-axis of the curve y = f{x), as OA, OB, Fig. 47, must represent the roots off(x). Exercises 1. From the curve y = x^ sketch the curves j/ — 4 = x^; i/ = 4x^; ^y = x^; y = (x - 4)2. „ ^, , x' . , X* , (x — 3)2 2. Sketch y = -i^;y = ^^ - z] V = -2 - ^' , but sec 0° = L 110 ELEMENTARY MATHEMATICAL ANALYSIS [§54 In a similar way the functions of 90° may be investigated. Tiie following table gives the variation of the functions as the angle varies from 0° to 90°, from 90° to 180°, etc.: Angle From 0° to 90° From 90° to 180° Prom 180° to 270° From 270° to 360° Sin Cos Tan Cot Sec Oto + 1 + Ito to +«> + 0O to + 1 to + 00 + CO to + 1 + Ito Oto - 1 -o= to to - oo — oo to — 1 + 1 to + oo Csc The student is to supply the results for the last two coltimns. 54. Fundamental Selations. The trigonometric functions are not independent of each other. Because of the relation x^ + j/^ = a'-, it is possible to compute the numerical or absolute values of the remaining five functions when the value of any one of the six is given. This may be accompHshed by means of the fundamental formulas derived below. Divide the members of the equation by a? Then or, a:^ + 2/2 = 1, sin" e + cos2 e = I. Likewise divide (1) through by x^; then ' + ©"=©" or sec2 = 1+ tan2 B. Also divide (1) through by y'^; then cgc^ ? = 1 + cot= 9. or (1) (2) (3) (4) THE CIRCLE AND THE CIRCULAR FUNCTIONS 111 Also, since we obtain and likewise y a X a tan e = cot e = y sin 6 cos d cos 6 (5) (6) Formulas (2) to (6) are the fundamental relations between the six trigonometric functions. They must be committed to memory by the student. = tan csc-=l + cot- FiG. 55. — Diagram of the relations between the six circular functions. sin cos The above relations between the functions may be illustrated by a diagram as in Fig. 55. The simpler, or reciprocal, relations are shown by the connecting lines drawn above the functions. The reciprocal equations and the formulas (2), (3), and (4) are sufficient to express the absolute or numerical value of any function of any angle in terms of any other function of that angle. The algebraic sign to be given the result must be properly selected in each case according to the quadrant in which the angle lies. Exercises All angles in the following exercises are supposed to be less than ninety degrees. 112 ELEMENTARY MATHEMATICAL ANALYSIS [§54 1. Sin e = 1/5. Find the values for the other five circular functions. Draw a right triangle whose hypotenuse is 5, whose altitude is 1 and whose base coincides with OX. In other words, make a = 5 and y = 1 in Fig. 56. Calculate x = v'25 — 1 = 2 \/6 and write down all of the functions from their definitions. 2. Cos 9 = 1/3. Find esc 9. Take a = 3 and a; = 1 in Fig. 56. Find y and then write down the function from its definition. 3. Tan 9 = 2. Find sin d. Take x = I and y = 2 in Fig. 56, calculate a, and then write down the function from its definition. O X A Fig. 56. — Triangle of reference for B and for complement of S. 4. Sec e = 10. Find esc 6. Take a = 10 and x = 1 and compute y. 6. Find the values of all functions of 9 if cot 6 = 1.5. 6. Find the functions of 9 if cos 8 = 0.1. 7. Find the values of each of the remaining circular functions in each of the following cases: (a) sin e = 5/13. (d) tan e = 3/4. (6) cos e = 4/5. (e) sec 9 = 2. (g) tan 6 = m. {h) sin e = Vc" + d' (c) sec = 1.25. (/) tan e = 1/3. Show that the following equalities are correct: 8. tan d cos 9 = sin 6. 9. sin e cot B sec 9 = 1. 10. (sin 9 + cos 9)2 = 2 sin 9 cos 9 + 1. 11. tan 9 + cot 9 = sec 9 esc 9. 12. Express each trigonometric function in terms of each of the others; i.e., fill in all blank spaces in the following table; §54] THE CIRCLE AND THE CIRCULAR FUNCTIONS 113 sin cos tan cot sec cso sin sin 1 esc cos cos 1 sec tan tan 1 cot cot 1 tan cot sec 1 cos sec CSC 1 sin CSC Fig. 57. — Diagram for exercise 13. The following exercises refer to angles <360° of any quadrant: 13. If sin 9 = — 3/4 and tan B is positive, find the remaining five functions. Hint: Since sin e is negative and tan 9 is positive, the angle 9 is in the third quadrant. See Fig. 57. 8 114 EaiEMENTARY MATHEMATICAL ANALYSIS [§55 14. If C08 9 = 12/13 and sin 9 ia negative, find the remaining five functions of 6. 15. If tan e = — \/3 and cos B is negative, find the remaining func- tions of e. 16. If cos 9 = — 1/3 and sin 9 is positive, find the remaining functions. 17. If tan 9 = 5/12 and sec 9 is negative, find the remaining functions of 9. 18. If sin 9 = 3/5 and tan 9 is negative, find the jemaining func- tions of 9. Pl(k,h) P lh,/c) Fig. 58. — Triangles of reference for complementary angles. 65. Functions of Complementary Angles. Angles are said to be complementary if their sum is 90°. Angles are said to be supplementary if their sum is 180°. Let be an angle in the first quadrant, and draw the angle (90° -0) of terminal side OPi, as shown in Fig. 58. Let P and Pi lie on a circle of radius a. Let the coordinates of the point P be {h, k), then Pi is the point {k, h). Hence PiDi/OPi = h/a = sin (90° -5). But from the triangle PDO, h/a = cos 8. Hence Likewise, sin (90° — 6) = cos 6 tan (90° - 0) = cot e sec (90° - d) = esc e These relations explain the meaning of the words cosine, co- tangent, cosecant, which are merely abbreviations for comple- §56] THE CIRCLE AND THE CIRCULAR FUNCTIONS 115 merit's sine, complement's tangent, etc. Collectively, cosine, cotangent, and cosecant are called the co-functions. Likewise, from Fig. 58, cos (90° — 6) = siad cot (90° - 0) = tan e CSC (90° — d) = sec d Later it will be shown that the above relations hold for all values of d, positive or negative. 56. Graph of the Sine and Cosine. In rectangular coordinates we can think of the ordinate y of a point as depending for its value upon the abscissa or x of that point by means of the equation y = sin X, provided we think of each value of the abscissa laid off on the Z-axis as standing for some amount of angular mag- nitude. Therefore the equation y = sinx must possess a graph Y A e C p ^ - - 7 ^ - - i. -- -- - - - - - - - - - - - - - A / s ^ \\ A / « -TT A / 1/ s p s. /v B D,D/^ 1 s / A s / y / \ / y / \ / y y ^_ - - - - — ^ - - - - ^ - -- -q |h - - - - V s - - - - - -^ 2 ^ - - = X L. _ _ _ „ _ _ _ _ _ a 5 ^^ 2-1 r- _ _ „ ^ _ _ ^ ^ T- ^ _ _ ^ -> Fig. 59. — Construction of the sinusoid. in rectangular coordinates. In order to produce the graph of y = sin X, it is best to lay off the angular measure x on the X- axis in such a manner that it may conveniently be thought of in either radian or degree measure. If we suppose that a scale of inches and tenths is in the hands of the student and that a graph is required upon an ordinary sheet of unruled paper of letter size (8|- X 11 inches), then it will be convenient to let 1/5 inch of the horizontal scale of the X-axis correspond to 10° or to ir/18 radians of angular measure. To accomplish this, the length of one radian must be 1.15 inches (i.e., 18/5t inch), which length must be used for the radius of the circle on which the arcs of the angles are laid off. Hence, to graph y = sin x, 116 ELEMENTARY MATHEMATICAL ANALYSIS [§56 draw at the left of a sheet of (unruled) drawing paper a circle of rddius 1.15 inches, as the circle OP5B, Fig. 59. Take as the origin and prolong the radius BO for the positive portion OX of the X-axis. Divide OL into 1/5-uich intervals, each corre- sponding to 10° of angle; eighteen of these correspond to the length IT, if the radius BO (1.15 inches) be the unit of measure. Next divide the F-axis proportionately to sin x in the following manner : With a pair of bow dividers, or by means of a protractor, divide the semicircle into eighteen equal divisions as shown in the figure, thus making the length of each small arc exactly 1/5 inch. The perpendiculars, or ordinates, dropped upon OX from each point of division, divided by the radius, are the sines of the corresponding angles. Draw lines parallel to OX through each point of division of this circle. ' These cut the F-axis at points Ai, Ai, . Then if the radius of the circle be called unity, the distances OAi, OA 2, OA3, . . are respectively the sines of the angles OBPi, OBP2, OBP3, These are the successive ordinates corresponding to the abscissas already laid off on OL. The curve is then constructed as follows : First draw vertical lines through the points of division of OX; these, with the horizontal lines already drawn, divide the plane into a large number of rec- tangles. Starting at and sketching the diagonals (curved to fit the alignment of the points) of successive "cornering" rec- tangles, the curve OCNTL is approximated, which is the graph oi y — sin x. This curve is called the sinusoid or sine curve. The curve is of very great importance for it is found to be the type form of the fundamental waves of science, such as sound waves, vibrations of wires, rods, plates and bridge members, tidal waves in the ocean, and ripples on a water surface. The ordinary progressive waves of the sea are, however, not of this shape. Using terms borrowed from the language of waves, we may call C the crest, TV the node, and T the trough of the sinusoid. It is obvious that as x increases beyond 27r'', the curve is re- peated, and that the pattern OCNTL is repeated again and again both to the left and to the right of the diagram as drawn. Thus it is seen that the sine is a periodic function of period 2t' or 360°. \For lack of room only a few of the successive points Pi, P2, P3, , , , of (iivigign ef the quadrant OPjPf are actually lettered in Fig. 59, §67] THE CIRCLE AND THE CIRCULAR FUNCTIONS 117 The small rectangles lying along the X-axis are nearly squares. They would be exactly equilateral if the straight Hne OAi was equal to the arc OPi. This equality is approached as near as we please as the number of corresponding divisions of the circle and of OX is indefinitely increased. In this way we arrive at the notion of the slope of a curve in mathematics. In this case wfe say that the slope of the sinusoid at is + 1 and at A'^ is — 1, and at L is + 1. We say that the curve outs the axis at an angle of 45° at and at an angle of 315° (or — 45° if we prefer) at N. The slope at C and at T is zero. The, curve y = a sin x is made from y = sin x by multiplying all of the ordinates of the latter by a. The number a is called the amplitude of the sinusoid. 57. The Cosine Curve. In Fig. 60 let the angles COPi, COP2, COP3, etc., be laid off from the position of the F-axis OY as initial side. Then if the radius of the circle be called unity, the dis- Y ^Z), D D ¥- s; f^^ \ \ R/\ \ \ \ N ' ^ 1 /\ \\\\ D. S / / ^\\\\ s / / ^>A\\\\ s / \ A L / A, 7r \ or / H TT 27r ? S / ? \ \ / \ s / / _ _ _ _ — — — ^ — — — ^ — _ _ — — T Y 1 t- Fig. 60. — Construction of the cosine curve. tances ODj, OD2, OD3, . . are respectively the cosines of the angles COP\, COP^, COP3, . If the distances laid off on the Z-axis represent the measures of the successive angles COPi, COP2, then the curve shown in the figure has the equation y = cos X. The construction shows that the curve is exactly the same as the sine curve of Fig. 59 except that the origin for the cosine curve is under the crest while in the sine curve the origin is at a node. If the origin be taken at 0' in Fig. 59 the curve may be called the cosine curve. 118 ELEMENTAEY MATHEMATICAL ANALYSIS [§68 In Fig. 62 the curve ABODE is the cosine curve y = cos x. The other curve is the sine curve y = sin x. 68. The Sine of a Negative Angle. In Fig. 61 the full drawn curve represents the graph for y = sin x. The graph for y = sin (— x) (1) r FiQ. 61. — The relation between y = aia x and y = sin (— s). may be obtained by rotating the graph for y = sin x, 180° about the F-axis, by Theorem I on Loci. This gives the dotted curve of Fig. 61. But from the properties of the sinusoid, the dotted curve is the reflection in the Z-axis of the curve drawn in full, hence the equation of the dotted curve may also be written — 2/ = sin X. (2) Hence, from (1) and (2) sin (— x) = —sin x. (3) 69. Complementary Angles. Fig. 62 shows the curves for y = cos X and for y = sin x. By the properties of these curves Fig. 62. — Comparison of the sine and cosine curves. it is obvious that the cosine curve may be regarded as the sine curve translated x/2 units to the left. That is, the cosine curve y = cos X (1) has also the equation y = sin (a; + I) • (2) §60] THE CIRCLE AND THE CIRCULAR FUNCTIONS 119 Since this curve (the cosine curve) is symmetrical about the Y- axis, its equation remains unchanged if we change x to {—x), by Theorem I on Loci. Hence the cosine curve has also the equation y = sin ( - ^ + |\ = smi^-x\- (3) Comparing (1) and (3) we see that we have proved for all values of X that sin ( x| = cosx. (4) By comparing (1) with (2) we see that sin (s + *) ~ ^°^ ^' this fact is, however, niuch less useful than that represented by equation (4). Exercises From 'the curves for y = sin x and y = sin( — x), Fig. 61 shows that: 1. sin {x — v) = sin {—x) and hence sin (tt — s) = sin x. From the curves for y = cos x and y =^ sxixx, Fig. 62, shows that: 2. sin X = cos {x — s). 3. cos x = sin {x — fir). 4. cos {x + -fir) = sin x. 5. cos (.—x) = cos x. 60. Trigonometric Functions of Negative Angles. We have already shown, (3) §57, that sin (— x) = — sin x. (1) Also from Fig. 62, since the cosine curve is symmetrical about the y-axis, cos (— x) = cos X. (2) Dividing the members of (1) by the members of (2) we find tan (— x) = tan x. (3) 61. Odd and Even Fimctions. A function that changes sign but retains the same numerical value when the sign of the argu- ment is changed is called an odd function. Thus sin x is an odd function of x, since sin (—x) = —sin x. Likewise x^ is an odd function of x, as are all odd powers of x. The graph of an odd function of a; is symmetrical with respect to the origin ; that is, 120 ELEMENTARY MATHEMATICAL ANALYSIS [§62 if P is a point on the curve, then if the line OP be produced backward through a distance equal to OP to a point P', then P' also lies on the curve. The parts of 2/ = x' in the first and third quadrants are good illustrations of this property. A function of x that remains unaltered, both in sign and numerical value, when the argument is changed in sign, is called an even function of x. Examples are cos x, x', x^ — 3x*, The graph of an even function is symmetrical with respect to the y-axis. Most functions are neither odd nor even, but mixed, like x^ + sin X, x^ -\- x', and x + cos x. Exercises 1. Is sin's an odd or an even function of x1 Is tan'a; an odd or an even function of x7 2. Is the function sin x + 2 tan x an odd or an even'functfon? Is sin X + cos x an odd or an even function of a;? 62. The Defining Equations Cleared of Fractions. The student should commit to memory the equations defining the trigonometric functions when cleared of fractions. In this form the equations are quite as useful as the original ratios. They are written: y = a sin 6 y = x tan 6 a = x sec 6 X = a cos 6 X = J cot d a = y esc 9 As applied to the right angled triangle, these three sets of equa- tions may be stated in words as follows: Either leg of a right triangle is equal to the hypotenuse multiplied by the sine of the opposite, or by the cosine of the adjacent, angle. Either leg of aright triangle is equal to the other leg multiplied by the tangent of the opposite, or by the cotangent of the adjacent, angle. The hypotenuse of a right triangle is equal to either leg multiplied by the secant of the angle adjacent, or by the cosecant of the angle opposite that leg. These statements should be committed to memory. 63. Orthographic Projection. In elementary geometry we learned that the projection of a given point P upon a given line or plane is the foot of the perpendicular dropped from the given point §63] THE CIRCLE AND THE CIRCULAR FUNCTIONS 121 upon the given line or plane. Likewise if perpendiculars be dropped from the end points A and B of any line segment AB upon a given line or plane, and if the feet of these perpendiculars be called P and Q, respectively, then the line segment PQ is called the projection of the line AB. Also, if perpendiculars be dropped from all points of a given curve AB upon a given plane MO, the locus formed by the feet of all perpendiculars so drawn is called the projection of the given curve upon the plane MO. To emphasize the fact that the projections were made by. using perpendiculars to the given plane, it is customary to speak of them as orthogonal or orthographic projections. Pig. 63. — Orthographic projection of line segments. The shadow of a hoop upon a plane surface is not the ortho- graphic projection of the hoop unless the rays of light from the sun strike perpendicular to the surface. This could only happen in our latitude upon a suitable non-horizontal surface. The shortening, by a given fractional amount, of a set of parallel line segments of a plane may be brought about geometric- ally by orthographic projection of all points of the line segments upon a second plane. For, in Fig. 63, let AiBi, A^Bi, A3B3, etc., be parallel line segments lying in the plane MN. Let their projections on any other plane be Aid', AiC^', Ai'C-/, etc., respectively. Draw A\.C\ parallel to Ai'Ci' and Aid parallel to 122 ELEMENTARY MATHEMATICAL ANALYSIS [§63 Aj'Cj', etc. Then since the right triangles AiBiCi, AiB^Ct, AiBaCz, etc. are similar, AiBi A2B2 AsBs AiCi A2C2 AiC Call this ratio a. It is evident that a>l. Substitute the equals: A/d' = AiCi, Aj'Ca' = A^d, etc. i'hen AiBi _ A2B2 _ A3B3 _ , _a Ai'Ci' ~ Ai'Ci' " As'c ~ ' ~r The numerators are the original lii\e segments; the denominators are their projections on the plane MO. The equality of these fractions shows that the parallel lines have all been shortened in the ratio a: I. The above work shows that to produce the curve y = (x/a)", (o < 1), from 2/ = a;" by orthographic projection it is merely neces- sary to project all of the abscissas oi y = x" upon a plane passing through YOY' making an angle with OX such that unity on OX projects into a length a on the projection of OX. To produce the curve y = ax" (a < 1) from y = x" hy orthographic projection it is merely necessary to project all of the ordinates oi y = x" upon a plane passing through XOX' making an angle with OY such that unity on OY projects into the length a on the projection of OY. To lengthen all ordinates of a given curve in a given ratio, 1 : a, the process must be reversed; that is, erect perpendiculars to the plane of the given curve at all points of the curve, and cut them by a^lane passing through XOX' making an angle with OY such that a length a (,a> 1) measured on the new K-axis projects into unity on OF of the original plane. In Fig. 50 the projection of OP in any of its positions, such as OPi, OP2, OP3, ■ ■ ., is ODi, OD2, OD3, . . . , or is the abscissa of the point P. Thus for all positions X = a cos 6. The sign of x gives the sign, or sense, of the projection. In each case is said to be the angle of projection. This definition of projection is more general in one respect than that discussed above. By the present definition the projection of a line is negative if 90° < 9 < 270° (read, "if 6 is greater than 90° §64] THE CIRCLE AND THE CIRCULAR FUNCTIONS 123 but is less than 270°"). This concept is important and essential in expressing a component of a displacement, of a velocity, of an acceleration, or of a force. The cosine of 6 might have been defined as that proper fraction by which it is necessary to multiply the length of a line in order to produce its projection on a line making an angle d with it. Exercises 1. A stretched guy rope 75 ft. long makes an angle of 60° with the horizontal. What is the length of the projection of the rope on a horizontal plane? What is the length of the projection of the rope on a vertical plane? 2. Find the lengths of the projections of the line through the origin and the point (1, -y/s) upon the OX and OF axes, if the Une is 12 inches long. 3. A line 8 inches long makes an angle of 45° with the X-axis. What is the length of its projection on the X-axis? 4. A velocity of 20 feet per second is represented as the diagonal of a rectangle the longer side of which makes an angle of 30° with the diagonal. Find the components of the velocity along each side of the rectangle. 5. Show that the projections of a fixed hne OA upon all other lines drawn through the point are chords of a circle of diameter OA . See Fig. 66. 6. Find the projection of the side of a regular hexagon upon the three diagonals passing through one end of the given side, if the side of the hexagon is 20 feet. 64. Polar Coordinates. In Fig. 64, the position of the point P may be assigned either by giving the x and y of the rectangular coordinate system, or by giving the vectorial angle 6 and the distance OP measured along the terminal side of 6. Unlike the distance o used in the preceding work, it is found conven- ient to give the line OP a sense or direction as well as length; such a line is called a vector. In the present Case, OP is known as the radius vector of the point P, and it is usually symbolized by the letter p. The vectorial angle 6 and the radius vector p are together called the polar coordinates of the point P, and the system used in locating the point is known as the system of polar 124 ELEMENTARY MATHEMATICAL ANALYSIS [§65 co5rdinates. In Fig. 64 the point P' is located by turning from the fundamental direction OX, called the polar axis, through an angle 6 and then stepping backward the distance p to the point P'; this is, then, the point (— p, 9). P' has also the coordinates (p, 02), in which 6^ = + 180°; likewise Pi is (+ p', di) and P'l is (— p', Bi). Thus each point may be located in the polar system of coordinates in two ways, i.e., with either a positive or a negative radius vector. If negative values of B be used, there are four ways of locating a point without using values oi B> 360°. In giving a point in polar coordinates, it is usual to name the radius vector first and then the vectorial angle; thus (5, 40°) means the point of radius vec- tor 5 and vectorial angle 40°. 65. Polar Coordinate Paper. Polar coordinate paper (form MZ) is prepared for the con- struction of loci in the polar system. A reduced copy of a sheet of such paper is shown in Fig. 65. This plate is grad- uated in degrees, but a scale of radian measure is given in the margin. The radii proceeding from the pole meet the circles at right angles, just as the two systems of straight lines meet each other at right angles in rect- angular coordinate paper. For this reason, both the rectangular and the polar systems are called orthogonal systems of coordinates. We have learned that the fundamental notion of a function implies a table of corresponding values for two variables, one called the argument and the other the function. The notion of a graph implies any sort of a scheme for a pictorial representation of this table of values. There are three common methods in use: the double scale, the rectangular coordinate paper, and the polar paper. The polar paper is very convenient in case the argument is an angle measured in degrees or in radians. Since in a table of values for a functional relation we need to consider both positive and negative values for both the argument and the function, it is necessary to use on the Polar coordinates. §65] THE CIRCLE AND THE CIRCULAR; FUNCTIONS 125 polar paper the convention already explained. The argument, which is the angle, is measured counter-clockwise if positive and clockwise if negative from the line numbered 0°, Fig. 65. The function is measured outward from the center along the terminal side of the angle for positive functional values and outward from the center along the terminal side of the angle produced backward through the center for negative functional values. In this scheme it appears that four differ- ent pairs of values are represented by the same point. This is made FiQ. 65. — Polar coordinate paper. clear by the points plotted in the figure. The points Pi, Pi, Pa, Pt are as follows: Pi: (6, 40°); (6, - 320°); (- 6, 220°); (- 6, - 140°). Ps: (10, 135°); (10, - 225°); (- 10, 315°); (-10, - 45°). Pa: (5, 230°); (5, - 130°); (- 5, 50°); (- 5, - 310°). Pi-. {,&, 330°); (6, - 30°); (- 6, 150°); (- 6, - 210°). The angular scale cannot be changed, but the functional scale can be changed at pleasure. In case the vectorial angle is given in radians, the point may 126 ELEMENTARY MATHEMATICAL ANALYSIS be located on polar paper by means of a straight edge and the marginal scale on form M3. The point 0, Fig. 65, is called the pole and the line OA, the polar axis. Exercises 1. Plot upon polar coordinate paper the following: (a) (0.1, 30°;; (6) (0.2,40°); (c) (0.6,120°); (d) (0.8,-30°); (e) (1.2,300°); (/) (0.7, - 47°). Let 10 cm. = 1 unit for p. 2. Plot upon polar coordinate paper the following: (a) (1.3, 45°); (6) (11.1, 137°); (cj (9.2, - 47°); (d) (8.5, - 216). Let 1 cm. = 1 unit for p. 3. Plot upon polar coordinate paper the following: (a) (10, C); (6) (9, D; (c) (8.2, 1.6'); (d) (12, 3.2"-). Let 1 cm. = 1 unit for p. 4. Explain why the locus for p = 3 is a circle with center at the pole and radius equal to three units. 5. Draw the loci for p = 5 and p = 7. 6. Explain why the locus 8 = J tt is a straight line passing through the pole and making an angle of 45° with the polar axis. Explain why this locus is indefinite in extent and does not terminate at the pole. 7. Draw loci for: 9 = | tt, and d = — \ir. 8. Plot the locus for p = 9, if 9 is measured in radians. Use 2 cm. as the unit for p. 66. Graphs of p = a cos 9 and p = a sin 0. These are two funda- mental graphs in polar coordinates. The equation p = o cos 6 states that p is the projection of the fixed length a upon a radial line proceeding from and making a direction angle 6 with a, or, in other words, p in all of its positions must be the side adjacent to the direction angle in a right triangle whose hypotenuse is the given length o. (See §62 and Fig. 66.) It must be remem- bered that the direction angle d is always measured from the fixed direction OA. Hence, to construct the locus p = a cos 6, proceed as follows: Draw a number of radical lines from 0, Fig. 66. Project upon each of these the constant length OA, or a. These projections are then radius vectors for p = a cos and a curve drawn through their end points gives the required locus. Thi locus is a circle since P is always at the vertex of a right triangle standing on the fixgd hypotenuse a, and therefore the point P is on the semicircle AOP; for, from plane geometry, a right triangle is always inscribable in a semicircle. §66] THE CIRCLE AND THE CIRCULAR FUNCTIONS 127 When 6 is in the second quadrant, as 62, Fig. 66, the cosine is negative and consequently p is negative. Therefore the point P2 is located by measuring backward through 0. Since, however, P2 is the projection of a through the angle 62 (see §63), the angle at P2 must be a right angle. Thus the semicircle OP2A is described as d sweeps the second quadrant. When 6 is in the third quadrant, as ds, the cosine is still negative and p is measured backward to describe the semicircle APiO a second time. As 6 sweeps the fourth quadrant, the semicircle OP2A is described the second time. Thus the graph in polar coordinates Fig. 66. — The graph of p = a cos e. of p = a cos d is a circle twice drawn as 6 varies from 0° to 360°. Once around the circle corresponds to the portion ABC of the "wave" y = a cos x, in Fig. 61. The second time around the circle corresponds to the portion CDE from trough to crest of the cosine curve. Trough and crest of all the successive "wave lengths" correspond to the point A, the nodes to the point 0. The polar representation of the cosine of a variable bj^ means of the circle is more useful and important in science than the Cartesian representation by means of the sinusoid. The ideas here presented should be thoroughly mastered by the student. The graph of p = a sin 6 is also a circle, but the diameter is the line OB making an angle of 90° with OA, as shown in Fig. 67. Since p = a sin 6, the radius vector must equal the side lying opposite the angle 6 in a right triangle of hypotenuse a, if 128 ELEMENTARY MATHEMATICAL ANALYSIS ['§67 0° < e < 90°. Since angle AOPi = angle OBPi, the point Pi is the vertex of any right triangle erected on OB, or a, as a hypote- nuse. The semicircle BP2O is described as increases from 90° to 180°. Beyond 180° the sine is negative, so that the radius vector p must be laid off backward for such angles. Thus P3 is the point corresponding to the angle 63 of the third quadrant. As 6 sweeps the third and fourth quadrants the circle OP1BP2O is described a second time. Therefore the graph of p = asiad is the circle tiince drawn of diameter a, and tangent to OA at 0. The first time around the circle corresponds to the crest, the second time around corresponds to the trough of the wave or sinusoid drawn in rectangular coordi- nates. corresponds to the nodes of the sinusoid and B to the maximum and minimum points, or to the crests and troughs. We have seen that the graph of a function in polar coordinates is a very different curve from its graph in rect- angular coordinates. Thus the cosine of a variable if graphed in rectangular coordinates is a sinusoid; but if graphed FiQ. g7_ xhe graph of i^ polar coordinates it is a circle (twice p = a sine. drawn) . There is in this case a very great difference in the ease with which these curves can be constructed; the sinusoid requires an elabo- rate method, while the circle may be drawn at once with com- passes. This is one reason why the periodic, or sinusoidal rela- tion, is preferably represented in the natural sciences by polar coordinates. Exercises 1. Show that if — o is negative, p = — a cos 9 is a circle, diameter a, with center to left of the pole §a units. 2. Show that if — a is negative p — — a sin is a circle, diameter o, with center below the pole 50 units. 67. Graphical Table of Sines and Cosines. The polar graphs of p = a sin and p = a cos 9 furnish the best means of construct- ing graphical tables of sines and cosines. The two circles passing §68] THE CIRCLE AND THE CIRCULAR FUNCTIONS 129 through shown on the polar coordinate paper, form M3, Fig. 65, are drawn for this purpose. A supply of this coordinate paper should be in the hands of the student. If the diameter of the sine and cosine circles be called 1, then the radius vector of any point on the lower circle is the cosine of the vectorial angle, and the radius vector of the corresponding point on the upper circle , is the sine of the vectorial angle. Thus, from the diagram of form M3, we read cos 45° = 0.707; cos 60° = 0.500; cos 30° = 0.866. These results are correct to the third place. Exercises 1. /From coordinate paper, form M3, find the values of the following : (a) cos 36°; (b) cos 62°; (c) cos 126°; (,d) sin 81°; (e) sin 25°; (/) sin 226°. 68. Graphical Table of Tangents and Secants. Referring to Fig. 65, it is obvious that the numerical values of the tangents of angles can be read off by use of the uniform scale bordering the polar paper, form M3. The scale referred to lies just inside of the scale of radian measure, and is numbered 0, 2, 4, Thus to get the numerical value of tan 40° it is merely necessary to call unity the side OA of the triangle of reference OAP, and then read the side AP = 0.84; hence tan 40° = 0.84. To the same scale (i.e., OA = 1) the distance OP = 1.31, but this is the secant of the angle AOP, whence sec 40° = 1.31. By use of the circles we find sin 40° = 0.64 and cos 40° = 0.76. In case we are given an angle greater than 45° (but less than 135°) use the horizontal scale through B. Starting from B as zero the distance measur/ed on the horizontal scale is the cotangent of the given angle. The tangent is found by taking the reciprocal of the cotangent. Exercises Find the unknown sides and angles in the following right triangles. The numerical values of the trigonometric functions may be taken from the polar paper. The vertices of the triangles are supposed to be lettered A, B, C with C at the vertex of the right angle. The small letters a, b, c represent the sides opposite the angles of the same name. See also table of Natural Trigonometric Functions at end of the book. 9 130 ELEMENTARY MATHEMATICAL ANALYSIS [§68 By angle of elevation of an object ia meant the angle between a horizontal line and a line to the object, both drawn from the point of observation, when the object lies above the horizontal line. The simi- lar angle when the object lies below the observer is called the angle of depression of the object. The solution of each of the following problems must be cheeked. The easiest check is to draw the triangles accurately to scale on form Ml, measuring the unknown sides and angles. 1. When the altitude of the sun is 40°, the length of the shadow cast by a flag pole on a horizontal plane is 90 feet. Find the height of the pole. Outline of Solution. Call height of pole a, and length of shadow b. Then A = 40° and B = 50°. Hence, o = 6 tan 40°. Determining the numerical value of the tangent from the polar paper, we find a = 90 X 0.84 = 75.6 ft., which result, if checked, is the height of the pole. To check, either draw a figure to scale, or compute the hypotenuse c, thus : c = 90 sec 40° From the polar paper find sec 40°. Then c = 90 X 1.31 = 117.9 Since a^ + b' = c', we have c' - b^ = a'', or (c - 6) (c + 6) = a'. Hence, if the result found be correct, (117.9 - 90) (117.9 + 90) = (75.6) ^ 5800 = 5715. These results show that the work is correct to about three figures, for the sides of the triangle are proportional to the square roots of the numbers last given. 2. At a point 200 feet from, and on a level with, the base of a tower the angle of elevation of the top of the tower is observed to be 60°. What is the height of the tower? 3. A ladder 40 feet long stands against a building with the foot of the ladder 15 feet from the base of the wall. How high does the ladder reach on the wall? 4. From the top of a vertical cliff the angle of depression of a point §68] THE CIRCLE AND THE CIRCULAR FUNCTIONS 13.1 on the shore 150 feet from the base of the cliff is observed to be 30°. Find the heiglit of the cliff. 6. In walking halt a mile up a hill, a man rises 300 feet. Find the angle at which the hill slopes. If the hill does not slope uniformly the result is the average slope of the hill. 6. A line 3.5 inches long makes an angle of 35° with OX. Find the lengths of its projections upon both OX and OY. 7. A vertical cliff is 425 feet high. From the top of the cliff the angle of depression of a boat at sea is 16°. How far is the boat from the foot of the chff? 8. The projection of a line on OX is 7.5 inches, and its projection on OY is 1.25 inches. Find the length of the line, and the angle it makes with OX. 9.- A battery is placed on a cliff 510 feet high. The angle of depres- sion of a floating target at sea is 9°. Find the range, or the horizontal distance of the target from the battery. 10. From a point A the angle of elevation of the top of a monument is 25°. From the point B, 110 feet farther away from the base of the monument and at the same elevation as A, the angle of elevation is 15°. Find the height of the monument above the line AB. 11. Find the length of a side of a regular pentagon inscribed in a circle whose radius is 12 feet. 12. Proceeding south on a north and south road, the direction of a church tower, as seen from a milestone, is 41° west of south. From the next milestone the tower is seen at an angle of 65° W. of S. Find the shortest distance of the tower from the road. 13. A traveler's rule for determining the distance one can see from a given height above a level surface (such as a plain or the sea) is as follows : " To the height in feet add half the height and take the square root. The result is the distance you can see in miles." Show that this rule is approximately correct, assuming the earth a sphere of raldius 3960 miles. Show that the drop in 1 mile is 8 inches, and that the water in the middle of a lake 8 miles in width stands lOf feet higher than the water at the shores. 14. Observations of the height of a mountain were taken at A and B on the same horizontal line, and in the same vertical plane with the top of the mountain. The elevation of the top at A is 52° and at B is 36°. The distance AB is 3500 feet. Find the height of the mountain. 16. The diagonals of a rhombus are 16 and 20 feet. Find the lengths of the sides and the angles of the rhombus. 132 ELEMENTARY MATHEMATICAL ANALYSIS 16. The equation of a line is y = f.r +10. Compute the shortest distance of this Une from the origin. 17. Find the perimeter and area of ABCD, Fig. 68. 18. Find BC and the total area of ABCD, Fig. 69. 69. The Law of the Circular Functions. It will be emphasised in this book that the fundamental laws of exact science are three in number, namely: (1) The power function expressed hy y = ax" where n may be either positive or negative; (2) the harmonic or periodic law y = aaia nx, which is fundamental to all periodically occurring phenomena; and (3) a law to be discussed in a sub- sequent chapter. While other important laws and functions arise in the exact sciences, they are secondary to those expressed by the three' fundamental relations. Fig. 68. — Diagram for Exercise 17. Fig. 69. — Diagram for Exercise 18. We have stated the law of the power function in the following words (see §34): In any power function, if x change hy a fixed multipk, y changes by a fixed multiple also. In other words, if x change by a constant factor, y will change by a constant factor also. Confining our attention to the fundamental functions, sine and cosine, in terms of which the other circular functions can be expressed, we may state their law as follows :i 1 Chapter XI is devoted to a diacuasion of theae fundamental periodic laws. §70] THE CIRCLE AND THE CIRCULAK FUNCTIONS 133 The circular functions, sin 6 and cos B, change periodically in value proportionally to the periodic change in the ordinate and abscissa, respectively, of a point moving uniformly on the circle a;2 + 2/2 = aK The use of the periodic law in the natural sciences is, of course, very different from that of the power function. The student will find that circular functions similar toy = a sin nx will be required in order to express properly all phenomena which are recurrent or periodic in character, such as the motion of vibrating bodies, all forms of wave motion, such as sound waves, light waves, electric waves, alternating currents and waves on water surfaces, etc. Almost every part of a machine, no matter how compli- cated its motions, repeats its original motions at stated intervals and these recurrent positions are expressible in terms of the circular functions and not otherwise. The student will obtain a very limited and unprofitable idea of the use of the circular functions if he deems that their principal use is in numerical work in solving triangles, etc. The importance of the circular functions lies in the power they possess of expressing natural laws of a periodic character. 70. Rotation of Any Locus. In §36 we have shown that any locus y = f{x) is translated a distance a in the x direction by substituting (x — a) for x in the equation of the locus. Likewise the substitution of (y — b) for y was found to translate the locus the distance b in the y direction. A discussion of the rotation of a locus was not considered at that place, because a displacement of this type is best brought about when the equations are ex- pressed in polar coordinates. If a table of values be prepared for each of the loci p = cos 8 (1) P = cos (9i - 30°) (2) as follows: 134 ELEMENTARY MATHEMATICAL ANALYSIS [§70 Equation 1 Equation 2 e p 9i P -30° 0.866 0° 0.866 -20° 0.940 10° 0.940 -10° 0.985 20° 0.985 0° 1.000 30° 1.000 10° 0.985 40° 0.985 20° 0.940 50° 0.940 30° 0.866 60° 0.866 40° 0.766 70° 0.766 60° 0.643 80° 0.643 60° 0.500 90° 0.500 and then if the graph of each be drawn, Fig. 70, it will be seen that the curves differ only in location and not at all in shape or size. If a value be given to 61 in the second equation which is 30° greater than a value given to d in the first equa- tion, the two values of p from equations (1) and (2) are equal. Thus, if AOP is the value given to S in equa- tion (1) and if AOP' = AOP + 30° is the value given to Oi'va. equation (2), then OP will equal OP'. Thus the point P' may be _,_„„. , , looked upon as having been J)iG. 70. — Kotation of the circle „i . ; j f „ .i ■ i o OAP [p =a cos 8] to the position obtamed from the pomt P OA'P' \p = a cos (e — 30°)]. by a positive rotation about of 30°. Thus the graph for p = cos {d - 30°) may be obtained from the graph for p = cos d by rotating it about the pole through an angle of 30°. The same reasoning will apply if {fi - a) be substituted for 6; in this case the locus of the first curve is rotated about the pole through an angle a, in the positive sense if a be positive, in a negative sense if a be negative. §71] THE CIRCLE AND THE CIRCULAR FUNCTIONS 135 By the same reasoning as used above, we see that if in the polar equation of any curve, d is replaced by (6 — a), the graph of the new equation is the graph of the original equation rotated about the pole through an angle a, but is otherwise unchanged. Thboeems on Loci XIV. If {d — a) be substituted for 6 throughout the polar equa- tion of any locus, the curve is rotated through the angle a. Note that the rotation is positive when a is positive and nega- tive when a. is negative. Exercises paper draw: a = cos i 1. Upon a sheet of polar coordinate p = cos (9 - 60°); p = cos (9 + 60°). 2. Upon a sheet of polar coordinate paper draw: p = sin B; p = sin (.9 - 30°j; p = sin [6 '+ 30°). 3. Upon a sheet of polar coordinate paper draw: p = cos 9; P = cos (fl — I) ; p = cos (9 + |j ; p = cos (9 — t). 71. Polar Equation of the Straight Line. In Fig. 71 let MN be any straight line in the plane and OT be the per- »^ pendicular dropped upon MN from the pole 0. Let the length of OT be a and let the direction angle of OT be a, where, for a given straight line, a and a are constants. Let p be the radius vector of any point P on the line MN and let its direction angle be d. Then, by definition, - = cos (6 — or). Fig. 71. — ^Equation of MN is a = p cos P (e-a). Therefore the equation of the straight line MN is a = p cos {d — a), (1) 136 ELEMENTARY MATHEMATICAL ANALYSIS [§72 for it is the equation satisfied by the (p, 6) of every point of the line. This is the equation of any straight line, for its location is perfectly general. The constants defining the line are the per- pendicular distance a upon the given line from 0, and the direction angle a of this perpendicular. The perpendicular OT, or a, is called the normal to the line MN, and the equation (1) is called the normal equation of the straight line. The equation of the circle shown in the figure is Pi = o cos {6 — a), (2) in which pi represents the radius vector of a point Pi on the circle. The relation pi p = a^, which can be deduced from (1) and (2), is interesting. Because of it, the circle is often called the inverse of the line MN with respect to the point 0. Exercises 1. Write the polar equation of the line tangent tp the circle p = 5 cos (9 — 30°) at the end of the diameter passing through the pole. 2. A line is 3 units distant from the pole and makes an angle of 45° with the polar axis. Write its polar equation. 3. Describe the curves p = 10 cos I * — 4) and 10 = p cos ( ^ ~ i) • Draw the following circles : 4. p = 3 COE 1 (e - 30°). 7. P = 2 sin (6 + 135°). 6. p = 3 cos {e + 120°). 8. p = ^cos{e + l)- 6. p = 2 sin (9 - 45°). 9. p = : 5 sin (1 - e) • 10. Show that p = a sin 9 is the locus p = cos 9 rotated 90° counter clockwise. 72. Relation between Rectangular and Polar Coordinates. Think of the point P, Fig. 72, whose rectangular coordinates are (x, y). If the radius vector OP be called p and its direction angle br I'alhd 0, then the polar coordinates of P are (p, 6). Then x and y fo: any position of P are the projections of p through the angle 6, and the angle (90° — d), respectively, or X = p cos d, (1) y = p sin 8. (2) §72] THE CIRCLE AND THE CIRCULAR FUNCTIONS 137 These are the equations of transformation that enable us to write the equation of a curve in polar coordinates when its equation in rectangular coordinates is known, or vice versa. Thus the straight line x = 3 has the equation p cos 6 = 3 in polar coordinates. The line x + y = 3 has the polar equation p cos d + p sin 6 = 3. The circle x^ + y^ = a^ has the equation p^cos" 9 + p2 sin^ e = o^ or or To solve equations (1) and (2) for 8, we write 6 = the angle whose cosine is -> P 6 = the angle whoser sine is -• P The verbal expressions "the „„„!„ „,!,„„„ „„„; ;„ " „i„ Fig. 72. — Rectangular and polar angle whose cosine is, etc., are coordinates of a point P. abbreviated in mathematics by the notations "arc cos," read "arc-cosine," and "arc sin," read "arc-sine," as follows: 6 = aic cos (x/p) (3) 9 = arc sin (y/p) (4) Dividing the members of (2) by the members of (1) we obtain y tan = -J which, solved for 6, we write = the angle whose tangent is y which may be abbreviated d = arc tan (y/x) and read "8 = the arc-tangent of y/x." The value of p in terms of x and y is readily written P = VxM^- (5) (6) 138 ELEMENTARY MATHEMATICAL ANALYSIS [§73 Exercises 1. Write the polar equation of x' + y' + 8x = 0. The result is p* + 8p cos fl = 0, or p = — 8 cos e. 2. Write the polar equations of (a) x' + y^ — 4j/ =0; (6) a;' + y' - 6x - iy = Q; (c) x^ +y'- Qy = 4. 3. Write the polar equations of {a) x + y = 1; (6) x + 2y = 1; (c) X + Vly = 2. 4. Write the rectangular equations of (a) p cos + p sin 9 = 4; (6) p cos e — 3p sin 9 = 6. 5. Write the polar equation of x* + 2y' — 4x = 0. 6. Write the rectangular equation of p = 2 cos 9 + 3 sin 9. Hint: Multiply both members of the equation by p, replace p' by (x' + j/2), p cos 9 by X, and p sin 9 by y. 7. Write the rectangular equation of p = 3 cos 9 — 2 sin 9. 8. Write the rectangular equation of p = 5 sin 9 — 3 cos 9. 73. Identities and Conditional Equations. It is useful to make a distinction between equalities like (a - x){a + x) = a' - x\ (1) which are true for all values of the variable x; and equalities like x^ -2x = 3, (2) which are true only for certain particular values of the unknown number. When two expressions are equal for all values of the variable for which the expressions are defined, the equality is known as an identity. When two expressions are equal only for certain particular values of the unknown number, the equality is spoken of as a conditional equation. The fundamental formula sin^ (j) + cos'' = 1 is an identity. 2 sin A + 3 cos ^ = 3.55 is a conditional equation. The symbol = is sometimes used to distinguish an identity; thus a' — a;' = (a — x){a'^ -\- ax + x''). The following illustrations and exercises contain problems both in the establishment of trigonometric identities and in the finding of the values of the unknown number from trigonometric conditional equations. §73] THE CIRCLE AND THE CIRCULAR FUNCTIONS 130 The truth of a trigonometric identity may be established by reducing each side to the same expression. In this work, however, the student will be required to transform the left-hand side by means of the fundamental relations (2) to (6), §64, until it is identically equal to the right-hand side. Facility in the establishment of trigonometric identities is largely a matter of skill in recognizing the fundamental forms and of ingenuity in performing transformations. All solutions of conditional equations should he checked. The following worked exercises will illustrate the method. Illustration 1: Show that (1 — sin u cos u) (sin u + cos u) ^ sin' u + cos' u. Taking the left-hand member (1 — sin u cos u) (sin u + cos u) = sin u + cos w — sin^ u cos u — sin u cos^ u = cos u {1 — sin^ m) + sin u (1 — cos' u) = cos u cos' u + sin u Bin' u = cos' u + sin' u. This last expression is the right-hand member of the given identity. Thus the identity is verified. Illustration 2: Show that sec' a; — 1 = sec' x sin' x. sec' a; — 1 = see's (1 -; — ) = sec'x (1 — cos' x) = see's sin's. \ sec' x/ Illustration 3: Solve for all values of x less than 360° ' 2 sin X + cos s = 2. Transposing and squaring we get cos' X =4 — 8sina;-|-4 sin' x. Since sin' x + cos' s = 1, 1 — sin' s = 4 — 8 sin x + 4 sin' x, 5 sin' a;-8sins + 3=0, sin s = 1 or 0.6 X = 90°, and 37° or 143° approximately. Check: 2 sin 90° -t- cos 90° = 2 -|- = 2 Check: 2 sin 37° -f- cos 37° = 1.2 + 0.8 = 2 Does 2 sin 143° + cos 143° = 1.2 - 0.8 = 0.4 = 2? The last value does not check. The reasons for this will be dis- cussed later in §98. Therefore the correct solutions are 90° and 37° approximately. 140 ELEMENTARY MATHEMATICAL ANALYSIS [§74 Exercises 1. Solve 6 cos" e + 5 sin 9 = 7 for all values of ff < 90°. Suggestion: Write 6(1 — sin* 9) + 5 sin 9 = 7 and solve the quadratic in sin 9. 6 sin" 9-5 sin 9 + 1 =0, or (3 sin 9 - 1)(2 sin 9 - 1) = 0. sin 9 = i or J = 19.6° approximately and 30°. The results should be checked. 2. Prove that for all values of 9 (except ir/2 and 3ir/2, for which the expressions are not defined) sec* 9 — tan' 9 = tan* 9 + sec* 9. 3. Show that ' sec' u — sin* u = tan* u + cos* u, for all values of the variable u except 90° and 270°, for which the expressions are not defined. 4. Find u, if tan u + cot u = 2. 6. Find sec 9, if 2 cos 9 + sin 9 = 2. 6. Show that sec a +1 tana tan a ~ sec o — 1 Hint: Multiply both numerator and denominator of the left-hand member by (sec a — 1). 7. Show that sec a + tan a = x sec a — tan a 8. Show that sin* a + sin* a tan* a = tan* a. 9. Show that (esc* a — 1) sin* a = cos* a. 10. Show that sin A _ 1 + cos A 1 — cos A ~ sin A 11. Show that 2 cos* m — 1 = cos* u — sin' u. 12. Show that cos' a — sin' a + 1 = 2 cos* a. 13. Show that sec* u + esc* u = esc* u sec* u. 14. Show that (tan a + cot o)* = sec* a esc* a. 16. Solve sec x — tan s + 1 = for all values of x less than 360°. 74. The Graph of p = a cos + b sin 0. Before reading this section the student should review exercises 6 and 7, §72. Let us §74] THE CIRCLE AND THE CIRCULAR FUNCTIONS 141 find the Cartesian equation for the curve whose polar equation is p = a cos + 6 sin 6, (1) where a and b are any constants, positive or negative. First multiply each member of (1) by p. ap cos B + bp sin 6 (2) Since p^ = x^ + y^, p cos B = x, and p sin = y, equation (2) beaomes x^ + y^ = ax + by. (3) Transposing and completing squares [■-ir-b-i]' + b' (4) Fio. 73. — The circles p = a cos e, p = b sin $, and p = a cos 9 + 6 sin e, or the circles OA, OB, and OC respectively. This is the Cartesian equation of a circle with center at the point (io, ib) and of radius iy/a' + 6^. The circle passes through the pole or origin since the coordinates (0, 0) satisfy the equation (3), and also passes through the point (a, 6), since these coordiT nates satisfy (3). Thus if upon the diameters of the circles p = o cos 6 and|P = 6 sin B, we construct a rectangle, the circle having a diagonal of this rectangle as a diameter, is the locus of p = aoosd + b sin B. See Fig. 73. 142 ELEMENTARY MATHEMATICAL ANALYSIS [§75 Exercises Draw the graphs for the following : 1. p = 2 cos 9 + 2 sin 9. 2. p = 3 cos 9 + 2 sin 9. 3. p = — 2 cos 9 + 2 sin 9. 4. p = — 3 cos 9 — 2 sin 9. 5. In Fig. 74 let a =2 and a = 30°. Find the equation of each of the four circles in the form p = a cos 9 + 6 sin 9. p=ac°^^ Fig. 74. — Diagram for Exercise 5. 75. Additive Properties. The shearing of a curve in a straight line, considered in §38, may be thought of as the addition of the ordinates of the curve and of the straight line, corresponding to a given value of the abscissa. This sum is the corresponding ordinate of the new curve. In the more general case the curve y = fW) + P'(x) may be constructed from the curves y = f(x) and y = F{x) by adding their ordinates. Thus the curve for y = x^-\ — ) Fig. 75, was constructed by adding the ordinates of the curves y = x^ and j/ = • In the same way the curve for p = f(d) + F{6) may be con- structed from the curves p = f{d), and p = F(,d) by adding the radius vectors corresponding to the same value of the. vectorial angle. Thus points on the circle p = 2 cos 6 + 3 sin 6, Fig. 73, may be located by adding (using the bow dividers) the radius §76] THE CIECLE AND THE CIRCULAR FUNCTIONS 143 vectors of p = 2 cos 9 and p = 3 sin 6. That is, OP = OPi + OPi for all positions of OP. I Exercises 1. Plot on polar coordinate paper the curve for p =3 cos 9 + 2 sin 9 making use of the circles p = 3 cos 9 and p = 2 sin 9. 2. Plot on polar coordinate paper the curve for p = cos 9+1 making use of the circles p = cos 9, and p = 1. Note that when 90° < 9 < 270°, the p for p = cos 9 is negative, and that the addition referred to above is algebraic addition. 3. Plot upon polar coordinate paper the curve for p = 1 + sin 9, making use of the circles p = 1, and p = sin 9. 4. Plot upon polar coordinate paper the curve for p = 2 cos 9 — 1. ,6. Plot upon polar coordinate paper the curve for p = cos 9 + 2. 76. Graph of y = tan x. If this graph is to be constructed on a sheet of ordinary letter paper, 8^ inches X U inches, it is desirable 144 ELEMENTARY MATHEMATICAL ANALYSIS [§76 to proceed as follows:* Draw at the left of the sheet of paper a semi- circle of radius 1.15 . . inches (that is, of radius = 18/5ir), so that the length of the arc of an angle of 10°, or ir/18, radians will be i of an inch. Take for the X-axis a radius COX prolonged, and take for the F-axis the tangent OY drawn through 0, as in Fig. r MA M' a' /T\J lL_______ /A W \/ Q rUj/T I _ _ _ E k//^___l^\ /_\ T^ZA Z 5 _-_ ^Z ^^ /^^5Z ^ z S^ ^ l^^—/ \ / K ^ fe:cr— -P -^^•- ^'^ -il^ ^ ^^^^ ' " "^ Z 3^ Z wxSc ~^, Z \ ^ I VV>^ _S^l "_Z___ ^\ t :_j____ \\ / '\ / \ \ / '^ w 2T-" Y' B N' B' N" Fig. 76. — Graphical construction of the curve of tangents y = tan X. For lack of room only a few of the points Si, St, ... Ti, Ti, . . .are lettered in the diagram. The dotted curve is s/ = cot x. 76. Divide the semicircle into eighteen equal parts and draw radii through the points of division and prolong them to meet OF in points Ti, Ti, Ti, Tt, . . . Then on the F-axis there is laid off a scale YY' in which the distances OTi, OT2, . . are propor- ' tional to the tangents of the angles OCSi, OCS2, . . . ; for the tangents of these angles are OTi/CO, Orj/CO, . . . and CO is the unit of measure made use of throughout this diagram. Draw horizontal lines through the points of division on OF and vertical > The student should understand the construction of Figs. 76 and 78, but it is opt necessary that be actually draw them. [§77] THE CIRCLE AND THE CIRCULAR FUNCTIONS 145 lines through J inch intervals on OX, thus dividing the plane into a large number of small rectangles. Starting at 0, t, 2fir, ... — IT, — 2ir, . and sketching the diagonals of con- secutive cornering rectangles, the curve oi y — tan x is approxi- mated. Greater precision may be obtained by increasing as desired the number of divisions of the' circle and the number of corresponding vertical and hprizontal lines. It is observed that the graph of the tangent is a series of similar branches, which are discontinuous for x = ir/2, — ir/2, (3/2)ir, — (3/2)ir, ... At these values of x the curve has vertical asymptotes, as shown at AB, A'B', in Fig. 76. If the number of corresponding vertical and horizontal lines be increased sufficiently, the slope of the diagonal of any rectangle gives a close approximation to the true slope of the curve at that point. It has already been noted that all of the trigonometric functions are periodic functions of period 2v. It is seen in this case, how- ever, that tan x has also the shorter period x; for the pattern M'N' of Fig. 76 is repeated for^each interval ir of the variable x. 77. Graph of cot x. In order to lay off a sequence of values of cot S on a scale, it is convenient to keep the denominator con- P» Ps Pi Pr, P. P^ Pi ^ ^ 7 m \M j£ 4 ^ ^ Pii Z>io DsDtD, O DiDi Dz Di Di Fig. 77. — Construction of a scale of cotangents. stant in the ratio (abscissa) /(ordinate) which defines the cotangent. The denominator may also, for convenience, be taken equal to unity. Thus, in Fig. 77, the triangles of reference DiOPi, D2OP2, for the various values of 6 shown, have been drawn so that the ordinates P\Di, PzDi, . are equal. If the constant ordi- nate be also the unit of measure, then the sequence ODi, OD2, OD3 , ODt, ODi, represents, ' in magnitude and sign, the cotan- gents of the various values of the argument d. Using ODi, OD2, ... as the successive ordinates and the circular measure of Q 10 146 ELEMENTARY MATHEMATICAL ANALYSIS [§78 as the successive abscissas, the graph oi y — cot x is drawn, as shown by the dotted curve in Fig. 76. The sequence ODi, OD2, Fig. 77 is exactly the same as the sequence OTi, OT2, . Fig. 76, but arranged in the reverse order. Hence, the graph of the cotangent and of the tangent are alike in general form, but one curve descends as the other ascends, so that the position, in the plane ZF, of the branches of the curve are quite different. In fact, if the curve of the tangents be rotated about 07 as axis and then translated to the right the distance ir/2, the curves would become identical. Therefore, for all values of x, tan (7r/2 — x) = cot x. (1) This is a result previously known. 78. Graph of y = sec x. Since sec 6 is the ratio of the radius divided by the abscissa of any point on the terminal side of the angle d, it is desirable, in laying off a scale of a sequence of values of sec d, to draw a series of triangles of reference with the abscissas in all cases the same, as shown in Fig. 78. In this figure the angles were laid off from CQ as initial line. Thus CTs/CSi = sec QCSi, or, if CSi be unity, the distances like CTt, laid off on CQ, are the secants of the angles laid off on the arc QSi,0 or laid off on the axis OX. The student may describe the manner in which the rectangles made by drawing horizontal lines through the points of division on CQ and the vertical Unes drawn at equal intervals aloiig OX, may be used to construct the curve. If the radius of the circle be 1.15 inches, what should be the length of Oir in inches? The student may sketch the locus oi y = esc x, and compare with the locus y — sec x. Exercises 1. Discuss from the diagrams, 59, 76, 78, the following statements: Any number, however large or small, is the tangent of some angle. The sine or cosine of any angle cannot exceed 1 in numerical value. The secant or cosecant of any angle is always numerically greater^ than I (,or at least equal to 1), §79] THE CIRCLE AND THE CIRCULAR FUNCTIONS 147 2. Show that sec (o ~ ^) ~ <'*° ^ ^°'' ^^ values of x. 3. If tan 9 sec 9 = 1, show that sin $ = KVs — 1) and find 9 by use of polar coordinate paper, Form M3. 4. Describe fully the following, locating nodes, troughs, crests, etc. : (a) y = sin [x -"^y (c) y = tan y> +lj' (b) y = cos (^J + 1) ' (d) y = tan (x + 1). r N A N' \\ / 1 \ T \\\ / ^ \ \ \ \ /p S '~ WW \ / _ ._ ^ « ®„ m 7 r M 2' '///I 7 < " /// / \ /■/ / / \ / / / \ I" B N' , N' Fig. 78. — Graphical construction of y' = sec x. 79. Increasing and Decreasing Functions. The meanings of these terms' have been explained in §27. Applying these terms to the circular functions, we may say that y = sin x, y = tan x, y = sec X are increasing functions for < a: < ■k/2. The co- functions, y = cos z, y — cot x, y = esc x, are decreasing func- tions within the same interval. 148 ELEMENTARY MATHEMATICAL ANALYSIS [§79 Exercises Discuss tlie following topics from a consideration of the graphs of the functions: 1. In which quadrants is the sine an increasing function of the angle? In which a decreasing function? 2. In which quadrants is the tangent an increasing, and in which a decreasing, function of its variable? 3. In which quadrants are the cos 0, cot 6, sec 9, esc 6, increasing and in which are they decreasing functions of 9? 4. Show that all the co-functions of angles of the first quadrant are decreasing functions. 1. Show that 2. Show that Miscellaneous Exercises tan' a . , T — r^ — r- — sin' a 1 + tan' a \/l — sin'a COS a y/\ — COS 3. Show that cot' a — cos' a = cot' a cos* o. 4. Show that s Vcsc' - 1* 6. Show that 6. Show that 7. Show that Vsec'a — 1 1 + tan' a __ sin' a 1 + cot' a. " cos' a 1 + cos a 8. Show that CSC a \\ — sin a , \'- — - — , = sec a — tan a. I + sin a sin a , 1 + cos a -\ ; = 2 CSC a COS a. cot a + tan a 9. Show that 1 — T r-L = sin u cos M. cot u + tan u 10. Show that CBO* M (1 -r cos* m) — 2 cot' Mai. §79] THE CIRCLE AND THE CIRCULAR FUNCTIONS 149 11. Find the distance of the end of the diameter of , p = 8 cos (9 - 60°) from the line OX. 12. If PI = a cos 9, and P2 = a sin 6, find pi — pi when = 60° and 0=5. 13. Find the polar equation of the circle x' + y' + Qx = 0: 14. For what value of 9 does p = 3.55, if p = 2 sin 9 + 3 cos 9? Result: 9 = 23° 30' and 43° 30'. Hint: Draw the circles p = 3.55 and p = 2 sin 9 + 3 cos 9 on polar coordinate paper and find the vectorial angles for the two points of intersection. This problem is the same as: "Solve the equation 2 sin 9 + 3 cos 9 = 3.55 for 9." 16. Solve graphically the equation 2 sin 9 + 3 cos 9 = 2. Hint: Draw on polar coordinate paper the curves p = 2 and p = 2 sin 9 + 3 cos 9. 16. Solve graphically the equation 4 cos 9 — 3 sin 9= 3.5. 17. Find sin 9 if esc 9= vV_±A'. a 18. A circular arc is 4,81 inches long. The radius is 12 inches. What angle is subtended by the arc at the center? Give result in radians and in degrees. 19. Certain lake shore lots are bounded by north and south lines 66 feet apart. How many feet of lake shore to each lot if the shore- line is straight and runs 77° 30' E. of N.7 20. If 2/ = 2 sin A + 3 cos A - 3.55, take A as 20°; as 23°; as 26° and find in each case the value of y. From the values of y just found find a value of A for which y is approximately zero. This process is known as "cut and try." 21. The line y = ^x ia to coincide with the diameter of the circle p = 10 cos (9 — a). Find a. 22. The line y = 2x is to coincide with the diameter of the circle p = 10 sin (9 + a). Find a. 23. To measure the width of the slide dovetail shown in Fig. 79, two carefully ground cylindrical gauges of standard dimensions are placed in the V'a at A and B, as shown, and the distance X carefully taken with a micrometer. The angle of the dovetail is 60°. Find the reading of the micrometer when the piece is planed to the required dimension MN = 4 inches. Also find the distance Y. (Adapted from "Machinery," N. Y.) 24. Sketch y = ix and y = sinx and then y = ix — sinx. 26. Sketch the curve y = cos .-c + 2 sin x, making use of the curves y = cos X and ^ = 2 sin x. 150 ELEMENTARY MATHEMATICAL ANALYSIS [|79 26. Find the maximum value of the function given in exercise 25. Hint: Find the maximum value of p in the graph of p = cos 9 + 2 sin e. 27. Find the maximum value of 2 cos x — 3 sin x. 28. Since p = cos 9 + 2 sin fl is a circle passing through the pole, the equation may be put in the form p =' a cos (9 — a). Find a and a. Result: o = VS and a = 63° 20' approximately. 29. A circle is inscribed in a 30°, 60° right triangle. Find the diame- ter of the circle (a) if the shorter leg of the triangle is 2 inches; (6) if the longer leg is 2 inches; (c) if the hypotenuse is 4 inches, (d) Find the length of the sides of the triangle if the diameter of the inscribed circle is 2 inches. Fig. 79. — Diagram to Exercise 23. 30. A circle is inscribed in a 45° right triangle. Find the diameter of the circle if the legs of the triangle are 4 inches. 31. The center of the circle p'= 10 cos (9 — a) Ues on the Ijne Zx — 2y = \. Find two possible values for a. 32. The center of the circle p = 10 sin (9 + a) lies on the line X — 22/ = 6. JFind two possible values for a. 33. Write the Cartesian equations for: (a) p = 2 cos 9 + 3 sin 9. (b) p = 2 cos 9 — 5 sin 9. (c) p = 2 sin 9 — 5 cos 9. 34. Find the co6rdinates of the center and the radius for: (o) x» + 2/2 - 2x - 4i/ + 4 = (d) 2x2 + 2y^ + 3x + by= (5) x« + 2/' + 2x + 42/ + 4 = (e) 3x2 + ^yi _ gj. - ■y/2y = 10 (c) x' + !/» + 3x - 42/ = (/) x2 + ^2 + 7x- \^Zy = 25 §79] THE CIRCLE AND THE CIRCULAR FUNCTIONS 151 36. Which circles of exercise 34 pass through the origin? 36. Write the equation of a line passing through the origin and the center of the circle x' + y^ — 3x — 5y = 6. 37. Write the equation of a Une parallel to3x —2y = Q and passing through the center of x' + y'' — Sx — 2y = 0. CHAPTER V THE ELLIPSE AND HYPERBOLA 80. The Ellipse. If all ordinates of a circle be shortened by the same fractional amount of their length, the resulting curve is called an ellipse. For example, in Fig. 80, the middle points of the positive and negative ordinates of the large circle were marked and a curve drawn through the points so selected. The result is the ellipse ABA'B'A. If a;2 + 2/2 = a* (1) is the equation of a circle, then x^ + {myY = o^ (2) in which m is any constant > 1, is the equation of an ellipse; for substituting my for y divides all of the ordinates by m, by Theorem IX on Loci, §28. Fig. 80.- -Construction of an ellipse. Dividing both members of (2) by a' we obtain. -i + '-iV = 1. (3) Let 6' be written in place of — ^- Equation (3) becomes m'- V + h = ^ (4) which is the standard form of the equation of an ellipse. 81. Orthographic Projection of a Circle. The ellipse may also be looked upon as the orthographic projection of the circle. 152 §81] THE ELLIPSE AND HYPERBOLA 153 Let ABCD, Fig. 81a, be a circle with a radius o. Let AOC, Fig. 816, be an end elevation of the same circle. Rotate this circle about BOD as an axis through an angle, /J, to the position A"OC". Project the rotated circle upon its original plane, into the curve A'BC'D. We shaU show that A'BC'D is an ellipse. Take any point P upon the original circle. It rotates into the point P", and P" projects into P'. The equation of the circle is x^ + y^ = o", where y = MP. To get the equation for the curve A'BC'D replace MP by its equal MP'/cos /?. (See Fig. 81b.) Whence, ' + (MP'y cos''/3 Since MP' is the ^/-coordinate of P', x^ + y2 cos'/3 A P / ] ^ p P' /^ ~~?' ^ \ n ^\ o V a ^\ VV / / y b " ~ a Fig. 81. — The ellipse considered as the orthographic projection of a circle. or V' o' a^oos'jS Replacing o cos jS by 6(= OC), the equation of an ellipse. 1, As a consequence of the above, it ia seen that the shadow cast on 164 ELEMENTAKY MATHEMATICAL ANALYSIS [§82 the floor by a circular hoop held at any angle in the path of vertical rays of light is an ellipse. If the abscissas of a circle be lengthened by amounts propor- tional to their lengths, the resulting curve is an ellipse. Let x^ + y^ — V be the equation of the circle. Then 2 + 2/2 = b2 is the equation formed by lengthening all abscissas in the raticf 1 : m, TO > 1. Dividing by Ji^ and replacing m'-h'^ by a}, we obtain a^ "^ 62 ^• Thus the ellipse of Fig. 80 could have been formed by doubling all of the abscissas of the circle BD'B'D. Hence we see that if all parallel chords of a circle are lengthened or shortened by an amount proportional to their length, an ellipse is formed. If the deformation takes place in chords parallel to either the X- or F-axis the equation is of the form (4) of the last section, called the symmetric equation of the ellipse. The diameter of the circle from which the ellipse may be formed by shortening parallel chords is called the major axis of the ellipse. Thus AA', or 2a, Fig. 80 is the major axis of ABA'B'. The diameter of the circle from which the ellipse could have been formed by lengthening parallel chords is called the minor axis of the ellipse. Thus BB', or 26, Fig. 80 is the minor axis of ABA'B'. The point of intersection of the axes is called the center of the ellipse. One-half of the major and minor axes are called, respectively, the semi-major and semi-minor axes of the ellipse. The points A and A' are called the vertices of the ellipse. 82. Ejcplicit Form of Equation. The equation of the ellipse o" ^ 62 ^'■' when solved for y may be put in the important form .y = + - Va^-x^ (2) The equation of the circle x^ + y^ = a^ solved for y is y = ± Va^ - x2- (3) §83] THE ELLIPSE AND HYPERBOLA 155 Equations (2) and (3) are in a form very useful for many purposes. It is easy to see that (2) states that the ordinates of the ellipse are the fractional amount hfa of the ordinates of the circle -(3) . The definition of the term function permits us to speak of j/ as a function of x, or of a; as a function of y, in cases like equation (1) above; for when x is given, y is determined. To distinguish this from the case in which the equation is solved for y, as in (2), y, in the former case, is said to be an implicit function of x, and in the latter case, y is said to be an explicit function of x. 83. Section of a Cylinder. If a circular cylinder be cut by a plane, the section of the cylinder is an ellipse. For, select any diameter of a circular section of the cylinder as the X-axis. Let a plane be passed through this diameter making an angle a with the circular section. Then if ordinates (or chords perpen- dicular to the common X-axis) be drawn in each of the two planes, all ordinates of the sec- tion made by the cutting plane can be made from the ordinates of the circular section by multi- plying them by sec a. Hence any plane section of a cylinder is an ellipse. 84. Parametric Equations of the Ellipse. Let ABA'B'A, Fig. 82, be an ellipse whose semi-major axis is a and whose semi-minor axis is 6. Upon AA' and BB' as diameters construct circles. These circles are called, respectively, the major and minor auxiliary circles. From the origin, draw any radius vector, as QP2P1, making an angle d with the positive direction of the axis of x. Through P2 and Pi draw lines parallel, respectively, to the X- and F-axes, and let P be their point of intersection. It will be shown that P is a point upon the ellipse. Let the coordinates of P be a; and y. Then -A construction of the ellipse. X = a cos (1) 156 ELEMENTARY MATHEMATICAL ANALYSIS [§84 and or and Then y = b sin 6, (2) - = cos 8, a f =sine. ^ + f" = cos^ fl +sin2 e = 1, which shows that P is upon the ellipse. Equations (1) and (2) are called parametric equations of the ellipse. 6 is called the variable parameter, or the eccentric angle. The method used above of locating points upon the ellipse constitutes one of the best practical methods of constructing an ellipse when its axes are known. For, by it a large number of points upon the ellipse may be easily, located and a smooth curve drawn through them. If the abscissa and ordinate of any point of a curve are ex- pressed in terms of a third variable, the pair of equations are called the parametric equations of the curve. Thus, X = U y = t + l are the parametric equations of a certain straight line. Its ordinary equation y = ix + l can be found by eliminating the parameter t. Exercises 1. Write the equation of the ellipse formed by diminishing the lengths of all ordinates oix' + y^ =4 by one-half of their length. 2. Write the equation of the ellipse formed by diminishing the' lengths of all ordinates oi x' + y' =4 by one-third of their lengths. 3. Write the equation of the ellipse formed by lengthening all ordinates of the circle x' + y^ = 16 by one-third of their length. 4. Write the equation of the ellipse formed by lengthening all abscissas of the circle x' + ^' = 1 by one-fourth of their length. §84] THE ELLIPSE AND HYPERBOLA 157 6. Write the equation of the ellipse whose semi-axes are 4 and 3. 6. Construct accurately an ellipse whose semi-axes are 3 inches and 2 inches. 7. Construct accurately an ellipse whose parametric equations are x = 3 cos e, and y = 2sm e. 8. Write the parametric equations of an ellipse whose semi-axes are 6 and 10. 9. Draw a curve whose parametric equations are x = cos 6, and y = sin e. 10. Find the major and minor axes for the following : (c) 4x' + 25y' = 100 (d) 25a;2 + 4y' = 100 11. Find the axes of the ellipse k'x^ + h'y^ = hV. 12. Write the equation of the ellipse whose major and minor axes are 10 and 6, respectively. 13. Find the axes of the elUpse whose equation is 2/ = ± i V36 — a;^ [Note that o must be 6.] 14. Write the parametric equations of the ellipse y = + IVSl -x2. [a must be 9; b = f X 9 = 6.] 16. Discuss the curve a; = ± I v'4 - 2/2. 16. Discuss the following curves by comparing them with a;2 + 2/" = 1- 4x2 -1-2/2 = 1 \x^ H- 2/2 = 1. 17. Write the Cartesian equation of the curves whose parametric equations are: . , Fa: = 2 cos 9 , . Tx = 6 cos 6 i \ \^ ~ V^^ cos 9 ^°'' ly = sin e ^ ' ly =2 sin S ^'^' ly = V2 sin B. 18. What locus is represented by the parametric equations X = 2t + 1 2/ = 3« -f- 5? 19. What curve is represented by the parametric equations X = 2 -H 6 cos e and 2/ = 5 -1- 2 sin e? 158 ELEMENTARY MATHEMATICAL ANALYSIS 20. Show that the curve X = 3 + 3 cos e 2/ = 2 + 2 sin 9 is tangent to the co6rdinate axes. Rg. 83.— a mechanical cons. ru.tion of the ellipse. See Exercise 23. 21. The circle x' + y^ = 36 is picjocted upon a plane. Find an equation of the projection if the angle between the plane and the plane of the circle is 30°. 22. A right circular cylinder is cut by a plane making an angle of 60° with the axis of the cylinder. Find an equation of the curve of intersection, if the radius of the cylinder is 6 units. Fig. 84. — Theory of the common "ellipsograph" or elUptic trammel. See Exercise 24. 23. The line AB, Fig. 83, whose length is (a + 6) moves in such a way that the ends A and B always lie on the X- and K-axes, respect- ively. Show that the point P describes an ellipse. 24. The- edge of a straight ruler, NMP, Fig. 84, is marked so that §84] THE ELLIPSE AND HYPERBOLA 159 PM = b and PN = a. It is moved keeping M and N always on AA' and BB', respectively. Show that P describes an ellipse. The elliptic "trammel" or "ellipsograph" is constructed on this principle by use of adjustable pins on PMN and grooves on AA' and BB'. 25. Draw a semicircle of radius a about the center C, Fig. 85, and produce a radius to such that CTO = a + 6. From C draw any number of lines to the tangent to the circle at T. From draw hnes meeting the tangent at the same points of TN. At the points where the lines from C cut the semicircle, draw parallels to CT. Show that Fig. 85. — A graphical construction of an eUipse. See Exercise 25. the points of meeting of the latter with the lines radiating from O determine points on an ellipse, with center at and semi-axes equal to a and b. Hint ; OD = SM = a cos 9. From the triangles OPD and ONT OD _PD TN b From the triangles CNT and CMS SM{=OD) _ SC{==a sine) TN ~ a ' 160 ELEMENTARY MATHEMATICAL ANALYSIS [§85 Hence PD =b sin e. 86. Origin at a Vertex. Equation (4) §80, equation (2) §82, and equations (1) and (2) §84 are the most useful forms of the equa- tions of the ellipse. It is obvious that the ellipse may be translated by the methods already explained to any position in the plane. The ellipse with center at (A, k) and its axes parallel to the coordinate axes has the equation {x - hy {y - kr . ,.s a2 -r 52 - ^' ^^) Of special importance is the equation of the ellipse when the origin is taken at the left-hand vertex. This form is best obtained from equation (2), §82, by translating the curve the distance a in the X direction. Thus, y=± -Vo» - (x - ay, u or y^ = — X -„ x^, a a^ ' or, letting 21 stand for the coeflBcient of x, 2/2 = 2lx -~^x^ = 2lxil - x/2a). . (2) For small values of x, x/2a is very small as compared with 1 and the ellipse nearly coincides with the parabola y^ = 2lx. 86. Theorem. Any equation of the second degree, lacking the term xy and having the terms containing x^ and y' both present and wUh coefficients 0/ like signs, represents an ellipse with axes parallel to the coordinate axes. This is readily shown by putting the equation ax^ + hy' + 2gx + 2fy + c = (1) in the form (1) of the preceding section. The procedure is as follows: a{x' + 2^^x) +h[y' + 2f^y)^-c. (2) a(x^ + 2lx + Q +&(2/^ + 2(. + |) = f + f-c. (3) §87] THE ELLIPSE AND HYPERBOLA 161 Let M stand for the expression in the right-hand member of (3) ; then we get a b This shows that (1) is an ellipse whose center is at the point and which is constructed from the circles whose cen- \ a' hi ters are at the same point and whose radii are the square roots of the denominators in (4). The major axis is parallel to OX or OY according as a is less or greater than 6. The case when the locus is not real should be noted. Compare §43. iLLtrsTRATiON: Find the center and axes of the ellipse a;2 _|- ^yi _|_ 6j; _ 8?/ = 23. Write the equation in the form x'^ + Qx + 4j/2 -Sy = 23. Complete the squares a" + 6x + 9 + 4y2 - 8?/ + 4 = 36. Rewriting (a + 3)' + 4(!/ - l)' = 36. (x + 3)' , {y - ly ^ 36 "^ 9 This is seen to be an ellipse whose center is at the point (—3, 1) and whose semi-axes are 6 and 3. 87. Limiting Lines of an Ellipse. It is obvious from the equation y = +-^/o''-x^ that a; = a and x =. — a are limiting lines beyond which the curve cannot extend; that is, x cannot exceed o in numerical value without y becoming imaginary. The same test may be applied to equations of the form a;2 + 4x -h 92/2 - 6?/ -(- 4 = 0. 162 ELEMENTARY MATHEMATICAL ANALYSIS [§87 Solving for y in terms of x 3y = l + Vl-{x + 2)2. The values of y become imaginary when (x + 2)2>I, or a: + 2>+lor<-l, or x> - 1 or < - 3. These, then, are the limiting lines in the x direction. Finding the limiting lines in the y direction in the same way, the rectangle within which the ellipse must lie is determined. In cases . like the above the actual process of finding the limiUng lines and the location of the center of the ellipse is best carried out by the method of §86. Illustration: Find the coordinates of the center, the length of the axes, and the equation of the limiting lines of x' + ix + Qy^ - Qy = 4. Completing the squares, (X + 2y + 9(2/ - \Y = 9, or (X + 2)' (y - \y _ g— + J 1- The center of the ellipse is at the point ( — 2, |), its semi-axes are 3 and 1. It may be constructed by translating the ellipse -„ + y = 1, two units to the left and \ unit up. Hence the limiting lines are a; = + 3 — 2 and v = + 1 + 3, or x = 1, x = — 5, ?/ = f , and y--l Exercises Find the lengths of the semi-axes, the coordinates of the center, and the equations of the limiting lines for the seven following loci and translate the curves so that the terms in x and y disappear, by the method of §87. 1. x^ - 6x -I- 42/2 + 82/ = 5. 2. 2/2 - 8i/ + 4x2 -h 6 = 0. §87] THE ELLIPSE AND HYPERBOLA 163 8. 12x« - 48a; + 3y' + 6y - 13. 4. x^ + %2 - 12a; + 6y =■ 12. 5. 4a;« + y' -12x + 12 2/ - 2 = 0. 6. x' +2y' - X - V2y = 1/2. 7. Show that a;^ — 4a; + 4^^ ^ 82/ + 4 = is an ellipse. 8. Show that x' + 4a; + Qy' — 6y = passes through the origin. 9. Discuss the curves : ,.,£■+!! = , wM:-' 10. Compare the following parabolas with the standard parabola y = a;2 by means of the appropriate Theorems on Loci: k (a) y = 2px2 (c) j/ = — x^ (h) y 2pa;2 (d) y 2px^ + 6. What are the roots of the last function? 11. Write the symmetrical equation of the ellipse if its parametric equations are: X = (3/2) cos e y = (2/3) sin e. 12. Discuss the curve y^ = (18/5)x - (9/25)x2. 13. Find the center of the curve y^ = 2x (6 — x). 14 Write the parametric equations for the following :_ (a) x^ + 3y' =4; (6) 2x^ + 5y^ = 6; (c) 5x= + y^ = 7. 15. Write the parametric equations for x' +2x + 4;/2 - l&y + 13 = 0. Hint ; The equation may be put in the form (x + ir , (y - 2)' Since x = 2 cos 6 and y = am. 9 are parametric equations for x^ v'^ -7 +Y = 1, x=2cosfl — 1 and 2/ = sin 9 + 2 are parametric equations of the given ellipse. 16. Write parametric equations for the following : (a) x' — 2x + 9yi - 6x = 0; (6) 4x2 + 4x + j/2 - 2?/ = 5. (c) 3;2 _ 4^ ^ ^2 + gj/ = 3. 164 ELEMENTARY MATHEMATICAL ANALYSIS 88. The Rectangular Hyperbola. In §34 the graph of xy =■ k, where fc is a constant, was called a rectangular, or equilateral, hyperbola. It was observed that the X- and F-axes are asymp- totes of the curve. We shall now find the equation of the equi- lateral hyperbola when rotated about the origin through an angle of ( — 45°). For convenience let k be represented by jo^. Since this is a positive number, the curve will appear in the first and third quadrants, as shown by the curve RPS, Fig. 86. Fig. 86. — The rectangular hyperbolas 2 xy — a' and x' — y' = a'. Let P be any point on the original curve 2xy = aK (1) Let P' be this poiat after rotation. Let OD' = x and let B'P' = y. OB = EP = E'P' = D'K - D'H = OD' cos 45° - D'P cos 45° = iV2 {X - y). (2) DP = OE = O'E' = OK + KE' = OD' cos 45° + D'P cos 45° ^W2(.x + y). , (3) But OD is x and DP is y in equation (1). Substituting then §89] THE ELLIPSE AND HYPERBOLA 165 l\/2(,x — y) and i\/2{x + y) for x and y, respectively, in equa- tion (1), we obtain x2 - y2 = a2, (4) the equation of the curve R'P'S', or the equation of the rectangu- lar hyperbola 2xy = a^ after it has been rotated (— 45°) about the origin. The equation of the asymptotes of the curve x'^ — y'' = a'' are y = -\- x and y = — x. The curve for xy = k is sometimes called the equilateral hyperbola referred to its asymptotes as axes. 89. Parametric Equations. The parametric equations of the rectangular hyperbola x^ — y^ = a^ are X = a sec 8, (1) and y = a tan e. (2) For, dividing (1) and (2) by a, squaring, and subtracting, -, - ^ = sec" e - tan^ 9 = 1, which is the same as equation (4) above. Exercises 1. Find the equations of the following curves after rotation about the origin through an angle of — 45°. (o) 2xy = 1; (6) xy = 1; (c) xy = 4; (c) xy = f; (d) xy = 3; (e) xy - 2 = 0. 2. Show that y^ — x^ = a^ is the equation of the curve 2xy = a' after rotation about the origin through an angle of + 45°. 3. Show that x^ — y^ = a^ is the equation of the curve 2xy = — o' after rotation about the origin through an angle of -|- 45°. 4. Find the equations of the curves given in exercise 1 after rota- tion about the origin through an angle of -|- 45°. 6. Find the equations of the asymptotes for x* — y^ — 2x + 4y = 7. Hint: Completing the squares {X - 1)» - (2/ - 2)2 = 4 166 ELEMENTARY MATHEMATICAL ANALYSIS Since the assnnptotes for i' — ^' — 4 are y •• ± x, the asymp- totes for (X - 1)2 - (V - 2)2 = 4 are 2/ - 2 = ± (a; - 1), or y = X + 1 and y +x = 3. 6. Find the equations of the asymptotes and sketch the curves for (a) x' - y' +2x + ^y = 4; (6) 2x^ - 22/2 + 4x - 81/ = 0. 90. The Hyperbola of Semi-axes a and b. The ellipse was defined as the curve produced by lengthening or shortening all ordinates of the circle x^ + y' = a', an amount proportional to their lengths. Attention has been called to the fact that such a curve results also from the orthographic projection of the circle, or from taking the section of a right circular cylinder by a plane. The parametric equations of the circle are X = a cos 6, and y = a sin d; and the parametric equations of the ellipse derived from this circle as described above are X = a cos d, and y = b sin 6. Let us define the hyperbola as the curve obtained from the equilateral hyperbola, x^ — y^ = a^, by shortening or lengthening all ordinates by an amount proportional to their lengths. Its equation is then obtained by replacing y in x'^ — y^ = o' (1) by my. Hence we have for the equation of the hyperbola x^ — {myy = a^. a^ By dividing by a^ and replacing — ^ by h'^, we obtam m a" b' .-L--^, (2) §91] THE ELLIPSE AND HYPERBOLA 167 the symmetrical form of the equation of the hyperbola. It is easily shown that X = a sec d, (3) and y = b tan e. (4) are parametric equations of the hyperbola whose Cartesian equa- tion is given by (2). For from (3) and (4) we obtain -' -|-, = sec^e - tan^e = 1. Note that the parametric equations of the equilateral hyper- bola, X = asecd and y — a tan 6 bear the same relation to the parametric equations of the hyperbola, that the parametric equations of the circle bear to the parametric equations of the ellipse. It is seen that if the ordinates of the asymptotes to the equilateral hyperbola are affected in the same way as the ordinates of the curve itself, i.e., if the asymptotes are considered as part of the locus transformed, they are still the asymptotes to the hyperbola after the transformation. The equations of the asymptotes of the equilateral hyperbola are y == ± X. By the transformation they become my = + X, a or, smce ™ ~ ft' y=±^x, (5) which are the equations of the asymptotes of x^ 7/^ _ ;^2 ~ p - 1- 91. Construction of the Hyperbola. To construct the hypsr- bola draw two concentric circles of radii OA = a and OB = 6, as in Fig. 87. Divide each circumference, by means of bow 168 ELEMENTARY MATHEMATICAL ANALYSIS [§91 dividers, into the same number of convenient intervals. Lay ofif, on XOX', distances equal to a sec 5 by drawing tangents at the points of division on the circumference of the a-circle; also lay off distances equal to 6 tan 6 on the vertical tangent to the 6- circle by prolonging the radii of the circle through the points of division of the circumference. Draw horizontal and vertical lines through the points of division of MN and XX', respectively, dividing the plane into a large number of rectangles. J i r 1 —Is/ N T G \x \, y yi \ Y ^ 1 / /y^ V^ \ 1 t - ^ M \ \A.' w A D / [ / \ / \ ■A \ y. y f bVV' \ "vV Y ^; Fig. 87.— The hyperbola xP-ja?- - ^fjV = 1. The point of intersection of the vertical and horizontal line corresponding to the same value of 5 is a point on the hyperbola. The curve may be drawn by starting from the points A and A' and sketching the diagonals of successive rectangles. In the above construction, there is no reason why the diameter of the 6-circle may not be greater than that of the o-circle. The line A A' = 2a is called the transverse axis, the line BB' = 26 is called the conjugate axis, the points A and A' are called the vertices, and the point is called the center of the hyperbola. §91] THE ELLIPSE AND HYPERBOLA 169 Solving the equation (2) §90, for y, the equation of the hyperbola may be written in the useful form y = + ~ Vx^^T^. (1) Compare this equation with the equation of the ellipse, (2) §82. It is easy to show that the vertical distance PG, Fig. 87, of any point of the curve from the asymptote G'G can be made as small as we please by moving P outward on the curve away from 0. Write the equation of the hyperbola in the form 2/1 = V^^^^, (2) and the equation of the asymptote GG' in the form 2/2 = ^s. (3) Then PG? = 2/2 - 2/1 = ^ (x - Va;^ - a-') (4) Multiply both numerator and denominator in equation (4) by X + Vx^ - a^. PC — ~ =^ (5) 0-X + Vx^ - ffl^ Now, as X increases in value without limit the right side of (5) approaches zero. Whence PG = Oa,sx = a, Exercises 1. Write the symmetrical equation of the hyperbola from the parametric equations x = 5 sec 6, y = 3 tan 8. 2. Find the Cartesian equation of the hyperbola from the relations X = 7 sec e, 2/ = 10 tan 8. Note that the graphical construction of the hyperbola holds if 6 > u. 3. What curve is represented by the equation 25 16 170 ELEMENTARY MATHEMATICAL ANALYSIS [§92 4. What curve is represented by the equation y = ^Vs' — o'? 5. Write the equation of a hyperbola having the asymptoteB y = ±. (3/4) X, and transverse axis = 24. 6. Show that the curves x^ + 6i 4?/ + 4 = and (.X + 3)2 -{y + 2Y = l are the same, and show that each is a hyperbola. Fig. 88. — Conjugate hyperbolas. 7. What curve is represented by the equations X = h + as&a e 2/ = A; + 5 tan e? 8. Discuss the curve x' — 8x — 2y' — 12y = 6 . 92. Conjugate Hyperbolas. Let GAJ'JA'G', Fig. 88, be the hyperbola whose equation is 6^ 1. (1) Its transverse axis is A A' = 2a and its conjugate axis is BB' = 26. *92] THE ELLIPSE AND HYPERBOLA 171 Its asymptotes O'G and J' J are the diagonals, produced, of the rectangle constructed upon A' A and B'B as sides. The hyperbola GBJG'B'J', having B'B as transverse axis and A A' as conjugate axis and G'G and J' J as asymptotes is called the conjugate of GAJ'JA'G'. If Y'OY were the Z-axis and if X'OX were the Y-axis the equa- tion of the hyperbola GBJG'B'J' would be 0= 1 (2) By the above supposition we have interchanged x and y. Hence, to get the true equa- tion we must interchange x and y in equation (2). Therefore the equation of the hyperbola conjugate to x^/a^ — 2/Y62 = 1 is r (3) Fig. 89 shows a family of pairs of conjugate hyper- bolas. Fig. 89. — A family of conjugate pairs of hyperbolas with common asymptotes. (An interference pat- tern made from a glass plate under compression. From R. Strauble, "TJeber die Elsticitats-zahlen una moduln des Glases." Wied. Ann. Bd. 68, 1899, p. 381.) Exercises 1. Sketch on the same pair of axes the four following hyperbolas and their asymptotes: a) a;2 - ^2 = 25 (3) x' 25 ■ -1- (2) a;2 _ 2,8 = _ 25 Find the axes of the hyperbola (4) i,y -- X2 25 ■ = ± 2/'_ 9 1. 2. iVx«- 64. 3. Sketch the curves: a;2 4 9 = 1 and 4 ~ ■'i- - — 1 172 ELEMENTARY MATHEMATICAL ANALYSIS [§92 4. Sketch the curves: and 16 r 16 t 9 = 1. 6. Write the equation of the hyperbola conjugate to 2/ = ± f y/x' - 64. 6. Compare the graphs of: 2/ = + f Vx^ 3 7. Show that 3x^ Fig. 90. — Diagram for Exercise 13. 64 2/ = + f Vx' - 16 y = ± i Vx^ - 4 2/ = ± i Vx^ - 1 y = ±1 Va;' - 1/16 2/ = + f Vx^ - 42/' — 7a; + 52/ + 2 = is a hyperbola. Find the position of the center and of the vertices. The vertices locate the so-called "limiting Unes" of the hyperbola. Write the equations of the asymptotes. 8. Show that a;' - 4a;, - ^y^ + 42/ = 4 is a hyperbola. Find the coordinates of its center, the equations of its asymp- totes, and the equations of its limiting hnes. 9. Discuss the graphs: x^ -y^ = 1 and 2/2 -x ■>■ = 1. = 2, and find the 10. Discuss the graph 16a;2 — y'^ — 40a; — hmiting lines. 11. Write the equation conjugate to ^ _^ - 1 4 16 12. Write the equation conjugate to a;' - 2a! - 2/' - 62/ = 24. 13. A difficult problem : Prove that if a circular cylinder be cut by §92] THE ELLIPSE AND HYPERBOLA 173 a plane at an angle of 45° to the axis of the cylinder, and if then the surface of the cylinder be unrolled into a flat surface, the curved boundary of the surface is a sinusoid. Thus if a stove pipe be cut at an angle of 45° to its axis, and if then the sheet metal be unrolled into a flat sheet, the bounding curve is a sinusoid. In Fig. 90 only one quarter of the cylinder is shown. If P be any point on the section of the cylinder made by the cutting plane, and if the length of the arc AD be called e and the distance DP be called y, the problem is to show that y = sin 8, provided the radius of the cylin- der be called 1. If the angle of cutting be different from 45°, the equation of the curve is of the form y = bain 6, where b = tan BOC. F{x, y) = xy sia - (2) CHAPTER VI SINGLE AND SIMULTANEOUS EQUATIONS 93. Notation of Functions. It has been pointed out that the symbols f(,x), F{x), 4>(,x), ^{x), etc., are used to denote functions of X. Likewise the symbols f{x, y), F(x, y), (x, y),^ix, y), etc., are used to represent functions of two arguments x and y. For example, f{x, y) in a particular problem may be used to stand xy for the function / , , „ ■ . We may indicate this fact by writing V a;2 +y^ Again we may abbreviate the function of x and y, xy sin -> by the symbol F(x, y). This abbreviation can be indicated by writing The equation F{x, 2/) = (3) indicates that y is a function of a;; y is a function of x expressed implicitly. If equation (3) were solved for y giving V^Kx), (4) J/ is a function of x expressed explicitly. Equations (3) and (4) represent the same functional relation between x and y. Thus x^ + 2/2 — a'' = shows that y is & function of x but the functional relation is expressed implicitly. If the equation be solved for y, giving y = ± ^a^ — x"^, the same functional relation between x and y holds, but now 2/ is an explicit function of x. In the same problem or discussion the symbols f{x), f(y), f{u), or /(t)) denote the same functional form although the ftrguments may differ. If ^(") = V#+t' 174 §94] SINGLE AND SIMULTANEOUS EQUATIONS 175 f{v) means the same function but with every x replaced by y, thus m - ^ Again, if in a particular problem or discussion f{x) =x^ + 2x-l, then fiy) = y^ + 2y - 1, /(2) = 2= + 2-2 - 1 = 7, f(- 1) = (- 1)^ + 2(- 1) - 1 = - 2, /(O) = + - 1 = - 1. Exercises 1. URx) mx' + 3x+2, find/(2/);/(3);/tO);/(-l);and/(-2). 2. If /(a;) ^ s' + 2x^ + X, find A-1); /(O); /(+1J; f(z); Al/v); and /(<2). 3. liF(e) =sinfl, findf(,r/2);Ftx);F(0);F(x/6);F(V3)andF(|j-) 4. If*>(fl) s tanS, find¥>(0);¥>(7r/6);«>lir/3);#>(ir'/2);¥'(7r)and*>(|ir). 5. If /(I, 2/) ^ -==^' find/(2, 1);/ (0, 2);/ fe «) and/(m, n). V a;'' + y' Hint: To find /(2, 1) replace x by 2, and ?/ by 1 in the given func- tion of X and y. 94. A poljmomial in x of the nth degree is defined as aox" + aia;""' + UiX"'^ + + a„_ia; + On, where the symbols, ao, ai, 02, . . ., stand for any real con- stants whatsoever, positive or negative, integral or fractional, rational or irrational, and where n is any positive integer. If none, of the coefficients are zero the number of terms in a polynomial in x of the nth degree is (n + 1). In what follows in this chapter /(re) is supposed to stand for a polynomial in x. 96. The Remainder Theorem. Let f{x) = aax" + Oia;"-^ + ajx"-^ H- . . -|- On-iX + On. (1) Then /(r) =aor» + aif-^ -|- a^r''-'^ + -f (h-\r + a„. (2) By subtracting (2) from (1), /(*) ~ /('■) — "oCa;" — r") + a\{x''~^ — r"~^) -f + o^-i(s - r). (3) 176 ELEMENTARY MATHEMATICAL ANALYSIS The right-band side of this equation is made up of a series of terms containing differences of like powers of x and r, and, hence, by the well-known theorem in factoring,' each binomial term is exactly divisible by {x — r). The quotient of the right-hand side of (3) by {x — r) may be written out at length, but it is sufficient to abbreviate it by the symbol Q{x) and write fix) -fir) or :/^=QW+;^- (5) Equation (5) shows that f(r) is the remainder when f(x) is divided by (a; — r). Thus we have the Remainder Theorem : If a polynomial in x be divided by (a; — r), the remainder which does not contain x is obtained by writing, in the given function, r in place of x. This theorem shows, for example, that the remainder of the division (.t' - 6x2 + iix _ 6) -f. (x - 4) is 43 - 6(4)2 + n(4) - 6, or 6; also that the remainder of the division (x' - 6x2 + iia; _ 6) 4- (x -I- 1) is (- 1)3 - 6(- 1)2 -1^ 11(- 1) -■ 6 = - 24. The theorem enables one to write the remainder without actually performing the division. Exercises Without performing the division find the remainder of the following divisions : 1. (x2 + 3x - 2) -=- (x - 1). 2. (x' + 3x2 + 2x -1) ^ (x - 2). 3. {x* + 4x= + 3x2 _ 62; - 1) -^ (I + i). 4. (x' - 3x2 + 2x - 1) -^ (x -2). 6. (x2 + 3x + 2) + (x -h 1). 6. (x2 + 3x + 2) -i- {x + 2). 96. Factor Theorem. From equation (5) of the preceding section, we see that if fir) is zero, the remainder of the division of > See Appendix, Chapter XV, p. 159. §96] SINGLE AND SIMULTANEOUS EQUATIONS 177 fix) by (x — r) is zero, or /(a;) is exactly divisible by (x — r), i.e. (x — r) is a factor of f(x). Thus we have the Factor Theorem: If a polynomial in x becomes zero when r is written in the place of x, {x — r) is a factor of the polynomial. This means, for example, that if 3 be substituted for x in the function x' — 6a;^ + Ha; — 6 and if the result 3^ - 6(3)^ + 11(3) - 6 is zero, then {x - 3) is a factor of x^ — Qx'^ + lis — 6. The value r of the argument x that causes the function to take on the value zero has already been named a root or a zero of the function. ' The factor theorem may, therefore, be stated in the form : A polynomial in x is exactly divisible by (x — r) where r is any root of the polynomial. The familiar method of solving a quadratic equation by factoring is nothing but a special case of the present theorem. Thus, if x^ - 5x + Q = 0, (x - 2)(x - 3) = 0; and the roots are x = 2 and x = 3. The numbers 2 and 3 are such that when substituted in x^ — 5x + 6 the expression is zero; and the factors of the expression are x — 2 and ic — 3 by the factor theorem. Exercises 1. Tabulating the cubic polynomial /(a;) = x' — 6a;* -|" H^ —6, we obtain: X -3 - 2 - 1 - 1 1.5 2 2.5 3 4 fix), -120, -60, -24, -6, 0, +0.375, 0, - 0.375, 0, 6 What is the remainder when the function is divided by .r — 4? By I + 2? By a; + 3? By a; - 1.5? By a: - 3? Name three factors of the above function. 2. Find the remainder when a;* — 5x^ + 12.1;^ + 4a; — 8 is divided by a; - 2. 3. Show by the remainder theorem that x" + a" is divisible by X + a when n is an odd integer, but that the remainder is 2o" when n is an even int^er. 4. Without actual division, show that x* — ix' — 7x — 24 is divisible by a; — 3. 6. Show that a* + a^ — ab' — ¥ is divisible by o — b. 12 178 ELEMENTARY MATHEMATICAL ANALYSIS [§97 6. Show that (x + l)Ha; - 2) - 4(a; - l)(a: - 5) + 4 is divisible by I - 1. 7. Show that &x^ - 3x* - 5a;' + 5a;' - 2x - 3 is divisible by x + 1. 8. Show that (b - c){h + c)' + (c - a){c + a)' + (o - b)(a + b)' is divisible by (5 — c)(_c — a) {a — b). Hint: First consider the function as a polynomial in 6; then as a polynomial in c; and then as a polynomial in a. 9. Show that (6 - c)' + (c - o)' + (o - 6)' is divisible by (6— c){,c — a)(fl — b). 97. An Equation with Given Roots. The factor theorem enables us to build up a polynomial having given roots. If, for example, 1, 2, and 3 are roots, 2; — 1, x — 2, and a; — 3, are factors of the poly- nomial. Hence (x — l)(x — 2) (x — 3), or x' — 6x' + llx — 6 is a factor of the polynomial. Introducing another factor k, which does not contain x, cannot introduce another root, as a, for k can- not contain the factor (x — a). For the same reason, multiplying the equation x' — 6x^ -)- llx — 6 = by fc, when k does not contain x, cannot introduce roots, or solutions, in the equation. On the other hand if the equation be multiplied by a function of x, roots of the equation may be introduced or removed. For, clearly, if the multiplier contains the factor (x — a), the root a will be introduced; and if the multiplier contains the factor (x — 1) in its denominator, the factor (x — 1) will be divided out from both numerator and denominator, if it is a factor of the numerator and the root 1 wiU be removed from the function. Exercises Build up polynomial equations having the following numbers for roots: 1. 1, 3, and 4. 2. -1, 2, and -3. 3. 0, 2, and -1. 4. 1, 0, 0, and 2. 98. Legitimate and Questionable Transformations. If one equation is derived from another by an operation which has no effect one way or another on the solution, it is spoken of as a legitimate transformation ; if the operation is of such a nature that it may ha.ve an effect upon the roots, it is called a questionable §98] SINGLE AND SIMULTANEOUS EQUATIONS 179 transfonnation, meaning thereby that the effect of the operation requires examination. In performing operations on the members of equations, the effect on the solution must be noted, and proper allowance made in the result. It cannot be too strongly emphasized that the test for any solution of an equation is that it satisfy the original equation. "No matter how elaborate or ingenious the process by which the solution has been obtained, if it do not stand this test it is no solution; and, on the other hand, no matter how simply obtained, provided it do stand this test, it is a solution."^ By the principles or axioms of algebra, an equation remains true if we unite the same number to both sides by addition or subtraction; or if we multiply or divide both members by the same number, not zero; or if like powers or roots of both members be taken. But we have indicated in the preceding section that these operations may affect the number of roots of the equation. This is obvious enoHgh in the case already cited. Sometimes, however, the operation that removes or introduces roots is so natural and its effect is so disguised that the student is apt not to take due account of its effect. Thus, the roots of 3(x - 5) = x(x - 5) + x' - 25 (1) are — 1 and 5, for either of these when substituted for x will satisfy the equation. Dividing the equation through by a; —5, the resulting equation is 3 = a; + a; + 5. This equation is not satisfied by a; = 5. One root has disappeared in the transformation. It is easy to keep account of this if (1) be given in the form (a; - 5)(a; + 1) = 0, but the fact that a factor has been removed may be overlooked when the equation is written in the form first given. A very important effect upon the roots of an equation results from squaring both members. The student must always take proper account of the effect of this common operation. To il- lustrate, take the equation a; + 5 = 1 - 2a;. (2) ' Chrystftl's Algebra, 180 ELEMENTARY MATHEMATICAL ANALYSIS It is satisfied only by the value a; = — f . Now, by squaring both sides of the equation, we obtain a;'' + lOs + 25 = 1 - 4x + ix', which is satisfied by either a; = 6 or a; = — |. Here obviously, an extraneous solution has been introduced by the operation of squaring both members. It is easy to show that squaring both members of an equation is equivalent to multiplying both sides by the sum of the left and right members. Thus, let any equation be represented by L(x) = R^x) (3) in which L(x) represents the given function of x that stands on the left-hand side of the equation and R{x) represents the given function of x that stands on the right-hand side of the equation. Squaring both sides, [Lix)]^ = [B(,x)V. Transposing, [L(,x)r - [R\x)V = 0, factoring, [Lix) + R(x)] mx) - Rix)] = 0. But (3) may be written L{x) - R{x) = 0. Thus, by squaring the members of equation (3) the factor L(x) + R(x) has been introduced. The sum of the left- and right-hand members of (2), above, is 6 — a;. Hence, squaring both sides of (2) is equivalent to the introduction of this factor, or thq operation introduces the root 6, as already noted. As another example, suppose that it is required to solve sin a cos a = \ (4) for a < 90°. Substituting for cos a, Equation (4) becomes sin aVl — sin" a. - \, (5) squaring sin^ a(l — sin'' a) = y^, §98] SINGLE AND SIMULTANEOUS EQUATIONS 181 completing the square sin* a. — sin'' a-\-\ = j-^-. Hence, sin a = + Vi + i \/3 =• ± 0.9659 or ± 0.2588. Only the positive values satisfy (4); the negative values were introduced in squaring (5). If, however, the restriction a < 90° be removed, so that the radical in (5) must be written with the double sign, then no new solutions are introduced by squaring. Among the common operations that have no effect on the solu- tion are multiplication or division by known numbers, or addition or subtraction of like terms to both members; none of these intro- duce factors containing the unknown number. Taking the square root of both numbers is legitimate if the double sign be given to the radical. Clearing of fractions is legitimate if it be done so as not to introduce a new factor. If the fractions are not in their lowest terms, or if the equation be multiplied through by an expression having more factors than the least common multiple of the denominators, new solutions may appear, for extra factors are probably thereby introduced. Hence, in clearing of fractions, the multiplier should be the least common denominator and the fractions should be in their lowest terms. This, however, does not constitute a sufficient condition, therefore iAe only certainty lies in checking all results. Exercises Suggestions: It is important to know that any equation of the form oa;2» + bx'' + c = can be solved as a quadratic by finding the two values of a;". Fre- quently equations of this type appear in the form dx' + ex~^ = f. Likewise any equation of the form aj(x) + 6V7(S) + c = can be solved as a quadratic by finding the two values of VjCi) and 182 ELEMENTARY MATHEMATICAL ANALYSIS then solving the two equationa resulting from putting ■\/f(x) equal to each of them. One of these usually gives extraneous solutions. These two tjrpes occur in the exercises given below. Since operations which introduce extraneous solutions are often used in solving equations, the only sure test for the solution of any equation is to check the results by substituting them in the original equation. Take account of all questionable operations in solving the following equations: 3a; 6,9 + 7.- Note : 3 la not a, root. ' X -3 a; + 3a;-3 2. {x^ + 5x + 6)/{x - 3) + 4a; - 7 = - 15. 3. 3(a; - 5){x - l){x - 2) = (x - 5)(x + 2)(.x + 3). Note : Divide by (a; — 5), but take account of its effect. 4. x'/a + ax = x''/h + 6a;. 6. oa;(ca; — 36) = 5o(36 — ex). 6. a;' — Ji* = n — a;. 7. (a; - 4)» + (a; - 5)» = 31[(a; - 4)^ - (a; - 5)'^]. Divide by (a; - 4) + (a; - 5) or 2x - 9. a;^ — 3a; 1 8. ■ _ . — h 2 + _ - =0. If the fractions be added, multi- plication is unnecessary. There is only one root. 9. X = 1 - Va;' - 7. 10. V a: + 20 - Va; - 1 - 3 = 0. 11. \/l5/4 + a; = 3/2 + -y/g. 12. 20a;/ V 10 a; - 9 - VlOx - 9 = IS/VlOa; -9+9. 13. — ;= , = ;;. Consider as a proportion and take y/x- y/x-Z ^-^ by composition and division. 14. a;^_+ 5/2 = (13/4)x>^. 16. y/ x^ - 2y/x + a; = 0. Divide by y/~x. 16. 2V'a;2 -5x + 2 - x' + 8x = 3x - 6. Call a;^ - 5a; + 2 = u'. 17. 4a;2 - 4a; + 20\/2a;2 - 5a; + 6 = 6a; + 66. 18. x-^ - 2a;-i = 8. 22. 8x^ - Sx'^ = 63. 19. x^^ - 5a;^i +4 = 0. 23. (x - a)" - 3(x - a)-» = 2. 20. 110a;-* + 1 = 21a;-2 24. 2a;^ - 3a;^ + x = 0. 21. Vx + 4x-J^ = 5. 99. Intersection of Loci. In §41 it was shown that the coordi- nate of the points of intersection of two loci could be found by solving the equations of the loci considered as simultaneous equations. SINGLE AND SIMULTANEOUS EQUATIONS 183 Let all terms of an equation be transposed to the left-hand member, rendering the right-hand member zero. Let this left- hand member be abbreviated by u. The equation then takes the form M = 0. (1) In a similar way let a second equation be put in the form « = 0. (2) Fig. 91 . — Intersections curves. Let the graphs for equations (I) and (2) be represented in Fig. 91. The coordinates of any point on curve (1) make u equal to zero. The coordinates of any point on curve (2) make v equal to zero. Consider the graph of u + kv = Q, (3) where h is any constant. The co- ordinates of a point of intersection of the u and v curve satisfy equa- tion (3). For, these coordinates make u zero and they make v zero, then they make u + kv zero.. Fur- ther, the coordinates of a point on the u curve which is not on the v curve do not satisfy equation (3). For these coordinates make u zero but do not make v zero, then they do not make u + kv zero. Similarly the coordinates of a point on the v curve which is not on the u curve do not satisfy equation (3). Hence the graph of (3) passes through all points of intersection of the u and V curves but does not intersect these curves in any other points. Thus to find the coordinates of the points of intersec- tion of the u and v curve we may solve (1) and (3) or (2) and (3) as simultaneous equations. The locus of the equation WW = (4) is the M and v curves considered as a single locus. For, the coordi- nates of a point on the u curve make u zero, then they make uv zero. Similarly the coordinates of a point on the v curve make 184 ELEMENTARY MATHEMATICAL ANALYSIS uv zero. The coordinates of a point neither upon the u curve nor upon the v curve make neither u nor v zero, then they cannot make uv zero. Hence the locus of (4) consists of all points on the u and v curve but of no other points. To find the points of intersection of the circle x^ + y' = 25 and the straight line x + y = 7 yre solve the equations by the usual method, as follows: x^ + y' = 25\ (5) X +y = 7j (6) The graphs are a circle and a straight line, as shown in (1), Fig. 92. Squaring the second equation, the system becomes x^+y'> = 25\ (7) X' + 2xy + 2,2 = 49 / (8) The second equation represents the two straight lines shown in (2) Fig. 92. The effect of squaring has been to introduce two extraneous solutions corresponding to the points Ps and Pt. For, eCiuation (8) may be written {x + y + 7)ix + y — 7) =0 while (6) from which it was derived is x + y — 7 = 0. Multiplying (7) by 2 and subtracting (8) from it, the last pair of equations becomes x^ -2xy + y* = l\ (6) x' + 2xy + v" = 49 J (7) which gives the four straight lines of Fig. 92, (.4). Taking t^e square root of each member, but discarding the equation x + y + 7 = 0, because it corresponds to the extraneous solutions introduced by the questionable operation, we have: x-y = ±l\ (8) (9) -2/= ±1\ + y =7 / By addition and subtraction we obtain the results: (10) X = S\ y=4J a; =4"! 2/ = 3/ (11) represented by the intersections of the lines parallel to the axes shown in Fig. 92, (5). §99] SINGLE AND SIMULTANEOUS EQUATIONS 185 This is a good illustration of the graphical changes that take place during the solution of simultaneous equations of the second degree. The ordinary algebraic solution consists, geometrically, in the succes- sive replacement of loci by others of an entirely different kind, but all passing through the points of intersection (as Pi, Pa, Fig. 92) of the original loci. The final locii are straight lines parallel to the axes. FiQ. 92. — Graphic representation of the steps in the solution of a certain set of simultaneous equations. Exercises Find the coordinates of the points of intersection of the following pairs of equations; sketch curves representing all equations involved in the solution: 1. xy = 1 3x - 5y = 2 2. X' + y' = 5 y^ = 4x Hint fob Ex. 4: Let u = x^andw 3. x' + X = 4j/* 3j + 6j/ = 1 4. x2 + 2/2 = 9 a;2 - yi = 4 = 1/2. Solve for u and v. 186 ELEMENTARY MATHEMATICAL ANALYSIS [§100 Solve graphically the following : 6. x' + y' = 25 6. x' + y^ = 25 X +y =2 x^ +y^ + 2x -6y + 6 =0 7. y = x^ + X — I xy = 1. 100. Quadratic Systems.' Any linear-quadratic systena of simultaneous equations, such as y = mx + k ax'' + hy^ + 2hxy -\- 2gx + 2fy + c = can always be solved analytically; for y may readily be eliminated by substituting from the first equation into the second. A system of two quadratic equations may, however, lead, after elimination, to an equation of the third or fourth degree; and, hence, such equations cannot, in general, be solved until the solutions of the cubic and bi-quadratic equations are known. A single illustration will show that an equation of the fourth degree may result from the elimination of an unknown number between two quadratics. Thus, let x^ — y = 5x a;2 + j/2 = 10. From the first, y = x^ — 5x. Substituting this value of y in the second equation, and performing the indicated operations, we obtain a;4 _ lOx' + 26s^ - 10 = 0. WhUe, in general, a bi-quadratic equation results from the process of elimination from two quadratic equations, there are special cases of some importance in which the resulting equation is either a quadratic equation or a higher equation in the quadratic form. Two of these cases are: (1) Systems in which the terms containing the unknown num- bers are homogeneous; that is, systems in which the terms con- ^ A large part of the remainder of this chapter can be omitted if the students have had a good course in algebra in the secondary school. §101] SINGLE AND SIMULTANEOUS EQUATIONS 187 taining the unknown numbers are all of the second degree with respect to the unknown numbers, such, for example, as x'' — 2xy = 5 3x^ - lOy^ = 35. (2) Systems in which both equations are symmetrical; that is, such that interchanging x and y in every term does not alter the equations; for example x' + y^ - X - y = 78 xy + X + y = 39. 101. Unknown Terms Homogeneous. The following work illustrates the reasoning that will lead to a solution when applied to any quadratic system all of whose terms containing x and y are of the second degree. Let the system be x^ — xy = 2 2x^ + 2/2 = 9. (1) Divide each equation by x'^ (or y'^), then 1 - iy/x) = 2/x^ 2 + (y/xr = Vx'. (2) Since the left members were homogeneous, dividing by x' renders them functions of the ratio (y/x) alone; call this ratio m. Then equations (2) contain only the unknown numbers m and x^. The latter is readily eliminated by subtraction, leaving a quad- ratic for the determination of m. When m is known, substituting in (2) determines x, and the relation y = mx determines the corresponding values of y. The above illustrates the principles on which the solution is based. In practice, it is usual to substitute y = mx at once, and then eliminate x' by comparison; thus, from the substitution y = mxin (1), we obtain x' - mx^ = 2 2x' + mV = 9. (3) 188 ELEMENTARY MATHEMATICAL ANALYSIS [§101 Thence, Whence, or a;2 = 2/(1 - m) x^ = 9/(2 + m'). 2/(1 -m) = 9/(2 + m^), 2m' + 9m = 5. \ \ Y \ \ \ \ 4 / V y// \ 3 // u\ 2 \^ 1 y/ X' / J ' JC -4 -3 - 2 /-I ^^Z' % 2 3 4 / // A - a = - Vs Vi" // •4 \ \ ' \ \ Y 1 (4) (5) (6) Fig. 93. — Solutions of a set of simultaneous quadratics given graph- ically by the coordinates of the points of intersection of an ellipse and hyperbola. Factoring, whence. Hence, (2m - l)(m 4- 5) = m = 1/2 or — 5. X = + 2 or + (l/3)-v/3 2/ = + 1 or + (5/3)v'3. (7) (8) (9) §102] SINGLE AND SIMULTANEOUS EQUATIONS 189 These solutions should be written as corresponding pairs of values as follows: X = 2 X. = -2 X = (1/3)V3 a; = - (l/3)\/3 y = l v=-l t/='-(5/3)V3 y= (5/3)^3 This system can readily be solved without the use of the mx sub- stitution by merely solving the first equation fpr y and substituting in the second. Graphically (See Fig. 93), the above problem is equivalent to finding the intersections of the curves : x(x - y) = 2 (V2x)' + y' = 9 The first is a curve with the two asymptotes x = and x — y = 0. That these lines are asymptotes is readily seen if the equation be 2 put in the form y = x If a; is positive, y is less than x, or the curve is below the line y = x. If x is negative y is greater than x, or the curve-is above the line y = x. As x increases in numerical value, 2 - approaches zero and the curve approaches the line y = x. As a; approaches zero, y increases without limit. As a matter of fact, the curve is a hyperbola, although proof that such is the case cannot be given until the method of rotating any curve about the origin has been explained. The second curve is obviously an ellipse generated from a circle of radius 3 by shortening the abscissas in the ratio ■y/2 : 1. The two curves intersect at the points: X =2 - 2 0.557 . . . -0.557 ... 2/ = 1 - 1 - 2.887 ... +2.887 ... The auxiliary lines, y = ^x and y = — 5x, made use of in the solution are shown by the dotted lines. 102. Symmetrical Systems. Simultaneous quadratics of this type are readily solved analytically by solving for the values of the binomials x -\- y and x — y. The ingenuity of the student will usually, show many short cuts or special expedients adapted to the particular problem. The following worked examples point oat some of the more common artifices used. 1. Solve x + y =Q (1) xy = 5. (2) 190 ELEMENTARY MATHEMATICAL ANALYSIS [§102 Squaring (1) x^ + 2xy + y^ = 36. (3) Subtracting four times (2) from (3) x'- - 2xy + 2/= = 16. Whence But from (1) Therefore a; = 5 2/ = l 2. Solve X -y = X +y = and X2 + 2/2 : ± 4. 6. = 34 a; = 1 2/ =5. (1) xy ■■ Adding two times (2) to (1) = 15 (2) x' > + 2xy + y' = 64. (3) Subtracting two times (2) from (1) X Whence, from (3) and (4) 2 - 2xy + X +y = 2/2=4- ±8 (4) Therefore X - V = ±2. X = 5 X =3 a; = - 5 ' X = -3 ^ = 3 y = 5 y = -3 y = -S The hyperbola and circle j by the student. 3. Solve represented by (1) and (2) should be drawn x> + y^ = 72 (1) X +y = Cubing (2) a;3 4- 3a;2j, ^ ^xy' = 6. ' + y' = 216. (2) (3) Subtracting (1) and dividing by 3 4 whence, since xy(x + y) X +y = 48, = 6 (4) we have y = 8. (5) §102] SINGLE AND SIMULTANEOUS EQUATIONS 191 From (2) and (5) proceed as in example 1, and find 1 = 4 , X = 2. r> and , y = 2 2/ = 4 Otherwise, divide (1) by (2) and proceed by the usual method. 4. Solve a;2 + SI/ = ^ (a; + J/) (1) y-'-'rxy = ^- (x + v). (2) Adding (1) and (2) {x + yy - 6{x + y) '= 0, (3) whence, X + 2/ = or 6. (4) Now, because x + y is a factor of both members of (1) and (2), the original equations are satisfied by the unlimited number of pairs of values of x and y whose sum is zero, namely, the coordinates of all points on the line x -\- y = Q. Dividing (1) by (2), we get x/y = 7/11. This, and the line x -\- y = &, from (4), give the solution: y = T/Z y = 11/3. Graphically, the equation (1) is the two straight lines i; {x-7/3){x + y) =0. Equation (2) is the two straight lines (2/ - n/3){x+y) =0. These loci intersect in the point (7/3, 11/3) and also intersect every- where on the line x + y = 0. Exercises 1. Show that 3-2 + J/2 = 25 X + y = 1 has a solution, but that there is no real solution of the system a;2 + j/2 = 25 X +y = U. 192 ELEMENTARY MATHEMATICAL ANALYSIS [§103 2. Do the curves Do the curves 3. Solve x' + y' = 25 xy — 100, intersect? a;2 + 2/2 = 25 xy = 12, intersect? (x^ + y'){x + y) = 272 x^ + y^ + x'+ y = 42. Note : Call x^ + y^ = u, and x + y = v. 4. Show that there are four real solutions to x^ -\- y^ - \2 = X -\-y xy-\-S = 2{x +y). 5. Solve x'' -\- y^ -\- x + y = li xy = 6. 103. Graphical Solution of the Cubic Equation. The roots of a cubic x' + ax' + j3x + 7 = (where a, /3, and 7 are given known numbers) may be determined graphically as explained in §40. Another method of solving the cubic equation graphically will now be given. The roots of the equation x^ + ax^ + ^x + y = (1) are the JST-intercepts for the graph of y = x^ + ax' + ^x + y. (2) If we replace x in equation (2) by (x — k), where fc is a constant, the equation (2) becomes y = ix-ky+ a(.x - ky + /3(a; - k) + y, or y = x^ + {a- Zh)x' + (j3 - 2ak + ^k')x -{¥ - al 0. 104. Method of Successive Approxunations. It must be re- membered that the graphic methods of solving numerical equa- tions by finding one or both coordiaates of points of intersection of graphs, gives results only approximately correct. The degree of accurately depends upon the scale of the drawing and upon the accuracy with which the graphs are constructed. The results thus obtained may be used as a first approximation to the solution by a method illustrated below Suppose that it is required to find to four decimal places one root of x' — X — 1 = 0. See §103 and Fig. 95. The graphic method gives X = 1.32. This is the first approximation. A second approximation is found as follows : Substituting 1.32 for x in y = x' — X — \ J (1) gives — 0.0200 for y. This shows that 1.32 is not the exact value for y. Substituting 1.33 for a; gives 0.0226 for y. Put these results in tabular form X y p Q 1.32 1.33 -0.0200 +0.0226 Differences 0.01 0.0426 This shows that the X-intercept of the graph of the given equation is between the points P and Q, Fig. 96. Thus a root of x' — a; — 1 is greater than 1.32 and less than 1.33. Now reason as follows: The actual root lies between 1.32 and 1.33, and the zero value of y corresponds to it. This zero is approximately 200/426 of the way between the two values of y. Hence if the curve be nearly straight between x = 1.32, and x = 1.33, the desired value of x is approxi- mately 200/426 of the way between 1.32 and 1.33 or it is x = 1.3247 approximately. This value is probably correct to the fourth decimal §104] SINGLE AND SIMULTANEOUS EQUATIONS 197 place. The next step will show that this result is correct to four decimal places. To find a third approximation we build another table of values: 1.3247 1.3248 Differences 0.0001 y -0.0000766 +0.0003499 0.0004265 Fig. 96. — Method of approximation to a root of an equation. Reasoning as before, we get x = 1.324718 which is very likely true to the last decimal place. The above method is applicable to an equation like exercise 8 above. In fact it is the only numerical method that is applicable tn such cases. Exercises Find correct to four decimal places the roots of: 1. x' -ix + 10 = 0. 2. X' - 12x -8=0. See Exercises 1 and 2, §103. CHAPTER VII PERMUTATIONS AND COMBINATIONS; THE BINOMIAL THEOREM ] 105. Ftmdamental Principle. If one thing can be done in n different ways and another thing can be done in r different ways, then both things can be done together, or in succession, in n Xr different ways. This simple theorem is fundamental to the work of this chapter. To illustrate, if there be 3 ways of going from Madison to Chicago and 7 ways of going from Chicago to New York, then there are 21 ways of going from Madison to New York. To prove the general theorem, note that if there be only one way of doing the first thing, that way could be associated with each of the r ways of doing the second thing, making r ways of doing both. That is, for each way of doing the first, there are r ways of doing both things; hence, for n ways of doing the first there are n X r ways of doing both. Illustrations: A penny may fall in 2 ways; a common die may fall in 6 ways; the two may fall together in 12 ways. In a society, any one of 9 seniors is eligible for president and any one of 14 juniors is eligible for vice-president. The number of tickets possible is, therefore, 9 X 14 or 126. I can purchase a present at any one of 4 shops. I can give it away to any one of 7 people. I can, therefore, purchase and give it away in any one of 28 different ways. A product of two factors is to be made by selecting the first factor from the numbers a, b, c, and then selecting the second factor from the numbers x, y, z, u, v. The number of possible products is, therefore, 15. If a first thing can be done in n different ways, a second in r different ways, and a third in s different ways, the three things can^be done in n X r X s different ways. This follows at once from the fundamental principle, since we may regard the first 198 §106] PERMUTATIONS AND COMBINATIONS 199 two things as constituting a single thing that can be done in nr ways, and then associate it with the third, making nr X s ways of doing the two things, consisting of the first two and the third. In the same way, if one thing can be done in n different ways, a second in r different ways, a third in s, a fourth in t, etc., then all can be done together inn X r X s X t "different ways. Thus, n different presents can be given to x men and a women in (x + a)" different ways. For the first of the n presents can be given away in (x + a) diiferent ways, the second can be given away in (x + a) different ways, and the third in (a; + a) different ways and so on. Hence, the number of possible ways of giving away the n presents to {x + a) men and women is (a; + a){x + a)(x + a) to n factors, or {x + a)". Exercises 1. A building has 6 exits. In how many ways can a person leave the building and enter by a different door? 2. A car has five seats. In how many different ways may three people be seated, each occupying a different seat? 3. In how many different ways may 3 presents be given away to 10 people? 106. Definitions. Every distinct order in which objects may be placed in a line or row is called a permutation, or an arrangement. Every distinct selection of objects that can be made, irrespective of the order in which they are placed, is called a combination, or group. Thus, if we take the letters a, b, e, two at a time, there are six arrangements, namely, ab, ac, ba, be, ca, cb, but there are only three groups, namely, ab, ac, be. If we take the three letters all ,at a time, there are six arrange- ments possible, namely, abc, acb, boa, baa, cab, eba, but there is only one group, namely, abc. Permutations and combinations are both results of mode of selection. Permutations are selections made with the understand- ing that two selections are considered as different even though they differ in arrangement only; combinations are selections made with the understanding that two selections are not considered as different, if they differ in arrangement only. 200 ELEMENTARY MATHEMATICAL ANALYSIS [§107 In the following work, products of the natural numbers like 1X2X3; 1X2X3X4X5; etc. are of frequent occurrence. These products are abbreviated by the sjonbols 3\, 5 Land read "factorial three," "factorial five" respectively. 107. Formula for the Number of Permutations of n Different Things Taken All at a Time. We are required to find how many possible ways there are of arranging n different things in a line. Lay out a row of n blank spaces, so that each may receive one of these objects, thus: I 1 I I 2 I I 3 I I 4 I I 5 I . . . MlJ In the fijst space we may place any one of the n objects; therefore, that space may be occupied in n different ways. The second space, after one object has been placed in the first space, may be occupied in (n — 1) different ways; hence, by the fundamental principle, the two spaces may be occupied in n(n — 1) different ways. In like manner, the third space may be occupied in (n — 2) different ways, and, by the same principle, the first three spaces may be occupied in n(n — l)(n — 2) different ways, and so on. The next to the last space can be occupied in but two different ways, since there are but two objects left, and the last space can be occupied in but one way by placing therein the last re- maining object. Hence, the total number of different ways of occupying the n spaces in the row with the n objects is the product n(n - l)(n -2) . . 3-2-1, or n!. If we use the symbol Pn to stand for the number of permutations of n things taken all at a time, then we write P„ = n! (1) 108. Formula for the Nxmiber of Permutations of n Things Taken r at a Time. We are required to find how many possible ways there are of arranging a row consisting of r different things, §108] PERMUTATIONS AND COMBINATIONS 201 when we may 8ele(}t the r things from a larger group of n different things. For convenience in reasoning, lay out a row of r blank spaces, so that each of the spaces may receive one of the objects, thus: \ 1 I I 2 \ 3 j . . . i r-1 I I r \ In the first space of the row, we may place any one of the n objects; therefore, that space may be occupied in n different ways. The second space, after one object has been placed in the first space, may be occupied in (w — 1) different ways; hence, by the fun- damental principle, the two spaces may be occupied in n{n — 1) different ways. In like manner, the third space may be occupied in (n — 2) different ways; hence, the first three may be occupied in n{n — !)(«■ — 2) different ways, and so on. The last, or rth, space can be occupied in as many different ways as there are objects left. When an object is about to be selected for the rth space, there have been used (r — 1) objects (one for each of the (r — 1) spaces already occupied). Since there were n objects to begin with, the number of objects left is n — (r — 1), orn — r + 1, which is the number of different ways in which the last space in the row may be occupied. Hence, the formula: P„,. = n(n - i)(n - 2) (n - r + i), (1) in which P„,r stands for the number of permutations of n things taken r at a time. This formula, by multiplication and division by (n — r) ! becomes : _ n(n - 1) . . . (w - r + l)(n - r){n - r - 1) . . . 3-2-1 ""■ ~ {n-r){n-r-l). 3-21 n' or P.,. = , v.- (2) ' (n — r) ! ^ ' This formula is more compact than the form (I) above, but the fraction is not in its lowest terms. Formula (1) is easily remembered by the fact that there are just r factors, beginning with n and decreasing by one. Thus we have Pio,7 = 10X9X8X7X6X5X4. 202 ELEMENTARY MATHEMATICAL ANALYSIS [§109 Exercises , 1. How many permutations can be made of six things taken all at a time? 2. How many different numbers can be made with the five digits 1, 2, 3, 4, 5, using each digit once and only once to form each number? 3. The number of permutations of four things taken all at a time bears what ratio to the number of permutations of seven things taken all at a time? 4. How many arrangements can be made of eight things taken three at a time? 5. How many arrangements can be made of eight things taken five at a time? 6. How many four-figure numbers can be formed with the nine digits 1, 2, 9 without repeating any digit in any number? 7. How many different signals can be made with seven different flags, by hoisting them one above another five at a time? 8. How many different signals can be made with seven different flags, by hoisting them one above another any number at a time? 9. How many different arrangements can be made of nine ball players, supposing only two of them can catch and one pitch? 10. How many different ways may the letters of the word algebra be written, using all of the letters? 109. Formula for the number of combinations, or groups, of n different things taken r at a time. It is obvious that the number of combinations, or groups, con- sisting of r objects each that can be selected from n objects, is less than the number of permutations of the same objects taken r at a time, for each combination or group when selected can be arranged in a large number of ways. In fact, since there are r objects in the group, each group can be arranged in exactly r\ different ways. Hence, for each group of r objects, selected from n objects, there exists r! permutations of r objects each. There- fore, the number of permutations of n things taken r at a time, is r! times the number of combinations of n objects taken r at a time. Calling the unknown number of combinations x, we have xXrl = P„„ = , ^" ,, , {n — r)\ or solving for x ^ n\ r!(n — r)! §109] PERMUTATIONS AND COMBINATIONS 203 This is the number of combinations of n objects taken r at a time, and may be symbdiized C 5J (I) This fraction will always reduce to a whole number. It may be written in the useful form P _ n{ n - l){n - 2) . . . (n - r + 1) ,„. ^"" ~ 1X2X3. r ' ^''' It is easily remembered in this form, for it has r factors in both the numerator and the denominator. Thus for the number of combinations of ten things taken four at a time we have four factors in the numerator and denominator, or „ ^ 10 X 9 X 8 X 7 ^">'' 1X2X3X4 ■ Exercises 1. Howmany different products of three each can be made with the five numbers a, 6, c, d, e, provided each combination of three factors gives a different product. 2. How many products can be made from nine different numbers, by taking six numbers to form each product? 3. How many products can be made from nine different numbers, by taking four numbers to form each product? 4. How many different hands of thirteen cards each can be held at a game of whist? 6. A building has 5 entrances. In how many ways can a. person enter the building and leave by a different door? 6. In how many ways can a child be named, supposing that there are 400 different Christian names, without giving it more than three names? 7. In how many ways can a committee of three be appointed from six Italians, four Frenchmen, and seven Americans provided each nationality is represented? 8. There are five straight lines in a plane, no two of which are par- allel; how many intersections are there? 9. There are five points in a plane, no three of which are coUinear; how many lines result from joining each point to every other point? 10. In a plane there are n straight lines, no two of which are parallel ; how many intersections are there? 204 ELEMENTARY MATHEMATICAL ANALYSIS [|110 11. In a plane there are n points, no three of which are collinear; how many straight lines do they determine? 12. In a plane there are n. points, no three of which are collinear, except r, which are all in the same straight line; find the number of straight lines which result from joining them. 13. In how many ways can seven people sit at a round table? 14. In how many ways can seven beads of different colors be strung so as to form a bracelet? 15. How many different sums of money can be formed from a dime, a quarter, a half dollar, a dollar, a quarter eagle, a half eagle, and an eagle? 110.* The Arithmetical Triangle. In deriving by actual mul- tiplication, as below, any power of a binomial x + a from the preceding power, it is easy to see that any coeflSicient in the new power is the sum of the coefficient of the corresponding term in the multiplicand and the coefficient preceding it in the multiplicand. Thus x' + 3ax^ + So^a; + a' X + a X* + 3ax^ + 3aV + a^x ax' + 3aV + 3a'x + a* x'^ + Aax' + &aV + Aa'x + a\ or, retaining coefficients only, we have 1+3+3+1 1^ 1 1+3+3+1 1+3+3+1 1+4+6+4+1 from which the law of formation of the coefficients 1, 4, 6, . . . is evideAt. Hence, writing down the coefficients of the powers of a; + o in order, we have §in] PERMUTATIONS AND COMBINATIONS 205 Powers CoefScients ] L 2 3 4 5 6 7 8 9 10 11 ] 1 ] I 1 . 2 ] L 2 1 3 ] I 3 3 1 4 ] L 4 6 4 1 5 ] L 5 10 10 5 1 6 L 6 15 20 15 6 1 7 ] I 7 21 35 35 21 7 1 8 t 8 28 56 70 56 28 8 1 9 ] I 9 36 84 126 126 84 36 9 1 10 L 10 45 120 210 252 210 120 45 10 1 In this triangle, each number is the sum of the number above it and the number to the left of the latter. Thus 84 in the 9th line equals 56 + 28, etc. The triangle of numbers was used previous to the time of Isaac Newton for finding the coefficients of any de- sired power of a binomial. At that time it was not suspected that the coefficients of any power could be made without first obtaining the coeflBcients of the preceding power. Isaac Newton, while an undergraduate at Cambridge, showed that the coefficients of any power could be found without knowing the coefficients of the preceding power; in fact, he showed that the coefficients of any power n of a binomial were functions of the exponent n. The above triangle of numbers is known as the arithmetical triangle or as Pascal's triangle. 111. Binomial Expansion. The demonstration of the binominal theorem may be based upon the following law of multiplication: The product of any number of •polynomials is the aggregate of all the possible partial products which can be made by taking one term and only one from each of the polynomials. This statement is merely a definition of what is meant by the product of two or more polynomials. (See Chapter XV, §305.) Thus, {x + a){y + b){z + c) = xyz ■+ ayz + bxz + cxy + abz + box + cay + abc 206 ELEMENTARY MATHEMATICAL ANALYSIS [§111 Each of vthe eight partial products contains a letter from each parenthesis, and never two from the same parenthesis. The number of terms is the number of different ways in which a letter can be selected from each of the three parentheses. In the present case this is, by §105, 2X2X2 = 8. Let it be required to write out the value of (x + a)", where x and o stand for any two numbers and n is a positive integer. That is, we must consider the product of the n parentheses (x + a)(x + a){x + a) (x + a), by the distributive law stated above. First. Take an x from each of the parentheses to form one of the partial products. This gives the term x" of the product. Second. Take an a from the first parenthesis with an x from each of the other (n — 1) parentheses. This gives aa;""' as another partial product. But if we take a from the second paren- thesis and an x from each of the other (n — 1) parentheses, we get ax"-'- as another partial product. Likewise by taking a from any of the parentheses and an x from each of the other (n — 1) paren- theses, we shall obtain aa;»~' as a partial product. Hence, the final product contains n terms like ax"~', or, adding these, we obtain nax""^ as a part of the product. Third. We may obtain a partial product like a^x^~'^ by taking an a from any two of the parentheses, together with the x's from each of the other (n —2) parentheses. Hence, there are as many partial products like o^a;»"^ as there are ways of selecting two a's, from n parentheses; that is, as many ways as there are groups, or combinations, of n things taken two at a time, or n{n — 1) r2 Hence, — - a^a;""^ is another part of the product. 1 '^ Fourth. We may obtain a partial product like a'a;"~' by taking an a from any three of the parentheses together with the a;'s from each of the other (» — 3) parentheses. Hence, there are as many partial products like a'a;»-' as there are ways of selecting three o's from n parentheses, that is, as many ways as there are combina- §111] PERMUTATIONS AND COMBINATIONS 207 tions of n things taken three at a time, or V9^ Hence, TY^ a^x"-' is another part of the product. In general, we may obtain a partial product like a'x"'' (where r is an integer < n) by taking an a from any r parentheses together with the x's from each of the other (w — r) parentheses. Hence, there are as many partial products Uke a'x"~' as there are ways of selecting r a's from n parentheses; that is, as many ways as there are combinations of n .things taken r at a time, or -r-T — '■ — Ti' Hence, -7-7 — '- — rr a'x"'' stands for any term r] {n — r)l ' r\ (n ^ r)l •' in general in the product (x + o)". Finally, we may obtain one partial product like a" by taking an a from each of the parentheses. Hence, a" is the last term in the product. Thus we have shown that / I \ 1 11 n(n — I ) , , , (x + a)" = X" + nax"-! -\ — ^ a^x""^ + . . . 1-2 + r!(n°-r )l^''^""^+ ' +"^"- ^^^ This is the binomial formula of Isaac Newton. The right-hand side is called the expansion or development of the power of the binomial. It is obvious that the expansion of (x — a)" will differ from the above only in the signs of the alternate terms containing the odd powers of a, which, of course, will have the negative sign. Exercises 1.. Expand {u + Sy)^. Here x = u and a = 3y. By the formula we get u^ + bu^iSy) + I0u\3y)' + lOu^iSyy + 5u{Zyy + {3y)K Performing the indicated operations, we obtain u^ + 15u'y + 90u'y^ + 270u^y^ + i05uy* + 2i3yK Expand each of the following by the binomial formula : 2. (r«- ■ 2y. 3. (3b - -iy. 4. (c + xy. 6. (2a;!! -x)\ 6. (1- ay. 7. (-X + 2ay. 14. (x^ + x^y. 16. (o-»- - lyiy. 16. (\/^ - -yaby. 208 ELEMENTARY MATHEMATICAL ANALYSIS [§112 8. (i + xy. 9. (62 - c^y. 10. (3o + iy. 11. (5d - 3yy. 12. (3a;»_- 1)*. 13. (Vo + .!;)«. 17. (a + [X + 2/1)'. 18. (a + 6 - ?/)'• 19. (a;2 + 2oa; + a^y. 112. Binomial Expansion .for Fractional and Negative Exponents. It is proved in the Calculus that /•. , \ 1 , , n{n — 1) , , n(m — l)(n — 2) , , (1 ± a;)» = 1 ± na; + -^^-^j — -x'' ± — ~ a;' + . . . is true for fractional and for negative values of n, provided x is less than 1 in absolute value. The number of terms in the expan- sion is not finite, but is unlimited. By the above formula, we have V 1 -\- X = i. + (2) X -\ 21 ^ + " 3] x' + . . . = 1 + (i) X - (ij X^ + (Vff) X' - (tI^)^ 1 2 If --1 this becomes V f = 1 + i - ^T + \\rs ~ "JT^" + • ■ Therefore, using five terms of the expression \A|= 2048 ~ 1.2241 approximately. The square root, correct to five figures, is really 1.2247. Thus the error in this case is less than one-tenth of 1 percent if only five terms of the series be used. The degree of accuracy in each case is depend- ent both upon the value of n and upon the value of x. Obviously, for a given value of n, the series converges for small values of x more rapidly than for larger values. As another example, suppose it is required to expand (1 — x)~'. By the binomial theorem (1 - x)-i = 1 + (- 1)( - x) + ~ ^ ^~ ,^ ~ ^\ - xy + -'^-'-l^^-'-'h -^y+... = l+x+x'+x^+. . . §113] PERMUTATIONS AND COMBINATIONS 209 If five terms of the series be used, the error is -^ f or a; «= i, or about 6 percent. 113. Approximation Fonnulas. If x be very small, the expan- sion of (1 + a;)« = 1 + ns + -^-^1 — x^ + ■ ■ is approximately (1 + a;)" - 1 + nx, (1) since x^, x' and all higher powers of x are much smaller than x. Thus, using the symbol ^ to express "approximately equals," we have, for example (1.01)3 = 1.03. For, (1 + 1/100)5 _ 1 +3/100. The true value of (1.01)' is 1.030301, so that the approximation is very good. Likewise (i - x)" ^ I — nx, (2) if X be small. If X, y, and z be small compared with unity, the following ap- proximation formulas hold : (i-+x)(i+y)^ i+x-l-y, (3) f^-i+x-y, (4) (i-|-x)(i-hy)(i + z)=T= i4-x-f-y-hz. (5) The approximation formulas are proved as follows : (1 -|- x) (1 +y) = l+x + y + xy^l+x + y, for a;?/ is small compared to x and y. ,. I V = 1 + X — y + , , = 1 + a; — y, for the fraction is small compared to x and y. 1 + x) (1 + y) {1 + z) ^ {1 +x + y) (1 + z) ^ 1 + X + y + z Exercises 1. Explain the following approximation formulas, in which |x| < 1 14 210 ELEMENTARY MATHEMATICAL ANALYSIS [§113 Vl - X "5= (1 +x)-i^ (1 + x)-i =F (1 +x^)i === 2. Compute the approximate numerical value of the following : (a) (1.03) i (d) (1.05) i (6) (1.02) (1.03) (e) 1.02/1.03 (c) (1.01)(1.02)/(1.03)(1.04). 3. The formula for the period of a simple pendulum is T =WT7i- For the value of gravity at New York, this reduces to T = 6.253' in which I, the length of the pendulum, is measured in inches. This pendulum beats seconds when I = (6.253)=i or 39.10 inches. What is the period of the pendulum if I be lengthened to 39.13 inches? Hint: T = 6.253 ^ - "6:253" - 6:253^^ + ^^^ VT (1 + h/2l). 6.253 Take I = 39.10, and h = 0.03. Then ?" = 1 +■ 0.03/78.20 = 1.00038. A day contains 86,400 seconds. The change of length would, there- fore, cause a loss of 32.8 seconds per day, if the pendulum were attfiched to a clock, §114] PERMUTATIONS AND COMBINATIONS 211 4, On the ocean how far can one see at an elevation of h feet above its surface? Call the radius of the earth o( = 3960 miles), and the distance one can see d, which is along a tangent from the point of observation to the sphere. Since h is in feet, and since a + Toon! d, and a are the sides of a right triangle, we have (o + ^/5280)'' = d' + a\ or "[ ' + sis] ■-''■+«■■ Expanding the binomial by the approximation formula we have .[ '+mk] ='' + < d2 = 2a;i/5280 = 2 X 3960^/5280 -¥, or d = Vp where d is expressed in miles and h in feet. See §68, exercise 13. 5. By what percent is the area of a circle altered if its radius of 100 cm. be changed to 101 cm.? 6. By what percent is the volume of a sphere, |-7ro', altered if the radius be changed from 100 cm. to 101 cm.? 7. If the formula for the horse power of a ship is I.H.P. = „„-, i where S is speed in knots and D is displacements in tons, what in- crease in horse power is required in order to increase the speed from fifteen to sixteen knots, the tonnage remaining constant at 5000? What increase in horse power is required to maintain the same speed if the load or tonnage be increased from 5000 to 5500? 114.* Graphical Representation of the Coefficients of any Power of a Binomial. If we erect ordiaates at equal intervals on the X-axis proportional to the coeflBcients of any power of a binomial, we find that a curve is approximated, which becomes very striking as the exponent is taken larger and larger. In Fig. 97 the ordinates are proportional to the coefficients of the 999th power of {x + a). The drawing is due to Quetelet. 212 ELEMENTARY MATHEMATICAL ANALYSIS [§114 The limit of the broken line at the top of the ordinates in Eig. 97 is, as n is increased indefinitely, a beU-shaped curve, known as Fig. 97. — Graphical representation of the values of the binomial coefficients in the 999th power of a binomial. The middle coeflScients are taken equal to 5, for convenience, and the others are expressed to that scale also. the probability curve. In treatises on the Theory of Probability, it is shown that the equation of the curve is 2/=ae~*^^ CHAPTER VIII PROGRESSIONS 116. An Arithmetical Progression or an Arithmetical Series, is any succession of terms such that each term differs from that immediately preceding by a fixed number called the common difference. The following are arithmetical progressions: (1) 1, 2, 3, 4, 5. (2) 4, 6, 8, 10, 12. (3) 32, 27, 22, 17, 12. (4) 2i, 3i 5, 6i 7i. (5) (u - v), u, {u + v). (6) a, a + d, a + 2d, a -\- 3d, . . . The first and last terms are called the extremes, and the other terms are called the means. Where there are but three numbers in the series, the middle number is called the arithmetical mean of the other two. To find the arithmetical mean of the two numbers a and 5, proceed as follows: Let A stand for the required mean; then, by definition A — a = b — A, whence A - ^ + ^ Thus, the arithmetical mean 6f 12 and 18 is 15, for 12, 15, 18 is an arithmetical progression of common difference 3. By the arithmetical mean, or arithmetical average, of several numbers is meant the result of dividing the sum of the numbers by the number of the numbers. It is, therefore, such a number 213 214 ELEMENTARY MATHEMATICAL ANALYSIS [§116 that if all numbers of the set were equal to the arithmetical mean, the sum of the set would be the same. The general arithmetical progression of n terms is expressed by: Number of term: 12 3 4 . n Progression: a, (a + d), (a + 2d), (a + 3d), . . . (a + [n — 1] d) Here a and d may be any algebraic numbers whatsoever, integral or fractional, rational or irrational, positive or negative, but n must be a positive integer. When the common difference is nega- tive, the progression is said to be a decreasing progression ; other- wise, it is an increasing progression. From the general progression written above, we see that a for- mula for the nth term of any arithmetical progression may be written I = a -H (n - i)d, (1) in which I stands for the nth term. Formula (1) enables us to obtain the value of any one of the num- bers, I, a, n, d, when the other three are given. Thus: (1) Find the 100th term of 3 4- 8 -h 13 -I- . . . Here a = 3, d = 5, n = 100. Therefore ' Z = 3 + 99 X 5 = 498. (2) Find the number of terms in the progression 5 + 7 -I- 9 + . . . + 39. Here a = 5, d = 2,1 = 39. Therefore 39 = 5 -t- (ra - 1)2, or n = 18. (3) Find the common difference in a progression of fifteen terms in which the extremes are f and 425. Here u, = ^,1 = 42^, n = 15, whence 42| = J -F (15 - l)d, or d = 3. 116. The Sum of n Terms. If s stands for the sum of n terms of an arithmetical progression, and if the sum of the terms be §116] PROGRESSIONS 215 written first in natural order, and again in reverse order, we have s = a + (a + d) + (a + 2d) + + (a + [n - 1] d), (1) s = 1+ {I - d) + {I - 2d) + . . + Q -In- l]d). (2) Adding (1) and (2), term by term, noting that the positive and negative common differences nullify one another, we obtain 2s = (a + Z) + (a + Z) + (a + + . ■ ■ + (a + l), (3) or, since the number of terms in the original i5rogression is n, we may write 2s = n{a + I), or s = n(a + l)/2. (4) If the value for I, from (1) §115, be substituted in formula (4) it becomes s = n [2a + (n - i)d]. (5) In equation (4), (a + Z)/2 is the average of the first and nth terms. The formula (4) states, therefore, that the sum equals the number of the terms multiplied by the average of the first and last. An arithmetical progression is a very simple particular instance of a much more general class of expressions known in mathematics as series. A series is any sequence of terms formed accord- ing to some law, such as: (x + 1) + (x + 2y+ {x + sy +. . . x + 3x^ + 5x^+ . . . cos X + cos 2x + cos 3x -\- . . . It is only in a very limited number of cases that a short expression can be found for the sum of n terms of a series. An arithmetical progression is one of these cases. Formula (4) enables us to find the value of any one of the numbers s, n, a, I, when the values of the other three are given. Thus: (1) Find the number of terms in an arithmetical progression in which the first term is 4, the last term 22, and the sum 91. Here a = 4, Z = 22, s = 91, whence, 91 = ra(4 + 22) /2, or n = 7. 216 ELEMENTARY MATHEMATICAL ANALYSIS [§116 The two formulas, (1) §116 and (4) §118, contain five letters; ' hence, if any two of them stand for unknown numbers, and the values of the others are given, the values of the two unknown numbers can be found by the solution of a system of two equations. Thus : (2) Find the number of terms in a progression whose sum is 1095, if the first term is 38 and the difference is 5. Here ■ s = 1095, a = 38, and d = 5, whence, I = 38 + {n - 1)5, (6) 1096 = n(38 + l)/2. (7) From (6) / = 33 + 5n. (8) From (7) 2190 = 38ra + nl. (9) Substituting the value of / from (8) in (9), we get 2190 = 71n + 5nK (10) Solving this quadratic equation, we find n = 15, or - 29.2. The second result is inadmissible, since the number of terms cannot be either negative or fractional. Exercises Solve each of the following: 1. Given, o = 7, d = 4, n = 15; find 2 and s. 2. Given, a = 17,1 = 350, d = 9; find n and s. 3. Given, a = 3, n = 50, s = 3825; find I and d. 4. Given, s = 4784, a = 41, d = 2; find Zand n. 5. Given, s = 1008, d = 4, Z = 88; find a and n. 6. Find the sum of the first n even numbers. 7. Find the sum of the first n odd nvmibers. 8. Insert nine arithmetical means between —7/8 and + 7/8. 9. Sum (o + 6)2 + [a" + ¥) + (,a -byton terms. 10. Find the sum of the first fifty multiples of 7. 11. Find the amount of $1.00 at simple interest at 5 percent for 1920 years. 12. How long must $1.00 accumulate at 3| percent simple interest until the total amounts to $100? 13. How many terms of the progression 9 + 13 + 17 + . . must be taken in order that the sum may equal 624? How many terms must be taken in order that the sum may exceed 750? §117] PROGRESSIONS 217 14. Show that the only right triangles whose sides are in arithmetical progression are those whose sides are proportional to 3, 4, and 6. 117. A geometrical progression or a geometrical series is any succession of terms such that each term is the product of the preceding term by a fixed factor called the ratio. The following are examples: (1) 3, 6, 12, 24, 48. (3) 1/2, 1/4, 1/8, 1/16, 1/32. (2) 100, -50, 25, -12i (4) a, ar, ar\ ar\ ar* . . The geometrical mean G of two numbers, a and 6, is a number- such that a, G, 6 is a geometrical progression. By definition G/a = b/G, whence, G^ = ab, or G = Vab. Thus, 4 is the geometrical mean of 2 and 8. The arithmetical mean of 2 and 8 is 5. The geometrical mean of n positive num- bers is the value of the nth root of their product. Thus the geo- metrical mean of 8, 9, and 24 is -?/ 8 X 9 X 24 = 12. 118. The nth Term and the Sum of n Terms. If a represents the first term and r the ratio of any geometrical progression, the progression may be written: Number of term: 123 4 .. n— 1 n. Progression: o, ar, ar^, ar', . . ar"'^, ar"~^. Therefore, representing the nth term by I, we obtain the simple formula 1 = ar»-i. (1) Representing by s the sum of n terms of any geometrical pro- gression, we have s = a -\- ar + ar^ + . . . + ar" ~^ + ar" ~ ^, or, s = ail+r + r^+ . . + r"-^ + r"-^). But, by a fundamental theorem in factoring, ^ the expression in the 1 See Appendix, Chapter XV. 218 ELEMENTARY MATHEMATICAL ANALYSIS [§118 (2) parenthesis is the quotient of 1 — r» by 1 — r. Hence, a(i — r») Another form is obtained by introducing I by the substitution or»-' = I, a — rl which gives s = — — — (3) Formula (1), or (2), enables one to find any one of the four numbers involved in the equations when three are given. The two formulas (1) and (2) considered as simultaneous equations enable one to find any two of the five numbers a, r, n, I, s, when the other three are given. But if r be one of the unknown numbers, the equations of the system may be of a high degree and beyond the range of Chapter VI unless solved by graphical means. If n be an unknown number, an equation of a new type is introduced, namely, one with the unknown number appearing as an exponent. Equations of this type, known as exponential equations, will be treated in the chapter on logarithms. The following examples illustrate cases in which the resulting single and simultaneous equations are readily solved. (IJ Insert three geometrical means between 31 and 496. Here a = 3l,l = 496, and n = 6. Hence, 496 = 31 X r' r* = 16, or r = ± 2. Consequently the required means are either 62, 124, and 248, or — 62, + 124, and - 248. (2) Find the sum of a geometrical progression of five terms, the extremes being 8 and 10,368. Here a = 8,1 = 10,368, and n = 5. Hence, 10,368 = 8r* (1) §118] PROGRESSIONS 21,9 aad 8 = (10,368r - 8)/{r - 1). (2) From the first, r = 6 whence, from the second, s = 12,440. (3) Find the extremes of a geometrical progression whose sum is 635, if the ratio be 2 and the number of terms be 7. Here s = 635, r- = 2, and n = 7. Hence, I = a2«, (3) 635 = (2/ - a). (4) Substituting I from (3) in (4), we get 635 = 128 o - a. Hence, a = 5, and I = 320. (4) The fourth term of a geometrical progression is 4, and the sixth term is 1. What is the tenth term? Here ar^ = 4, (5) and ar^ = 1. (6) Dividing (6) by (5) we obtain r 2 _ 1 i, or r = + i Therefore, from (5), a = 4^/r^ = +32. Then the tenth term is ± 32(+ \y = tV. Exercises 1. Find the sum of seven terms of 4 + 8 + 16 + . . . 2. Find the sum of - 4 + 8 - 16 + . . . to six terms. 3. Find the tenth term and the sum of ten terms of 4 — 2 + 1 ■ 4. Find r and s; given a = 2,1 = 31,250, Ji = 7. 220 ELEMENTARY MATHEMATICAL ANALYSIS [§119 6. Insert two geometrical means between 47 and 1269. 6. Insert three geometrical means between 2 and 3. 7. Insert seven geometrical means between o' and 6*. 8. Show that the quotient (o" — 6»)/(o — 6) is a geometrical progression. 9. Sum x"""^ + x"~' y + a;""' y' + . . to n terms. 10. Sum a;"~i — a^~' y + s""' y' — ■ ■ . to w terms. 11. Sum a + ar~^ + ar~' + . . . to n terms. 12. If a, b, c, d, ' . . . are in geometrical progression, show that a'^ + 6', 6* + c^, c^ + (i^ . . are also in geometrical progression. 13. If any numbers are in geometrical progression, show that their differences are also in geometrical progression. 14. A man agreed to pay for the shoeing of his horse as follows: 1 cent for the first naU, 2 cents for the second nail, 4 cents for the third nail, and so on until the eight naUs in each shoe were paid for. What did the last nail cost?. How much did he agree to pay in all? 119. Compound Interest. Just as the amount of principle and interest of a sum of money at simple interest for n years is ex- pressed by the (n + l)st term of an arithmetical progression, so, in a similiar way, the amount of any sum at compound interest for n years is represented by the (n + l)st term of a geometrical pro- gression. Thus, the amount of $1.00 at compound interest at 4 percent for twenty years is given by the expression 1(1.04)2". The amount of p dollars for n years at r percent is K' + i5-o)"- The present value of $1.00, due twenty years hence, estimating compound interest at 4 percent, is 1/(1.04)2". The value of $1.00, paid annually at the beginning of each year into a fund accumulating at 4 percent compound interest, is, at the end of twenty years (1.04)1 + (104)'' + . . . (1.04)2", which is the sum of the terms of a geometrical progression of twenty terms. §120] PROGRESSIONS 221 Problems of this character in compound interest, in compound discount, and in the more complicated problems that proceed therefrom, are basal to the theory of annuities, life insurance, and depreciation of machinery and structures. The computation of the high powers involved necessitates the postponement of such problems until the subject of logarithms has been explained. 120. Infinite Geometrical Progressions. If the ratio of a geometrical progression be a proper fraction, the progression is said to be a decreasing progression. Thus, 1 1 i i 1 cnA ill 1 ■I) 2) i! 8) iw> ana 3, s, jt, ^t are decreasing progressions. If we increase the number of terms in the first of these progressions the sums will always be less than 2; but the difference (2 — s) will become and remain less than any pre-assigned number. Definition: A constant, a, is called the limit of a variable, t, if, as t runs through a sequence of numbers, the difference (a — t) becomes, and remains, numerically smaller than any pre-assigned number. By definition, 2 is, therefore, the limit of the first of the above progressions. The sum of n terms of this particular progression should be written down by the student for a number of successive values for n, thus: Number of terms: 1, 2, 3, 4, 5, ... 10, Sum: 1, 1 + i 1 + f , 1 + I, 1 + li . . 1+Ui, The nth term differs from 2 by only l/2»-i. It is easy to show that the sum of every decreasing geometrical progression approaches a fixed limit as the number of terms becomes infinite. Write the formula'^ in the form If we suppose that r is a proper fraction and that n increases with- s = a — or» 1 - - r ar' 222 ELEMENTARY MATHEMATICAL ANALYSIS [§121 out limit, then r» can be made less than any assigned number; for, the value of any power of a proper fraction decreases as the ex- ponent of the power increases. As the other parts of the second fraction in (1) do not change in value as n changes, the fraction as a whole can be made smaller than any number that can be assigned. Hence, we write limit n— 00 b]'Th <^' The left-hand side is read: "The limit of s as n becomes infinite." The symbol = means: "approaches" or "becomes." Exercises As n = 00 , find the limit of each of the following : 1. * - i + 1 - tV + • • Here a = -^jT = — -3, 1 whence, limit s = — j-r = f . 1 ~ ( "2) 2. 0.3333 . . Here a = -,%, r = ,Vi 3 whence, limit s = — = \ 3. 9-6+4- ' ^.\-\+-h---- 4. 0.272727 ... 7. 4 4- 0.8 + 0.16 + . . . 5. 0.279279279 . . . 8. Express the number 8 as the sum of an infinite geometrical progression whose second term is 2. 121.* Graphical Representation. Note that all the essentials of a geometrical progression may be studied if we assume the first term to be unity, for the number a occurs only as a single constant multiplier in each term, and also occurs in the same manner in the formulas for I and s. To represent the geometrical series 1 + r + r'' . . + r"-' graphically, lay off OM = 1 on OY, OSi = 1 on OX, SiPi = r on the unit line, and draw MP^. Draw the arc P11S2 and erect P2S2. Draw the arc P2'S2 and erect PzSs. Continue this con- struction until the perpendicular P„iS„ is erected. The series of trapezoids OMPiSi, S1P1P2S2, SJ'^iPiSi, .. . , S,_iP„_iP„S„ §121] PROGRESSIONS 223 are similar and, since PiSi = r X OM, it follows that P2S2 = rPiSiyPiSa = rP^Si, . , P„S„ = rF„_iS„_i. Hence we have: OM = OSi =1 PkSi = S1S2 = r .'. 0 1. Fig. 98 shows the series whose ratio is r- = 1.2. Fig. 99 shows the series whose ratio is 0.8. Y U M P, P2 Pa 1 ^ \ ,.X r^ f* Pi ' 1 ^ n NJTs —- — - — .___^ O Si S, Sa Si So L Fig. 99. — Graphical construction of the sum of a G. P. r < 1. The line MPi has the slope (r - 1) in Fig. 98 and the slope — (1 — r) in Fig. 99. In each case the F-intercept is 1. Its 1-2/ equation is, in both cases, y = (r In both figures, when y = P^S, l)x + 1, ora; = 1 -r r", X = OSn. Substituting these values for x and y, we get for the sum of n terms, 1 — r" Fig. 98 shows that when the number of terms is 1 -r 224 ELEMENTARY MATHEMATICAL ANALYSIS [§122 allowed to increase without limit, the, sum OSn also increases with- out limit. Fig. 99 shows that when the number of terms is made to increase without limit, the sum 0S„ approaches OL as a limit. Now the value of OL is the value of x when ^ = 0. Hence the limit of the sum of the progression, or OL, is -t-^ — Consult also §9, problem 6, exercise 3 and Figs. 15, 16. 122.* Harmonical Progressions. A series of terms such that their reciprocals form an arithmetical progression are said to form an harmonical progression. The following are examples: C1^ 1 1 1 1 ('■) 2: 3> 4! T- (2) 1, T, T) TT- (3) l/{x-y),l/x, l/(x + tj). (4) i 1, - 1, - i (5) 4, 6, 12. (6) 1/a, l/(a + d), 1/ia + 2d), . . Although harmonical progressions are of such a simple character, no simple expression has been found for the sum of n terms. Our knowledge of arithmetical progressions enables us to find the value of any required term and to insert any required number of harmonical means between two given extremes, as in the examples below. (1) Write six terms of the harmonical progression 6, 3, 2. We must write six terms of the arithmetical progression, ^, ^, ^. The common difference of the latter is ^, so that the arithmetical pro- gression is §, §, §, f , ^, 1, and the harmonical progression is 6, 3, 2, 1.5, 1.2, 1. (2) Insert two harmonical means between 4 and 2. We must insert two arithmetical means between ^ and -^; these are ^ and -1%, whence the required harmonical means are 3 and 2.4. 123.* Harmonical Mean. The harmonical mean of two numbers is found as follows: Let the two numbers be a and 6 and let H stand for the required mean. Then we have 1/H - 1/a = 1/6 - 1/H. That is, 2/H = 1/a + 1/6 = (a -I- 6) /ab. Hence, • H = 2ab/(a -1- b). (1) §124] PROGRESSIONS 225 Thus the harmonical mean of 4 and 12 is 96/(4 + 12) = 6. By the harmonical mean of several numbers is meant the reciprocal of the arithmetical mean of their reciprocals. Thus the har- monical mean of 12, 8, and 48 is 13i-t- 124. * Relation between A, G, and H. As previously found, A= {a+ 6)/2, G= V^,H = 2ah/{a + b). Hence, AH = ab, and, since ab = C, AH = G\ or G = VaH. (1) Exercises 1. Continue the harmonical progression 12, 6, 4. 2. Find the difference (1.8 + 1.2 4- 0.8 + . to 8 terms) - (1.8 + 1.2 + 0.6 + . . . to 8 terms). 3. If the arithmetical mean between two numbers be 1, show that the harmonical mean is the square of the geometrical mean. Questions and Exercises for Review of Chapters I to VIII 1. Define scale; uniform scale; non-uniform scale; arithmetical scale; algebraic scale; double scale. 2. Define constant; variable. 3. Define function; increasing function; decreasing function; even function; odd function. 4. Give illustrations of even functions; of odd functions. 6. Express the area, A, of an equilateral triangle as a function of the length, X, of its sides. 6. Express the volume, V, of a right circular cone as a function of its altitude h. The radius of the base is 10 inches. 7. A strip of tin L feet long and 40 inches wide is made into a gutter with rectangular cross section, by bending up an equal portion of each side. Express the cross section, y, of the gutter as a function of the breadth, x, of the amount of tin turned up. Show that the maximum cross section is 200 square inches. 8. A strip of tin 24 inches square has an equal square cut from each corner. The rectangular projections are then turned up to form a tray 15 226 ELEMENTARY MATHEMATICAL ANALYSIS [§124 with square base and rectangular sides. If x is the side of the square cut out show that 4x(12 — x)* is the function representing the volume of the tray. 9. In a triangle whose sides are 6, 8, and 10 feet is inscribed a rec- tangle the base of which lies in the longest side of the triangle. Ex- press the area, A, of the rectangle as a function of its altitude, h. 10. A ladder 20 feet long leans against the vertical wall of a house. Express the area, A, of the triangle formed by the ladder, the wall, and the horizontal ground, as a function of the distance, x, of the foot of the ladder from the wall. 11. Find graphically the values of the following: (a) (31.6) (7.21); (6) f^; (c) (1.36)'; (d) ~-y 12. Describe the method of representing the position of points on a plane by the rectangular, or Cartesian, system of coordinates. Define axes; origin; abscissa; ordinate; quadrant. How are the quadrants numbered? 13. What is meant by the graph, locus, or curve, of an equation? 14. What is meant by the equation of a curve, graph, or locus of a point. 15. Which of the following points are on the curve Zx -\-2y = 4: (a) (2, -1); (6) (3, 1); (c) (-4, 8); (d) (0, 0). 16. Find the distance of each of the following points from the origin : (a) (1, 3); (6) (-2, 3); (c) (2, -3); (d) (-3, -2). 17. Show that, for all values oi m, y = mx is a straight line passing through the origin. 18. Show that the equation of any straight line passing through the origin is of the form y = mx. 19. Find the equation of a straight line passing through the origin and the point (—3, 5). 20. Show that, for all values of m and b,y = mx + 6 is the equation of a straight line. 21. Show that the equation of any straight line is of the form y = mx + b. 22. Find the equation of a straight line passing through the points (1, 3) and (-2, 5). 23. Define slope of a straight line. 24. Define K-intercept, and X-intercept, of a straight line. 25. Find the slope, s-intercept, and 2/-intercept, for the following: (a) 3x +2y = 6; (6) x - 2y = 5; (c) 2y - 3x = 7. 26. Define X-, and K-intercepts of a curve. §124] PROGRESSIONS 227 27. Write the equations of a line if: (a) F-intercept is 3 and slope is 2, (6) y-intereept is 1 and slope is —2, (c) y-intercept is —2, and slope is 5, id) X-interoept is 3 and slope is 2, (e) X-intercept is —2 and slope is 3, (J) X-intercept is J and slope is — ^, (g) X-intercept is 2 and i/-intercept is 3. 28. What is meant by curve of the parabolic type? 29. What is meant by curves of the hyperbolic type? 30. What is the parabola? 31. What is the equilateral, or rectangular, hyperbola? 32. What is the cubical parabola? 33. What is the semi-cubical parabola? 34. When is a curve symmetrical with respect to the X-axis; with respect to the y-axis; with respect to the line x = y; with respect to the line y = — x; with respect to the origin? Give equation of two curves for each of the cases considered above. 35. Sketch, y = x^; y = \x^; y = 2x\ 36. Sketch y'' = x; y^ = Jx; y' = 2x. 37. Sketch y = x'; y = - x'. 1 2 1 38. Sketch y = -; V = - ^: y = 2i' 39. Sketch x'^ = y'; x' = y'^. 40. Sketch y = x^; y = — x'. 41. Define rational equation, empirical equation. 42. Write the equation of the curve y = x^ — 3x, after it is translated (a) two units to the right; (6) three units to the left; (c) one unit up; (d) five units down; (e) one unit to the left and two units down. 43. Find the coordinates of the vertex of : (o) y = x'' + 2x; (6) y = x' - 2x + 3; (c) y = 3x^ + 6x; (d) y = 6x - 3x^ + 2. 2; _]_ 3 44. Show that y = — 3^ is an hyperbola. Write the equation of its asymptotes. 45. What is meant ty shearing notion? 46. Show that shearing the curve y = ax' in the line y = mx, is equivalent to translating the original curve. Find the coordinates of the vertex of the translated curve. 47. What is meant by the roots of a function? 228 ELEMENTARY MATHEMATICAL ANALYSIS [§124 48. Find the roots of: (a) *2 + 2x - 3; (6) x^ - 3x; (c) 3x^ + 2x - 6. 49. Write the equation of the curve j/^ = x' — x' when reflected in : (a) the X-axis; (6) the F-axis; (c) the line x = y; (d) the line X = — y. 50. The roots of a function correspond to what points on the graph of the function? 52. Write the equation of a circle, radius o, center at the origin; center at the point (h, k). 53. Show that x' + y' + 2gx + 2/v + d = is a circle. 54. Find the coordinates of the center and the length of the radius of: (a) x' + y' - 2x - ^y + 1 = 0; {d) 2x' + 2y^ + 3x + by = 0; (6) x2 + ^2 + 2x + 42/ + 1 =0; (e) Sx^ + S?/^ - 6x - V2y = 10; (c) x2 + 2/2 + 3x - 42/ = 0; (/) x^ + 2/' + 7x - VZy = 25. 66. Which circles of exercise 54 pass through the origin? 66. Write the equation of the circle if i (o) the radius is 5 and the center is at (1, 2) ; ' (6) the radius is 6 and the center is at ( — J^, 2) ; (c) the radius is 10 and the center is at ( — 2, — 3) ; (d) it passes through the origin and the center is at (1, 1) ; (e) it passes through the origin and the center is at ( — 2, 3); (/) it passes through (1, 2) and the center is at (—2, 3). 57. Write the equation of a line passing through the origin and the center of the circle ' x2 + 2/2 - 2x + 32/ = 5. 58. Write the equation of a circle passing through the point (2, 3) and through the center of the circle x2 + 2/^ - 3x - 22/ = 0. 69. Show that if two straight lines are mutually perpendicular, the slope of one is the negative reciprocal of the slope of the other. 60. Show that {x - a)^ + y^ = a^ and (x - ZaY + y^ = a' are tangent to each other. 61. Find analytically the coordinates of the points of intersection of x2 + 2/2 — 4x — 92/ = 9 and y — ^ x + 1. 62. Find approximate solutions for exercise 61 by drawing the curves on squared paper. 63. Solve graphically the simultaneous equations x^ + y' - 2x ~ 4:y = 4: -£2 + 2/2 + 4x - 42/ = 0. 64. Define degree'; radian. 66. Define the six circular functions. §124] PROGRESSIONS 229 66. Express the following as radians: (o) 45°; (6) 90°; (c) 180°; (d) 135°; (e) 225°; (f) 60°; (g) 30°; (h) 300°; (i) 270°; U) 315°; (ft) 120°; (l) 160°; (ra) 216°; (n) 310°. 67. Express the following radians as degrees: (a) i^; (fe)i^; (c) |^; (rf) |^; (e) |^; (/) 3; (?) 2. 68. How many revolutions per minute are 10 radians per second? 5 7r radians per second? fir radians per second? 69. A car is running at the rate of 30 miles per hour. Its 36-inch tire is revolving at the rate of how many radians per second? 70. A shaft rotates at the rate of 15,000 revolutions per minute. What is its angular velocity in radians per second? 71. Give the values of the circular functions of: (a) 30°; (6) 60°; (c) 45°. 72. Give the algebraic signs of the functions of an angle in the first quadrant; in the second quadrant ; in the third quadrant; in the fourth quadrant. 73. Give the functions of the following angles: (o) 120°; (6) 135°; (c) 150°; (d) 210°; (e) 225°; (J) 240°; {g) 300°; (h) 316°; W 330°; (j) 0°; (k) 90°; (I) 180°; (m) 270°; (n) 360°. 74. Find the functions of a if : (a) sin a = f and cos a is negative; (6) sin a = f and cos a is positive; (c) sin a = -f and tan a is positive; (d) sin a = f and tan a is negative; (e) tan a = 2 and cos a is negative; (/) tan a = — 3 and sin a is positive; (g) sec or = 5 and tan a is negative. 75. Which of the circular functions are even functions? Which are odd functions? 76. Show that cos a = sin (^ — a) . 77. Draw the graph oi y = sin x. 78. Show that the curve for y = cos x may be obtained from the curve for y = sinx by translating it ■ir/2 units to the left. 79. Show that sin^ a + cos^ a = 1. 80. Show that sec^ a = 1 + tan" a. 81. Show that esc" a = 1 + cot" a. 82. Show that tan a = cos a nn on ii i J cos a 83. Show that cot a = -^ • sm a 230 ELEMENTARY MATHEMATICAL ANALYSIS [§124 84. Express sin a; in terms of: (a) cos x; (6) tan x; (c) cot x; (d) sec x; (e) esc x. 86. Express cos x in terms of: (a) sin x; (6) tan x; (c) cot x; {d) sec x; (e) esc x. 86. Express tan x in terms of: (a) sin x; (6) cos x; (c) cot a:; (d) sec x; (?) esc a;. 87. The longer leg of a plot of land in the form of a 60° right triangle is 80 rods. Find the area of the plot in acres. 88. A plot of land in the form of a 60° right triangle contains 72 acres. Find the length in rods of each side of the triangle. Hint : Let x represent the number of rods in the length of the shorter leg. 89. The shorter side of a rectangle is 100 feet, the diagonal is 200 feet. Find the length of the longer side. 90. Explain how points may be located in a plane by means of polar coordinates. 91. Define pole; polar axis; radius vector; vectorial angle. 92. Draw curves for: (a) p = 1; (6) p = 2; (c) p = 3; (d) p = 5; (e) e = 0; (/) e =,r/4; (g) e = 7r/3; {h) e = T-/2; 6 = 2. Hint: fl is measured in radians. 93. What curve is represented by p = a cos 9? Prove. 94. What curve is represented by p = 6 sin 9? Prove. 95. Draw on a sheet of polar coordinate paper curve for the following : (a) p = 2 cos e; (6) p = — 2 cos 9; (c) p = 2 sin 9; (d) p = — 2 sin 9. 96. Prove that p = a cos 9 + 6 sin 9 is a circle. 97. Draw curves for the following : (a) p = 2 cos 9 + 3 sin 9; (6) p = 3 cos 9 — 2 sin 9; (c) p = — 2 cos 9 + sin 9; (d) p = — 3 cos 9 — 3 sin 9. 98. Draw the circles p = 1 and p = cos 9 and from them plot the graph for p = 1 + cos 9. 99. Plot curve for the following equations : (a) p = 1 + sin 9; (6) p = 1 — sin 9; (c) p = 2 + cos 9; (d) p = 1 - 2 cos 9. §124] PROGRESSIONS 231 100. From a sheet of polar coordinate paper, form M3, find values for the following : (a) sin 30°; (6) cos 30°; (c) sin 45^; (d) cos 45°; (e) tan 45°; (/) cos 10°; (g) sin 116°; (h) cos 216°; (i) sin 127°; (j) tan 37°; (fc) sin 227°; {I) cos 316°. 101. Show that when the curve for p = f{8) is rotated about the pole through an angle a, its equation becomes p = f{e — a). 102. State fourteen "Theorems on Loci." 103. Find the polar equation of a straight line. 104. The center of the circle p = 10 sin (9 — a) lies on the line X — y = 3. Find a. 105. The center of the circle p = 10 cos (0 — a) lies on the line 3x - 2y = 1. Find a. 106. The center of the circle p = 5 sin {$ + a) lies on the line X - 2?/ = 6. Find a. 107. Write the Cartesian equations for: (a) p = 2 cos e + 3 sin 6; (6) p = 3 cos 9 — 2 sin $; (c) p = 2 sin 9 — 3 cos $. 108. Solve analytically 2 = 2 cos 9 — 3 sin 9 for all values of 8 between 0° and 360°. 109. Solve graphically the equation given in exercise 108. 110. Sketch a curve for y = - — 2x. " X 111. Sketch a curve for y = ^ -j- sin x. 112. A circle is inscribed in a 30°, 60° right triangle. Find the diameter of the circle if the shorter leg of the triangle is 4 inches; if the longer leg of the triangle is 6 inches; if the hypotenuse of the triangle is 10 inches. Find the lengths of the three sides of the tri- angle if the radius of the inscribed circle is 6 inches. 113. A circle is inscribed in a 45° right triangle. Find the diameter of the circle if the legs of the triangle are each 4 inches in length. 114. A circle is circumscribed about a 30°, 60° right triangle. Find the radius of the circle if the hypotenuse of the triangle is 10 inches. 116. Write the polar equation for x^ - y^ = a2(x2 + 2/2)2. 116. Define an ellipse; major axis; minor axis. 117. Give parametric equations for an ellipse. 232 ELEMENTARY MATHEMATICAL ANALYSIS [§124 118. Find the coordinates of the center and the lengths of the semi- axes of the ellipse X = 3 + 2 cos a y = 2 — svD. a. 119. Show that every section of a right circular cylinder by a plane is an ellipse. 120. Show that the projection of a circle upon a plane is an ellipse. 121. Define an equilateral, or rectangular,\ hyperbola. Define an hyperbola. 122. Give parametric equations for an hyperbola. 123. Define the axes, the center, and the asymptotes of an hyperbola. 124. Find the coordinates of the center, the lengths of the semi- axes, and the equations of the asymptotes for x' - y^ + 2x - ^y = 11. 126. Find the equation for the curve of sy = 4 when rotated about the origin through an angle of —45° 126. Define conjugate hyperbolas. 127. Write the equation of the hyperbola conjugate to x'i — y'' — X -\- iy = 11. 128. Sketch the curve with asymptotes for X = 3 + 2 sec a 2/ = 1 — 3 tan a. 129. Write the equation of the curve formed when the circle x2 -|- j/2 = o' is sheared in the Une y = x. Sketch the curve. 130. Write the equation of the curve formed when the hyperbola x^ V^ -J — ^ = 1 is sheared in the line y = x. Sketch the curve. 131. State and prove the remainder theorem. 132. State and prove the factor theorem. 133. Without performing the division, find the rehiainder of (s' - 2x2 + 3 - 1) -H (a; + 2). 134. Explain what is meant by questionable and legitimate transformations. 136. Explain a method of finding approximately the roots of a cubic equation. 136. Find the equation of the straight line passing through the points of intersection oi x^ + y' + 2x + 4y — 11 — and X- + y^ - 2x - 2y = 0. §124] PROGRESSIONS 233 137. What are the equations of the coordinate axes? 138. What is the locus of x^ = 4? of y' = 4? of a^ = 2/«? of o2a;2 = 62!/2? 140. Solve { „ , .", „ . 141. Define series. 142. Define arithmetical progression; geometrical progression; har- monical progression. 143. Define arithmetical mean; geometrical mean. 144. Derive formulas for I and s of an arithmetical progression. 146. Derive formulas for I and s of a geometrical progression. 146. Define an infinite geometrical progression. • 147. Derive the formula for the sum of an infinite geometrical progression. 148. Find the value of 0.273273273 . . 149. A debt of $10,000 is to be paid in ten years. An equal amount is paid at the end of each year. Find this amount if the indebtedness draws interest at 5 percent. 150. An equal amount of money is deposited at the end of each year for twenty years as a sinking fund to replace a piece of machinery valued at $10,000. How much must be deposited at the end of each year, if the deposits draw 4 percent compound interest. CHAPTER IX THE LOGARITHMIC AND THE EXPONENTIAL FUNCTIONS 125. Historical Development. The almost miraculous power of modern calculation is due, in large part, to the invention of logarithms in the first quarter of the seventeenth century by a Scotchman, John Napier, Baron of Merchiston. This invention was founded on a very simple and obvious principle, that had been quite overlooked by mathematicians for many genera- tions. Napier'sinventionmay be explained as follows:^ Let there be an arithmetical and a geometrical progression which are to be associated together, as, for example, the following : 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024 Now the product of any two numbers of the second line may be found by adding the two numbers of the first progression above them, finding this sum in the first Une, and finally taking the num- ber lying under it ; this^Iatter number is the product sought. Thus, suppose the product of 8 by 32 is desired. Over these numbers of the second line stand the numbers 3 and 5, whose sum is 8. Under 8 is found 256, the product desired. Now since but a limited variety of numbers is offered in this table, it would be useless in the actual practice of multiplication, for the reason that the particular numbers whose product is desired would probably not be found in the second line. The overcoming of this obvious obstacle constitutes the novelty of Napier's inven- tion. Napier proposed to insert any number of intermediate terms in each progression. Thus, instead of the portion 0, 1, 2, 3, 4 1, 2, 4, 8, 16 1 Merely the fundamental principles of the invention, not historical details, are given in what follows. For a very brief course in logarithms, only §§131-144 need be taken. 234 §126] LOGARITHMIC AND EXPONENTIAL FUNCTIONS 236 of the two series we may wi-ite 0, I 1, U, 2, 2i 3, 3), 4 1, \/2, 2, Vs, 4, a/32, 8, ^128, 16 by inserting arithmetical means between the consecutive terms of the arithmetical series and by inserting geometrical means between the terms of the geometrical series. Let these be computed to any desired degree of accuracy, say to two decimal places. Then we have the series A. P. G.P. 0.0 1.00 0.5 1.41 1.0 2.00 1.5 2.83 2.0 4.00 2.5 5.66 3.0 8.00 Again inserting arithmetical and geometrical means between the terms of the respective series we have: A. P. G.P. 0.00 1.00 0.25 1.19 0.50 1.41 0.75 1.69 1.00 2.00 1.25 2.38 1.50 2.83 1.75 3.36 2.00 4.00 2.25 4.76 By continuing this process each consecutive three figure number may finally be made to appear in the second column, so that, to this degree of accuracy, the product of any two such numbers may be found by the process previously explained. The decimal points of the factors may be ignored in this work, as for example, the product of 2.38 X 14.1 is the same as that of 238 X 14.1 236 ELEMENTARY MATHEMATICAL ANALYSIS [§126 . except in the position of the decimal point. The correct position of the decimal point can be determined by inanection after the significant figures of the product have been obtained. Using the above table we find 2.38 X 14.1 = 33.6. The above table, when properly extended, is a table of loga- rithms. As geometrical and arithmetical progressions different from those given above might havo been used, the number of possible systems of logarithms is indefinitely great. The first column of figures contains the logarithms of the numbers that stand opposite them in the second column. Napier, by this process, said he divided the ratio of 1.00 to 2.00 into "100 equal ratios," by which he referred to the insertion of 100 geometrical means between 1.00 and 2.00. The "number of the ratio" he called the logarithm of the number, for example, 0.75 opposite 1.69, is the logarithm of 1.69. The word logarithm is from two Greek words meaning " The number of the ratios." In order to produce a table of logarithms it was merely necessary to compute numerous geometrical means; that is, no operations except multi- plication and the extraction of square roots were required. But the numerical work was carried out by Napier to so many decimal places that the computation was exceedingly difficult. The news of the remarkable invention of logarithms induced Henry Briggs, professor at Gresham College, London, to visit Napier in 1615. It was on this visit that Briggs suggested the ad- vantages of a system of logarithms ia which the logarithm of 10 should be 1, for then it would only be necessary to insert a sufficient number of geometrical means between 1 and 10 to get the logarithm of any desired number. With the encourage- ment of Napier, Briggs undertook the computation, and in 1617, published the logarithms of numbers from 1 to 1000 and, in 1624, the logarithms'of numbers from 1 to 20,000, and from 90,000 to 100,000 to fourteen decimal places. The gap between 20,000 and 90,000 was filled by a Hollander, Adrian Vlacq, whose table, published in 1628, is the source from which nearly all the tables since published have been derived. 126. Graphical Computation of the Terms of a Geometrical Progression. Draw the lines y = x and y = rx, Fig. 100. From the point (1, r) on ?/ = rx draw a horizontal line io y = x, thence §127] LOGARITHMIC AND EXPONENTIAL FUNCTIONS 237 a vertical line toy = rx, etc., thereby forming the "stairway" of line segments between y = x and y = rx a,s shown in Fig. 100. Then the points, N, P, Q, etc., have the ordinates r, r'\ r^, etc., as required, for, to obtain the ordinate of P, or PD, the value of x used was OD = r, hence P is the point on y = rx for a; = r, or y PD Likewise Q is by construction the point on y rx for X = r^, hence the y of the point Q = r X r' = r^, etc. 8 T 6 Y U / h / 1 4 q/ :/i 7 3 ./ '// /• 2 N / ^ / >r' M / >■>■- ---' ■ r r— " 4— ,- ]rr- r^ J c a-2-1012345 Fig. 100.^ — Graphical construction of the successive terms of a G. P. In the diagram r =3/2, and the curve isy = (3/2)*. / The points P, Q, etc., are now carried horizontally to points whose abscissas are, respectively, 2, 3, 4, etc., thus giving points on the curve for y = r*. 127. Graphical Computation of Logarithms. In Fig. 100 the termis of a geometrical progression of first term 1 and ratio IN = r are represented as ordinates arranged at equal intervals along OX. Fig. 100 is drawn to scale for the value of r = 1.5. Fig. 101 is a similar figure drawn for r = 2, in which a process is used for locating intermediate points of the curve, so that the locus may 238 ELEMENTARY MATHEMATICAL ANALYSIS [§127 be sketched with greater accuracy. The lines y =• x and y = rx (in this case y = 2x) are drawn, and the "stairway" constructed as before (See §126). Vertical lines drawn through a; = —2, —1, 0, 1, 2, 3, . . . and horizontal lines drawn through the hori- zontal tread of each step of the stairWay divides the plane into a large number of rectangles. Starting at M and sketching the diagonals of successive cornering rectangles the smooth curve -. -1 1 2 3 - . Fig. 101. — Graphical construction of the curve y = 2". MNP is obtained. Intermediate points of the curve are located by doubling the number of vertical lines by bisecting the distances between each original pair, and then by increasing the number of horizontal lines in the following manner: Draw the line y = s/r x (in the case of the Fig. 101, y = V'2 x). At the points where this line cuts the vertical risers of each step of the "stairway" (some of these points are marked .A, B, C in the diagram) draw a §127] LOGARITHMIC AND EXPONENTIAL FUNCTIONS 239 new set of horizontal lines. Each of the original rectangles is thus divided into four smaller rectangles. Starting at M and sketching a smooth curve along the diagonals of successive cornering rec- tangles, the desired graph is obtained. By the use of the straight line y = -s/r x another set of intermediate points may be located, and so on, and the resulting curve thus drawn to any degree of accuracy required. In explaining this process, the student will show that the method of construction just used consists in the doubling of the number of horizontal lines of the figure by the successive insertion of geometrical means between the terms of a geometrical progression, while at the same time the number of vertical lines is successively doubled by the insertion of arithmet- ical means between the terms of an arithmetical series. Thus the graphical work of construction of the curve corresponds to the successive insertion of geometrical and arithmetical means in the two series discussed in §125. As explained above, the ordinate y of any point of the curve MNP of Fig. 101 is a term of a geometrical progression, and the abscissa x of the same point is the corresponding term of an arithmetical progression. Since, when y is given, the value of x is determined, we say, by definition, that a; is a function of y (§6). This particular functional relation is so important that it is given a special name: x is called the logarithm of y, and the statement is abbreviated by writing X = logy, but to distinguish from the case in which some other geometrical progression might have been used, the ratio of the progression may be written as a subscript, thus X = logr?/, which is read "x is the logarithm of y to the base r." The ratio of the geometrical progression, or r, is called the base. If we assume that the process of locating the successive sets of intermediate points by the construction of successive geometrical means will lead, if continued indefinitely, to the generation of the curve MNP without breaks or gaps, then we may say that in the equation X = lOgry, ; (1) 240 ELEMENTARY MATHEMATICAL ANALYSIS [§127 the logarithm is a function of y defined for all -positive values of y and for all halites of x. It is seen at once from the method of construction used in Fig. 101 that the values of y at a; = 1, 2, 3, 4, ... , are respectively y = r, T^, r-', r*, . . , and the values oi y a.t x = 1/2, 3/2, 5/2, . . . , are y = r^, r^, r^, . . . , respectively, and similarly for other intermediate values of x. In other words, the equation connecting the two variables x and y may be written y = r^ (2) Thus, when the values of a variable x run over an arithmetical progression {of first term 0) while the corresponding values of a variable y run over a geometrical progression {of first term 1), the relation between the variables may be written in either of the forms (1) or (2) above. Equation (2) is called an exponential equation and y is said to be an exponential function of x, while in (1) x is said to be a logarithmic function of y. The student has fre- quently been called upon in mathematics to express relations between variables in two different or "inverse" forms, analogous to the two forms y = r' and x = logr?/. For example, he has written either y = x^, 01 X = ± y/y; and either y = X ^ 01 X = y^ The graph of a function is of course the same whether the equation be solved for x or solved for y. Exercises 1. Write the following equations in logarithmic form: (a) y = l(y; (d) u = 5'; (b) y = 3'; (e) z = o"; 1 (c) y = a^; if) u = 1.1'. > As a matter of fact, both the arithmetical and the geometrical methods given above define the function only tor rational values of x; that is, the only values of X that come into view in the process explained above are whole numbers and intermediate rational fractions like 2|, 2j, 2f, 2^j, 2j|, . . . §128] LOGARITHMIC AND EXPONENTIAL FUNCTIONS 241 2. Write the following equations in exponential form : (o) X = logio y; (d) u = log, y; (b) X = logs y, (e) t = logs z; (c) X = logs y; (/) u = loga x. 3. Find the values of the following : (a) logio 100; (d) log2 64; (6) logs 25; (e) log, 81; (c) logs 27; if) log ^16. 128. The Subtangent of the Exponential Curve. The student is expected to construct the curves described in the following exercises by the method of §127. The inch or 2 cm. may be adopted as the unit of measure; the curves should be drawn on plain paper within the interval from x = — 2toa;=-j-;2. If tangents' be drawn to the curves at x = — 2, — 1, 0, 1, 2, it will be noted, as nearly as can be determined by experiment, that the several tangents to any one curve cut the X-axis at the same constant distance to the left of the ordinate of the point of tangency. This distance is called the subtangent of the curve. This distance is greater than unity if r = 2 and less than unity if r = 3. The value of the base for which the subtangent is exactly unity is later shown to be a certain irrational or incommensurable number, approximately 2.7183 . , represented in mathematics 1 It is not easy to draw accurately the tangent to a curve at a given point. A number of instruments have been designed to assist in drawing tangents to curves. One of these, called a "Radiator," will be found listed in most catalogs of drawing instruments. Another instrument consists of a straight edge provided with a vertical mirror as shown in Fig. 102. When the straight edge is placed across Fig. 102. — Mirrored ruler for drawing the normal (and hence the ' tangent) to any curve. a curve the reflection of the curve in the mirror and the curve itself can both bii seen and usually the curve and image meet to form a cusp or angle. The straight edge may be turned, however, until the image forms a smooth continuation of the given curve. In this position the straight-edge is perpendicular to the tangent and the tangent can then be accurately drawn. See Gramberg, Technische Messungen, 1911. 16 242 ELEMENTARY MATHEMATICAL ANALYSIS [§129 by the letter e, and called the Naperian base. This number, and the number ir, are two of the most important and fundamental constants of mathematics. Exercises Draw the following curves on plain paper using 1 inch or 2 cm. as the unit of measure; make the tests referred to in the second paragraph of §128. 1. Construct a curve similar to Fig. 101, representing the equation X = log2 y, from a;=— 2toa; = +2, and draw tangents at a; = — 1, X = 0, X = 1, X = 2. > 2. Construct the curve whose equation is a; = logs y from a; = — 2 to a; = +2, and draw tangents at a; = — 1, x = 0, a; = 1, x = 2. 3. Construct the curve whose equation is x = logs.? y, and show by trial or experiment that the tangent to the curve at x = 2 cuts the X- axis at nearly x = 1, that the tangent at x = 1 cuts the X^xis at nearly x = 0, that the tangent at x = cuts the X-axis at nearly x = — 1, etc. 4. Draw the curve x = logo.s y and show that it is the same as the reflection of X = log2 y in the mirror x = 0. Note: The student must remember that the experimental testing of the properties of the tangents to the curves called for above does not constitute mathe- FiQ. 103. matical proof of the usual deductive sort famUiar to him. The experimental tests have value, however, in preparing the student for a rigorous in- V estigation of these same properties when taken up in the calculus. 129. Slope of the Exponential Curve. Let MP, Fig. 103, be any exponential curve, y = r. By the slope of the curve at P we mean the slope of the tangent TP at P. We have just shown experimentally that the length of the subtangent TD is constant for all positions of the point P on the curve y = r". We can then write slope of curve at P = ^^^ = j> (1) 1 U K where k is the constant length of TD. We can also write slope of curve at P = cy, (2) where c = t • k §130] LOGARITHMIC AND EXPONENTIAL FUNCTIONS 243 From (2) we can conclude that: The slope of an exponential curve at a given point is proportional to the ordinate at that point. At the point (0, 1) the slope is c = t- As we have seen, the value oik {= TD) depends on the value of r in the equation of the curve, y = r". For some values of r it is less than 1, for others greater than 1. We have defined e as that value of r for which k = TD = 1. This is equivalent to defining e as that value of r for which the curve y = r" has the slope unity at the point (0, 1) . Later we shall adopt this definition of e. Since c = t = 1 for the curve y = e", it follows from (2) that for this particular curve of the family of curves y = r", the slope at any point is equal to the ordinate. The reasoning of this section is based on the experimentally de- termined result that for a given exponential curve the subtangent is of constant length. 130.* The Exponential Function. The expression a", where a is any positive number except 1, has a definite meaning and value for all positive or negative rational values of x, for the meaning of numbers affected by positive or negative fractional exponents has been fully explained in elementary algebra. The process outlined above likewise defines logrS for aU rational values of x, but not for irrational values of x, such as -\/2, VB, etc. As a matter of fact the expression a' has, as yet, no meaning assigned to it for irrational values of x; thus 10^^ has no meaning by the definitions of exponents previously given, for \/2 is not a whole number, hence 10"^^ does not mean that 10 is repeated as a factor a certain number of times; also \/2 is not a fraction, so that 10"^^ cannot mean a power of a root of 10. But if any one of the numbers of the following sequence : 1, 1.4, 1.41, 1.414, 1.4142, 1.41421, . . be used as the exponent of 10, the resulting power can be com- puted to any desired number of decimal places. For example, IQi" is the 141th power of the 100th root of 10; to find the 100th root we may take the square root of 10, find the square root of 244 ELEMENTARY MATHEMATICAL ANALYSIS [§131 this result, then find its 5th root, finally finding the 5th root of this last result. If the various powers be thus computed to seven places we find: 10'* =25.11887 10'" =25.70396 101" « =25.94179 101.4142 =25.95374 101.41421 = 25.95434 . 101.414213 = 25.95452 . . . 101.4142135 = 25.95455 . Now the sequence of exponents used in the first column is found by extracting the square root of 2 to successive decimal places. The sequence in the second column approaches a limit. This limit is taken hy definition as the value of 10'^''. In general, if x is an irrational number, a" is defined as the limit of a sequence of numbers, o^', a'^', . . . , a^. . . , the exponents xi, xi, . ., Xn,. . . being a sequence of rational mumbers approaching a; as a limit. It thus appears that if a and y are any given positive numbers, there is a number x, rational or irrational, which satisfies the equa- tion a' = y. The expression a' is called the exponential func- tion of X with "base a. 131. Definitions. — In the exponential equation a' = y: The number a is called the base. The number y is called the exponential function of x to the base a, and is sometimes written y = expoX. The number x is called the logarithm of y to the base ,a, and is written x = logay- Thus in the equation a" = y, x may be called either the exponent of a or the logarithm of y. The two equations, y = a' X = logay, express exactly the same relations between x and y; one equation is solved for x, the other is solved for y. The graphs are identical, just as the graphs oi y = x^ and x = ± \/y are identical. 132. Common Logarithms. In the equation 10"° = y, x is §133] LOGARITHMIC AND EXPONENTIAL FUNCTIONS 245 called the common logarithm of y. It is also called the Brigg's logarithm of y. Thus, the common logarithm of any number is the exponent of the power to which 10 must be raised to produce the given number. Thus 2 is the common logarithm of 100, since 10" = 100; likewise 1.3010 will be found to be the con^mon logarithm of 20 correct to 4 decimal places, since lO^-'"" = 20.00. 133. Systems of Logarithms. If in the exponential equation y = a', where a is any positive number except 1, different values be assigned to y and the corresponding values of x be computed and tabulated, the results constitute a system of logarithms. The number of different possible systems is unlimited, as abeady noted in §125. -As a matter of fact, however, only two systems have been computed and tubulated; the natural, or Naperian, or hyperboUc, system, whose base is the incommensurable number e, approximately 2.7182818, and the common, or Brigg's, system, whose base is 10. The letter e is set aside in mathematics to stand for the base of the natural system. Natural logarithms of all numbers from 1 to 20,000 have been computed to 17 decimal places. The common logarithms are usually printed in tables of 4, 5, 6, 7 or 8 decimal places. It will be found later that the graphs of all logarithmic functions of the form x = logo y can be made by stretching or by contract- ing in the same fixed ratio the ordinates of any one of the logarith- mic curves. That is, the logarithms of one system can be ob- tained from those of another system by multiplying by a constant factor. For this reason numerical tables in more than one system of logarithms are unnecessary. In the following pages the common logarithm of any number n wiU be written log n, and not logu n; that is, the base is supposed to be 10 unless otherwise designated; In x for logeS and Ig x for logic X are also used. Exercises Write the following in logarithmic notation: 1. 103 = 1000. 6. e" = y. 2. 10-3 ^ 0.001. 7. 10»" = 1.7783. 3. 10» = 1. 8. lO»Mio = 2. 4. IP = 121. 9. oi = a. 5. 16«" = 2. 10. 10i°8io!' = y. Express the following in exponential notation : 246 Ea^EMENTARY MATHEMATICAL ANALYSIS [§134 11. logu 4 = 0.6021. 16. log-^^iOO = I 12. log 10000 = 4. 17. logj7(l) = -ll. 13. log 0.0001 = - 4. 18. logic 10 = i- 14. logs 1024 = 10. 19. log 1 = [O. 16. log. o = 1. 20. logal = [o. 134. Graphical Table. function defined by the two progressions whose use was suggested In Fig. 104 is shown the graph of the 10 N / / / / / / L -i ^f 10 N / or / A r = 10 / / y / / / y" y ^ ^ ^ 1 1 2 3 4 5 6 7 8_ 9fl [o Fig. 104. — The curve L = logioiV. by Briggs to Napier, and which are referred to in the last para- graph of §125. By inserting means three times between and 1 in the arithmetical progression and between 1 and 10 in the geometrical progression, we get- A. P. or logarithms G. P. or numbers Exponential form of G. P. 0.000 1.000 lOO-oO" 0.125 1.334 X00.126 0.250 1.778 100.260 0.375 2.371 100.375 0.500 3.162 IQO.eoo 0.625 4.217 100. 6S6 0.750 5.623 100.760 0.875 7.499 100.»76 1.000 10.000 IQi.ooa §135] LOGARITHMIC AND EXPONENTIAL FUNCTIONS 247 If we let L stand for the logarithm of the number N, the functional relation is obviously L = logioiV, or iV = 10^. The curve, Fig. 104, may now be used as a graphical table of logarithms from which the results can be read to about 2 decimal places. The logarithms of numbers between 1 and 10 may be read directly from the graph. Thus, logio 7.24 = 0.860. If the logarithm is between and 1, the number is read directly from the graph. Thus if the logarithm is 0.273, the number is 1.87. If we multiply the readings of the A/^-scale by 10", we must add n to the readings on the L-scale, for lO^A'' = 10^+". If we divide the readings on the A''-scale by 10", we must subtract n from the readings on the L-scale, for N/10" = 10^ ~". This fact enables us to read the logarithms of all numbers from the graph, and conversely to find the number corresponding to any logarithm. Thus we have, log 72.4 = 1.860, log 724 = 2.860, log 0.724 = 0.860 - 1, log 0.0724 = 0.860 - 2. If the logarithm is 1.273, the number is 18.7. If'the logarithm is 2.273, the number is 187. If the logarithm is 0.273 - 1, the number is 0.187. If the logarithm is 0.273 - 2, the number is 0.0187. We observe that the computation of a three place table -oi logarithms would not involve a large amount of work. Such a table has actually been computed in drawing the curve of Fig. 104. The original tables of Briggs and Vlacq involved an enor- mous expenditure of labor and extraordinary skill, or even genius in computation, because the results were given to fourteen places of decimals. 135. Properties of Logarithms. The following properties of logarithms follow at once from the general properties or laws of exponents. (1) The logarithm of 1 is in all systems. For a" = 1, that is, logal = 0. In Fig. 101, note that the curve passes through (0, 1). (2) The logarithm of the base itself in any system is 1. For a^ = o, that is, log„a = 1. In Fig. 101, by construction N is always the point (1, r), where r is the ratio of the first or funda- mental progression in which means are inserted; in the present notation, this is the point (1, a). 248 ELEMENTARY MATHEMATICAL ANALYSIS [§136 (3) Negative numbers have no logarithms. This follows at once from Fig. 101. In Figs. 100, 101, and 104, note that the curves do iiot extend below the X-axis. Note: While negative numbers have no logarithms, this does not prevent the computation of expressions containing negative factors and divisors. Thus to compute (287) X (- 374), find (287) X (374) by logarithms and give proper sign to the result. 136. Logarithm of a Product. Let n and r be any two positive numbers, and let logo n = X and logo r = y. (1) Then, by definition of a logarithm, §131, n — a' and r = a". (2) Hence, nr = a" a" = a''*^ Therefore, by definition of a logarithm, §131, logo nr = X + y, or, by (1), log. nr = loga n + log. r. (3) Hence, the logarithm of the product of two numbers is equal to the sum of the logarithms of those numbers. In the same way, if log. s = «, then nrs = a"^-'-', that is, log. nrs = log. n + log. r + log. s. Exercises Find the results by the formulas and check by the curve of Fig. 104. 1. Given log 2 = 0.3010, and log 3 = 0.4771; find log 6; find log 18. 2. Given log 5 = 0.6990 and log 7 = 0.8451; find log 35. 3. Given log 9 = 0.9542, find log 81. 4. Given log 386 = 2.5866 and log 857 = 2.9330; find the logarithm of their product. 6. Given log llx = 1.888 and log 11 = 1.0414; find log x. 137. Logarithm of a Quotient. Let n and r be any two positive numbers, and let log. n = X and log. r = y. (1) §138] LOGARITHMIC AND EXPONENTIAL FUNCTIONS 249 From (1), by the definition of a logarithm, n = O' and r = a". Hence, n/r = a" -T- a" = a''". Therefore, by definition of a logarithm, loga {n'/r) = X - y, or by (1) logo (n/r) = logo n - log, r. (2) Hence, the logarithm of the quotient of two numbers equals the logarithm of the dividend minus the logarithm of the divisor. Exercises Find the results by the formulas and check by the curve of Fig. 104. 1. Given log 6 = 0.6990 and log 2 = 0.3010; find log (5/2); find log 0.4. 2. Given log 63 = 1.7993, and log 9 = 0.9542; find log 7. 3. Given log 84 = 1.9243 and log 12 = 1.0792; find log 7. 4. Given log 1776 = 3.2494 and log 1912 = 3.2815; find log 1776/1912; find log 1912/1776. 5. Given log a;/12 = 0.4321 and log 12 = 1.0792, find log x. 138. Logarithm of any Power. Let n be any positive number and let logo n = X. (1) From (1), by the definition of a logarithm, n = a". Raising both sides to the pth power, where p is any number what- soever, UP = a"'. Therefore, by definition of a logarithm, logo (w) = px, or, by (1), logo (n^) = p logon. (2) Hence, the logarithm of any power of a number equals the logarithm of the number multiplied by the index of the power. 250 ELEMENTARY MATHEMATICAL ANALYSIS [§139 The above includes the two cases: (1) the finding of the logarithm of any integral power of a number, since, in this case p is a positive integer; and (2) the finding of the logarithm of any root of a number, since, in this case, p is the reciprocal of the index of the root. Exercises 1. Given log 2 = 0.3010; find log 1024; find log V2; find log y ^. 2. Given log 1234 = 3.0913; find log Vl234; find log -^/i^Si. 3. Given log 5 = 0.6990; find log 53 ; find log sl. 4. Show that log (11/15) + log (490/297) - 2 log (7/9) = log 2. 6. Find an expression for the value of x from the equation 3' = 567. Solution: Take the logarithm of each side; X log 3 = log 567. But log 567 = log (3< X 7) = 4 log 3 + log 7. Therefore X log 3 = 4 log 3 + log 7, or X = 4 + (log 7)/(log 3). 6. Find an expression for x in the equation 5' = 375. 7. Given log 2 = 0.3010 and log 3 = 0.4771, find how many digits in 6'°. 8. Find an expression for x from the equation 3» X 2»+i = -v/si^. 9. Prove that log (75/16) - 2 log (5/9) + log (32/243) = log 2. 139. Characteristic and Mantissa. The common logarithm of a number is always written so that it consists of a positive decimal part and an integral part which may be either positive or negative. Thus log 0.02 = log 2 - log 100 = 0.3010 - 2. Log 0.02 is never written — 1.6990. When a logarithm of a number is thus arranged, special names are given to each part. The positive or negative integral part is called the characteristic of the logarithm. The •positive decimal part is called the mantissa. Thus, in log 200 = 2.3010, 2 is the characteristic and 3010 is the mantissa. In log 0.02 = 0.3010 — 2, (— 2) is the characteristic and 3010 is the mantissa. Since log 1=0 and log 10 = 1, every number lying between 1 §139] LOGARITHMIC AND EXPONENTIAL FUNCTIONS 251 an4 10 has for its common logarithm a number between and 1 ; that is, the characteristic is 0. Thus log 2 = 0.3010, log 9.99 = 0.9996, log 1.91 = 0.2810. Starting with the equation log 1.91 = 0.2810, we have, by §136, log 19.1 = log 1.91 + log 10 = 0.2810 + 1 = 1.2810, log 191 = log 1.91 + log 100 = 0.2810 + 2 = 2.2810, log 1910 = log 1.91 + log 1000 = 0.2810 + 3 = 3.2810, etc. Likewise, by §137, log 0.191 = log 1.91 - log 10 = 0.2810 - 1, log 0.0191 = log 1.91 - log 100 = 0.2810 - 2, log 0.00191 = log 1.91 - log 1000 = 0.2810 - 3, etc. Since thexharaoteristic of the common logarithm of any number having its first significant figure in units place is zero, and since moving the decimal point to the right or left is equivalent to multiplying or dividing by a power of 10, or equivalent to adding an integer to or subtracting an integer from the logarithm, (§134): (1) the value of the characteristic is dependent merely upon the position of the decimal point in the number; (2) the value of the mantissa is the same for the logarithms of all numbers that differ only in the position of the decimal point. In particular, we derive therefrom the following rule for finding the characteristic of the common logarithm of any number: The characteristic of the common logarithm of a number equals the number of places the first significant figure of the number is removed from units' place, and is positive if the first significant figure stands to the left of units' place and is negative if it stands to the right of units' place. Thus, in log 1910 = 3.2810, the first figure 1 is three places from units' place and the characteristic is 3. In log 0.0191 = 0.2810 — 2, the first significant figure 1 is two places to the right of units' place and the characteristic is — 2. A computer in determining the characteristic of the logarithm of a number first points to units' place and counts zero, then passes to the next place and counts one and so on until the first significant figure is reached. 252 ELEMENTARY MATHEMATICAL ANALYSIS [§140 Logarithms with negative characteristics, Uke 0.3010 — 1, 0.3010 — 2, etc., should be written in the equivalent form 9.3010 - 10, 8.3010 - 10, etc. Exercises , 1. What numbers have for the characteristic of their logarithm? What numbers have for the mantissa of their logarithms? 2. Find the characteristics of the logarithms of the following num- bers: 1234, 5, 678, 910, 212, 57.45, 345.543, 7, 7.7, 0.7, 0.00000097, 0.00010097. 3. Given that log 31,416 = 4.4971, find the logarithms of the foUowmg numbers: 314.16, 3.1416, 3,141,600, 0.031416, 0.31416, 0.00031416. 4. Given that log 746 = 2.8727, write the numbers which have the foUowmg logarithms: 4.8727, 1.8727, 7.8727 - 10, 9.8727 - 10, 3.8727, 6.8727 - 10. 140. Logarithmic Tables. A table of common logarithms con- tains only the mantissas of the logarithms of a certain convenient sequence of numbers. For example, a four place table will con- tain the mantissas of the logarithms of numbers from 100 to 1000; a five place table will usually contain the mantissas of the logarithms of numbers from 1000 to 10,000, and so on. Of course it is unnecessary to print decimal points or characteristics. A table of logarithms should contain means for readily obtaining the logarithms of numbers intermediate to those tabulated, by means of tabular differences and proportional parts.' The tabular differences are the differences between successive mantissas. If any tabular difference be multiplied successively by the numbers 0.1, 0.2, 0.3, . . . , 0.8, 0.9, the results are called the proportional parts. Thus, from a four place table we find log 263 = 2.4200. The tabular difference is given in the table as 16. If we wish the logarithm of 263.7, the proportional part 0.7 X 16 or 11.2 is added to the mantissa, giving, to four places, log 263.7 = 2.4211. This process is known as interpolation. Corrections of this kind are made with great rapidity after a 1 The student is supposed to have Slichter's Four Place Tables, Macmillan A Co., New York. The edition printed on three sheets of heavy manilla paper per- forated to lit in notebook is preferred. See also tables at end of this book. §140] LOGARITHMIC AND EXPONENTIAL FUNCTIONS 253 little practice. It is obvious that the principle used in the correction is the equivalent of a geometrical assumption that the graph of the function is nearly straight between the successive values of the argument given in the table. The corrections should invariably be added mentally and all the work of interpolation should be done mentally if the finding of the proportional parts by mental work does not require multiplication beyond the range of 12 X 12. To make interpolations mentally is an essential practice, if the student is to learn to compute by logarithms with any skiU beyond the most rudimentary requirements. A good method is the following: Suppose log 13.78 is required. First write down the characteristic 1 ; then, with the table at your left, find 137 in the number column and mark the corresponding mantissa by placing your thumb above it or your first finger below it. Do not read this mantissa, but read the tabular differ- ence, 32. From the p. p. table find the correction, 26, for 8. Now return to the mantissa marked by your finger, and read it increased by 26, i.e., 1393; then place 1393 after the characteristic 1 previously written down. The accuracy required for nearly aU engineering computations does not exceed 3 or 4 significant figures. Four figure accuracy means that the errors permitted do not exceed 1 percent of 1 percent. Only a small portion of the fundamental data of science is reliable to this degree of accuracy.^ The usual meas- urements of the testing laboratory fall far short of it. Only in certain work "in geodesy, and in a few other special fields of engineering, need more than four place logarithms be used. Exercises Knd the logarithms of the following : 1. 136. 4. 375.S 7. 2.758. 2. 752. 5. 217.6 8. 762,700. 3. 976. 6. 17.62 9. 0.1278. ^ Fundamental constants upon wMch much of the calculation in applied science must be based are not often known to four figures. The mechanical equivalent of heat is hardly known to 1 percent. The specific heat of superheated steam is even less accurately known. The tensile, tortional, and compressive strength of no structural material would be assumed to be known to a greater accuracy than the above-named constants. Of course no calculated result can be more accurate than the least accurate of the measurements upon which it depends. 254 ELEMENTARY MATHEMATICAL ANALYSIS [§U1 141. Anti-logarithms. If we wish to find the number which has a given logarithm, it is convenient to have a table in which the logarithm is printed before the number. Such a table is known as a table of anti-logarithms. It is usually not best to print tables of anti-logarithms to more than four places; to find a number when a five place logarithm is given, it is preferable to use the table of logarithms inversely, as the large number of pages required for a table of anti-logarithms is a disadvantage that is not com- pensated for by the additional convenience of such a table. Exercises From a four place table of anti-logarithms, find the numbers cor- responding to the following logarithms: 1. 2.7864. 2. 3.1286. 3. 1.8152. 4. 9.6278 - 10. 5. 8.1278 - 10. . 6. 6.1785 - 10. 142. Cologarithms. Any computation involving multiplica- tion, division, evolution, and involution may be performed by the addition of a single column of logarithms. This possibility is secured by using the cologarithms, instead of the logarithms, of aU divisors. The cologarithm, or complementary logarithm, of a number n is defined to be (10 — log n) — 10. The part (10 — log n) can be taken from the table just as readily as log n, by subtracting in order all the figures of the logarithm, including the characteristic, from 9, except the last figure, which must be taken from 10. The subtraction should, of course, be»done mentally. Thus log 263 = 2.4200, whence colog 263 = 7.5800 - 10. In like manner colog 0.0263 = 1.5800. It is obvious that the addition of (10 — log n) — 10 is the same as the subtraction of log n. The convenience arising from this use may be illustrated as follows : Suppose it is required to find x from the proportion 37.42 :x ::647 : v'0.S82! We then have 2 log 37.4 = 3.1458 (1/2) log 0.582 = 9,8825 - 10 colog 647 = 7.1891 _ 10 log X = 0.2174 X = [1. 650]. §143] LOGARITHMIC AND EXPONENTIAL FUNCTIONS 255 It is a good custom to enclose a computed result in square brackets. 143. Arrangement of Work. All logarithmic work should be arranged in a vertical column and should be done with pen and ink. Study the formula in which numerical values are to be substituted and decide upon an arrangement of your work in the vertical column which will make the additions, subtractions, etc., of logarithms as systematic and easy as possible. Fill out the vertical column with the names and values of the data before turning to the table of logarithms. This is called blocking out the work. The work is not properly blocked out unless every entry in the work as laid out is carefully labelled, stating exactly the name or value of the magnitude whose logarithm is taken, and unless the computation sheet bears a formula or statement fully explaining the purpose of the work. Computation Sheet, Form M7, is suitable for general logarithmic computation. Illustration 1. Find the weight in pounds of a circular disk of steel of radius 2.64 feet and thickness 0.824 inch, if the specific gravity of the steel be 7.86. Formula : Call r the radius in feet and t the thickness in inches. Take 64.48 as the weight of one cubic foot of water. Then the weight in pounds w is given by w =-nrH Substituting this result in equation (2), we have slope at P = e'. (4) 262 ELEMENTARY MATHEMATICAL ANALYSIS [§146 This expresses the fact that the slope oi y = e' at any point is e", or is the ordinate y of that point, a fact that was first indicated experimentally in §129. Later an approximate value, 2.7183, will be found for e. 146. Comparison of the Curves y. = v and y = e*. In Fig. 105 the slope oiy = e' atP is given by DP measured by the unit OM. The distance TD, the subtangent, is constant for all posi- tions of the point P- We shall prove two theorems. 1. The curve for y = r" can be made from y = e' by multiplying all of the abscissae of the latter by a constant. There is a number m such that e"* = r'. Hence y = r' may be written y = (e"')' = e""'. Now this curve is made from 2/ = e* by substituting mx for X, or by multiplying all of the abscissas of y = e' hy 1/m. 2. The slope ofy = r' at any point is a constant times the ordinate of that point. The curve y = r" can be made from y = e' hy mul- tiplying all of the abscissas of the latter by 1/m. Therefore the side TD of the triangle PDT va. Fig. 105 will be multiplied by 1/m, the other side DP remaining the same. T) o IS 1 17 7\ 1 « a u 1^ H s 3 14 B IS B A n 11 in 1 7 -n s / 1 4 / \ 3 / H = log. X 1 - -^ — ^ x=ey .^ -4 -3 -2 -1 -1 / !fs ? 3 4 5 6 7 r 9 1112 .3 14 -9, f ^ - 2/.log, *| -S — _ ^ -4 ^' e' -el Fig. 106. — Exponential and logarith- mic curves to the natural base e = log,, x 2.7183. Hence the slope of the curve, or DP/TD, will be multiplied by m, since the denominator of this fraction is multiplied by 1/m. Hence, the slope oiy = r' at any point is m times the ordinate of that point, where m satisfies the equation e" = r. The curve y = e-' is (See §25) the curve y = e'' reflected in the y-axis. This curve, as well as the curve y = log, x and its symmetrical curve, are shown in Fig. 106. Sometimes the curve y = e" Ss. called the exponential curve and the curve y = log. x §146] LOGARITHMIC AND EXPONENTIAL FUNCTIONS 263 is called the logarithmic curve. This distinction, however, has little utility, as the equation of either locus can be expressed in either notation. The notation y = In x is often used to indicate the natural loga- rithm of X and the notation y = Ig x, or y = log x, is used to stand for the common logarithm of x. Table IV. The following table of powers of e is useful in sketching exponen- tial curves. eo.2 = 1.2214 ei^ - = 1 6487 e-0.2 = 0.8187 e»-^ = 1.4918 eM = = 1 3956 g-0,4 = 0.6703 e»« = 1.8221 e!-4 = = 1 2840 g-0.6 = 0.5488 e»' = 2,2256 g-0.8 = 0.4493 e = 2.7183 e-' = 0.3679 e^ = 7.3891 e-'- = 0.1353 e' = 20.0855 e-3 = 0.0498 e* = 54.5982 e-" = 0.0183 \ / \ \\ . / / ?n= ^ fc: m = w % ^fe -2-10 12 Fig. 107. — A family of exponentials, y = e" Exercises 1. Draw the curve y = e'- + e~^. Show that y is an even function of X, that is, that y does not change when the sign of x is changed. 2. Draw the curve y = e" — e'". Show that y is an odd fun " 264 ELEMENTARY MATHEMATICAL ANALYSIS [§147 tion of X, that is, that the function changes sign but not absolute value when the sign of x is changed. 3. Draw the graphs oi y = e", and y = e~''. 4. Draw the graphs of y = e*/*, and y = e~'/'. 6. Compare the curves: y = e*/*, y = e*''*, y = e', y = e'". 6. Sketch the curves y = 1', y = 2', y = Z', y = i", y = 5', y = &', y = 8',y = 10"^, from a;=-3toa;=+3. 147. Change of Base and Properties of the Exponential Curv& Consider the curves y = e' and y = a', Fig. 108, where a > e. For a given y = OH, the abscissas HP\ and ffiPz are log, y and logaj/, respectively. It has been shown (§146) that the curve y = a' can be obtained from the curve y = e* by multiplying the abscissas of the latter curve by — , where m is the number such that ■ a. (1) T "/ * , r »/ "/ 4/ y B * 2=^ Jt> .yp ^ /> 7^P\ In other words. That is, EPi = - EP^. m log. y = - log' y- (2) ■•^ As soon as m is known we have a means of changing from a systeni of ^"*- I08--Co°iparison of logarithms with base e to one with base V = er and y = a". ° . a. The number — is called the modulus m of the logarithmic system whose base is a. The modulus of the common system of logarithms is represented by M. It is the value of — where m satisfies e"> = 10, or TO = log, 10, (3) which is equation (1) for a = 10. That is. Hence, e'^ = 10, or e = 10". M = logio e = 0.4343. (4)' §147] LOGARITHMIC AND EXPONENTIAL FUNCTIONS 265 Hence, if N represents any number, logio N = 0.4343 log. N, (5) log.N = — - — logioN = 2.3026 log, N. (6) 04343 From the definition of m and M, 100.«43 = g^ (7) g2.3026 = 10. (8) Incidentally it should be noted that, since M = —' from (3) and (4), ^°S^°" = 1^' ^^^ A remarkable property of the logarithmic curve appears from comparing the curves y = a' and y = a''*'''. The second of these curves can be derived from y = a' by translating the latter curve the distance h to the left. But y = a*+'' may be written y = a'^a^, from which it can be seen that the new curve may also be considered as derived from y = a' hy multipljang aU ordinates oiy = a" by a*. Translating the exponential curve in the negative x-direction is the same as multiplying all ordinaies by a certain fixed number, or is equivalent to a certain orthographic projection of the original curve upon a plane through the X-axis. Changing the sign of h changes the sense of the translation and changes elongation to shortening or vice versa. Exercises 1. Compare the curve y = e" with the curve y = 10*. 2. Graph the logarithmic spiral p = e>,6 being measured in radians. Note : The radian measure in the margin of Form MZ should be used for this purpose. 3. Graph p = e-«. 4. The pressure of the atmosphere is given in millimeters of mer- cury by the formula y = 760.e-»'/'i""' where the altitude x is measured in meters above the sea level. Pro- 266 ELEMENTARY MATHEMATICAL ANALYSIS [§148 duce a table of pressure for the altitudes x =0; 10; 50; 100; 200; 300; 1000; 10,000; 100,000. 5. From the data of the last problem, find the approximate pressure at an altitude of 25,000 feet. 6. Show that the relation of exercise 4 may be written X = 18,421 (log 760 - log y). 7. Determine the value of the quotient j for the following values of x : 2, 3, 5, 7. 8. How large is e"""', approximately? 9. What is the approximate value of lO"""!? 148. Logarithmic Double Scale. The relation between a number and its logarithm can be shown by a double scale of the sort discussed in §§3 and 10. Such a scale is shown in Fig. 109. It may be constructed as follows: First con- struct the uniform scale A, in which the unit distance — 1 is shown divided into 100 equal parts. Opposite 0.3010 ( = log 2) of the A- scale place a division line on the 5-scale marked by the number 2. Opposite 0.4771 (= log 3) of the A-scale place a division line of the B- scale marked by the number 3. Likewise op- posite 0.6021 (= log 4) of A place 4 on B; op- posite 0.6990 (= log 5) of A place 5 on B; etc. Intermediate points on B are similarly located — for example the 2.1 mark on B should be placed opposite 0.3222 (= log 2.1) on A. The non-uniform scale B is called a loga- rithmic scale, for the lengths measured along it are proportional to the logarithms of the natural numbers. The double scale of Fig. 109 may obviously be used as a table of logarithms. Thus from it we may read log 7.1 = 0.85; log 3.3 = 0.52; log 1.5 = 0.175. Since log 10a; = 1 + log x, it follows that, if the scales A and B, Fig. 109, were extended another unit to the right, this second s — 3 a .a a ho o = i! §149] LOGARITHMIC AND EXPONENTIAL FUNCTIONS 267 unit would be identical to the first one, except in the attached numbers. The numbers on the A-scale would be changed from 0.0, 0.1, 0.2, . .,1.0 to 1.0, 1.1, 1.2, . ,, 2.0, while those on the non-uniform, or S-scale, would be changed from 1, 2, 3, . . ., 10 to 10, 20,30, . ., 100. Passing along this scale an integral number of unit intervals corresponds thus to change of characteristic in the logarithms, and to change in the position of the decimal point in the numbers. ■ It is not, however, necessary to construct more than one block of this double scale, since we are at liberty to add an integer n to the num- bers of the uniform scale, provided at the same time we multiply the numbers of the non-uniform scale by 10". In this way we may obtain any desired portion of the extended scale. Thus, we may change 0.1, 0.2, 0.3, . ., 1.0 on X to 3.1, 3.2, 3.3, . . ., 4.0, by adding 3 to to each number, provided at the same time we change the numbers 1, 2, 3, 4, . ., 10 on the S-scale to 1000, 2000, 3000, 4000, . ., 10,000 by multiplying them by 10^. If n be negative (say — 2) we may write, as in the case of logarithms, 8.0 — 10, 8.1 — 10, 8.2 — 10, . . ., 9.0 - 10, or, more simply, - 2, - 1.9, - 1.8, - 1.7, ., — 1.0, changing the numbers on the non-uniform scale at the same time to 0.01, 0.02, 0.03, . ., 0.10. Exercises Read the following from the double scale. Fig. 109. 1. log 5.5 2. log 2.4 3. log 1.9 4. log 71 6. anti-log'O.74 6. anti-log 0.38 7. anti-log 1.38 8. anti-log 2.38 149. The Slide Rule. By far the most important apphcation of the non-uniform scale ruled proportionally to log z, is the com- puting device known as the slide rule. The principle upon which the operation of the slide rule is based is very simple. If we have two scales' divided proportionally to log x (A and B, Fig. 110), so arranged that one scale may slide along the other, then slid- ing one scale (called the slide) until its left end is opposite any desired division of the first scale, selecting any desited division of the slide, as at R, Fig. 110, and taking the reading of the original scale beneath this point, as N, the product of, the two factors whose logarithms are proportional to AB and BR can be read 268 ELEMENTARY MATHEMATICAL ANALYSIS [§149 X directly from the lower scale at N. For AN is, by construction, the sum oi AB and BR, and since the scales were laid off proportionally to log x and marked with the numbers of which the distances are the logarithms, the process described adds the loga- rithms mechanically, but indicates the results in terms of the numbers themselves. By this device all of the operations commonly carried out by use of a logarithmic table may be per- formed mechanically. Full description of the use of the shde rule need not be given in de- tail at this place, as complete instructions are found in the pamphlet furnished with each slide rule. A very brief amount of individual instruction given to the student by the instruc- tor will insure the rapid acquirement of skill in the use of the instrument. In what follows, the four scales of the slide rule are designated from top to bottom of the rule, hy A, B, C, D, respectively. The ends of the scales are called the indices. AH|Ordinary 10-inch sUde rule should give results accurate to three significant figures, which is ac- curate enough for most of the purposes of applied science. An exaggerated idea sometimes prevails con- cerning the degree of accuracy required by work in science or in applied science. Many of the fundamental constants of science, upon which a large number of other results depend, are known only to three decimal places. In such cases greater than three figure accuracy is impossible even if desired. In other cases greater accuracy is of no value even if possible. The real deside- ratum in computed results is, first, to know by a suitable check thai the work of compiUation is correct, and, second, to know to what order or degree of ac- curacy both the daia and the resuU are dependahle. The absurdity of an undue number of decimal ^-2!* » j3 a §149] LOGARITHMIC AND EXPONENTIAL FUNCTIONS 269 places in computation is illustrated by the original tables of loga- rithms, which if now used would enable one to compute from the radius of the earth, the circumference to 1/10,000 ■part o/ an inch. The following matters should be emphasized in the use of the slide rule : (1) All numbers for the purpose of slide rule computation should be considered as given with the first figure in units place. Thus 517 X 1910 X 0.024 should be considered as 5.17 X 1.19 X 2.4 X 10^ X 10' X 10~^ The result should then be mentally approxi- mated (say 24,000) for the purpose of locating the decimal point, and for checking the work. (2) A proportion should always be solved by one setting of the slide. (3) A combined product and quotient like aXhXcXd ■ rXsXt should always be solved as follows: Place runner on a of scale D; set r of scale C to a of scale D; runner to fe of C; s of C to rurmer; runner to c of C; t of C to runner; runner to d of C; find on D the significant figures of the ' result. (4) The runner must be set on the first half of A for square roots of numbers having an odd number of digits, and on the second half of A for the square roots of other numbers. (5) Use judgment so as to compute results in most accurate manner — thus instead of computing 264/233, compute 31/233 and hence find 264/233 = 1 + 31/233.^ (6) Besides checking by mental calculation as suggested in (1) above, also check by computing several neighboring values and graphing the results if necessary. Thus check 5.17 X 1.91 X 2.4 by computing both 5.20 X 19.2 X 2.42 and 5.10 X 1.90 X 2.38. 1 Show by trial that this* gives a more accurate result. 270 ELEMENTARY MATHEMATICAL ANALYSIS [§149 Exercises Compute the following on the slide rule: 1. 3.12 X 2.24; 1.89 X 4.25; 2.88 X 3.16; 3.1 X 236. 2. 8.72/2.36; 4.58/2.36; 6.23/2.12; 10/3.14. 3. 32.5 X 72.5; 0.000116 X 0.00135; 0.0392/0.00114. 4. 3,967,000 H- 367,800,000. g 78.5 X 36.6 X 20.8 , 6.64X42.6 8.75X5.25 ' „ .^■'^^J^?^^ 32.5 ' 32.3 Solve the proportion 6.46 X 57.5 X 8.55 3.26 X296 X 0.642' X : 1.72 = 4.14 : V^gh. where g = 32.2 andA^ = 78.2. o n , VlTl X 1.41 9. Compute 166.7X4.5 ' 10. The following is an approximate formula for the area of a seg- ment of a circle : <• A = h'/2c + 2ch/3, where c is the length of the chord and h is the altitude of the segment. Test this formula for segments of a circle of unit radius, whose arcs are 7r/3, ir/2, and tt radians, respectively. 11. Two steamers start at the same time from the same port; the first sails at 12 miles an hour due south, and the second sails at 16 miles an hour due east. Knd the bearing of the &st steamer as seen from the second {l) after one hour, (2) after two hours, and compute their distances apart at each time. The following exercises require the use of the data printed herewith. An "acre-foot" means the quantity of water that would cover 1 acre 1 foot deep. "Second-foot" means a discharge at the rate of 1 cubic foot of water per second. By the "run-off" of any drainage area is meant the quantity of water flowing therefrom in its surface stream or river, during a year or other interval of time. 1 square mUe = 640 acres. 1 acre = 43,560 square feet. 1 day = 86,400 seconds. 1 second-foot = 2 acre-feet per day, approximately. 1 cubic foot = 7J gallons, approximately. 1 cubic foot water = 62^ pounds water, approximately. 1 h.p. = 550 foot-pounds per second. 450 gallons per minute = 1 second-foot, approximately. §150] LOGARITHMIC AND EXPONENTIAL FUNCTIONS 271 Each of the following problems should be handled on the slide rule as a continuous piece of computation. 12. A drainage area of 710 square miles has an annual run-off of 120,000 acre-feet. The average annual rainfall is 27 inches. Find what percent of the rainfall appears as run-off. 13. A centrifugal pump discharges 750 gallons per minute against a total lift of 28 feet. Find the theoretical horse power required. Also daily discharge in acre-feet if the pump operates fourteen hours per day. 14. What is the theoretical horse power represented by a stream discharging 550 second-feet if there be a fall of 42 feet? 15. A district containing 25,000 acres of irrigable land is to be sup- plied with water by means of a canal. The average annual quantity of water required is Sf feet on each acre. Find the capacity of the canal in second-feet, if the quantity of water required is to be delivered uniformly during an irrigation season of five months. 16. A municipal water supply amounts to 35,000,000 gallons per twenty-four hours. Find the equivalent in cubic feet per second. 17. A single rainfall of 3.9 inches on a catchment area of 210 square mUes is found to contribute 17,500 acre-feet of water to storage reser- voir. The run-off is what percent of the rainfall in this case? 150. Semi-logarithmic Coordiaate Paper. Fig. Ill represents a sheet of rectangular coordinate paper, on which ON has been chosen as the unit of measure. Along the right-hand edge of this sheet is constructed a logarithmic scale LM of the type discussed in §148, i.e., any number, say 4, on the scale LM stands opposite the logarithm of that number (in the case named opposite 0.6021) on the uniform scale ON. Let us agree always to designate by capital letters distances measured on the uniform scales, and by lower case letters dis- tances measured on the logarithmic scale. Thus Y will mean the ordinate of a point as read on the scale ON, while y will mean the ordinate of a point as read on the scale LM. Moreover, we agree to plot a function, using logarithms of the values of the function as ordinates and the natural values of the argument, or variable, as abscissas. Let PQ be any straight line on this paper, and let it be required to find its equation, referred to the uniform a;-scale OL and the logarithmic 2/-scale LM. We proceed as follows : 272 ELEMENTARY MATHEMATICAL ANALYSIS [§150 The equation of this line, referred to the uniform Z-axis OL and the uniform Y-axis, ON, where is the origin, is Y = mx + B, m being the slope of the line, and B its F-intercept. ,Now, for the line PQ, m = 0.742 and B = 0.36, so that the equation of PQ is Y = 0.742a: + 0.36. (1) To find the equation of this curve referred to the scales LM and OL, it is only necessary to notice that y = log 2/ ;v Q I % 1.U /^ / 9 .8 .7 .6 y 7 / y 6 S Y / y y 3 < i2 IL .1 .2 .3 .4 .5 .6 .7 .8 .9 1.0^ Fig. 111.— The theory of the use of semi-logarithmic paper. so that we obtain log y = 0.742a; + 0.36. (2) The intercept 0.36 was read on the scale ON, and is therefore the logarithm of the number corresponding to it on the scale LM. That is, 0.36 = log 2.30. Substituting this value in equation (2) we obtain log V = 0.742a; + log 2.30, §150] LOGARITHMIC AND EXPONENTIAL FUNCTIONS 273 which may be written log y - log 2.30 = 0.742a;, log 2|o = 0-7*2.. }^M *=' ^ cS c =■ ° o"' w-° c / 1, / . / 8 / y 7 y y ' 6 = / b y /^ 4 ny / S / •^ S / y 4 2 2 1 0.1 n A L 0.1 0.2 0:3 0.4 0.5 o.< 0.7 0.8 0.0 \,0B Semi LosarTthmlc Paper Fig. 112, — Illustration of squared paper, form M5. The finer rulings of form M5 have, however, been omitted. Changing to exponential notation this becomes 2.30 10»' y = 2.30(10°'«»). (3) In general, if the equation of a straight line referred to the scales OL and ON is F = m + S, , (4) 18 274 ELEMENTARY MATHEMATICAL ANALYSIS [§150 its equation referred to the scales OL and LM may be obtained by replacing Y by log y and B by log 6 in the manner described above, giving log y = mx + log 6, (5) which, as above, may be reduced to the form y = bio"*. (6) This is the general equation of the exponential curve. Hence: Any exponential curve can he represented by a straight line, provided ordinates are read from a suitable logarithmic scale, and abscissas are read from a uniform scale. Fig. 112 represents the same line PQ, y = (2.30)10°'"', as Fig. 111. The two figures differ only in one respect; in Fig. Ill the rulings of the uniform scale ON are extended across the page, while in Fig. 112 these rulings are replaced by those of the scale LM. Coordinate paper such as that represented by Fig. 112 is known as semi-logarithmic paper. It affords a convenient coordinate system for work with the exponential function. Every point on PQ (Fig. 112) satisfies the exponential equation y = 2.30(10" '^2-). Thus, in the case of the point R, 3.98 = 2.30(10'''")'''2» = 2.30(10»-238). The slope of any line on the semi-logarithmic paper may be read or determined by means of the uniform scales BC and AB oi form M5. The scale AD of form M5 is the scale of the natural loga- rithms, so that any equation of the form y = e"" can be graphed at once by the use of this scale. Thus, the line y = e"''(Fig. 113) passes through the point A or (0, 1), and a point on BC op- posite the point marked 1.0 on AD. Note that 1.0 on scale AD, - 2.718 on the non-uniform scale of the main body of the paper, and 0.4343 on the scale BC aU fall together, as they should. To draw the line y = 10""', the corner D of the plate may be taken as the point (0, 1). On the line drawn once across the sheet representing y = 10"*, y has a range between 1 and 10 only.* §150] LOGARITHMIC AND EXPONENTIAL FUNCTIONS 275 To represent the range of y between 10 and 100, two or more sheets of form M5 may be pasted together, or, preferably, the continua- tion of the line may be shown on the same sheet by suitably chang- ing the numbers attached to the scales AB and BC. Thus Fig. 113 shows in this manner y = IW". Il 1 1 1 1 1 1 1 1 1, ,,,, 9\ /,* \ / / ! 8 \ / I R \ / 7 N, / / / ' S, / . 6 V / / g \, 1 / / 5 \ 1 / / 6 V / / 1 4\ / 1 4 / / a \ / / 8 ^ / 3 1 1 t / \ / ^ a / A >> ■i ^ / / / I I t / / ^ < \ s 1 ^ / ^ '^ 1 1 1 1 1 1 1 1 \ \ \ .1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 l.OB Fig. 113. — Seini-logarithmic coordinate paper. The dotted line gives two sections oi y= 10^"^. Remember that on semi-logarithmic paper the line y = bio""^ (7) passes through the point (0, 6) with slope m. Note that f = lo^C* - ») (8) passes through the point (a, 6) with slope m. Illtjstration 1. Draw the curve ^x = log ^y on semi-logarithmic paper. This is the curve y = 2(105"^). This curve passes through the point (0, 3) with slope |, hence can readily be drawn. 276 ELEMENTARY MATHEMATICAL ANALYSIS [§150 _ lUiTTBTBATiON 2. Draw the curve y = 3(10'^""*^]. From (8) above it is seen that this curve passes through the point (2, 3) with slope 2. Illustration 3. Plot the following data upon semi-logarithmic paper and find, if possible, the equation connecting the x- and y- values. 10 8 I ^ ^ 4 S 2 1 ^ • > ^ ^ ^ ^ 0.2 0.3 Fig. 114.- 4 5 0.6 0.7 0.8 as -Diagram for Illustration 3. X y 0.2 3.18 0.4 3.96 0.6 5.00 0.8 6.30 The points plotted upon semi-logarithmic paper he on a straight line as shown in Fig. 114. Hence, it is possible to find the equation connecting x and y. The equation of this straight line is Y = \x^r log 2.51, §151] LOGARITHMIC AND EXPONENTIAL FUNCTIONS 277 or or log V = hx -\- log 2.51, '• y _ ^ log 2.51 2' y 2.51 = 10' V = 2.51(10»), an empirical equation connecting the x- and j^-values of the table. Exercises On semi-logarithmic paper draw the following: 1. y = 10", y = W, y = 10»^. 2. y = 10-==, y = IQ-'i*, y = 10-»«. 3. y = e^', y = e". i. y = e"", y = e~^'. 5. Sx = log y, (l/2)a; = log y. 6. Find an empirical equation connecting the x- and the y-values given in the accompanying table. X y 0.2 0.4 , 0.6 0.8 5.8 3.4 2.6 On semi-logarithmic paper draw the following: ^. y = 10'/2, y = io»/io. ■ 8. Graph y ^ 2(10)' and | = 10'-'. 161.* The Compound Interest Law. Computation of e. The law expressed by the exponential curve was called by Lord Kelvin the compound interest law and since that time this name has been .generally used. It is recalled that the exponential curve was drawn by using ordinates equal to the successive terms of a geometrical progression which are uniformly spaced along the 278 ELEMENTARY MATHEMATICAL ANALYSIS [§151 X-axis. Since the amount of any sum at compound interest is given by a term of a geometrical progression, it is obvious that a sum at compound interest accumulates by the same law of growth as is indicated by a set of uniformly spaced ordinates of an expo- nential curve; hence the term "compound interest law," from this superficial view, is appropriate. The detailed discussion that follows will make this clear: Let $1 be loaned at r percent per annum compound interest. At the end of one year the amount is (1 -|- r/100). At the end of two years the amount is (1 -|- r/100)'', ■ and at the end of t years it is (1 + r/100)'. If interest be compounded semi-annually, instead of annually, the amount at the end of t years is (1 -|- r/200)''', and if compounded monthly the amount at the end of the same period is (1 +-r/\2QQy^' or if compounded n times per year y= {1 + r/lOOn)"', where t is expressed in years. Now if we find the limit of this expression as n is increased indefinitely, we will find the amount of principle and interest on the hypothesis that the interest was compounded conlinuously . For convenience let r/lOOn = 1/m. Then 2/ = (1 + 1/m)-'/'»», (1) where the limit is to be taken as m or n becomes infinite. Calling (1 + l/uY = f(u) (2) and expanding by the binomial theorem for any integral value of u we obtain r/- \ II fi / \ i w(w — 1) 1 , f{u) = 1 + u{l/u) + \2 ^ + • ■ • = 1 -I- 1 + (1 - l/w)/2! -I- (1 - 1/m)(1 - 2/u)/3\ -I- ... (3) In the calculus it is shown that the limit of this series as u becomes infinite is the limit of the series l-hl + l/2!-M/3!-h ... (4) The limit of this series is easily found; it is, in fact, the Naperian base e. It is shown in the calculus that the restriction that u shall be an integer may be removed, so that the limit of (3) may be found when m is a continuous variable. §152] LOGARITHMIC AND EXPONENTIAL FUNCTIONS 279 It is easy to see that the Umit of (4) is > 2^ and < 3. The sum of the first three terms of the series (4) equals 2i; the rest of the terms are positive, therefore e>2^. The terms of the series (4), after the first three, are also observed to be less, term for term, than the terms of the progression: (1/2)2 + (1/2)3 + (5) But this is a geometrical progression the limit of whose sum is 1/2. Therefore (3) is always less than 2| + 5, or 3. The value of e is readily approximated by the following computation of the first 8 terms of (4): 2.00000 = 1 + 1 0.50000 = 1/2! 0.16667 = 1/3! 0.04167 = 1/4! 0.00833 = 1/5! 0.00139 = 1/6! 0.00020 = 1/7! Sum of 8 terms = 2.71826 The value of e here found is correct to four decimal places. Returning to equation (1) above, the amount of $1 at r percent compound interest compounded continuously is y = e"/""- (6) Thus $100 at 6 percent compound interest, compounded annually, amounts, at the end of ten years, to y = 100(1.06)"' = $179.10. The amount of $100 compounded continuously for ten years is y = 100e«-6= $182.20 The difference is thus $3.10 152. Logarithmic Increment. The compound interest law is one of the important laws of nature. As previously noted, the slope or rate of increase of the exponential function y = ae'" at any point is always proportional to the ordinate or to the value 280 ELEMENTARY MATHEMATICAL ANALYSIS [§152 of the function at that point. Thiis when in nature we find any function or magnitude that increases at a rate proportional to itself we have a case of the exponential or compound interest law. The law is also frequently expressed by saying, as has been re- peatedly stated in this book, that the first of two magnitudes varies in geometrical progression while a second magnitude varies in arith- metical progression. A famUiar example of this is the increased friction as a rope is coiled around a post. A few turns of the hawsers about the bitts at the wharf is sufficient to hold a large ship, because as the number of turns increases 'In arithmetical progression, the friction increases in geometrical progression. Thus the following table gives the results of experiments to de- termine what weight could be held up by a one-pound weight, when a cord attached to the first weight passed over a round peg the number of times shown in the first column of the table: Average logarithmic increment = n = number of turns of the cord on the peg w = weight juBt held in equilibrium by one-pound weight Logs of preceding numbers d = logarithmic increment 1 1.6 3.0 5.1 8.0 14.0 23.0 0.204 0.477 0.708 0.903 1.146 1.362 1 li 2 21 3 0.273 0.231 0.195 0.243 0.216 0.23 If the weights sustained were exactly in geometrical progression, their logarithms would be in arithmetical progression. The test for this fact is to note whether the differences between logarithms of successive values are constant. These differences are known as logarithmic increments or in .case they are negative, as loga- rithmic decrements. In the table the logarithmic increments fluctuate about the mean value 0.23. The equation connecting n and w is of the form w = 10'"'"' or n = m log w §153] LOGARITHMIC AND EXPONENTIAL FUNCTIONS 281 By graphing columns 1 and 3 on squared paper, the value of m is determined and we find w = lO"-^'", or n = 2.2 log w. Another way is to graph columns 1 and 2 on semi-logarithmic paper. An interesting example of the compound interest law is Weber's law in psychology, which states that if stimuli are in geometrical progression, the sense perceptions are in arithmetical progression. 163. Modulus of Decay, Logarithmic Decrement. In a very large number of cases in nature a function obeying the "compound interest" law appears as a decreasing function rather than as an increasing function, so that the equation is of the form y = ae~>"', (1) where ( — 6) is essentially negative, b is the modulus of decay, or the logarithmic decrement, corresponding to an increase of x by unity. The following are examples of this law: (1) If the thickness of panes of glass increase in arithmetical pro- gression, the amount of light transmitted decreases in geometrical progression. That is, the relation is of the form L = oe-«, (2) where t is the thickness of the glass or other absorbing material and L is the intensity of the light transmitted. Since when t = the light transmitted must have its initial intensity, Lo, (2) becomes L = Loe-«. (3) The constant 6 must be determined from the data of the problem. Thus, if a pane of glass one unit thick absorbs 2 percent of the incident light, U = 100, Z, = 98 for « = 1, and 98 = 100e-», or log 98 - log 100 = - 6 log e. Therefore 6 = j^j^ = 0.02 The light transmitted by ten panes of glass is then iio = 100e-"'f»»«) = 100e-»-2, 282 ELEMENTARY MATHEMATICAL ANALYSIS [§153 or, by the table of §146, Lio = 100/1.2214 = 82 percent (2) Variation in atmospheric pressure with the altitude is usually expressed by Halley's Law, p = 760e-*/8»»», where h is the altitude in meters above sea level and p is the atmos- pheric pressure in millimeters of mercury. See §147, Exercises 4, 5, 6. (3) Newton's law of cooling states that a body surrounded by a medium of constant temperature loses heat at a rate proportional to the difference in temperature between it and the surrounding medium. This, then, is a case of the compound interest law. If 6 denotes temperature of the cooling body above that of the surround- ing medium at any time t, we must have e = ae~K The constant a must be the value of 8 when i = 0, or the initial tem- perature of the body. Exercises 1. A thermometer bulb initially at temperature 19°.3 C. is exposed to the air and its temperature B observed to be 14°.2 C. at the end of twenty seconds. If the law of cooling be given by e = ffoe"", where t is the time in seconds, find the value of 6 and 6. Soltjtion: The condition of the problem gives 9 = 19.3 when < = 0, hence, Bo = 19.3. Also, 14.2 = 19.3e-206. This gives log 19.3 - 20b log e = log 14.2, from which 6 can be readily computed. 2. If IJ percent of the incident light is lost when Ught is directed through a plate of glass 0.3 cm. thick, how much light would be lost in penetrating a plate of glass 2 cm. thick? ■ 3. Forty percent of the incident light is lost when passed through a place of glass 2 inches thick. Find the value of a in the equation L = LoB'"', where t is thickness of the plate in inches, L is the percent of light transmitted, and Lo = 100. 4. As I descend a mountain the pressure of the air increases each foot by the amount due to the weight of the layer of air 1 foot thick. As the density of this layer is itself proportional to the pressure, show that the pressure as I descend must increase by the compound inter- est law. 6. Power is transmitted in a clock through a train of gear wheels §154] LOGARITHMIC AND EXPONENTIAL FUNCTIONS 283 n in number. If the loss of power in each pair of gears is 10 per- cent, draw a curve showing the loss of power at the nth gear. Note: The graphical method' of §121, Figs. 98, 99, may appro- priately be used. 6. Given that the intensity of light is diminished 2 percent by passing through one pane of glass, find the intensity / of the light after passing through n panes. 7. The temperature of a body cooling according to Newton's law .^fell from 30° to 18° in six minutes. Find the equation connecting the temperature of the body and the time of cooling. 154. Empirical Curves on Semi-logarithmic Coordinate Paper. One of the most important uses of semi-logarithmic paper is in determining the functional relation between observed data, when such data are connected by a relation of the exponential form as already indicated in §160. Suppose, for example, that the following are the results of an experiment to determine the law connecting two variables x and y : 0.04 0.18 0.36 0.51 0.685 0.833 0.97 5.3 4.4 3.75 3.1 2.6 2.33 1.9 If the equation connecting x and y is of the exponential form, the points whose coordinates are given by corresponding values of x and y in the table will lie in a straight line on semi-logarithmic paper, except for such errors as may be due to inaccuracies in the observations. Plotting the points on semi-logarithmic coordinate paper, we find that they lie nearly on the line PQ (Fig. 115). Assuming that, if the data were exact, the points would lie exactly on this line,' we may proceed to determine the equation of this line as approximately representing the relation between x and y. It is easy to find the equation of such a line referred to the uni- ^ We would not be at liberty to make such an assumption if tte variation of the points away from the line was of a character similar to that represented by the dots near the top of Fig. 115. These points, although not departing greatly from the line shown; depart from it systematically. That is, they lie below it at each end , and above it in the center, seeming to approximate a curve (such as the one shown dotted") more nearly than the line. The points arranged about the line PQ depart as far from that line as do the points above the higher line, but they do not depart systematically, as if tending to lie along a smooth curve. When points arrange themselves as at the top of Fig. 115, one must infer that the relation connecting the given data is not exponential in character. 284 ELEMENTARY MATHEMATICAL ANALYSIS [§154 form scale AB and BC of form M5. We may imagine that all rulings are erased and replaced by extensions of the uniform AB scale, as in Fig. 111. The equation of the line PQ is then Y = mx + B, (1) where m is the slope, and B is the T-intercept. Now, for PQ, O-Jlf f-. iiiiiiiii IIIIIIIII rjmiiii ^Vs, 9: s 8 "^v 8 : N, - 7 ^<=r , ^V C ^*4 V P N, *^^[) : ^^ 4 ^^^^ 4 : ■~~^v^ 1 3 ;^ 8 " ^ -^ 1 ^v ■^^2 = 1 1 : " "" A L »'' 0.2 0.3 0.4 O.S 0.6 0.7 0.6 0.0 1.0B Semi Logarithmic Paper Fig. 115. — Empirical equations determined by use of form Mb. m = — 0.447 and B = 0.730 = log 5.37. Equation (1) becomes, therefore y = - 0.447a; + log 5.37 or, replacing Y by log y, in order to refer the curve to the scales AB and LM, log y - log 5.39 = - 0.447a:, whence y = s.ascio-""'*) (2) §155] LOGARITHMIC AND EXPONENTIAL FUNCTIONS 285 If it is desired to express the relation to the base e instead of base 10, we may note 10 = e2-''026 (§147, equation (8)), or, sub- stituting in (2), y = 5.39 (e2-303)-o.447x = 5.39 e-i»"« (3) The same result could have been obtained directly by determin- ing the slope of PQ from the uniform scale AD at the left of Form Mb. 155. Change of Scale on Semi-logaritbmic Paper. A sheet of semi-logarithmic paper, form Af5, is a square. If sheets of this paper be arranged "checker-board fashion" over the plane, then the vertical non-uniform scale will be a repetition of the scale LM, Fig. 115, except that the successive segments of length LM must be numbered 1, 2, 3, . ,9 for the original LM, then 10, 20, 30, , 90 for the next vertical segment of the checker-board, then 100, 200, 300, , 900, for the next, etc. It is obvious, therefore, that the initial point A of a sheet of semi-logarithmic paper may be said to have the ordinate 1, or 10, or 100, etc., or 10~^, lO"'', etc., as may be most convenient for the particular graph under consideration. The horizontal scale being a uniform scale, any values of x may be plotted to any convenient scale on it, as when using ordinary squared paper. However, if the hori- zontal unit of length (the length AB, form Mb) be taken as any value different from unity, then the slope m of the line PQ drawn on the semi-logarithmic paper can only be found by dividing its apparent slope by the scale value of the side AB. That is, the correct value of m in y — &10""' is, in all cases, _ apparent slope of PQ scale value of AB The "apparent slope" of PQ is to be measured by applying any convenient uniform scale of inches, centimeters, etc., to the horizontal and vertical sides of a right triangle of which PQ is the hypotenuse. 286 ELEMENTARY MATHEMATICAL ANALYSIS [§156 Exercises 1. A thermometer bulb initially at temperature 19°.3 C. is exposed to the air and its temperature e noted at various times t (in seconds) as follows: t 20 40 60 80 100 120 19.3 14.2 10.4 7.6 5.6 4.1 3.0 Plot these results on semi-logarithmic paper and test whether or not e follows the compound interest law. If so, determine the value of So and 6 in the equation 6 = SolO"". Note that the last point given by the table, namely t = 120, 6 — 3.0, goes into a new square if the scale AB be called 0—100. If the scale AB be called 0—200 then all entries can appear on a single sheet of form Af5. 2. Graph the following on semi-logarithmic paper: n \n 1 li 2 2h 3 w 1.6 3.0 5.1 8.0 14.0 23.0 Show that the equation connecting n and wis w = lO"'"". Sttggestion: The scale AB, form Mb, may be called — 5 for the purpose of graphing n. 3. Graph the following on semi-logarithmic paper, and find the equation connecting n and w. n 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 w 2.60 3.41 4.45 5.75 7.56 9.85 13.0 16.6 4. A circular disk is suspended in a horizontal plane by a fine wire at its center. When at rest the upper end of the wire is turned by means of a supporting knob through 30° The successive angles of the torsional swings of the disk from the neutral point are then read at the end of each swing as follows: Swing number 1 ■ 2 3 4 6 6 7 26°.4 23°.2 20°.5 18°.0 15°.9 14°.0 12°.3 Angle Show that the angle of the successive swings follows the compound interest law and find in at least two different ways the equation con- necting the number of the swing and the angle. Show also by the slide rule that the compound interest law holds. [tt> - SOlO"-"""] 166. The Power Function Compared with the Exponential^ Function. It has been emphasized in this book that the funds- §156] LOGARITHMIC AND EXPONENTIAL FIJNCTIONS 287 mental laws of natural science are three in number, namely: (1) the parabolic law, expressed by the power function y = ax" where n may be either positive or negative; (2) the harmonic or periodic law, y = asin nx, which is fundamental to all periodically occurring phenomena; and (3) the compound interest law dis- cussed in this chapter. While there are other important laws and functions in mathematics, they are secondary to those expressed by these fundamental functions. The second of the functions above named wUl be more fully discussed in the chapter on waves. The discussion of the compound interest law should not be closed without a careful comparison of power functions and exponential functions. The characteristic property of the power function y = ax" (1) is that as x changes hy a constant factor, y changes by a constant factor also. Let y = ax" = f(x). (2) Let X change by a constant factor m, so that the new value of x is mx. Call y' the new value of y. Then y' = a{mx)" = f{mx). (3) That is, y' a(mx)" ,., — = -^^ — '- = m", (4) y ax" which shows that the ratio of the two y's is independent of the value of X used, or is constant for constant values of m. Another statement of the law of the power function is: As .t increases in geometrical progression, y, or the power function, in- creases in geometrical progression also. r Let m be nearly 1, say 1 + t^, where r is the percent change in x and is small, then we have y' K^' + m) '^K^)' - = ' ,, ;""' = ^ ^^^^^ = (1 + r)" =F 1 + nr (5) y fix) ax" by the approximation formula for the binomial theorem (§113). 288 ELEMENTARY MATHEMATICAL ANALYSIS [§166 Hence, d' + m)-^^^^ y fix) 100 (6) f^)=nr. (7) 100 y The left-hand member is the percent change in y or infix). The number r is the percent change in the variable a;. Therefore (7) states that for small changes of the variable the percent of change in the function is n times the percent of change in the variable. Let the exponential function be represented by y = ae'' = Fix). (8) As already noted in the preceding sections, increasing x by a con- stant term increases y, or the function, by a constani factor. Thus y' F{x + h) aeoi'^") ' y Fix) ~ oe»' ~ ' ^ ' which is independent of the value of x, or is constant for constant h. The expression e'* is the factor by which y or, the function, is in- creased when X is increased by the term, or increment, h. See §147. In other words, as x increases in arithmetical progression, y, or the exponential function, increases in geometrical progression. The percent of change is [Fix + /i) - Fix)- 100 ^ = 100 [e'* - 1], (10) Fix) which is constant for constant increments h added to the variable x. If X change by a constant percent from a; to a; ( 1 + t?^) , it will be found that the percent change in the function is not constant, but is variable. The above properties enable one to determine whether measure- ments taken in the laboratory can be expressed by functions of either of the types discussed; if the numerical data satisfy the test that if the argument change by a constant factor the function also changes by a constant factor, then the relation may be repre- sented by a power function. If, however, it is found that a change §157] LOGARITHMIC AND EXPONENTIAL FUNCTIONS 289 of the argument by a constant increment changes the function by a constant factor, then the relation can be expressed by an equation of the exponential type. We have already shown how to determine the constants of the exponential equation by graphing the data upon semi-logarithmic paper. In case the equation representing the function is of the form y = ae^'' + c, (11) then the curve is not a straight line upon semi-logarithmic paper. If tabulated observations satisfy the condition that the function less (or plus) a certain constant increases by a constant factor as the argument increases by a constant term, then the equation of the type (11) represents the function and the other constants can readily be determined. The determination of the equations of curves of the parabolic and hyperbolic type is best made by plotting the observed data upon logarithmic coordinate paper as explained in the next section. 157. Logarithmic Coordinate Paper. If coordinate paper be prepared on which the uniform x and y scales are both replaced by non-uniform scales divided proportionately to log x and log y, respectively, then it is possible to show that any curve of the para- bolic or hyperbolic type when drawn upon such coordinate paper will be a straight line. This kind of squared paper is called logarithmic paper, and is illustrated in Fig. 116. To find the equation of a line PQ on such paper, we imagine, as in the case of semi-logarithmic paper, that aU rulings are erased and replaced by continuations of the uniform scales ON and MN, on which the length ON or MN is taken as unity. Denoting, as before, distances referred to these uniform scales by capital letters, we may write as the general equation of a straight line Y = mX + B. (1) In the case of the line PQ, m = 0.505, B = 0.219, and hence Y = 0.505X + 0.219. But, Y = log y, X = log X, where y and x represent distances 19 290 ELEMENTARY MATHEMATICAL ANALYSIS [§157 measured on the scale LM and LO respectively, and 0.219 = log 1.65. Hence log y = 0.505 log X + log 1.65 or log y — log 1.65 = 0.505 log .r. ar 1 10 AT 1 10 u Q ^ ^ ^ ^ y y ^ ^ ^ y 5 ^ ^ ^ 4 p N : 2 L 1 »0 1 2 3 4567S9 10 Single LogarLtlimic, Scale of Oommon LoearUlims In Margins Fig. 116. — Logarithmic coordinate paper, form itf4. The finer rulings of form Af 4 are not reproduced. This may be written in the form logj^ = log a;" whence or y — T.0.505 1.65 ~ "" • y = 1.65a;<'-=''^ (2) I §157] LOGARITHMIC AND EXPONENTIAL FUNCTIONS 291 In general, if £ = log 6, we may write the equation (1) in the form y — bx'" (3) If the straight line on logarithmic paper passes through the point (1, 1) its Cartesian equation is Y = mX, (4) or, referred to the logarithmic scales, log y = m log X — log a;"", or y = X". (5) If the straight line on logarithmic paper passes through the point (a, 6) with slope m, its equation referred to the logarithmic scales is (6) I = [;]■ On logarithmic paper, form Mi, the numbers printed in the lower and in the left margin refer to the non-uniform scale in the body of the paper. By calling the left-hand lower corner the point (1,10), (10, 10), (10, 1), (10, 100), (1,100) or (100, 100), . . . , instead of (1,1), these numbers may be changed to 10, 20, 30, . , or to 100, 200, 300, . . . , etc. If the range of any variable is to extend beyond any of the single decimal intervals, 1—10, 10—100, 100—1000, . . . , the "multiple paper," form MQ, may be used, or several straight lines may be drawn across form JW4 corresponding to the value of the function in each decimal interval, 1 — 10, 10 — 100, . . ., so that as many straight lines will be required to represent the function on the first sheet as there are intervals of the decimal scale to be represented. However, if the exponent m in i/ = bx" be a rational number, say n/r, then the lines required for all decimal intervals will reduce to r different straight lines. One of the most important uses of logarithmic paper is the de- termination of the equation of a curve satisfied by laboratory data. If such data, when plotted on logarithmic paper, give a straight line, an equation of the form (6) above satisfies the observations and the equation is readily found. The exponent m is determined by measuring the slope of the line with an ordinary 292 ELEMENTARY MATHEMATIQAL ANALYSIS [§157 uniform scale. The equation of the line is best found by noting the coordinates of any one point (o, 6) and substituting these and the slope m in equation (6). Illustration 1. Construct the semi-cubical parabola y= 2x1 on logarithmic paper. The result is a straight line of slope -| cutting the line LM, Fig. 116, at the point marked 2. 100? ^ ■» 9 ~ I 90 80 70 60 EO 40 30 20 £__________.£/ E / D F 2 3 4 5 6 78910 20 30 40 5060 K ? M Fig. 117. — Multiple logarithmic paper. 100 Illustbation 2. Find an empirical equation connecting the x and y of the accompanying table. X y X y 5 1.0 20 16.4 7 2.0 30 37.0 9 3.3 40 65.0 15 9.2 50 100.0 These points are shown plotted on the multiple logarithmic paper in Fig. 117, as the line PQ. The slope of this line is found to be 2. §157] LOGARITHMIC AND EXPONENTIAL FUNCTIONS 293 Substituting for a and 6 in (fi) the coSrdinates of any of the points on the line, for example (5, 1), we get y. = M ' 1 \6/ or 2/ = 25^- Exercises Draw the following on single or multiple logarithmic paper, forms M4or Af6: 1. y = x,y =2x,y =3x,y = ix. i. y = x^^, y = x^^, ■.^ 2. y = X, y = x^, y = x\ y ^ x*. 5. y = 2x', y = Ja;', A = itrK Z. y = 1/x, y = l/x\ y = 1/xK 6. Find the empirical equation connecting x and y of the following table. X y X y 1.6 3.05 6.5 6.40 2.5 3.92 7.5 6.85 3.5 4.65 8.5 7.25 4.5 5.30 9.5 7.70 5.5 5.82 7. Find the empirical equation connecting x and y of the following table. X y X y 1.2 2.15 2.0 5.90 1.3 2.50 2.3 7.80 1.5 3.85 2.5 9.30 1.7 4.30 8. Find the empirical equation connecting x and y of the following table. X y X y 1.5 10.0 4.5 3.30 2.0 7.5 5.0 2.98 2.5 6.0 6.0 2.49 3.0 5.0 7.0 2.12 3.5 4.25 8.0 1.87 4.0 3.73 9.0 1.65 294 ELEMENTARY MATHEMATICAL ANALYSIS [§157 Draw the following on single or multiple logarithmic paper as best suits the particular example. Carefully, label the scales and indicate the true numerical value of the division points. Use common sense values of the variables — ^for example in exercise 16 do not graph for speed over 30 knots. 9. p = 0.003«^, where p is the pressure in pounds per square foot on a flat surface exposed to a wind velocity of v miles per hour. Suggestion: The "common sense" range for v is from w = 10 to V = 100. c ' 1 G, .2 .; 1 ,' F .5 r .6 r .8 1 9 iB f 9 \ \ / 8 / / 7 \( A 6 f V / \^ 5 / \ / \ / ^ <, / ^ K 3 Sj -.5 > \ n s N Iv . o E 1 \-.2 C H -.1 A 1 i I a < i 5 r 8 9 lOo Fig. 118.— Diagram for Exercise 10. 10. Find the equ^,tions in rectangular coordinates of the lines EF and GH of Fig^ 118. 11. V = c-\/rs for c = 110 and r = 1. 12. / = y/2gh for g = 32.2. 13. C = E/B where E = 110 volts. 14. s = Igt^ where g = 32.2. 16. T = TrVZ/g, where g = 32.2. 16. p/po = (p/po)^*"', where po = 0.075, the weight of 1 cubic foot of air in pounds at 70° F. and at pressure po of 14.7 pounds per square inch. §158] LOGARITHMIC AND EXPONENTIAL FUNCTIONS 295 17. H = — p— , for D = 5000, 10,000, 15,000, 20,000, where C = 225, D is displacement in tons and /S is speed in knots. 18. H = -go", for N = 100, 200, 300, 400, 500, 600, 700, 800, 900, 1000. d is the diameter of cold rolled shafting in inches. The line should be graphed for values of d between d = 1 and d = 10. 19. F = O.OOOSilWBN', where N is revolutions per minute, R is radius in feet, W is weight in pounds, and F is centrifugal force in pounds. '• /'^ ; ;.::;:p ' y 1 0.9 0.8 = E — -7 2 '',' ;';■: ::* ...... ;: : : : ^ T 0.0 ^ y ;2 ^^- ;':';; ; 12 .' 1 Sj . .' . ^^ c' t' ■' % 0.R i ■=;-::^ = !■■ ^ ;=-: ="^:^ is .::. E ~ 7.^ "2*^ - Ji. : . J^ ^ — ^^> ;^:: : : -■^. , • ^ <-ii > < <' 0.1 a09 : : 1 y ;;;:: ;:; ',-; 1 1 i::: : _: = 0.07 ■p 2 z r = -^ ,. ,. z - i'! ::!I - 0.04 X-'' 2^: :. I_ « a2 0.3 a4 £=I*iigthofCreBtinJBet ^=Heaa on Creat in Feet g =BlB0liai^e In Second Feet ae 0.8 1.0 3 3 4 6 6 7 8 910 * 0.7 0.9 Discharge over Trapezoidal Wiex Fig. 119. — A weir formula graphed on multiple logarithmic pa per. 20. g = 3.37M^ for L = 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0. See Fig. 119. O.SSF'*^ 21. H = — jj-yjj — , where V is the velocity of water in feet per second under the head of H feet per 10,000 feet in clean cast-iron pipe of diameter d feet. See Fig. 120. 158. Slims of Exponential Functions. Functions consisting of the sum of two different exponential functions are of frequent occurrence in the application of mathematics, especially in elec- trical science. Types of fundamental importance are e" + e"" and e» — e-" which are so important that the forms (e" + e~")/2 and (e" — e~")/2 have been given special names and tables of 296 ELEMENTARY MATHEMATICAL ANALYSIS [§158 their values have been computed and printed. The first of these is called the hyperbolic cosine of u and the second is called the hyperbolic sine of u ; they are written in the following notation : cosh w = (6« + e~")/2, sinh li = (e« — e~")/2. Triction Head in Feei per IDOO Ft. of Pipe Not«: For opoD ooadults, multlplj IlydraUllo Radlut bj 4 to get Equivalent SUmetsT. Diagram givu noarly Boms reaults aa KuttWB Fonnulm Kith n=.011. Fur old or foul pipes multlplj required head bj 1.4& f>00 ^ ^^^ o' divide diagram veloolt; b; I>20 to 1,28 for V= 2 to & feet per Beoond . j g For Bubb pipes ffsO-fiO^lia Diagram, of Flow In Clean Oaat Iron or Wrought Iron Pipes Baaed on the Formula, H, in Feet per 1000 Feet = 0,38 K"^- FiG. 120. — A compHcated example of the use of multiple logarithmic paper, Form MQ. From Transactions Am. Soc. C. E., Vol. LI. If X = a cosh u and y= a sinh u, then squaring and subtracting x^ — y^ = a2(cosh'' u — sinh^ u) =..p + 2 + e-2" 2 + e--' 4 4 Therefore the hyperbolic functions x= a cosh u, and y = a sinh u '1 §158] LOGARITHMIC AND EXPONENTIAL FUNCTIONS 297 appear in the parametric equations of a rectangular hyperbola just as the circular functions X = a cos B, and y = asind appear in the parametric equations of the circle a;2 -)- j/2 = o^ 4 1 L 3.6 yi — \ 3 \ / \ 2.5 w ' I 2 f *^ 1,5 ^ ^1 ^ V' V • y e" N \5 l/ V -yie- ■ X ^ 1 — ' ^ -3.5 -3 -2. -2 -1. -1 H 6 1 1.5 2 2.S 3 3.5 / / l.S 1 -2 / 2.5 / -3 / 3.5 / -4 4.5 -5 Fig. 121. — The curves of the hyperboUc sine and cosine. The graphs of y = a cosh x and y = a sinh x were called for in exercises 1, 2, §146. They are shown in Fig. 121. The first of these curves is formed when a chain is suspended between two points of support; it is called the catenary. These two curves are best drawn by averaging the ordinates oi y = e' and y —e~', and the ordinates oi y = e'' and y = — e"'. Curves whose equations are of the form y = ae""* + be"' take on quite a variety of forms for various values of the constants. A good idea of certain important types can be had by a comparison of the curves of Figs. 122 and 123. 298 ELEMENTARY MATHEMATICAL ANALYSIS [§158 1.& (1) I .75 \w (1) 2^=e-*+o,Be-"'* (2) 3/=e"" .JO, (4) y=e:Z-ojseJ"' (6)2/=e"-i.Be *°* bA(s) w 5 P \ - 25 m \ N^ ^^ — ^ 1 1.5 Pig. 123. 2.6 §159] LOGARITHMIC AND EXPONENTIAL FUNCTIONS 299 The student should arrange in tabular form the necessary numerical work for the construction of the curves of Figs. 122 and 123. If the coefficient of the second exponent be increased in absolute value, the points of intersection with the F-axis remain the same, but the region of close approach of the curves to each other is moved along the curve y = e-' to a point much nearer the Y-axis as can be seen by comparing Fig. 123 with Fig. 122. 159. *Damped Vibrations. If a body vibrates in a medium like a gas or liquid, the amplitude of the swings are found to get smaller and smaller, or the motion slowly (or rapidly in some cases) dies out. In the case of a pendulum vibrating in oil, the rate of decay of the amplitude of the swings is rapid, but the ordinary rate of the decay of such vibrations in air is quite slow. The ratio between the lengths of the successive amplitudes of vibration is called the damping factor or the modulus of decay. The same fact is noted in case the vibrations are the torsional vibrations of a body suspended by a fine wire or thread. Thus a viscometer, an instrument used for determining the viscosity of lubricating oils, provides means for determining the rate of the decay of the torsional vibration of a disk, or of a circular cylinder suspended in the oil by a fine wire. The "amplitude of swing" is in this case the angle through which the disk or cylinder turns, measured from its neutral position to the end of each swing. In all such cases it is found that the logarithms of the successive amplitudes of the swings differ by a certain constant amount or, as it is said, the logarithmic decrement is constant. Therefore the amplitudes must satisfy an equation of the form A = ae~^ where A is amplitude and / is time. The actual motion is given by an equation of the form y = ae~^ sin ct, A study of oscillations of this type will be taken up more fully in the calculus. For the present it will suffice to graph a few examples. Let the expression be y = g-f/B sin t. (1) A table of values of t and y must first be derived. There are three ways of proceeding; (1) Assign successive values to t urespective of 300 ELEMENTARY MATHEMATICAL ANALYSIS [§169 the period of the eine (see Table V and Fig. 124). (2) Select for the values of t those values that give aliquot parts of the period 2t of the sine (see Table VI and Kg. 125). (3) Draw the sinusoid y = sin t carefuUy to scale by the method of §56; then draw upon the same V / '^ s . U>b / \ \ v'^ A r \ ^^1,0 1 1 12 4" t U > V [^ t ■ J 'i -' ■^13 0.B \ / 1.6 V \ -1.6 Fig. 124. — The curve y = e"*/' sin t. \ s*. / N >N t" / \ <1 r^ \ TT ITT /^ 'n lir 4T J ( V i 1 [/ 2 1 4 1 6J M U2 ^ 4 2 B 2 \ >^ ^ ^ rr"' / Fig. 125. — The curve y = e"*/' sin t. coordinate axes, using the same units of measure adopted for the sinus- oid, the exponential curve y = e~'/^; finally multiply together, on the slide rule, corresponding ordinates taken from the two curves, and locate the points thus determined. §159] LOGARITHMIC AND EXPONENTIAL FUNCTIONS 301 The first method involves very much more work than the second for two principal reasons: First, tables of the logarithms of the trigonometric functions with the radian and the decimal divisions of the radian as argument are not available; for this reason 57.3° must be multiplied by the value of t in each case so that an ordinary trigonometric table may be used; second, each of the values written in column (3) of the table must be separately determined, while if the periodic character of the sine be taken advantage of, the numerical values would be the same in each quadrant. TABLE V Table of the function y = e "^ sin t 1 2 3 4 5 t in radians log e-'/» = - (0.0869)i log sin ( or log sin 57.34 if ( is in degrees logy V 0.0 -0.0000 + 0.000 0.5 - 0.0434 9.6807 9.6372 + 0.434 1.0 - 0.0869 9.9250 9.8381 + 0.689 1.5 - 0.1303 9.9989 ' 9.8686 + 0.739 2.0 -0.1737 9.9587 9.7850 + 0.610 2.5 - 0.2172 9.7771 9.5599 + 0.363 3.0 - 0.2606 9.1498 8.8892 + 0.077 3.5 - 0.3040 9.5450 9.2410 -0.174 4.0 -0.3474 9.8790 9.5312 - 0.340 4.5 -0.3909 9.9901 9.5992 - 0.397 5.0 - 0.4343 9.9818 9.5475 - 0.353 5.5 - 0.4777 9.8485 9.3708 - 0.235 6.0 - 0.5212 9.4464 8.9252 - 0.084 6.5 - 0.5646 9.3322 8.7679 + 0.059 7.0 - 0.6080 9.8175 9.2095 + 0.162 7.5 - 0.6515 9.9722 9.3207 + 0.209 8.0 - 0.6949 9.9954 9.3005 + 0.200 8.5 - 0.7383 9.9022 9.1634 + 0.146 9.0 - 0.7817 9.6149 8.8332 + 0.068 9.5 - 0.8252 8.8760 8.0508 -0.011 10.0 - 0.8686 9.7356 8.8670 -0.074 10.5 - 0.9120 9.9443 9.0323 - 0. 108 11.0 - 0.9555 9.9999 9.0444 -0.111 11.5 - 0.9989 9.9422 8.9433 - 0.088 12.0 - 1.0424 9.7296 8.6872 -0.049 The second method, because of the use of aliquot divisions of the 302 ELEMENTARY MATHEMATICAL ANALYSIS [§159 period of tl^e Bine, such as ir/6 or ir/12 or r/l& or 5r/20, etc., possesses the advantage that the values used in column (3) need be found for one quadrant only and the values required in column (2) are quite as readily found on the slide rule as in the first method. TABLE VI Table of the function y = e""/* sin t 1 2 3 4 5 n = t in units of ir/6 radians log e- »"■/'" = - (0.0455)» log, ain nw/6 log!/ y 1 -0.0000 - 0.0455 0.000 + 0.450 9.6990 9.6535 2 -0.0910 9.9375 9.8465 + 0.702 3 - 0.1364 0.0000 9.8636 + 0.731 4 - 0.1819 9.9375 9.7556 + 0.570 5 - 0.2274 9.6990 9.4716 + 0.296 6 -0.2729 + 0.000 7 - 0.3184 9. ,6990 9.3806 - 0.240 8 - 0.3638 9.*9375 9.5737 - 0.375 _ 9 - 0.4093 0.0000 9.5907 -0.390 10 -0.4548 9.9375 9.4827 - 0.304 11 - 0.5003 9.6990 9.1987 -0.158 12 13 - 0.5458 -0.5912 0.000 + 0.128 9.6990 9.1078 14 - 0.6367 9.9375 9.3008 + 0.200 15 - 0.6822 0.0000 9.3178 + 0.208 16 -0.7277 9.9375 9.2098 + 0.162 17 -0.7732 9.6990 8.9258 + 0.084 18 19 - 0.8186 -0.8641 0.000 - 0.068 9.6990 8.8349 20 - 0.9016 9.9375 9.0279 -0.107 21 - 0.9551 0.0000 9.0449 - 0.111 22 - 1.0006 9.9375 8.9369 -0.087 23 - 1.0460 9.6990 8.6530 -0.045 24 - 1.0915 0.000 The third method is perhaps more desirable than either of the others if greater than two figures accuracy is not required. The curve can readily be drawn with the scale units the same in both dimensions, as is sometimes highly desirable in scientific applications. In Figs, 124 a^d 125 a, larger unjt ha? be^n used on the vertical §159] LOGARITHMIC AND EXPONENTIAL FUNCTIONS 303 scale than on the horizontal scale. In Fig. 125 the horizontal unit is incommensurable with the vertical unit. To draw the curve to a true scale in both dimensions it is preferable to lay off the coSrdinates on plain drawing paper and not on ordinary squared paper. Rec- tangular coordinate paper is not adapted to the proper construction and discussion of the sinusoid, or of curves, like the present one, that are derived therefrom. Curves whose equations are of the form y = je"*/' sin tor y = Se"'.'' sin t, etc., are readily constructed, since the constants i, 3, etc., merely multiply the ordinates of (1) by \, 3, etc., as the case may be. Likewise the curve y = e~'* sin ex is readily drawn since sin ex can be made from sin x by multiplying all abscissas of sin x by 1/c. CHAPTER X TRIGONOMETRIC EQUATIONS AND THE SOLUTION OF TRIANGLES A. FURTHER TRIGONOMETRIC IDENTITIES 160. The circle p = a cos d + h sin 0. In §74 an analytical proof was given of the fact that p = a cos fl + 6 sin 5 is the polar equation of a circle passing through the pole and having its center at the point (\a, 56). The demonstration there given should now be reviewed. Geometrical Explanation. The following geometrical dis- cussion should give the student a better under- standing of the important theorem of §74. We know (§66) that pi = a cos B is the polar equa- tion of a circle of diameter a, the diameter coinciding in direction with the polar axis OX; for example, the circle OA, Fig. 126. Like- wise, p2 = 6 sin is a circle whose diameter is of length 6 and makes an angle of -|-90° with the polar axis OX, for example, the circle OB, Fig. 126. Also, p = c cos {0 — 6) is a circle whose diameter c has the direction angle 5. See Theorem XIV on Loci, §70. We shall show that if the radius vectors corresponding to any value old in the equations pi = a cos d and pa = 6 sin 6 be added together to.make a new radius vector p, then, for all values of B, 304 Pig. 126. — Combination of the cir- cles p = a cos 6 and p = 6 sin 9 into a single circle p = a cos B + 6 sin e. §160] TRIGONOMETRIC EQUATIONS 305 the extre mity of p lies on a circle (the circle OC, Fig. 126) of di- ameter Vo^ + h'^. In other words we shall show geometrically that p = a cos & + 6 sin fl (1) is the equation of a circle. In Fig. 126, pi = a cos Q wiU be called the a-cirde OA; p^ = 6 sin 6 will be called the b-circle OB. For any value of the angle 6 draw radius vectors OM, ON, meeting the a- and 6-circles respec- tively at the points M and N. If P be the point of intersec- tion of MN produced with the circle whose diameter is the diagonal OC of the rectangle described on OA and OB, we shaU show that OM + ON = OP, no matter in what direction OP be drawn. Let the circle last mentioned be drawn, and project BC on OP. Since ONB and OPC are right angles, NP is the projection of BC {= a) upon OP. But OM also is the projection of a (= OA) upon OP. Hence NP = OM because the projections of equal parallel lines on the same line are equal. Therefore, for all values of d, NP = pi and OP = ON + NP = pi + pi, which is the fact that was to be proved. Designating the angle AOC by 5, the equation of the circle OC is by §70. P = Va^ 4- h'^ cos {6 - S) (2) The value of 5 is known, for its tangent is -• It should be observed that there is no restriction on the value oi 6. As the point P moves on the circle OC, the circumference is twice described as d varies from 0° to 360°, but the diagram for other positions of the point P is in no case essentially different from Fig. 126. The above reasoning and the diagram involve the restriction _ that both o and 6 are positive numbers. While it is possible to supplement the reasoning to cover the cases in which this restric- tion is removed, it is unnecessary as the analytical proof of §74 is applicable for all values of a and b. The equation of the circle OC in any position, that is, for any value of a and 6, positive or negative, may also be written in the form p = Va^ + b^sm{d + i) ' (3) in which e is the angle BOC in Fig. 126. The equation of the 20 306 ELEMENTARY MATHEMATICAL ANALYSIS [§160 circle OC has therefore been written in three different forms, namely equations (1), (2) and (3) above. Illustration 1. From the above we know that the equation p = 6 cos 9 + 8 sin 9 represents a circle. The diameter of the circle is Va* + ft" = VSM"^ = lOi so that the equation of the circle may also be written p = 10 cos (9 — i), where 5 is the angle whose tangent is - = ^ = 1.33. From a table of tangents S = 53° 8', so that the equation of the circle may be written p = 10 cos (9 — 5l3 °8'). Illustration 2. Write the equation of the circle p = cos 9 — -y/S sin 9 in the form p = c cos (9 — S) and in the form p = c sin (9 + e). Here a = 1, 6 = — -y/S, c = \/a* + 6« = 2. Hence C must be the point (1, — \/3) in the second quadrant. Then 5 = angle of second quadrant whose tangent is ( — -s/S/l), or 120°. Also « = — 30°. Hence the required equations are p = 2 cos (9 — 120°) and p = 2 sin (9 — 30°). The result of this section should also be interpreted when the vari- ables are x and y in rectangular coordinates, and not p and 9 of polar coordinates. Thus, y = a cos a; is a sinusoid with highest point or crest at a; =0, 2t, iar, . . . Likewise, y = b sin s is a sinusoid with crest at a; = y -«-' -~-' . . The above demonstration shows that the curve y = a cos X + b sin X is identical with the sinusoid y = Va' + 6" cos (a; - hi) = \/a'- + ¥ sin (a; + h) of amplitude \/a^ -\- b- and with the crest located at a; = hi, or at s — ^2, where hi is, in radians, the angle whose tangent is -> and hi is, in radians, the angle whose tangent is r- Exercises 1. Put the equation p = 2 cos 9 + 2v'3 sin 9 in the form p = c cos (9 — 8) and find the value of h. 2. Put the equation p = 4 cos 9 + \\/Z sin 9 in the form p = ccos (9 — 5). 3. Put the equation p = — 4 cos § — 4 sin 9 in the form p - c sin (9 + e), §161] TRIGONOMETRIC EQUATIONS 307 4. Put the equation p = 2-\/3 cos 6 + 2 sin fl in the form p = c cos (9 — 8). 6. Put the equation p = 3 cos 9 + 4 sin 9 in the form p = csin (9 + e). Put the same equation in the form p = c cos (9 — 'S). (S is the angle AOC, Fig. 126.) 6. Put the equation p = 5 cos 9 + 12 sin 9 in the form p = o sin (9 + e); also in the form p = c cos (9 — S). 7. Put the equation (x — 1)^ + (y — ly = 2 in. the form p = c sin (9 + a) and determine c and a. 8. Put the equation (a; + 1)^ + (2/ — \/3)' = 4 in the form p = c sin (9 — a) and determine c and a. 9. Put the equation (x + 1)^ + (y + Vs)^ = 4 in the form p = c sin (9 — a) and determine c and a. 10. Find the maximum value of cos 9 — \/3 sin 9, and determine the value of 9 for which the expression is a maximum. Suggestion : Call the expression p. The maximum value of p is the diameter of the circle p = cos 9 — -y/s sin 9. The direction angle of the diameter is the value of a when the equation is put in the form p = c cos (9 — a). 11. Find the value of 9 that renders p = f-v/S cos 9 — ^ sin 9 a maximum and determine the maximum value of p. 12. Find the maximum value of 3 cos t + isint. 161. Addition Fonnulas for the Sine and Cosine. From the preceding section, equations (1), (2) and (3), we know that the equation of the circle OC, Fig. 127, may be written in any one of the forms p = a cos 6 + 6 sin 0, (1) p = c sin(,e- e), (2) p = c cos (e-5). (3) Hence, for all values oi 9, d, and e, sin (6 — e) = - cos 5 + - sin 6, (4) cos (9 - 5) = -cose + - sin 9, (6) In each of these equations c = -^/a" + h^. The letters a and 6 stand for the co6rdinates of C irrespective of their signs or of the position of C, 308 ELEMENTARY MATHEMATICAL ANALYSIS [§161 Since (4) and (5) are true for all values of 6, they are true when » = 0° and when = 90°. First, Let = 0° in (4). Then a/c = sin (— e) = — sin e by §60, (6) -Second, let B = 90° in (4), Then 6/c = sin (90° - e) = cos « (7) Substituting (6) and (7) in (4) we have sin (5 — e) = sin cos e — cos 6 sin e (8) Fig. 127. — The circle p = c cos (9 — 5) or p = sin c (fl — e) used in the proof of the addition formulas. Note that e = 90° + « which is also true for negative angles, namely Si = 90° + ei. In like manner upon letting = and = 90° in succession in (5) we have - = cos (- 5) = cos 5, by §60. (9) = cos (90 — 5) = sin 5. Substituting (9) and (10) in (5) we obtain cos {8 — B) = cos 9 cos 6 + sin sin 5 (10) (11) §162] TRIGONOMETRIC EQUATIONS 309 Since these are, true for all values of S and e, put 5 = (-^Si) and e = ( — ei). Then by §60, these equations become sin (6 + ei) = sin cos ei + cos 6 sin €i (12) cos (0 + Si) = cos 6 cos 5i — sin 9 sin 5i (13) To aid in committing these four important formulas (8), (11), (12) and (13) to memory, it is best to designate in each case the angles by a and |3, and write (12) and (13) in the form sin (a + /3) = sin o: cos /3 + cos a sin /3 (14) cos (a + /3) = cos a cos |8 — sin a sin /3 (15) and also write (8) and (11) in the form sin (a — P) = sia a cos /3 — cos a sin /3 (16) cos (a — |3) = cos a cos /3 + sin q: sin j3 (17) The four formulas (14), (15), (16) and (17) must be committed to memory. They are called the addition fonnulas for the sine and cosine. The above demonstration shows that the addition for- mulas are true for all values of a and fi. By the above formulas it is possible to compute the sine and cosine of 75° and 15° from the following data: sin 30° = i sin 45° = iV2 cos 30° = i V3 cos 45° = iV2 Thus sin 75° = sin (30° + 45°) = sin 30° cos 45° + cos 30° sin 45° = HV2 + |\/3iV2 = iV^(V'3 + 1) Likewise sin 15° = sin (45° - 30°) = i-s/2(\/3 - 1) 162. Addition Formula for the Tangent. Dividing the mem- bers of (14) §161 by the members of (15) we obtain , , a\ sin (a -t- /3) sin a cos |8 -h cos a sin |3 ,, , tan (a + p) = -, — —37- = 5 ; ; — 5 ^1; cos (o -t- p) cos acosp — sm asmp Dividing numerator and denominator of the last fraction by cos a cos /3 sin a cos /3 cos a cos |8 tan ia + ^) = ^^E^^l_Jo[^^ (2) cos a cos fi _ sin a sin fi cos a cos ^ cos acosfi 310 ELEMENTARY MATHEMATICAL ANALYSIS [§163 or , , .. tan a + tan /3 ,„, tan (a + B) = — ^ (3) y I t^' I - tan<»tan|3 Likewise it can be shown from (16) and (17), §161, that . , „ tana — tan ^ , . tan (a - |3) = — t— r— ^ (4) ^ ' I + tanatan|3 Equations (3) and (4) are the addition formulas for the tangent. Exercises 1. Compute cos 75° and cos 15°. 2. Compute tan 75° and tan 15°. 3. Write in simple form the equation of the circle p = sin 6 + cos B. 4. Put the equation of the circle p = 3 sin 9 + 4 cos 6 in the form p = c sin (9 + 9i) and find, from the tables or by the slide rule, the value of ©i. 6. Derive a formula for cot (a + p). 6. Prove cos (s + t) cos {s — t) = cos'' s — sin^ t. 7. Express in the form c cos (a — b) the binomial 3 cos a + 4 sin o. 8. Express in the form c sin (a + 6) the binomial 5 cqs o + 12 sin a. 9. Find the coordinates of the maximum point or crest of the sinus- oid y = sin X + -\/3 cos x. [First reduce the equation to the form 2/ = c sin (a; + a)]. 163. Functions of Composite Angles. The sine, cosine, or tangent of the angles (90° — d), (90° + 6), (180° - 6), (180° + 6), (270° — 6), (270° + d) can be expressed in terms of functions of 6 alone by means of the addition formulas of §§161 and 162. Thus, write . sia (a + /?) = sin a cos /3 + cos a sin /3 (1) cos {a + fi) = cos a cos ;8 — sin a sin j3 (2) Put a = 180°, and /3 = + 0; then (1) and (2) become, re- spectively, sin (180° ± 0) = T sin e (3) cos (180° ±6) = ~ Gosd (4) TRIGONOMETRIC EQUATIONS 311 Also in (1) and (2) put a = 90°, and P = ±6, then (1) and (2) become, respectively, sin (90° ± 6) = cos 6 (5) cos (90° ± 6) = + sine ■ (6) By division of (3) by (4) and of (5) by (6), tan (180° ±6) = + tan 6», (7) tan (90° ± 6) = + cot d. (8) In a similar manner all of the results given in the following table may be proved to be true. (-ft./l) Pa AC ft, A) (A. ft) P(h.k) Pt(.-h,-h) P,(h.-k) (-ft.-ft) Pa P, (k.-h) A B Fig. 128. — An angle 9 combined with an even number of right angles, (A) and wijh an odd number of right angles, (B) . TABLE VII Functions of 6 Coupled with an Eeen or with an Odd Number of Bight Angles - e 90° -9 90°+ 9 180°- 9 180°+ 9 270°- 9 270°+ 8 sin — sin 9 cos 9 cos 9 sin 9 — sin 9 — cos 9 — cos B cos ' cos e sin 9 — sin 9 — cos 9 — cos 9 — sin B sin 8 tan — tan e cote — cot e — tan 9 tan 9 cot 9 - cot 9 AU of the above results can be included in two simple state- ments. For this purpose it is convenient to separate into different, 312 ELEMENTARY MATHEMATICAL ANALYSIS [§163 classes the composite angles that are made by coupling with an odd number of right angles, as (90° + fl), (ff - 90°), (270° - 6), (450° + 6), etc., and those composite angles that are made by coupling 6 with an even number of right angles, as (180° + 6) (180° - 6), (360° - 6), (- 6), etc. Note that is an even number, so that ( — 9) or (0° — 6) falls into the second class of composite angles. We can then make the following statements : Theorems on Functions of Composite Angles Think of the original angle 6 as an angle of the first quadrant: I. Any function of a composite angle made by coupling B {by addition or subtraction) with an even number of right angles, is equal to the same function of the original angle 6, with an algebraic sign the same as the sign of the function of the composite angle in its quadrant. II. Any function of a composite angle made by coupling (by addition or subtraction) with an odd number of right angles, is eqital to the co-function of the original angle B, with an algebraic sign the same as the sign of the function of the composite angle in its quadrant. For example, let the original angle be 6, and the composite angle be (180° + 8). Take any function of (180° + $), say tan (180° + 6), it is equal to + tan 6, the sign + being the sign of the tangent in the quad- rant of the composite angle (180° + 8), or third quadrant. Likewise cot (270° + 6) must equal the negative co-function of the original angle, or — tan 0, the algebraic sign being the sign of the cotangent in the quadrant of the composite angle (270° -|- 9), or fourth quadrant. In the above statements it has been assumed that the angle fl is an angle of the first quadrant. This is merely for the convenience of determining signs, for the results stated in itaUcs are true, no matter in what quadrant 9 may actually, he. Exercise Given sin 30° = J, cos 30° = JVS, tan 30° = iVS, cot 30° = \/3, find the sine, cosine, and tangent of each of the following angles by means of the above Theorems on Functions of Composite Angles: (a) 150°; ^b) 210°; (c) 240°; (d) 300°; (e) 330°; (/) 120°; (g) 60°; (h) -30°. §164] TRIGONOMETRIC EQUATIONS 313 164. Angle that a Given Line Makes with Another Line. The slope m of the straight line y = mx + b is the tangent of the direction angle, that is, the tangent of the angle that the line makes with OX. If Li and L^ are any two lines in the plane, the angle that Li makes with Lj is the positive angle through which L^ must he rotated about their point of intersection in order that Li may coincide with Li. Represent the direction angles of two straight lines y = miX + bi (1) y = mix. + hi (2) by the symbols di and 6^. Then, through the intersection of the lines pass a line parallel to the OX-axis, as shown in Fig. 129. Call the angle that the line Li makes with La) that is, the positive ^ '\> '> A V \- \ \L,. I Fig. 129. — The angle that a line Li makes with -La. J angle through which La, considered as the initial line, must be turned to coincide with the terminal position given by Li. If 9i > Bi, then 4> = Bi- 6^, but if fla > Oi, then = 180° - (Sj - di). In either case (by equations (7), §163, and (3), §60 tan = tan {di - d^). (3) That is, tan di — tan 82 ,,. (4) or tan (b = tan = 1 + tan 61 tan 62' nil — nia (5) I + niinia The condition that the given lines (1) and (2) are parallel is obviously that mi = ma (6) Thus, the lines y = 5x + 7 and y = 5x — 11 ar« parallel. 314 ELEMENTARY MATHEMATICAL ANALYSIS [§164 The condition that the given lines (1) and (2) are perpendicular to each other is that tan 4> shall become infinite; that is, that the denominator of (5) shall vanish. Hence the condition of perpen- dicularity is 1 + miW2 = 0, m: = - ^- (7) Therefore, in order that two lines may be perpendicular to each other, the slope of one line must be the negative reciprocal of the slope of the other line. Thus the lines y = %x — A and y = — fa; + 2 are per- pendicular. Exercises 1. Find the tangent of the angle that the first line makes with the second line of each set: [a) y = 2x + Z, y = x + 2, {h)y = Zx -Z, y = 2x + 1, (c) y = ix + 5, y = Zx - 1, id) y = lOx + I, y = Ux - 1, 2. Find the angle that the first line of each pair makes with the second: (a) y = X +5, y = - a; -|- 5. (6) 2/ = Ja; -H 6, y = - 2x. (c) 2/ = 2a; + 4, y = x + 1. (d) 2x+Zy = \, Ix =y = \. (e) 2i 4- 42/ = 3, 3a; 4- 62/ = 7. (/) 2x +Ay = 3, 6a; - 32/ = 7. 3. Find the angle, in each of the following cases, that the first line makes with the second: - (o) 2/= x/Vz +4, 2/ = V3x+ 2. (6) y = a;/\/3 -|- 1, y = VZx- 4. (c) y = y/Zx - 6, 2/ = s/Zx- Z. 4. Find the angle that 2i/ — 6a; -1- 7 = makes with y + 2x + 7=0 and also the angle that the second line makes with the first. §165] TRIGONOMETRIC EQUATIONS 315 166. The Functions of the Double Angle. The addition formulas for the sine, cosine, and tangent reduce to formulas of great importance for the special case fi = a. Thus sin (a + a) = sin a cos a + cos a sin a, or sin 2a = 2 sin a cos a. (1) Also cos (a + a) = cos a cos a — sin a sin a which can be written in the three forms: cos 2 a = cos^ a — sin^ a, (2) cos 2 a = 2 cos^ oi —I, (3) cos 2 o! = I — 2' sin^ a. (4) Forms (3) and (4) are obtained from (2) by substituting, respectively, sin^ a = 1 — cos'' a and cos^ a = 1 — sin^ a. Equations (3) and (4) are frequently useful in the forms : — , . I + cos 2a; ,g, (6) Again 2 • sin^ a = I — cos 2a 2 tan {a + a) = tan a + tan a 1 — tana tan a fan 2 tana ■•""'"" I - tan» a ^^^ 166. The Functions of the Half Angle. From (6) and (5) of §165 we obtain, after replacing a by u/2 and extracting the square root, sin (u/2) = ±v'(i — cos u)/2, (1) cos (u/2) = ± \/(i + cos u) /2 . (2) Dividing (1) by (2), we obtain * /- / ^ j_ 1 /i - COS u j^ I - cos u ^ sin u . ,„. tan (u/2) = ± V — ; = ± ^^ = ± — i (3) ^ ' ' » I + cos u sm u I + cos u Formulas (1), (2) and (3) have many important applications in mathematics. As a simple example, note that the functions of 15° 316 ELEMENTARY MATHEMATICAL ANALYSIS [§167 may be computed when the functions of 30° are known. Thus cos 30° = (1/2) VS therefore sin 15° = \/(l - cos 30°)/2 = V'l/2 - (1/4)^3- Also cos 15° = Vl/2 + (1/4)^3. Likewise by (3) tan 15° = L^^#^ = 2 - V3. Exercises 1. Compute sin 60° from the sine and cosine of 30°. 2. Compute sine, cosine, and tangent of 221°. 3. If sin X = 2/5, find the numerical value of sin 2x, cos 2x, and tan 2x. 4. Show by expanding sin (x + 2x) that sin 3a; = 3 sin a; — 4 sin 'x. , _, . „ 3 tan x — tan' x 0. Prove tan 3a; = — 5 5-: — ; 1—3 tan* X 6. Show that sin 29/sin e — cos 29/cos 9 = sec 8. 7. Show that Ism 2 + cos^) = 1 + sin e. 8. Show that cos 29(1 + tan 29 tan B) = 1. 9. If sin A = 3/5, calculate sin {A/2). 10. Prove that tan (7r/4 + 9) = ^ _ ^ g - 11. Prove that tan (ir/4 - 9)' = (1 - tan 9)/(l + tan 9). 12. Show that sec 9 + tan 9 = ^^— • cos 9 . » p., . , . 1 + 2 sin a cos a cos a + sin a 13. Show that - ■ x—- ;-= — = ■ — -. cos* a — sm* a cos a — sm o 14. Show that sec 9 + tan 9 = tan [i+g 16. Show that — ^—j — j :r-^ — tan A tan B. cot A + cot B 16. Prove that cos (s + t) cos {s — t) + sin (s + t) sin (s — i) = cos 2t. 167. Sums and Differences of Sines and of Cosines Expressed as Products. The following formulas, which permit the substi- tution of a product for a sum of two sines or of two cosines, are §167] TRIGONOMETRIC EQUATIONS 317 important in many transformations in mathematics, especially in the calculus. They are immediately derivable from the addition formulas. Thus, by the addition formulas (14) and (16), §161, we obtain sin (a + 6) + sin (a — 6) = 2 sin o cos 6. Likewise by subtraction of the same formulas sin (a + 6) — sin (a — b) = 2 cos a sin b. By the addition and subtraction, respectively, of the addition formulas for the cosine there results cos (a + 6) + cos (a — b) = 2 cos a cos 6. cos (a + 6) — cos {a — b) = — 2 sin a sin 6. These formulas can be written sin o cos b — 5 [sin (o + 6) + sin (a — 6)], (1) cos a sin 6 = | [sin (a + 6) — sin (o — b)], (2) cos a cos 6 = 2 [cos (a + b) + cos (a — 6)], (3) sin a sin 6 = — 5 [cos (a + 6) — cos (a — 6)]. (4) Represent (a + 6) by a and (a — b) by /?. Then o = (a + /3) /2 and b = (a - j8)/2 Hence the above formulas become sin a + sin j3 = 2 sm cos -' (5) • n a + . a — ,„, sm a — sin fl = 2 cos sm > (6) 22 cos a + cos /3 = 2 cos cos > (7) 2 2 cos a — cos /3 = — 2 sm sm ^- (8) 2 2 The principal use of these formulas is in certain transformations in the Calculus. A minor use is in adapting certain formulas to logarithmic work by replacing sums and differences by products. These formulas should not be committed to memory. They can be derived in a moment when needed by recalling their 318 ELEMENTARY MATHEMATICAL ANALYSIS [§168 connection with the addition formulas. Formula (2) is really- contained in formula (1). For by (1) cos a sin 6 = sin 6 cos a = 5 [sin(& + a) + sin (6 — a)] = 5 [sin (a + 6) — sin (a — 6)], since sin(— B) = — sin & Exercises Express as the sum or difference of sines or cosines: 1. sin 5x cos 2x. 6. sin 3x sin 7x. 2. cos 3a; sin 7x. 7. cos 3a; cos 8x. 3. cos 4a; cos x. 8. cos 5a; sin 2x. 4. sin 5x sin 2x. 9. sin 3a; cos lOx. 6. sin 3x cos 5x. 10. cos 2x cos 6x. 168.* Graph of y = sin 2x, y = sin nx, etc. Since the substi- tution of nx for X in any equation multiplies the abscissas of the curve by 1/n, or («>!) shortens, or contracts, the abscissas of all points of the curve in the uniform ratio n : 1, the curve y = sin 2x must have twice as many crests, nodes, and troughs in a given interval of x as the sinusoid y = sin x. The curve y = sin 2x is therefore readily drawn from Fig. 59 as follows: Divide the axis OX into twice as many equal intervals as shown in Fig. 59 and draw vertical lines through the points of division. Then in the new diagram there are twice as many small rectangles as in the original. Starting at and sketching the diagonals (curved to iit the alignment of the points) of successive cornering rectangles, the curve y = sin 2x is constructed. It is, of course, the ortho- graphic projection of J/ = sin x upon a plane passing through the F-axis and making an angle of 60° (the angle whose cosine is 1/2) with the x2/-plane. The curve y = cos 2x is sunilarly con- structed. In each of these cases we see that the period of the function is t and not 2ir. 169.* Graph of p = sia20,p — cos 20, etc. The curve p = cos 6 is the circle of diameter unity coinciding in direction with the axis OX. We have already emphasized that as d varies from 0° to 360° the circk is twice drawn, so that the curve consists of two §170] TRIGONOMETRIC EQUATIONS 319 superimposed circular loops. Now p = cos 2d wiU be found to consist of four loops, somewhat analogous to the leaves of a four- leafed clover, but each loop is described but once as 6 varies from 0° to 360°. The curve p = cos 36 is a three-looped curve, but each loop is twice drawn as S varies from 0° to 360°. Also p = cos 116 has eleven loops, each twice drawn, while p = cos 126 has twenty- four loops, each one described but once, as 6 varies from 0° to 360°. The curves p = cos 2 6, p = sin 39, p = sin 6/2 should be drawn by the student upon polar coordinate paper. 170.* Graph of y = sln^x, and y = cos^x. The graphs y = sin' x and y = cos'' x have important applications in science. The following E Y A s P^-^? ■'^.^-L — — — — — ci._4V. \ / N rr-l M N^ / N ff|--| - / s / "^ \ \3^ 1 1 / S^ / "— ^^"^-^ \ / ~~—-^^^~^ <. f^ ^"^^^r y / / H Fig. 130. — The graph oi y = cos^ x. graphical method offers an easy way of constructing the curves and it illustrates a number of important properties of the functions involved. We shall first construct the curve y = cos' x. At the left of a sheet of 8 J X 11-inch paper draw a circle of radius 36/57r ( = 2.30) inches, (OA, Fig. 130). Lay off the angles 9 from OA, as initial line, correspond- ing to equal intervals (10°) of the quadrant APE as shown in the figure. Let the point P mark any one of these equal intervals. Then dropping the perpendicular AB from A upon OP, the dis- tance OB is the cosine of 6, if OA be called unity. Dropping a perpendicular from B upon OA, the distance OC is cut off, which is equal to OB' or cos' e, since in the right triangle OB A, OB' — OCOA = OC-1. Making similar constructions for various values of the angle e, say for every 10° interval of the arc APE, the hne OA is divided at a number of points proportionally to cos' e. Draw horizontal lines 320 ELEMENTARY MATHEMATICAL ANALYSIS [§171 through each point of division of OA. Next divide the axis OX into intervals equal to the intervals of 8 laid off on the arc APE. Since the radius of the circle OA was taken to be (SB/Sir) inches, an interval of 10° corresponds to an arc of length 2/5 inch, which therefore Inust be the length of the equal intervals laid off on OX. Through each of the points of division of OX draw vertical lines, thus dividing the plane into a large number of small rectangles. Starting at A and sketching the diagonals of successive cornering rectangles, the locus ARS oi y = cos' x is constructed. From Fig. 130, it is seen that B always lies at the vertex of a right- angled triangle of hypotenuse OA. Thus as P describes the circle of radius OA, B describes a circle of radius OA/2. Therefore the curve ABS is related to the small circle ABO in the same manner that the curve of Fig. 59 is related to its circle; consequently the curve ARS of Fig: 130 is a sinusoid tangent to the X-axis. Thus the graph y = cos' a; is a cosine curve of amplitude 1/2 and wave length or period IT, lying above the X-axis and tangent to it. B. PLANE TRIANGLES: CONDITIONAL EQUATIONS 171. Law of Sines. The first of the conditional equations per- taining to the oblique triangle is a proportion connecting the sines Fig. 131. — Derivation of the law of sines and the law of cosines. of the three angles of the triangle with the lengths of the respec- tive sides lying opposite. Call the angles of the triangle A, B, C, and indicate the opposite sides by the small letters a, b, c, respec- tively. From the vertex of any angle, drop a perpendicular p upon the opposite side, meeting the latter (produced if necessary) §172] TRIGONOMETRIC EQUATIONS 321 at D. Then, by the properties of right triangles, we have, in either Fig. 131 (1) or 131 (2), p = c sin DAB. (1) From A BDC, p = a sin C. (2) But, sin DAB = sin A, Fig. 131 (1) = sin (180° - A), Fig. 131 (2) = sin A. Therefore p = c sin A = » sin C, (3) or a/sin A = c/sin C. (4) In like manner, by dropping a perpendicular from A upon a, we can prove b/sin B = c/sin C. (5) Therefore a/sin A = b/sin B = c/sin C (6) Stated in words, the formula says: In any oblique triangle the sides are proportional to the sines of the opposite angles. Geometrically: Calling each of the ratios in (6) 2B, it is seen from Fig. 131 (2) that R is the radius of the circumscribed circle since c/ sin C = 2R can be deduced from the triangle BAE. Similar construction can be made for the angle B or A. 172. Law of Cosines. From plane geometry we have the theo- rem: The square of any side opposite an acute angle of an oblique triangle is equal to the sum of the squares of the other two sides di- minished by twice the product of one of those sides by the projection of the other side on it. Thus, in Fig. 131 (1), o2 = 6'2 + c^ - 2bd. (1) But - d = c cos A. Therefore a^ = b^ + c'^ - 2bc cos A, (2) Likewise we learn from geometry that the square of any side oppo- site an obtuse angle of an oblique triangle is equal to the sum of the squares of the other two sides increased by twice the product of one of 21 322 ELEMENTARY MATHEMATICAL ANALYSIS (§172 those sides by the projection of the other on it. Thus, in Fig. 131 (2) , a2 = 62 + c2 + 2bd (3) ' But d = c cos DAB = c cos (180 — A) = — c cos A. Therefore (3) becomes a2 = b'' + c^ - 2bc cos A. (4) This is the same as (2), so that the trigonometric form of the geo- metrical theorem is the same whether the side first named is oppo- site an acute or opposite an obtuse angle. In the same way we may show that, in any^ triangle b2= c2-|-a2 -2cacosB, (5) c2 = a^-l-b^ - 2ab cosC. (6) Independently of the theorem from plane geometry, we note from Fig. 131 (1) a^ = (b - dy -f- p2 = (6 _ dy + c'' - d^ = 62 + (.2 _ -2,bd = fe'' -h c^ - 26c cos A. From 131 (2) o" = (6 -I- dy + p2 = (6 -h dy -I- c2 - d^ = 6=' -I- c2 -I- 2bd = 62 + 0^-1- 26c cos DAB = 62-)-c' - 26c cos A, since DAS = 180° - A and cos (180° - A) = - cos A. Second Phoop: Since any side of an oblique triangle is the sum of the projections of the other two sides upon it, the angles of projection being the angles of the triangle, we have a = b cos C -|- c cos B, b = c cos A -f- a cos C, (7) c = a cos B -h b cos A. Multiply the first of these equations by a, the second by 6, the third by c, and subtract the second and third from the first. The result is a^ — b^ — c^ — ah cos C -\- ca cos B — 6c cos A — ab cos C — ca cos B — be cos A = — 26c cos A, or a^ = 6^ -h c'' — 26c cos A. §173] TRIGONOMETRIC EQUATIONS 323 173. Law of Tangents. An important relation results if we take formula (5) §171 by composition and division. First write the law of sines in the form sin A sin jB' (1) Then, by composition and division, the sum of the first anteced- ent and consequent is to their difference as the sum of the second antecedent and consequent is to their difference; that is a + h _ sin A + sin B , ,„, a — & sin A — sin B Expressing the sums and difference on the right-hand side of (2) as products by means of the formulas (5) and (6) of §167, we obtain a + b ^ 2 sin i(,A + B) cos i(A - £) ', a-h 2 cos K^ + B) sin |(A - B) or simplifying and replacing the ratio of sine to cosine by the tan- gent, we obtain (3) a-l-b ^ tan \{k + B) . a - b tan J(A - B) In like manner it follows that b 4- c tan |(B 4- C). b - c tan KB - C) 4- a _ tan_|(C_+A) — a tan KC — A) (4) Fig. 132. — Geometrical (R\ derivation of law of tan- gents. Expressed in words: In any triangle, the sum of two sides is to their difference, as the tangent of half the sum of the angles opposite is to the tangent of half of -their difference. Geometrical Proof: From any vertex of the triangle as center, say C, draw a circle of radius equal to the shortest of the two sides of the triangle meeting at C, as in Fig. 132. Let the circle meet the side a&t R and the same side produced at E. Draw AE and AR. Call the angles at A, a, and /3, as shown. Then BE = a + 6 and BR = a - b. Also a + P = A, 324 ELEMENTARY MATHEMATICAL ANALYSIS [§174 and /. CRA = + B (the external angle of a triangle RAB is equal to the sum of the two opposite interior angles), or a - = B. Therefore a = lU + B), P = i{A- B). Draw RS \\ to EA. ZEAR = ZARS = 90°. By similar triangles BE/BR = AE/'SR ^AE . 8R AR ■ AR But BE = a + b and BR = a - b, while AE . ,SR.„ -r-p- = tan a and -j-k = tan p. AR AR Therefore a+6 ^ tan KA + -B). inerelore ^ _ ^ ^^^ ^^^ _ ^^ 174. The following special formulas are readily deduced from the sine formulas and are sometimes useful as check formulas in computa- tion. They are closely related to the law of tangents. From the proportion ' a:b:c = sin A: sin B:sin C by composition , c _ sin C 0+6 sin A + sin B Now by §165 (1) and §167 (5) this may be written c 2 sin jC cos jC + 6 ~ 2 sin UA + B) cos i(A - B)' Since C = 180° - (A + B), therefore C/2 = 90° - |(A + B), and cos C/2 = sin UA + B). c sin iC cos |(A + B) ,,, a + b cos i(A — B) cos i(A — B; In like manner it can be proved that c _ sin i(A + B) ,2) a - b sin |(A - B) Both (1) and (2) can be readily deduced geometrically from Fig. 132. 176. The s-formulas. The cosine formula y a2 = 62 -}- c^ - 26c cos A §175] TRIGONOMETRIC EQUATIONS 325 can be written in the forms a^ = (b + cy - 26c(l + cos A), (1) a2 = (6 - c)2 + 26c(l - cos A), (2) by adding (+26c) and (—26c) to the right-hand member in each case. But now we know from §166, (1) and (2), that 1 + C0S A = 2 cos" (A /2), 1 - cos JL = 2 sm" (A/2). , Therefore (1) and (2) above become o= = (6 + c)2 - 46c cos" (A/2), (3) ' a" = (6 - cy + 46c sin" (A/2). (4) Writing these in the form i\i ibc sin" (A/2) = o"]- (6 - c^, (5) 46c cos" (A/2) = (6 + c)" - a", (6) and dividing the members of (5) by the members of (6), we obtain tan" (A/2) =1^1^. (7) Factoring the numerator and denominator we obtain tan" (A/2) - fi + ^Tu^T^ + l - (8) '(6 + c + o) (6 + c — a) Let the perimeter of the triangle be represented by 2s, that is, let a + 6 + c = 2s. Hence, subtracting 2c, 26, and 2a in turn, a + 6 — c = 2s — 2c (subtracting 2c), a — 6-|-c = 2s — 26 (subtracting 26), 6 + c — a = 2s — 2a (subtracting 2a). Therefore equation (8) becomes tan" (A/2) = (^ -/>)i^ ' <=) . (g) s(s — a) Let (s - a) (s - 6) (s - c) /s = r\ (10) 326 ELEMENTARY MATHEMATICAL ANALYSIS [§175 Then or ' Likewise tan^ (A/2) = r-V(s - aY, tan (A/2) = r/(s — a). tan (B/2) = r/(s - b), tan (C/2) = r/(s - c), (11) (12) (13) Fig. 133. — Geometrical derivation of the s-formulas. Geometrically: These formulas may be found by means of the diagram Fig. 133. Let the circle be inscribed in the triangle ABC; its center is located at the intersection of the bisectors of the internal angles of the triangle. Let its radius be r. ATi = ATt, BTt = BTz, CTi = CTi, and since 2s = a + 6 + c, it follows that one way of writing the value of s is s = BTi + TiC + ATi. §176] TRIGONOMETRIC EQUATIONS 327 Therefore ATi = s -a.' Hence it follows that tan (A/2) = r/(s - o). (14) Since this result is the same as (11) above, it proves that the r of equation (10) is the radius of the inscribed circle, and therefore proves that the radius of the inscribed circle may be expressed by the formula Us - a)is -6)(s -c) a fact that is usually proved in text books on plane geometry. 176.* Miscellaneous Formulas for Oblique Triangles. The fol- lowing formulas are given without proof. They are occasionally useful for reference, although no use will be made of them in this book. The following notation is used: The three sides of the oblique triangle are named a, b, c, and the angles opposite these A, B, C, respectively. The semi-perimeter of the triangle is s, OT 2s = a + b + c. The radius of the circumscribed circle is B, that of the inscribed circle is r, and the radii of the escribed circles are Ta, n, r^ tangent, respectively, to the sides a, b, c of the given triangle. K stands for the area of the triangle. s = 4i? cos iA cos J-B cos §(7 (1) s — c — 4Rsin iA sin ^B cos iC (2) and analogs for s — a and s — b. r = iR sin JA sin iB sin iC (3) Tc = 4jB cos iA cos iB sin iC (4) and analogs for Ta and rt. Ta = s tan iA,n = s tan iB, r^ = s tan JC (5) 2K = ab sinC = be sin A = ca sin B (6) K = 2R'' sin A sin S sin C = |^ (7) K = Vsis -a) is- b) (s - c) (8) K = rs = ra(s — a) = n(s — 6) = r^is — c) (9) Z2 = rr^nr, (10) K^ = {s - a) tan iA = {s - b) tan iB = {s - c) tan |C (11) 328 ELEMENTARY MATHEMATICAL ANALYSIS [§177 C. NUMERICAL SOLUTION OF OBLIQUE TRIANGLES 177. An oblique triangle possesses six elements; namely, the three sides and the three angles. If any three of these six magnitudes be given (except the three angles), the triangle is determinate, or may be constructed by the methods explained in plane geometry; it will also be found that if any three of these six magnitudes be given, the other three may be computed by the formulas of trigonometry, provided, that the given parts include at least one side. It is convenient to divide the solution of triangles into four cases, as follows : I. Given two angles and one side. II. Given two sides and an angle opposite one of them. III. Given two sides and the included angle. IV. Given the three sides. The solution of these cases with appropriate checks will now be given. The best arrangement of the work of computation usually consists in writing the data and computed results in the left margin of a sheet of ruled letter paper (SJ inches X 11 inches) and placing the computation in the body of the sheet. Every entry should be carefully labeled and computed results should be enclosed in square brackets. AU work should be done on ruled paper and invariably in ink. Special calculation sheets (forms M2 and M7) have be'en prepared for the use of students. Neat- ness and systematic arrangement of the work and proper checking are more important thanr rapidity of calculation. 178. Computer's Rules. The following computer's rules are useful to remember in logarithmic work : Last Digit Even: When it becomes necessary to discard a 5 that terminates any decimal, increase by unity the last digit retained if it be an odd digit, but leave it unchanged if it be an even digit; that is, keep the last digit retained even. Thus log tt = 0.4971; hence write | log x = 0.2486. Also log sin 18° 5' = 9.4900 + (correction) 19.5 = 9.4920. Of course if the discarded figure is greater than 5, the last digit retained is increased by 1, whUe if the discarded figure is less than 5, the last digit retained is unchanged. 8179] TRIGONOMETRIC EQUATIONS 329 Functions or Angles in Second Quadbant: In finding from the table any function of an angle greater than 100° (but < 180°) replace the first two digits of the number of degrees in the angle by their sum and take the co-function of the result. The method is valid because it is equivalent to the subtraction of 90° from the angle. By §163 this always gives the cor- rect numerical value of the function. The algebraic sign should be taken into account separately. Thus, sin 157° 32' 7" = cos 67° 32' 7". In case of an angle between 90° and 100°, ignore the first digit and proceed in the same way. Thus, tan 97° 57' 42" = - cot 7° 57' 42" 179. Case I. Given two angles and one side, as A, B, and c. 1. To find €, use the relation A+B + C = 180°. 2. To find a and 6, use the law of sines, §171. 3. To check results, apply the check formula (1) or (2) §174. Illtjstkation: In an oblique triangle, let c = 1492, A = 49° 52', B = 27° 15'. It is required to compute C, a, b. The following form of work is self explanatory. It should be noted that the process of work and the meaning of each number entering the calculation is properly indicated or labeled in the work. Numerical Work Given To find o, b, and C. c = 1492 Formulas A = 49° 52' C = 180 - U + B) = = (102° 53') B = 27° 15' c sin A Work. " ~ sin C , _ c sin B sin C log sin A = log c = 9.8834 - 3.1738 ■ 10 log sin B = 9.6608 - - 10 ^ log sin C = 9.9889 - - 10 log o 3.0683 log 6 2.8457 a = [1170.] 6 [701.] Check. Check Formula c -b = : 791 a sin i(C + B) C + B = : 130° 8' c - h sin i(C - B) C - B ^ : 75° 38' 330 ELEMENTARY MATHEMATICAL ANALYSIS [§I80 i(C + B) = 65° 4' UC - B) = 37° 49' log a = 3.0683 log c - 6 = 2.8982 log r^b = 01701 log sin i {C + B) = 9.9575 - 10 log siD i (e - B) = 9.7875 - 10 Examples Find the remaining parts, given : 1. A = 47° 20', B = 32° 10', 2. B = 37° 38', C = 77° 23', 3. B = 25° 2', C = 105° 17', 4. C = 19° 35', A = 79° 47', Check a = 739. 6 = 1224. 6 = 0.3272. c = 56.47. 180. Case n. Given two sides and an angle opposite one of hem, as a, b, and A . Fig. 134. — Case II of triangles, for one, two, and impossible solutions. 1. To find B, use the law of sines, §171. 2. To find C, use the equation A +B + C = 180°. 3. To find c use the law of sines. 4. To check, apply the check formula (I) or (2), §174. When an angle as B, above, is determined from its sine, it admits of two values, which are supplementary to each other. There may be, therefore, two solutions to a triangle in Case II. The solutions are illustrated in Fig. 134. §180] TRIGONOMETRIC EQUATIONS 331 • In case one of the two values of B when added to the given angle A gives a sum greater than two right angles, this value of B must be discarded, and but one solution exists. If a be less than the perpendicular distance from C to c, no solution is possible. Illustration: Solve the triangle if a = 345, 6 = 534, and A = 25° 25'. The solution is readily understood from the following work. Numerical Work Given a = 345 6 = 534 To find c, B, and C. Formulas . „ b sin A A = 25° 25' a Work. C = 180 - (A + B) log 6 =2.7275 a Bin C log sin jl = 9.6326 - 10 sin A log a =2.5378 logsmB = 9.8223 - 10 B = [41°37'.l] or [138° 22'. 9] A+B =67°2'.l 163° 47'. 9 C = [112° 57'. 9] or [16°12'.l] logo =2.5378 2.5378 log sin C = 9.9641 - 10 9.4456 - 10 log sin A = 9.6326 - 10 9.6326 - 10 log c = 2.8693 2.3508 c = [740.1] [224.3] Check 6 sin i(C + A) ^ a _ sin K-B + C) c — a sm t c — a C + A C — A W + A) KC - A) log 6 log (c - o) logQ log sin KC + log sin KC — logO iC -A) = 395.1 = 138° 22'. = 87° 32'. = 69° 11' = 43° 46' = 2.7275 = 2.5967 = 0.1308 = 9.9707 - A) = 9.8400 - = 0.1307 A) 10 10 b — c sin |(B b -c B + C B -C UB + C) i(B - C) log o log (6 - c) logQ' Iogsini(B + C) logsini(fi-C) logQ' = 309.7 = 154° 35' = 122° 10'. 8 = 77° 17'. 5 = 61° 5'. 4 = 2.5378 = 2.4910 = 0.0468 = 9.9892 - = 9.9422 - = 0.0470 10 10 332 ELEMENTARY MATHEMATICAL ANALYSIS [§181 Examples Compute the unknown parts in each of the following triangles : , 1. a = 0.8, b = 0.6, B = 40° 15'. 2. o = 8.81, 6 = 11.87, A = 19° 9'. 3. 6 = 81.05, c = 98.75, C = 99° 19'. 4. c = 50.37, a. = 58.11, C = 78° 13'. 6. a = 1213, 6 = 1156, B = 94° 15'. 181. Case III. Given two sides and the included angle, as a,b,C. 1. To find A +B,useA +B = 180° - C. 2. To find A and B, compute (A — B)/2 by the law of tangents, §173, equation (4), then A = (A + B)/2 + (A - B)/2 and B = (A+ B)/2 - {A - B)/2. 3. To find c, use law of sines, §171. 4. To check, use law of sines. Illustration: Given a = 1033, 6 = 635, C = 38° 36'. Numerical Work Given To find c. A, and B. a = 1033 Formulas b = 635 A+B = 180 -C = 141° 24' C = 3*° 3^' tan UA-B) = ^-=-^ tanJCA + B) o'sin C tsin A Work a -b ' = 398 a + b = 1668 \{A+B) = 70° 42' log (a - 6) = 2.5999 logtanKA+'B) =0.4557 - log(o + 6) =3.2222 log tan 4U -B) = 9.8334 - 10 1{A -B) = 34° 16.3' A =[104° 58.3'] B = [36° 25.7'] logo =3.0141 log sin C = 9.7951 - 10 log sin A = 9.9850 - 10 logo =2.8242 c = [667.1] §182] TRIGONOMETRIC EQUATIONS 333 Check b sin C sin B log 6 =2.8028 log sin C = 9.7951 - 10 log sin B = 9.7737 - 10 logc =2.8242 c = [667.1] Examples Compute the unknown parts in each of the following triangles : 1. a =78.9, 6=68.7, C = 78° 10'. 2. c = 70.16, a = 39.14, B = 16° 16'. 3. 6 = 1781, c = 982.7, A = 123° 16'. 182. Case IV. Given the three sides. 1. To find the angles, use the s-formulas, §175, (11), and (13). 2. To check, use A + B + C = 180°. Illustration: Given a = 455, 6 = 566, c = 677, find A, B and C. Numerical Work (12) Given "Work. a = 455 6 = 566 c = 677 2s = 1698 s = 849 s — a = 394 s -6 = 283 s — c = 172 To find A, B and C. Formulas T tan iA = ' tan hB = tan JC = s - b r where r = ^'(^ - aKs - bHs - e). 2si = 1698 log (s-a) = 2.5955 log (s-b) = 2.4518 log (s - c) = 2.2355 logs =2.9289 logr" =4.3539 logr =2.1770 log tan JA = 9.5815 - 10 log tan IB = 9.7252 - 10 log tan iC = 9.9415 - 10 • Adding th« four numbera above this line cheoks the subtractions (> — a), {» -h), etc. 334 ELEMENTARY MATHEMATICAL ANALYSIS (§182 iA = 20° 53' JJS = 27° 58' JC = 41° 9' A = [41° 46'] B = [55° 56'] C = [82° 18'] Check. A+B+C = 180° Exercises Find the values of tlJe angles in each of the following triangles : 1. a = 173, 6 = 98.6, c = 230. 2. a = 8.067, 6 = 1.765, c = 6.490. 3. a = 1911, 6 = 1776, c = 1492. Miscellaneous Problems The instructor will select only a limited number of the following problems for actual computation by the student. The student should be required, however, to outline in writing the solution of a number of problems which he is not required actually to compute, and, when practicable, to block out a suitable check for each one of them. 1. From one corner P of a triangular field PQB the side PQ bears N. 10° E. 100 rods. QR bears N. 63° E. and PR bears N. 38° 10' E. Find the perimeter and area of the field. 2. The town B lies 15 miles east of A, C lies 10 miles south of A. X lies on the Hne BC, and the bearmg of AX is S. 46° 20' E. Find the distances from X to the other three towns. 3. To find the length of a lake (Fig. 135), the angle C = 48° 10', the side a = 4382 feet, and the angle B = 62° 20' were measured. Find the length of the lake c, and check. 4. To continue a line past an obstacle L, Pig. 136, the line BC and the angles marked at B and C were measured and found to be 1842 feet, 28° 15', and 67° 24', respectively. Find the distance CD, and the angle at D necessary to continue the line AB; also compute the distance BD. 5. Find the longer diagonal of a parallelogram, two sides being 69.1 and 97.4 and the acute angle being 29° 34'. What is the magnitude of the single force equivalent to two forces of 69.1 and 97.4 dynes respectively, making an angle of 29° 34' with each other? 6. A force of 75.2 dynes acts at an angle of 35° with a force F. Their resultant is 125 dynes. What is the magnitude of Fl §182] TRIGONOMETRIC " EQUATIONS 335 7. The equation of a circle is p = 10 cos 6. The points A and B on this circle have vectorial angles 31° and 54° respectively. Find the distance AB, (1) along the chord; (2) along the arc of the circle. 8. Knd the lengths of the sides of the triangle enclosed by the straight lines : e = 26° 115°; p cos (9 - 45°) = 50. Fig. 135. — Diagram for Problem 3. Fig. 136. — Diagram for Problem 4. 9. A gravel heap has a rectangular base 100 feet long and 30 feet wide. The sides have a slope of 2 in 5. Find the number of cubic yards of gravel in the heap. 10. A point B is invisible and inaccessible from A and it is necessary to find its distance from A. To do this a straight line is run from A to P and continued to Q such that B is visible from P and Q. The following measurements are then taken: AP, = 2367 feet; PQ = 2159 feet; APB = 142° 37'.3; AQB = 76° 13'.8. Find AB. 11. To determine the height of a mountain the angle of elevation of the top was taken at two stations on a level road and in a direct line with it, the one 5280 yards nearer the mountain than the other. The angles of elevation were found to be 2° 45' at the further station and 3° 20' at the nearer station. Find the horizontal distance of the mountain top from the nearer station and the height of the mountain above it. Use S and T functions. 12. Explain how to find the distance between two mountain peaks Ml and Af 2, (1) when A and B at which measurements are taken are in the same vertical plane with Mi and M^; (2) when neither A nor B is in the same vertical plane with Mi and M2. 13. The sides of a triangular field are 534 yards, 679 yards, and 474 yards. The first bears north, and following the sides in the order here given the field is always to the left. Find the bearing of the other two sides 'and the area. 336 ELEMENTARY MATHEMATICAL ANALYSIS [§182 14. From a triangular field whose sides are 124 rods, 96 rods, and 104 rods a strip containing 10 acres is sold. The strip is of uniform width, having as one of its parallel sides the longest side of the field. Knd the width of the strip. 16. Three circles are externally mutually tangent. Their radii are 5, 6, and 7 feet. Find the area and perimeter of the three-cornered area enclosed by the circles and the length of a wire that will enclose the group of three circles when stretched about them. 16. To find the distance between two inaccessible objects C and D, Kg. 137, two points A and B are selected from which both objects are visible. The dis- 137_ Diagram for tS'^ce AB is found to be 7572 feet. Problem 16. The following angles were then taken: ABD = 122° 37' BAC = 80° 20' ABC = 70° 12' BAD = 27° 13' Knd the distance DC and check. 17. A circle of radius o has its center at the point (pi, 9i). Knd its equation in polar coordinates. (Use law of cosines.) 18. A surveyor desired the distance of an inaccessible object from A and B, but had no instruments to measure angles. He measured AA' in the Une AO, BB' in the line BO; also AB, BA', and AB'. How did he find OA and OB? 19. From a point A a distant object C bears N. 32° 16' W. and from B the same object bears N. 50° W. AB bears N. 10° 39' W. The distance AB is 1000 yards. Knd the distance AC. 20. The angle of elevation of a mountain peak is observed to be 19° 30'. The angle of depres.sion of its image reflected in a lake 1250 feet below the observer is found to be 34° 5'. Find the height of the mountain above the observer and the horizontal distance to it. (See Fig. 138.) 21. One side of a mountain is a smooth eastern slope inclined at an angle of 26° 10' to the horizontal. At a station A a vertical shaft is sunk to a depth of 300 feet. From the foot of the shaft two horizontal tunnels are dug, one bearing N. 22° 30' E. and the other S. 65° E. These tunnels emerge at B and at C respectively. Find the lengths of the tunnels and the lengths of the sides of the triangle ABC. 22. A rectangular field ABCD has side AB = 40 rods; AD = 80 rods. Locate a point P in the diagonal AC so that the perimeter of the triangle APB will be 160 rods. {Hint: Express perimeter as a function of angle at P.) §182] TRIGONOMETRIC EQUATIONS 337 8. Find the area enclosed by the lines y = k' y = \/3 x, and the Fig. 138. — Diagram for Problem 20. circle x' — lOs + ^^^ = 0. (Hint: Change to polar coordinates.) 24. The displacement of a particle from a fixed point is given by d = 2.5 cos t + 2.5 sin t. What values of t give maximum and minimum displacements; what is the maximum displacement? 25. A quarter section of land is enclosed by a fence. A farmer wishes to make use of this fence and 60 rods of additional fencing in making a triangular field in one comer of the original tract. Find the field of greatest possible area. Show that it is also the field of maximum perimeter, under the conditions given. 26. A force Fi = 100 dynes makes an angle of 6° with the horizontal, and a second force Fi = 50 dynes makes an angle of 90° with Fi. Determine B so that (1) the sum of the horizontal components of Fi and Ft shall be a maximum; (2) so that the sum of the vertical com- ponents shall be zero. 27. Find the area of the largest triangular field that can be enclosed by 200 rods of fence, if one side is 70 rods in length. 22 338 ELEMENTARY MATHEMATICAL ANALYSIS [§182 28. Change the equation of the curve xy = I to polar coordinates, rotate through — 45°, and change back to rectangular coordinates. 29. A particle moves along a straight line so that the distance varies directly as (sin t + cos t). When t = 7r/4, the distance is 10. Find the equation of motion. 30. From the top of a lighthouse 60 feet.high the angle of depression of a ship at anchor was observed to be 4° 52'; from the bottom bf the lighthouse the angle was 4° 2'. Required the horizontal distance from the lighthouse to the ship and the height of the base of the lighthouse above the sea. 31. The Une AB runs north and south. The line AC makes an angle of 52° 8'. 6 with AB. Locate the Une BC perpendicular to AB so that the area ABC shall be 1 acre. 32. University Hall casts a shadow 324 feet long on the hillside on which it stands. The slope of the hillside is 15 feet in 100 feet, and the elevation of the sun is 23° 27' Find the height of the building. 33. To determine the distance of a fort A from a place B, a line BC and the angles ABC and BCA were measured and found to be 3225.5 yards, 60° 34', and 56° 10' respectively. Find the distance AB. 34. A balloon is directly over a straight level road, and between two points on the road from which it is observed. The points are 15,847 feet apart, and the angles of elevation are 49° 12' and 53° 29'. Find the height. 35. Two trees are on opposite sides of a pond. Denoting the trees by A and B, we measure AC = 297.6 feet, BC = 864.4 feet, and the angle ACB = 87° 43'. Find AB. 36. Two mountains are 9 and 13 miles respectively from a town, and they include at the town an angle of 71° 36'. Find the distance between the mountains. 37. The sides of a triangular field are, in clockwise order, 534 feet, 679 feet, and 474 feet; the first bears north; find the bearings of the other sides and the area. 38. Under what visual angle is an object 7 feet long seen when the eye is 15 feet from one end and 18 feet from the other? 39. The shadow of a cloud at noon is cast on a spot 1600 feet west of an observer, and the cloud bears S., 76° W., elevation 23°. Find its height. CHAPTER XI SIMPLE HARMONIC MOTION AND WAVES 183. Simple Harmonic Motion. In Fig. 139, x = 0T> = a cos DOM, where a is the radius of the circle. If now the point M is thought of as moving with constant or uniform speed on the circle, starting at A, or (which amounts to the same thing) if the radius OM is thought of as moving with constant angular velocity, say k radians per second, starting from OA, then angle DOM = kt and the position of the point D at time t is given by X = a cos kt, (1) where t is the time in seconds required for OM to move from posi- tion OA to position OM. Let us study the motion of the point D as M moves on the circle with constant speed. D starts at A and moves to the left with increasing speed until it arrives at ■ 0, where its speed begins to decrease, decreasing to at A'. Then the point moves to the right with increasing speed until it again passes through 0, after which its speed diminishes, becoming when it arrives at A . Then the whole motion is repeated. A body whose position on a straight line is given at any instant by an equation of the form (1), that is one which moves as the point D does, is said to describe simple harmonic motion. On account of the frequency with which this term will occur, we shall abbreviate it by the symbols S.H.M. Examples .339 Fig. 139. 340 ELEMENTARY MATHEMATICAL ANALYSIS [§183 of bodies that move approximately in this way are: The bob of a pendulum, a point in the prong of a vibrating tuning fork, a point in a vibrating violin string, the particles of air during the passage of a sound wave. The motion is oscillatory in character and repeats itself in definite intervals of time. The length of this interval can be easily found by considering the motion of the point M on the circle. The point D starting from any given position will return to this position moving in the same direction after an interval of time which is the time required for M to describe the circle, i.e., after 2ir/k seconds, the time in which the radius OM describes the angle 2ir radians at the rate of k radians per second. This time within which a body executing S.H.M. performs a complete oscillation is called the period of the S.H.M. It is denoted by T. Thus T = ^. (2) This expression can be obtained directly from the equation x = a cos kt by means of the fact that the cosine is a periodic function of period 2x. The period T is the amount by which t must be increased in order to increase the angle kt by the amount 27r. If t be increased by the amount 2ir/k, then kt is increased by 2x, because fc(t+^) =A;i + 2ir. The number of complete periods per second is ^ = T = .V (3) N is called the frequency of the S.H.M. 'if instead of counting time from the instant at which the auxiliary point M passed through A, we count it from the instant it passed through E, then ZEOM = kt, and it is clear that ZAOM = (kt — e) if e stands for the constant angle EOA. Then (1) becomes X = a cos (Jet — «). (4) The number a is called the amplitude, e is called the epoch angle, §184] SIMPLE HARMONIC MOTION AND WAVES 341 and (Jet — e) is called the phase angle of the S.H.M. represented by (4). In like manner the point D2, the projection of the point M upon the vertical diameter of the circle in Fig. 139, describes S.H.M. Its equation is 2/ = a sin (kt — e), (5) where time t is measured from the instant M passes through E. 184. Mechanical Generation of S.H.M. Fig. 140 illustrates a way in which S.H.M. may be described by mechanical means. rp ^ - B —3" \r\\ ^ ^. < ^4^^ ho. C K 1 ^ \ 2?__"5 =__ > s' —L 7L S '^ : - - - ,7 _ _ _ _ s_ . '-'-1 s, 7 ■S s :, , ^ €. S ., _ Si^: cj- 7 _S ^1 \ ii _^ c 1 S ' ' ^ ~-~. ■s ? v^ "5 7' \? rs M "--III II llml II 1 WyW II H ■^ |LJ_LiJ_LJ_LJ_ri I ^ 1 Pig. 140. — Mechanical generation of simple harmonic motion, and of a simple progressive wave. Let the uniformly rotating wheel OAB be provided with a pin M attached to its circumference and free to move in the slot of the cross-head as shown, the arm "of the cross-head being re- stricted to vertical motion by suitable guides G-G\- Then, as the wheel rotates, any point P of the arm of the cross-head describes simple harmonic motion in a vertical direction. The amplitude of the S.H.M. is the radius of the circle, or OB; its period is the time required for one complete revolution of the wheel. 342 ELEMENTARY MATHEMATICAL ANALYSIS [§18S Exercises 1. Find the periods of the following S.H.M. : {a) y = 3 sin 2t. (e) y = a sin (10< — 7r/3). (6) 2/ = 10 sin (1/2) « (/) t/ = o sin (2«/3 - 27r/5). (c) y = 7 cos 4<. ig) y = a sin (6< + c). {d) y = a sin 27r<. 2. Give the amplitudes and epoch angles in each of the instances given in exercise 1. 3. The bob of a second's pendulum swings a maximimi of 4 cm. each side of its lowest position. Considering the motion as rectilinear S.H.M., write the equation of motion.' ^ The term period is used differently in the case of a pendulum than in the case of S.H.M. The time of a swing is the period of a pendulum; the time of a awino- swang is the period of a S.H.M. Write the equation of motion of a pendulum of the same length which was released from the end of its swing 1/2 second after the first pendulum was similarly released. 4. A particle moves in a straight Une in such a way that its dis- placement from a fixed point of the line is given by d = 2 cos* t. Show that the particle moves in S.H.M., and find the amplitude and period of the motion. 6. A particle moves in a vertical circle of radius 2 units with angular velocity of 20 radians per second. Counting time from the instant the particle was at its lowest position, write the equation of motion of its projection (1) upon the vertical diameter; (2) upon the horizon- tal diameter; (3) upon the diameter bisecting the angle between the horizontal and vertical. 186. S.H.M. Record on Smoked Glass. If P, Fig. 140, be a tracing point attached to the vertical arm of the cross-head and capable of describing a curve on a piece of smoked glass, HK, which is moved to the right at constant speed, then when P describes S.H.M. in the vertical line OP, the curve NiCTNJ' traced on the plate HK is a sinusoid. For, if iVj be taken as origin, and if for convenience positive abscissas be measured to the left, the coordi- nates of P are X = Vt, and y = a sin {kt — e) §186] SIMPLE HARMONIC MOTION AND WAVES 343 where V is the linear velocity of the plate. Eliminating t between these two equations, y = asin yy - ej (1) the equation of a sinusoid. If the plate HK moves with the same velocity as the point M, we have V = ha and equation (1) becomes - = sm -, (2) a a the equation of an undistorted sinusoid.' 186.* Composition of Two S.H.M.'s at Right Angles. It is obvious that X = a cos ht represents a S.H.M. one quarter of a period in advance of a;' = a sin kt, since sin I fci + ^1 = cos kt. A pair of S.H.M.'s possessing this property are said to be in quadrature. (4) and (5), §183, may be said to be in quadrature. We have shown that if a point M, moving uniformly on a circle, be projected upon both the X- and 7-axes, two S.H.M.'s result. The phase angles of these two motions differ from each other by 7r/2. The converse of this fact, namely that uniform motion in a circle may be the resultant of two S.H.M.'s in quadrature, is easily proved, for the two equations of S.H.M. X = a cos kt y = a sin kt are obviously the parametric equations of a circle. Hence the theorem : Uniform motion in a circle may he regarded as the residtant of two S.H.M.'s of equal amplitudes and equal periods and differing by 7r/2 in phase angle. This important truth is illustrated by Fig. 141. Let the X- and 1 The student should note that ^ = sin - is of exactly the same shape as y = sm x, for multiplying both ordinates and abscissas of any curve by a is merely constructing the curve to a different scale. However, ^ = sin o is a distorted sinusoid, for the ordinates of y = sin x are multiplied by 3 while the abscissas are multiplied only by 2. 344 ELEMENTARY MATHEMATICAL ANALYSIS [§187 y-axes be divided proportionally to the trigonometric sine, as in Fig. 59. Through the points of division of the two axes draw lines per- pendicular to the axes, thus dividing the plane into a large number of small rectangles. Starting at the end of one of the axes, and sketch- ing the diagonals of successive cornering rectangles, the circle ABA'B' is drawn. If the same construction be carried out for the case in which the Y- axis is divided proportionally to 6 sin kt and in which the X-axis is divided proportionally to osin kt, the ellipse AiBiA'iB'i re- sults. These facts are merely a repetition of the statements made in §84. 187. Waves.— Let Fig. 142 represent a section obtained by passing at any instant a vertical plane perpendicular to the crests of a series of small waves on the surface of a body of water. The wavy line represents the ap- pearance of the surface at any instant. It is a fact that its equation is, in the case of small waves or ripples, ■ 1 , B -- -7- — ' — — — ^s-:: 7 ^ S / S / ^ , \ \ 1 \ \ 7 A 7 _ .__ — — — — ^2-: ■p ff\r — — — ^ V--S- — F — — — — g — 5, n' 1 Ai -- [ 1 , ■-: . = -^ 1 1 i--^-^- Ax 2/ = a sin he. (1) Fig. 142 represents the seo- ■^1 tion of the surface at any Fig. 141. — The circle and the instant, say t = 0. Now eUipse considered as generated by „„„i „,„„„„ „„„„ c a two S.H.M.'s in quadrature. ^'^.^f ^*^«3 °iove forward with a constant velocity, which we shall call Y. The wavy form is sinusoidal in section but of course it is not fixed, but keeps moving ahead. Hence the moving sinusoid of Fig. 140 may be looked upon as a repre- sentation of this kind of phenomena. The curve described on the moving plate UK of Fig.- 140, if referred to coordinate axes moving with the plate, is the sinusoid, or §187] SIMPLE HARMONIC MOTION AND WAVES 346 sine curve, whose equation is (1) above. If, however, we consider this curve as referred to the fixed origin Oi, then the moving sinusoid thus conceived is called a simple progressive sinusoidal wave or merely a wave. Under the conditions represented in Fig. 140, it is a wave progressing to the right with the uniform speed of the plate HK. At any single instant, the equation of the curve is y = a sin h{x - OiN), (2) where OiN is the distance that the node N has been translated to the right of the origin Oi. If V be the uniform velocity of translation of HK, then, OiN = Vt (3)1 Fig. 142. and the equation of the wave is y = asiah{x — Vt), or y = a sin Qix — hVt), or y = a sin (hy — kt), if k be put for hV, so that V = - (4) (5) Because of the presence of the variable t, (4) is not the equation of a fixed sinusoid, but of a moving sinusoid or wave. Applying the same terms used fbr S.H.M., the expression {hx — ht) is the phase angle, the expression (+ kt) is the epoch angle and a is the amplitude of the wave. See Fig. 143a and c. The expression Qix — kt) is a linear function of the variables 1 In what follows, t is not the time elapsed since itf , Fig. 140, was at A, as used in S183| but is the elapsed time since N was at 0i, These values of t differ by the time of half a revolution or by ir/k. 346 ELEMENTARY MATHEMATICAL ANALYSIS [§187 X and t. The sine or cosine of this function is called a simple harmonic fmiction of x and t. The wave form on the surface of water moves along with fixed velocity V. The particles of water, however, do not share in this b ^''S<^^'yr> <. X X X/ Fig. 143. — Wave forms, (a) of different amplitude; (5) of different wave lengths; (c) of different phase or epoch angles. forward motion. Each particle on the surface moves up and down in a vertical line as the wave form passes it. In fact we shaU now see that each particle describes S.H.M. in a vertical direction. To examine the motion of a single particle of water, we have §188] SIMPLE HARMONIC MOTION AND WAVES 347 only to regard x as constant, say x = Xi, in equation (4) above The displacement of this particle is then given by y = a sin (hxi — kt) or y = — a sin (M — hxi). That is y = a sin {kt — hxi — ir). (6) This is the equation of a S.H.M. whose period is T = 2T/k. The epoch angle is hxi + ir. This will be different for different par- ticles. This means that the phase angles of the S.H.M. of succes- sive particles differ, but they all oscillate up and down with the same period 2ir/k. 188. Wave Length. The wave length of a progressive wave is the distance from crest to crest or from trough to trough. It is the amount by which x must be increased in the equation of the wave in order that the angle (hx — kt) may be increased by 2ir. Hence the wave length, ^ = ¥- « 189. Period or Periodic Time. If we fix our attention upon any particular or constant value of x, and view the progressive wave as it passes the vertical line through this abscissa, the elapsed time from the passage of one crest to the next crest is called the period, or periodic time. It is readily seen to be the increment in t which changes the angle (hx — kt) by the amount 2ir. Hence the period The expression T is called the periodic time, or period, of the wave. It is the length of time required for the wave to move one wave length. To contrast wave length and period, think of a per- son in a boat anchored at a fixed point in a lake. The time that the person must wait at that fixed point (x constant) for crest to follow crest is the periodic time. The wave length is the distance he observes between crests at a given instant of time (t constant) . The number of periods per unit of time is called the frequency of the wave. Hence, if N represent the frequency of the wave, N = |'=|- (2, 348 ELEMENTARY MATHEMATICAL ANALYSIS [§190 190. Velocity or Rate of Propagation. The rate of movement V of the sinusoid on the plate HK, Fig. 140, is shown by equation (5), §187, to be k/h units of length per second. This is called the velocity of the wave or the velocity of propagation. The equa- tion of the wave may be written 2/ = o sin h{x — Yt). From equations (1) §188 and (1) §189 we obtain A; and since 7 = r, we have K k_L h~ t' V =^. (1) This equation is obvious from general considerations, for the wave moves forward a wave length L in time T, hence the speed of the wave must be m' 191. L and T Equation of a Wave. If we solve equations (1) §188 and (1) §189 for h and k respectively, and substitute these values of h and ifc in the equation 2/ = a sin {hx — kt) we obtain ■\i-a- From this form it is seen that the argument of the sine increases by 2ir when either x increases by an amount L or when t increases by the amount T. By use of (1), §190, the last equation may also be written ^- -' - .(2) a sm2^ y = asm L^^- Vt). 192. Phase, Epoch, Lead. Consider the two waves y . I27r\ = a sm y- (a; — Vt) y = a sin 2ir . j-ix- vt- E) a) (2) The amplitudes, the wave lengths and the velocities are the §192] SIMPLE HARMONIC MOTION AND WAVES 349 same in each, but the second wave is in advance of the first by the amount E (measured in linear units), for the second equation can be obtained from the first by substituting (x — E) for x, which translates the curve the amount E in the OX direction. In this case E is called the lead (or the lag if negative) of the second wave compared with the first. The lead is a linear magnitude measured in centimeters, inches, feet, etc. The terms phase and epoch are sometimes used to designate the time, or, more accurately, the fractional amount of the period required to describe the phase angle and epoch angle respectively. In this use, the phase is the fractional part of the period that has elapsed since the moving point last passed through the middle point of its simple harmonic motion in the direction reckoned as positive. See Fig. 143c. The tidal wave in mid-ocean, the ripples on a water surface, the wave sent along a rope that is rapidly shaken by the hand, are illustrations of progressive waves of the type discuseed above. Sound waves also belong' to this class if the alternate condensations and rarefactions of the medium be graphically represented by ordinates. The ordinary progressive waves observed upon a lake or the sea are not, however, progressive waves of this type. The surface of the water in this case is not sinusoidal in form, but is represented by another class of curves known in mathematics as trochoids. Exercises 1. Derive the amplitudOj the wave length, the periodic time, the velocity of propagation of the following waves : (a) y = a sin {2x — 3<). , > .„„ . 2w, _.. .. (b) y =5 sin (0.75a; - lOOOi). W V = 10° ^25^"^ ~ ^°' " ^^■ (c) 2/ = 10 sin (I - .*) . (/) 2/ = 100 sin(5x + 4t). 2,r (?) y = 0-025 sin ^(,x + 0), to the ordinates of a progressive wave y = asm {hx + /c<) (2) traveling to the left. Expanding the right members of (1) and (2) by the addition formula for the sine and adding y = 2a cos kt sin hx, (3) or in terms of L and T, §188 (1) and §189 (1), ' y = 2a cos (^) sin {~:j ■ (4) In Fig. 144, the origin is at and the X-axis is the Line of nodes ONX. If in equation (3) we look upon 2a cos kt as the vari- able amplitude of the sinusoid y = sin hx, we note that the nodes, of the sinusoid remain stationary, but that the amplitude 2a cos kt changes as time goes on. When t = 0, the sine curve has amplitude 2a and wave length 2ir/h. When t = ir/2k, or T/i, the sinusoid is reduced to the straight line y = 0. When t = ir/k, or T/2, the curve is the sinusoid y = — 2a sin hx which has a trough where the initial form had a crest, or vice versa. Exercises In the following exercises the height of the wave means the maxi- mum rise above the line of nodes. When a seiche is uninodal, the shores of the lake correspond to the points I and K, Fig. 144. When a seiche is binodal, the points / and H are at the lake shore. 1. From the equation of a stationary wave in the form y = 2a sin %rx/L cos 2-wtlT, show that K, Fig. 144, is at its lowest depth fori = r/2,,3r/2, 67/2, . 2. Henry observed a fifteen-hour uninodal seiche in Lake Erie, which was 396 kilometers in length. Write the equation of the prin- cipal or uninodal stationary wave if the amplitude of the seiche was 15 cm. 3. A small pond 111 meters in length was observed by Eridros to have a uninodal seiche of period fourteen seconds. Write the equation of the stationary wave if the ampUtude be o. 352 ELEMENTARY MATHEMATICAL ANALYSIS [§194 4. Forel reports that the uninodal longitudinal seiche of Lake Geneva has a period of seventy-three minutes and that the binodal seiche has a period of thirty-five and one-half minutes. The trans- verse seiche has a period of ten minutes for the uninodal and five minutes for the binodal. The longitudinal and transverse axes of the lake are 45 miles and 5 miles respectively. Write the equation of these different seiches. 5. A standing wave or uninodal seiche exists on Lake Mendota of period twenty-two minutes. If the maximum height is 8 inches and the distance .across the lake is 6 miles, write the equation of the seiche. 194.* Compound Harmonic Motion and Compound Waves. The addition of two or more simple harmonic functions of frequencies which are multiples of the frequency of a given first or fundamental harmonic, gives rise to compound harmonic motion. Thus, y = a sin fc< + & sin Zkt, corresponds to the superposition of a S.H.M. of period 2ir/3fc and amplitude 6 upon a fundamental S.H.M. of period 2ir/A; and amplitude a. To compound motions of this type, there correspond compound waves of various sorts, such as a fundamental sound wave with overtones, or tidal waves in restricted bays or harbors. The graphs of the curves y = sin X + sin 2x y = ainx + sin 3a are easily constructed. They may be drawn by adding the ordinates of the various sinusoids constructed on the same axis, as in Fig. 145. To compound the curves, first draw the component curves, say y = sin X and y = sin 3x of Kg. 145. Then use the edge of a piece of paper divided proportionally to sin x (that is, like the scale OB, Fig. 145) and use this as a scale by means of which the successive ordinates of a given X may be added. For example, to locate the point on the composite curve corresponding to the abscissa OD, Fig. 145, we must add DP and DQ. Hence place vertically at P the lower end of the paper scale just mentioned. The sixth scale division above P on this scale will then locate the required point M of the composite scale.' In Fig. 146 the curves: y = sin X + sin (2x + 27rn/16) y = sin 2x + sin (3x + 2)rre/16) I Note that if the method described be used, there is really no need of drawing the curve y = sin 3a:. If both curves are drawn, ordinates may conveniently be added with bow dividers. §194] SIMPLE HARMONIC MOTION AND WAVES 353 are shown for values of n = 0, 1, 2, . , IS in succession — that is, for successive phase differences corresponding to one-sixteenth of the wave length of the fundamental y = sin x. Fig. 145. — The curves y = sin x,y= sin 3x and the compound curve y = sin X + sin 3x. '^ Fifth Fig. 146. — The curves (o) j/ = sin x + sin {2x + 27rn/16) and (b) 2/ = sin 2a; + sm (3x + 2Tn/16), for n = 0, 1, 2, . . 15. {From Thomson and Tail.) Wave forms compounded from the odd harmonics only are espe- eially important, as alternating-current curves are of this type. See Fig. 147. 23 354 ELEMENTARY MATHEMATICAL ANALYSIS [§196 196.* Harmonic Analysis. Fourier showed in 1822 in his "Ana- lytical Theory of Heat" that a periodic single-valued function, say y — f(x), under certain conditions of continuity, can be represented by the sum of a series of sines and cosines of the multiple angles of the form y = ao + ai cos x + a^ cos 2x + Oa cos 3x + . . . + bi sin X +bi sin 2a; -|- 63 sin 3a; + . . . This means, for example, that it is always possible to represent the complex tidal wave in a harbor, by means of the sum of a number of simple waves or harmonics. The term harmonic analysis is given to the process of determining these sinusoidal components of a compound periodic curve. In §194 we have performed the direct operation of 50 V / / \ 26 / \ / s / \ S / ■ 1 ) 1 2 1 1 1 6H ) ! 2 2 4 2 i 2 i £ 3 ! 3 V \ > 25 \ /. V / 60 \ / ~" ' Fig. 147. — An alternating current curve. present. Only odd harmonics are finding the compound curve when the component harmonics are given. The inverse operation of finding the components when the compound curve is given is much more difficult, and its discussion must be post- poned to a later course. 196.* Connecting Rod Motion. If one end of a straight Une B be required to move on a circle while the other end of the line A moves on a straight Une passing through the center of the circle, the resulting motion is Icnown as connecting rod motion. The connecting rod of a steam engine has this motion, as the end attached to the crank travels in a circle while the end attached to the cross-head travels in a straight line. The motion of the end A, Fig. 148, of the connecting rod is approximately S.H.M. The approximation is very close if the §196] SIMPLE HARMONIC MOTION AND WAVES 355 connecting rod be very long in comparison with the diameter of the circle. A second approximation to the motion of the point A can be obtained by introducing the second harmonic or octave of the funda- mental. In Fig. 148, let the radius of the circle be a and the length of the connecting rod be I. The length of the stroke M'N is 2a, and the origin may conveniently be taken at the mid-point of the stroke, 0. When B is at E, A is at M and when B is at K, A is at A'^. Then MH = NK = I and OC = I. Now But and Hence X = CA - CO = CA - I = CD + DA - I. CD = a cos e DA = Vl^ - BD' = Vl' - a^ sin^ e. X = acose +1 Vl - (a^/l^) sin^ B - I (1) (2) (3) (4) Fig. 148. — Connecting rod motion. Approximating the radical by §113 (\/l — x = 1 — x/2) we obtain ^ , , / , a^ sin^ e\ , ,,, X = acos 9 +1 il 2p — ) ~ '• (^) Since sin^ 9 = (1 — cos 26) /2, we obtain X = a cos 9+27 "^"^ ^^ ~ 47' ^^^ which is approximately true as long as I is much greater than a. It is seen from the above result that the second approximation to connecting rod motion contains as overtone the octave, or second a* harmonic, ^j cos 29, in addition to the first or fundamental harmonic a cos 8. 356 ELEMENTARY MATHEMATICAL ANALYSIS [196 § Exercises 1. Draw the curve corresponding to equation (5) above if o' = 1.15 inches, and Z = 3 inches. 2. The motion of a slide valve is given by an equation of the form 3/ = oi sin (9 + e) + 02 sin (28 + 90°). Draw the curve if ai = 100, oj = 25, c = 40°, using 6 as the abscissa in rectangular coordinates. CHAPTER XII COMPLEX NUMBERS 197. ScaJes of Numbers. To measure any magnitude, we apply a unit of measure and then express the result in terms of numbers. Thus, to measure the volume of the liquid in a cask we may draw off the liquid, a measure full at a time, in a gallon measure, and conclude, for example, that the number of gallons is 125. In this case the number 12^ is taken from the arith- metical scale of numbers, 0, 1, 2, 3, 4, . . If we desire to meas- ure the height of a stake above the ground, we may apply a foot- rule and say, for example, that the height in inches above the ground is 12|, or, if the positive sign indicates height above the ground, we may say that the height in inches is -|- 12J. In this latter case the number -|- 12? has been selected from the algebraic scale of numbers . . . — 4, — 3, — 2, — 1, 0, + 1, + 2, + 3, + 4, . . The scale of numbers which must be used to express the value of a magnitude depends entirely upon the nature of the magnitude. The attempt to express certain magnitudes by means of numbers taken from the algebraic scale may sometimes lead, as every student of algebra knows, to meaningless absurdities. Thus a problem involving the number of sheep in a pen, or the number of marbles in a box, or the number of gallons in a cask, cannot lead to a negative result, for the magnitudes just named are arithmetical quantities and their meas- urement leads to a number taken from the arithmetical scale. The absurdity that sometimes appears in results to problems concerning these magnitudes is due to the fact that one attempts to apply the notion of algebraic number to a magnitude that does not permit of it. Science deals with a great many different kinds of magnitudes, the measurement of some of which leads to arithmetical numbers while the measurement of others leads to algebraic numbers; the remarkable fact is that two different number scales serve adequately to express magnitudes of so many different sorts.' The magnitudes of science 357 358 ELEMENTARY MATHEMATICAL ANALYSIS [§198 are so various in kind that one might reasonably expect that the variety of number systems required in the mathematics of these sciences would be very great. The arithmetical scale is used when we enumerate the number of gallons in a cask and say: 0, 1, 2, 3, . . . If we observe 3 gallons in the cask, and then remove one, we note those remaining and say tiDo; we may remove another gallon and say one, we may remove the last gallon and say zero; but now the magnitude has come to an end. The algebraic scale is used when we measure in inches the height of a stake above the ground and say three. We may drive the stake down an inch and say two; we may drive the stake another inch and say one; we may drive the stake another inch and say zero, or "level with the ground;" but, unUke the case of the gallons in the cask, we need not stop but may drive the stake another inch and say one below the ground, or, for brevity, minus one; and so on. Many of the magnitudes considered in science are completely ex- pressed by means of arithmetical numbers only; for example, such magnitudes as density or specific gravity; temperature;^ electrical re- sistance; quantity of energy; such as ergs, joules or foot-pounds; power, such as horse power, kilowatts, etc. All of the magnitudes just mentioned are scalar, as it is called; that is, they exist in one sense only — ^not in one sense and also in the opposite sense, as do forces, velocitiesj distances, as explained above. The arithmetical scale of numbers is therefore ample for their expression. The distraction, then, between an algebraic number and an arith- metical number is the notion of sense which must always be associated with any algebraic number. Thus an algebraic number not only answers the question "how many" but also affirms the sense in which that number is to be understood; thus the algebraic number -|- 12 J, if arising in the measurement of angular magnitude, refers to an angular magnitude of 12| units (degrees, or radians, etc.) taken in the sense defined as positive rotation. 198. Algebraic Number Not the Most General Sort. Algebraic numbers, although more general than arithmetical numbers, are themselves quite restricted. For, each algebraic number corre- sponds to a point of the algebraic scale. But for points not on the scale there corresponds no algebraic number. That is, the alge- braic scale is one-dimensional. It is thus seen that there is an ^ Temperature is an arithmetical quantity, since there is an absolute zero of temperature. Temperature does not exist in two opposite senses, but in a single sense. §199] COMPLEX NUMBERS 359 opportunity of enlarging our conception of number if we can re- move the restriction of one dimension — that is, if we can get out of the line of the algebraic scale and set up a number system such that one number of the system will correspond, for examiple, to each point of a plane, and such that one point of the plane will corre- spond to each number of the system. We will seek, therefore, an extension or generalization of the number system of algebra that will enable us to consider, along with the points of the algebraic scale, those points which lie without it. 199. Numbers as Operators. The extension of the number system mentioned in the last section may be facilitated by chang- ing the conception usually associated with symbols of number. The usual distinction in algebra is between symbols of number and symbols of operation. Thus a symbol which may be looked upon as answering the question "how many" is called a number, whUe a symbol which tells us to do something is called a symbol of opera- tion, or, simply, an operator. Thus in the expression -\/2) "v/ is a symbol of operation and 2 is a number. A symbol of operation may always be read as a verb in the imperative mood; thus we may read -s/x: "Take the square root of x." Likewise log x, and cos 9 may be read; "Find the logarithm of x," "Take the co- sine of 8." In these expressions "log" and "cos" are symbols of operation; they teU us to do something; they do not answer the question "how many" or "how much" and hence are not num- bers. Here we speak of -\/j log, cos, as operators ; we speak of x as . the operand, or that which is operated upon. It is interesting to note that any number may be regarded as a symbol of operation; by doing so we very greatly enlarge some original conceptions. Thus, 10 may be regarded not only as ten, answering the question "how many," but it may quite as well be regarded as denoting the operation of taking unity, or any other operand that follows" it, ten times; to express this we may write 10-1, in which 10 may be called a tensor (that is, "stretcher"), or a symbol of the operation of stretching a unit until the result obtained is tenfold the size of the unit itself. In the same way the symbol 2 may be looked upon as denoting the operation of doubling unity, or of doubling any operand that follows it; like- 360 ELEMENTARY MATHEMATICAL ANALYSIS [§199 wise the tensor 3 may be looked upon as a trebler, 4 as a quadrupler, etc. With the usual understanding that any symbol of operation operates upon that which follows it, we may write compound operators like 2-2-3-1. Here 3 denotes a trebler and 31 denotes that the unit is to be trebled, 2 denotes that this result is to be doubled and the next 2 denotes that this result is to be doubled. Thus representing the unit by a line running to the right, we have the following representation of the operators : The unit -> 3-1 -^^-^ 2-31 > > 2-2-31 T > > Notice the significance that should now be assigned to an expo- nent attached to these (or other) symbols of operation. The exponent means to repeat the operation designated by the operator; that is, the operation designated by the base is to be performed, and performed again on this result, and so on, the number of opera- tions being denoted by the exponent. Thus W means to perform the operation of repeating unity ten times (indicated by 10) and then to perform the operation of repeating the result ten times, that is, it means 10 (101). Also, 10' means 10[10(10-1)]. Like- wise log^ 30 means log (log 30) which, if the base be 10, equals log 1.4771, or finally 0.1694. An apparent exception- occurs in the case of the trigonometric functions. The expression cos'j; should mean, in this notation, cos (cos x), but because trigo- nometry is historically so much older than the ideas here ex- pressed, the expression cos'' x came to be used for (cos a;)', or (cos x) X (cos x), but cos~^ 6 means arc cos 6, not 1/cos 9. To be consistent with the notation of elementary mathematics, the expression \/4, looked upon as a symbol of operation, must denote an operation which must be performed twice in order to be equivalent to the operation of quadrupling; that is, such that (-\/4)^ = 4. Likewise i/i denotes an operation which must be' performed three times in succession in order to be equivalent to quadrupling. But we know that the operation denoted by 2, if performed twice, is equivalent to quadrupling; §200] COMPLEX NUMBERS 361 therefore \/4 = 2, etc. Just as 4^, 4', etc., may be called stronger tensors than a single 4, so -s/i, Vi, etc. may be called weaker tensors than the operator 4. 200. Reversor. The expression ( — 1), looked upon as a symbol of operation, is not a tensor, as it leaves the size unchanged of that upon which it operates. If this operator be applied to any magnitude, it will change the sense in which the magnitude is then taken to exactly the opposite sense. Thus, if 6 stands for six hours after, then ( — 1)(6) stands for six hours before a certain event, and ( — 1) is the sj'mbol of this operation of reversing the sense of the magnitude. Also if 6 stands for a line running six units to the right of a certain point, then ( — 1)(6) stands for a line running six units to the left of that point; so that ( — 1) is the symbol which denotes the operation of turning the straight line through 180°. As 2, 3, 4, when looked upon as symbols of operations, were called tensors, the operator ( — 1) may conveniently be designated a reversor. Exercises Show graphically the effect of the operations indicated in each of the following exercises. Take as the initial unit-operand a straight line 1/2 inch long extending to the right of the zero or initial point. Explaia each expression as consisting of the operand unity and symbols of operation — ^tensors, reversors, etc., which operate upon it, one after the other, in a definite order. 1. 2-3-1. 8. (Viy-i - 1)-1- 2. 3-3-1. 9. ( - l)s-22-31. 3. - 1-31. 10. 3-321. 4. 2'1. 11. ( - 1)'2-2«1. 6. VSI. 12. 3( - 1)V21. 6. (-v/2)^-l. 13. (\/2)-( - 1)"»-1. 7. -v/gVi-l. 14. Vl0-2-( - 1)1. 15. A tensor, if permitted to operate seven times in succession, will just double the operand. Symbolize this tensor. 16. A tensor, if permitted to operate five times in succession, will quadruple the operand. Symbohze this tensor. 362 ELEMENTARY MATHEMATICAL ANALYSIS [§201 201. Versors. The expression ■%/ — 1 cannot consistently, with the meaning abeady assigned to \/ and ( — 1), be looked upon as answering the question "how many," and therefore is not a number in that sense; yet if we consider \/ — 1 as a symbol of operation, it can be given a meaning consistent with the operators already considered. For if 2 is the operator that doubles, and \/2 is the operator that when used twice doubles, then since ( — 1) is the operator that reverses, the expression \/ — ^ should be an operator which, when used twice, reverses. So, as ( — 1) may be defined as the symbol which operates to turn a straight line through an angle of 180°, in a similar way we may define the expression ■%/ — 1 as « symbol which denotes the operation of turning a straight line through an angle of 90° in the positive direction. The restriction of positive rotation is inserted to make the definition unique. The symbols ( — 1) and ■y/ — 1 are not tensors. They do not represent a stretching or contracting of the operand. Their effect is merely to turn the operand to a new direc- tion; hence these symbols may be called versors, or "turners." 202. The Operator V^^. In Fig. 149 let a be any line. Then a operated upon by V - 1 (that is, V — 1 a) is a turned anti-clockwise through 90°, which gives OB. Now, of course, V — 1 ca n ope rate on V — 1 a ju st as well as on a. Then V — 1 ( V — 1 a), or P C, i s V — 1 g t urned positively through 90°. Similarly, V - UV - 1(V - 1 a)] is V^I i.s/'^l a) turned through 90°, etc. As we are at liberty to consider two turns of 90° as equivalent to one turn of 180°, therefore, \/ — 1 (V — 1 a) = ( — 1) o. Now OD = ( - 1) OS, OD = i- 1) (-v/^T a); but also 0D = B (\Rfa e a J C 9 J 'd Fig. 149.- -The integral powers of §203] COMPLEX NUMBERS 363 V^ ( - a), therefore, ( - 1) V^ a = V"^ ( - «)• Thus the student may show many like relations. The operator •%/ — 1 is usually represented by the symbol i and will generally be so represented in what follows. Exercises Interpret each of the following expressions as a symbol of operation: 1. 2, 3, 4, -1. ■2 3^23,4", (-1^ (-1)^ 3. V2,VZ,V- 1, \/'2, \/- 1. Select a convenient unit and construct each of the following expres- sions geometrically, explaining the meaning of each operator: 4. 2-3-5-1. 7. (-1)''V^^-1. 6. 2=-(-l)-l. 8. 2'-(-l)^-(\/- l)"-!. 6. 3V-1-21. 9. 3V - 1(-1)V -11. 203. Complex Numbers. An expression of the form a + hi is cdlled a complex number, since it contains a term taken from each of the following scales, so th.at the unit is not single but double or complex: - 3, ■- 2, - 1, 0, + 1, + 2, + 3, . - 3i, - 2i, - i, 0, + I, + 2i, + 3t, Any number belonging to the first scale is called a real nimiber, any number belonging to the second scale is called a pure imaginary. It is important to note that the only element common to the two series in this complex scale is 0. The explanation of the meaning of the symbol (a + hi) will be given in the following section. It will be shown in subsequent theorems that any expression made up of the sum, product, power or quotient of complex numbers may be put in the form a + hi, in which both a and 6 are re&l. 204. Meaning of a Complex Nimiber. Any real number, or any expression containing only real numbers, may be consid- ered as locating a point in a line. Thus, suppose we wish to draw the expression 2 + 5. Let be 364 ELEMENTARY MATHEMATICAL ANALYSIS [§204 the zero point and OX the positive direction. Lay off OA = 2 in the direction OX and at A lay off AB = 5 in the direction OX. Then the path OA + AB is the geometrical representation of 2+5. A B X Any complex number may be taken as the representation of the position of a point in a plane. For, suppose c + di is the complex number. Let 0, Fig. 150, be the zero point and OX the positive direction. Lay off OA = + c in the direction OX and at A erect di in the direction OY, in- stead of in the direction OX as in the last example. It is agreed to consider the step to the right, OA, followed by the step up- ward, AP, as the meaning of the complex number c + di^ Either the broken 'path OA + AP or the direct -path OP may he taken as the repre- smtation of c + di, and either path constitutes the definition of the sum of c and di. — di, and — c + di may be Fig. 150. — The geometrical con- struction of a complex number, c + di. di. In the same manner c constructed. The meaning of some of the laws of algebra as applied to imagi- naries may now be illustrated. Let us construct c + di + a + hi. The first two terms, c + di, give OA + AB, locating B (Fig. 151). The next two terms, a + hi, give BC + CP, locating P. Hence the entire expression locates the point P with reference to 0. Now if the original expression be changed in any manner allowed by the laws of algebra, the result is merely a different path to the same point. Thus: c + a +di + hiis the path OA, AD, DC, CP {c+a)+ {d + h)i is the path OD, DP a + di+ c + 6i is the path OE, EH, HC, CP a+di + hi + c is the path OE, EH, HF, FP, etc. §205] COMPLEX NUMBERS 365 The student should consider other cases. Is there any method of locating P with the same four elements, which the figure does not illustrate? 205. Laws. It can be shown by simple geometrical construc- tion that the operator i, as defined above, obeys the ordinary laws of algebra. We can then apply all of the elementary laws of algebra to the symbol i and work with it just as we do with any other letter. The following are illustrations of each law: r a c y ^ ^ ^ . 1 F G f •* ^-N •a I H B a c ts •■s ^ E ^ , . A D Fig. 151. — Illustration of the application of the laws of -algebra and the expression c + di + a + bi. CoMMUTATrvE Law: c-\-di-\-a + bi = c + a + di-\-bi = di-{-c + bi + a, etc. ai = ia, iai = iia = aii, etc. Thus the equation lO-s/ — 1 / - llO.or better, lOV - 1-1 \/ — l-lOl may be said to mean that the result of performing the operation of turning unity through 90° and performing upon , the result the operation of taking it ten times, is the same as the result of performing the operation of taking unity ten times and performing upon this result the operation of turning through 90°. AssociATivB Law: (c + di) + (a + bi) = c + {di + a) + bi, etc. {ab)i = a{bi) = abi, etc. DiSTEiBUTrvE Law: (a + b)i = ai + etc. 366 ELEMENTARY MATHEMATICAL ANALYSIS [§206 The expression -\/ — a, where a is an y num ber of the arith- metical scale, is defined as equivalent to \ / — l -o;that is, y/ — a - i\fa. For example, V — 4 = 2i, V —3 = i'\/^, etc. In what foUows it is presupposed that the student will reduce expressions of the form -y/ — ato the form i s/a before performing algebraic op- erations. From this it follows that y/ — a-^/ — b = — y/cA and not Vobl The relation -\/ — 4 = 2-\/ — 1 may be interpreted as follows : ( — 4) is the operator that quadruples and reverses; then •%/ — 4 is an operator which used twice quadruples and reverses. But 2-%/ — 1 is an operator such that tw o suc h operators quadruple and reverse. That is, V — 4 = 2\/ — 1. 206. Powers of i. We shall now interpret the powers of i by means of the new significance of an exponent and by the commu- tative, associative and other laws. First: i° or i° 1 = + 1 i^ = iH = i ^ i' .or i^ 1 = i %'• = iH = — \ j2 = _ ]^ j7 ~= m = — i i^ = iH = — i i' = m = + 1 i* = m^ = + 1 etc. etc. Whence it is seen that all even powers of i are either + 1 or — 1, and all odd powers are either i or — i. The student may reconcile this with Fig. 149. The zero power of i must be unity, for the exponent zero can only mean that the operation denoted by the symbol of operation is not to be performed at all; that is, unity is to be left unchanged; thus 10° or 10»-1 = 1, 2" = 1, log" x = x, sin" X = X, etc. Exercises Select as unit a distance 1/2 inch in length extending to the right and represent graphically each of the following expressions: 1. i + 2i' + 3i' + 4i* -f . 2. t + i« + i* + i« + i' + . 3. i + i* + e + i^ + i' + i'^ + . 4. i(i + i< + i' + i* + i' + ii2 + . . ). 5. i + i« -f- 1' + 2i^ + i* + t" + i' + 3i» + . . . §^07] ," COMPLEX NUMBERS 367 207. Conjugate Complex Numbers. Two complex numbers are said to be conjugate if they differ ohiy in the sign of the term containing \/ — 1." Such are x + iy and x — iy. Conjugate imaginaries have a real sum and a real product. For {x + yi) + {x — yi) = x + yi + x — yi, =. X + X + yi — yi = 2x. Likewise, applying the ordinary rules of algebra, {x + yi) (x — yi) = x^ — yH'' = a;^ + j/^ It is well to note that the product of two conjugate complex numbers is always positive and is the sum of two squares. This fact is very important and will be used frequently. Thus (3 - 4i)(3 + 4.1) = 3^'+ 42 = 25; (1 + i){l - i) = 2; (cos d + i sin 9) (cos d — i sin 6) = cos" 6 + sin" = 1; etc. 208. The sum, product, or quotient of two complex numbers is, in general, a complex number of the typical form a + bi. Let the two complex numbers be a; + yi and u + vi. (1) Their sum is (x + yi) + (u + vi) = (x + u) + {y + v)i by the laws of algebra. This last expression is in the form a + bi. (2) Their product is {x + yi) (u + vi) = x{u + vi)+ yi{u + vi) = xu + xvi + yui + yvi' = {xu — yv) + {xv + yu)i by the laws of algebra. This last expression is in the form a + bi (3) Their quotient is X + yi _ (x + yi){u — vi) u + vi (m + vi)(u — vi) By the preceding, the numerator is of the form a' + b'i. By §207, the denominator equals m" + «". Then the quotient equals a' + b'i a' b' . u^ + v^ m" + w" ' m" + »2 by distributive law. This last expression is of the form a + bi. 368 ELEMENTARY MATHEMATICAL ANALYSIS Exercises Reduce the following expressions to the typiqal form a + bi; the student must change every imaginary of the form -y/ — o to the form 1. V - 25 + V~^^ + V^^i2i - V^'ei - 6i. 2. (2V~^^ + 3v'^)(4\/"-^3 - 5V^^). 3. (x - [2 +3i]){x - [2 -3i]). 4. (-5 + 12V^T)^. 6. (vr+i)(-v/r^). 5. (3 - 4V^.)'. 7. (Ve"- V^~^)'. a 1 8. , . 12. 2 1 - i^ 13.; "■ S+V -2 • (1 - 0'- 10. , 'V _ 14. l^.^^A 11. 1 +V 15. (2 + sV^^n.^ i-i" 2 + v^n" ■ -„ o + a;i a — xi lb. ^ j ;• a — XI a + x% 209. If a complex number is equal to zero, the imaginary and real "parts are separately equal to zero. Suppose X + y \/ — 1 = 0, X and y being real numbers. Then x = — y V — 1. Now it is absurd or impossible that a real number should equal an imaginary, except they each be zero, since the real and imagi- nary scales are at right angles to each other and intersect only at the point zero. Therefore x = and y = 0. If two complex numbers are equal, then the real parts and the imaginary parts must be respectively equal. For if X + yi = u + vi then (x -«) + (?/ - v)i = 0. §2101 COMPLEX NUMBERS 369 Whence, by the above theorem, That is, X — u = and y — v= 0. a; = M and y = v. 210. Modulus. Let the complex number x + yihe constructed, as in Fig. 152, in which OA = x and AP = yi. Draw the line OP, and let the angle AOP be called 0. The numerical length of OP is called the modulus of the complex number x + yi. It is algebraically represented by -y/x^ + y^, or by the symbol \x + yi\. Thus, mod (3 + 4*) = V9 + 16 = 5. The student can easily see that two conjugate complex numbers have the same modulus. If 2/ = 0, the mod (x + yi) = \/^= \x\, where the vertical lines indicate that merely the numerical, or absolute, value of X is called for. Thus the modulus of any real number is the same as what is called the numerical, or absolute value, of the number. Thus mod (— 5) = 5. 211. Amplitude. In Fig. 152 the angle AOP or 6 is called the argument, or ampli- tude, or simply the angle, of the complex number x + yi. Putting r = \^x^ + y^ - mod {x + y Fig. 152. — Modulus and amplitude of a complex number. yi) = ;x +iy\, we have sin 6 = and cos 6 X r Therefore, .•B + ?/i = r cos + ir sin Q = r(cos 9 + i sin 9). (1) We have expressed the complex number x + yi in terms of its modulus and amplitude. The last member of (1) is called the polar fonn of the complex number {x + iy). To put 3 — 4i in this form, we have mod (3 - 4i) = \/9 + 16 = 5; sin 5 = ^ = - |; cos S = - = f ^ r 5 r 5 24 370 ELEMENTARY MATHEMATICAL ANALYSIS [§212 Therefore, The amplitude d is tan-' ( ~ o ) i ^^^ is in the fourth quadrant. Why? It is well to plot the complex number in order to be sure of the amplitude 6. It avoids confusion to use positive angles in all cases. For example, to change 3 — \/3 i to the polar form, plot the point (3, — \/3) and find from the triangle that r = 2 \/3 and 9 = 330°. Hence 3 - VS i = 2V3(cos 330° + i sin 330°). [< The ampUtude of all positive numbers is 0, and of all negative numbers is 180°. The unit expressed in terms of its modulus and amplitude is evidently l(cos + i sin 0). 212. Vector. The point P, Fig. 152, located by OA + AP, or X + yi, may also be considered as located by the line or radius vector OP; that is, by a line starting at 0, of length r and making an angle 6 with the direction OX. A directed line, as we are now considering OP, is called a vector. When thus considered, the two parts of the compound operator r (cos 5 + i sin 6) (1) receive the following interpretation : The operator (cos 6 + ism 6), which depends upon B alone, turns the unit Ijdng along OX through an angle 6, and may therefore be looked upon as a versor of rotative power 6. The versor (cos 6 -\- i sin 6) is often abbre- viated by the convenient symbol cWd. The operator r is a tensor, which stretches the turned unit in the ratio 1 : r. The result of these two operations is that the point P is locaited r units from in a direction making the angle 6 with OX. Thus, the operator (cos ^ + i sin 6) is simply a more general operator than i, but of the same kind. The operator i turns a unit through a right angle and the operator (cos -\- i sin 6) turns a unit through an angle B. If 6 be put equal to 90°, cos 6-\-i sin 6 reduces to i. §213] COMPLEX NUMBERS ' 371 For d = 0, cos 6 + i sin $ reduces to 1 6 = 90°, cos d + i sin d reduces to i 6 = 180°, cos 6 -\- i sin 6 reduces to — 1 6 = 270°, cos 6 + isin 6 reduces to — i Since 3 — 4i = 5(f — fi), the point located by 3 — 4i may be reached by turning the unit vector through an angle 8 = sin~'(— 4/5) = COS"' 3/5 and stretching the result in the ratio 1 :5. // a complex number vanishes, its modulus vanishes; and con- versely, if the modulus vanishes, the complex number vanishes. li X + yi = 0, then x = and y = 0, hy §210. Therefore, Vx^ + t/2 = 0. Also, if Va;^ + y^ = 0, then x^ + y^ = 0, and since x and y are real, neither x' nor y^ is negative, and so their sum is not zero unless each be zero. // two complex numbers are equal, their moduli are equal, but if two moduli are equal, the complex numbers are not necessarily equal. li X -i- yi = u + vi, then x = u and y = vhy §210. Therefore, V^^+^ = Vu^ + vK But if ■y/x'^ + y'^ = y/u^ + v^, obviously x"^ need not equal u^ nor y"^ = v'. 213. Sum of Complex Numbers. Let a given complex number locate the point A, Fig. 153, and let a second complex number locate the point B. Then if the first of the complex numbers be represented by the radius vector OA, and if the second complex number be represented by the radius vector OB, the sum of the two complex numbers will be represented by the diagonal OC of the parallelogram constructed on the lines OA and OB. This law of addition is the well-known .law of addition of vectors used in physics when the resultant of two forces or the resultant of two velocities, two accelerations, or two directed magnitudes of any kind, is to be found. The proof that the sum of the two complex numbers is repre- sented by the diagonal OC is very simple. Let the graph of the first complex number be ODi + DiA and let that of the second be OD2 -f- DiB. To add these, at the point A construct AE = ODi and EC = D^B. Then the sum of the two complex numbers is geometrically represented by OJ)^ + BiA + AE + EC, or by the 372 ELEMENTARY MATHEMATICAL ANALYSIS [§214 radius vector OC which joins the end points. Since, by construc- tion, the triangle AEC is equal to the triangle OD^B, AC must be equal and parallel to OB, and the figure OACB is a parallelogram. OC, which represents the required sum, is the diagonal of this parallelogram, which we were required to prove. Di Di Di Fig. 153. — Sum of two complex numbers. Exercises Mnd algebraically the sum of the following complex numbers, and construct the same by means of the law of addition of vectors. 1. (1 + 2i) + (3 + 4i). 4. (3 - 4i) - (3 + 4i). 2. (1 + i) + (2 + i). 5. (-2 + i) + (0 - ti). 3. (1 - i) + (1 + 2i). 6. (- 1 + i) + (3 + i) + (2 + 2i). 7. (2 - i) + (- 2 + i) + (1 + i)- 8. Find the modulus and ampHtude ^in degrees and minutes) of 2(cos 30° .+ i sin 30°) + (cos 45° + i sin 45°). 9. By the parallelogram of vectors, show that the sum of two con- jugate complex numbers is real. 10. If ij be the sum of the complex numbers Zi'= xi -|- iyi, Z8 = Xi -\-iyi, «a = Sa + Vii, etc., show that —R, zi, zj, 23, . . . form the sides of a closed polygon. 214. Product of Complex Nximbers. The product, of two or more complex ^umbers is a complex number whose modulus is'>the §214] COMPLEX NUMBERS 373 product of the moduli and whose amplitude is the sum of the ampli- tudes of the com,plex numbers. Let the complex numbers be ^1 = xi + y-ii = ri (cos Q\ + i sin 6i) Z2 = a;2 + 2/21 == rj (cos 02 + i sin ^2), etc. By actual multiplication, we get «i22 = rir-2 [(cos 01 cos 02 — sin 0i sin 02) + (sin 01 cos 02 + cos 0i sin di)i\ = rir^ [cos (0i + 02) + i sin (0i + di)] Whence it is seen that rir2 is the modulus of the product -and (01 + 02) is the amplitude. (2 + 2»)(v7+i) Fig. 154. — Product of two pomplex numbers. The above theorem is illustrated by Fig. 154. If the two given complex numbers be represented by their vectors OPi and OPt, their product will be represented by the vector OP3 whose direction angle is the sum of the amplitudes of the two given factors, and whose length OP3 is the product of the lengths OPi and OP2. The figure represents the product (2 + 2i) {y/z + i). Expressed in terms of modulus and ampUtude these may be written, -s/3'+ i = 2(cos 30° + i sin 30°) 2 + 21 = 2v^(cos 45° + i sin 45°) 374 ELEMENTARY MATHEMATICAL ANALYSIS [§215 Hence, ri = 2, rj = 2\/2, Bi = 30°, 6, = 45° Therefore (2 + 2i)(V3 + i) = 4v'2 (cos 75° + i sin 75°) Exercises Find the moduli and amplitudes of the following products, and construct the factors and products graphically. Take a positi-i/e angle for the amplitude in every case. 1. (1 + \/3t)(2-\/3 + 2i). 4. (1 + iy. _ 2. (2 + W3i){2 + 2i). 5. (2 - 2v'3i)(\/3 + Si). 3. (V3 + 3i)(2 - 2i). 6. (1 - iy. 7. (1 + i)\l - i)K 8. 2 (cos 15° + i sin 15°) X 3 (cos 25° + i sin 25°). Find numerical result by use of slide rule or trigonometric tables. 9. 2(cos 10° + i sin 10°) X (1/3) (cos 12° + i sin 12°) X 6(co3 8° +isin8°). 10. Find the value of 4-\/2 (cos 75° +isin75°) + (Vs + i). 216. Quotient of Two Complex Numbers. The quotient of two complex numbers is a complex number whose modulus is the quotient of the moduli and whose amplitude is the difference of the amplitudes of the two complex numbers. Let the complex numbers be Si = Xi + yii = ri(cos Q\-\- i sin 9i) Zi = Xi + yii = rjCcos di + i sin 62). We have zi _ ri(cos Bi + i sin gi)(cos dj — i sin 6i) 02 raCcos 02 + i sin S2)(cos 82 — i sin 0i) ^ ri[oos {di - 62) +i sin (9i - gg)] r2 (cos'' ^2 + sin^ 62) = -[cos (^1 -■^2) + i sin (^i - 62)]- r2 Whence it is seen that — is the modulus of the quotient and (.01 — 02) is the ampUtude. In Fig. 155, the complex number represented by the vector OPi when divided by the complex number represented by OP2 yields the result represented by OP3, whose length ri/rais found by dividing the length of OPi by the length of OP2, and whose, direction angle §216] COMPLEX NUMBERS 376 is the difference (ffi — 61) of the amplitudes of OPi and OPi. The figure is drawn to scale for the case: 5 (cos 60° + i sin 60°) 2 (cos 20° + i sin 20°) = (2.5) (cos 40° + i sin 40°) Fig. 155. — Quotient of two complex numbers. Exercises Find the quotient and graph the results in each of the following exercises. Always take ampUtudes as positive angles and if 9j > 61, take 9i + 360° instead of 9i. 1. (1 + \/3i) -^ (V2 + V2i). 3. (SVS -3i) -i- ( - 1 + \/3i). 2. (i + iVSi) -^ (^2 - V2i). 4. (1 - VSi) -h i. 6. 2(oos 36° + i sin 36°) -r- 5(cos 4° + i sin 4°;. 6. 1.2(cos 48° + i sin 48°) h- [2(cos 15° + i sin 15°) 3(cos 9° + i sin 9°;]. „ [4 + (4/3)-v/3i] (2 + 2Vdi) 8 + 8i 8. Express in terms of a, b, e, d, the ampUtude of (a + bi) + (c + di). I 216. De Moivre's Theorem. As a special case of §214 consider the expression (cos d + i sin S)» where n is a positive number. 376 ELEMENTARY MATHEMATICAL ANALYSIS [§216 This being the product of n factors like (cos e + i sin e), we write, by means of §214, (cos S + i sin 6) (cos + i sin S) = [cos(0+e+ . )+isin((9+e+ ...)], or (cos 5 + i sin 6)" = (cos nd + i sin nd), (1) which relation is known as De Moivre's theorem. De Moivre's theorem holds for fractional values of n. For, first consider the expression (cos e + i sin e)^^\ where the power 1/t of aos d -\- i sin B is, by definition, an operator such that the , SO that 4> — ~t Then (cos B + i sin 9)'/' = (cos tij) + i sin tij))^'^ = [(cos ^ + i sin + i sin

y27 (cos 76° - i sin 75°). 7. (1 + i)'. 3. -^125i. 8. (- 2 + 2i)^. 4. [cos 9° + i sin 9°]". 9. [(1/2)V3 - (l/2)i]'. 5. (S+VSi)'- -10. Find value of (-1 + V - 3)= + (-1 - V - 3)' by De Moivre's theorem. 11. Find the value of x^ - 2 z + 2 for x = 1 + i. 12. If ii = - 1/2 + (1/2)^^^ and J2 = - 1/2 - (1/2) V - 3, showthatjV = l.jV = l,ji' =J2,h^ =Ju3i^" =jV = l,jV"'^'=ii. 217. The Roots of Unity. Unity may be written 1 = cos + i sin 1 = cos 2ir + i sin 2ir 1 = cos 47r + i sin Air 1 = cos 6t + i sin 6ir and so on. By De Moivre's theorem the cube root of any of these is taken by dividing the amplitudes by 3. Therefore, from the above expressions in turn, there results Vi = cos + i sin = 1 •^1 = cos (27r/3) + i sin (27r/3) = cos 120° + i sin 120° = -l/2 + i(l/2) V3 Vl = cos (4ir/3) + i sin (47r/3) = cos 240° + i sin 240° _ = - 1/2 - i(l/2) V3 Vl = cos 67r/3 + i sin Qir/S = same as first, etc. 378 ELEMENTARY MATHEMATICAL ANALYSIS [§217 Therefore there are three cube roots of unity. Since these are the roots of the equation a;' — 1 = 0, they might have been found by factoring, thus x> - 1 = (x - 1) ix^ + x+ 1) = ix-l){x + 1/2 + i Vsi) (^ + 1/2-4 V3i) The three roots of unity divide the angular space about the point into three equal angles, as shown in Fig. 156. In the same way, it can be shown that there are four fourth roots, five fifth roots, etc., of unity and that the vectors representing them have modulus 1 and amplitudes that divide equally the space about 0. B \ ~^ 1 S f ^ o /\ D -ii\ / y' V 1 r j^ \ c ^ -^-^ Fig. 156. — The cube roots of unity. IlliTJStbatign 1. Find Vvl+3i Write \/3 + 3i in the form VS + 3i = 2-\/3(cos 60° + i sin 60°) Hence, by De* Moivre's theorem, WZ + 3i)^ = y/Vi (cos 30° + i sm 30°) = -v/T2[(l/2)^V3 + (l/2)i] = (1/2)1^108+ (1/2)^^12 i A second root can be found by writing VS + 3i = 2V'3 [cos (60° + 360°) - i ein (60° + 360°)] |218] COMPLEX NUMBERS 379 siace adding a multiple of 360° to the amplitude does not change the value of the sin and cosine. In applying De Moivre's theorem, there results (VS + 3i)>^ = ■>yi2 (cos 210° +i sin 210°) ' = V^12 [ - (l/2)v'3 - (l/2)i] Illttstration 2. Find the cube root of — \/2 + \/2 i. We write: - \/2 + i V2 = 2 (cos 135° + i cos 135°) = 2[cos (135° + Ji360°) + i cos (135° + 7i360°)]. in which n is any integer. Hence {-\/2+i V2)^ = \/2 [cos (45° + ?il20°) + i sin (45° + nl20°)]^ = ^ (cos 45° + i sin 45°) for n = = -^2 (cos 165° + i sin 165°) forra = 1 = -^2 (cos 285° + i sin 285°) for n = 2. These are the three cube roots of the given complex number. For « = 3 the first root is obtained a second time. Exercises Find all the indicated roots of the following: 1. (8 + SVsi)^. 6. (2 + 2i)^. 2. i^27(cos 75° + i sin 75°). 6. 32^. 3. -^iMi. 7. V/5I2. 4. ( - 2 + 2i)^. 8. Find to four places one of the imaginary 7th roots of + 1. Note: Cos 51° 25.7' + i sin 51° 25.7' = 0.6235 + 0.7818i. 218.* Irrational Numbers. A rational ntunber is a number that can be expressed as the quotient of two integers. All other real numbers are irrational. Thus V^j \/5j V^?, ir, e, are irrational numbers. An irrational number is always intermediate in value to two rational numbers which differ from each other by a number as small as we please. Thus 1.414, < •v/2 < 1.415 1.4142 < \/2 < 1.4143 1.41421 < V2 < 1.41422, etc. 380 ELEMENTARY MATHEMATICAL ANALYSIS [§218 It is easy to prove that \/2 cannot be expressed as the quotient of two integers For, if possible, let V2=l, (1) where a and 6 are integers and r is in its lowest terms. Squaring the members of (1) we have 2 =^. (2) This cannot be true, since 2 is an integer and a and 6 are prime to each other. An irrational number, when expressed in the decimal scale, is never a repeating decimal. For, if the irratiqn,al number could be expressed in that manner, the repeating decimal could be evalu- ated by §120 in the fractional form ^ _ ' which, by definition of an irrational number, is impossible. On the contrary, every rational number when expressed in the decimal scale is a repeating decimal. Thus 1/3 = 0.33 . . and 1/4 = 0.25000. . . The proof that ir and e are irrational numbers is not given in this book.^ The student should not get the idea that because irrational num- bers are usually approximated by decimal fractions, that the irrational number itself is not exact. This can be illustrated by the graphical construction of ■\/2. Locate the point P whose coordinates are (1, 1). Call the abscissa OD and the ordinate DP. Then OP = y/% OZ) = 1, and DP = 1. It is obvious that the hypotenuse OP must be considered just as exact or definite as the legs OB and DP The notion that irrational numbers are inexact must be avoided. The process of counting objects can be carried out by use of the primitive scale of numbers 0, 1, 2, 3, 4, . . . The other numbers made use of in mathematics, namely, (1) positive and negative numbers (2) integral and fractional numbers (3) rational and irrational numbers (4) real and imaginary numbers, ^ See Monographs on Modern Mathematics, edited by J. W. A. Young. §219] COMPLEX NUMBERS 381 may be looked upon as classes of numbers that permit the opera- tions siibtraction, division and evolution, to be carried out under all circumstances. Thus, in the history of algebra it was found that in order to carry out subtraction under all circumstances, negative numbers were required; to carry out division under all circum- stances, fractions, were required; to carry out evolution of arith- metical numbers under all circumstances, irrational numbers were required; finally to carry out evolution of algebraic numbers under all circumstances, imaginaries were required. It will be found that it wiE not be necessary to introduce any additional form of number into algebra; that is, the most general number required is a number of the form a 4- 6i, where a and 6 are positive or negative, integral or fractional, rational or irrational. This is the most general number that satisfies the following conditions: (a) The possibility of performing the operations of algebra and the inverse operations under all circumstances. (6) The conservation or permanence of the fundamental laws of algebra: namely, the commutative, associative, distributive, and index laws. Further extension of the number system beyond that of complex numbers leads to operators which do not obey the commutative law in multiplication; that is, in which the value of a product is dependent upon the order of the factors, and in which a product does not necessarily vanish when one factor is zero. Numbers of this kind the student may later study in the introduction to the study of electromagnetic theory under the head of "Vector ^ Analysis" or in the subject of "Quaternions.'' Such numbers or operators do not belong to the domain of numbers we are now studying. 219.* Simple Periodic Variation Represented by a Complex Num- ber. Fluctuating magnitudes exist that follow the law of S.H.M. although, strictly speaking, such magnitudes can be said to be "sim- ple harmonic motions" in only a figurative sense. For example we may think of the fluctuations of the voltage or amperage in an alter- nating current as following such a law. Thus if E represent the electromotive force or pressure of the alternating current, then the fluctuations are expressed by E = Eo sin oit 382 ELEMENTARY MATHEMATICAL ANALYSIS [§219 or by E = Eo sin 2irft, where/is the frequency of the fluctuation. Instead of S.H.M. such a variable is naiore accurately called a sinusoidal varying magnitude, although for brevity we shall often call it S.H.M. The graph in rec- tangular coordinates of such a periodic function is often called a "wave," although this term should, in exact language, be reserved for a moving periodic curve, such a.sy = a sin (hx — kt). If the polar representation p = a sin (ot — or ah + g- The circle on the vector OA is located or characterized equally well if the rectangular coordinates (c, d) of the end of the diameter of the circle be given. But the complex number c + diis represented by a vector which coincides with the diameter o of this circle. Hence we may represent the circle by the complex niunber c + di. Its modulus is a = ■\/c' + d'' and its amplitude is a + s. Therefore if in (1) we take o = \/c* + d-, at, = a and the variable angle at = 9, we can completely describe the S.H.M. by the complex number c + di. In the theory of alternating currents the sinusoidal varying current or voltage can conveniently be represented by a complex number, and that method of representing such magnitudes is in common use. One of the advantages of representing S.H.M. by a vector or by a complex number is the fact that two or more such motions of like periods may then be compounded by the law of addition of vectors. This method of find^g the resultant of two sinusoidal varying mag- nitudes of like periods possesses remarkable utihty and simplicity. To summarize, we may say: (o) A siniisoidal varying magnitude is represented graphically in §219] COMPLEX NUMBERS 383 polar coordinates by a vector, which by its length denotes the amplitude and by its direction angle with respect to OY denotes the epoch angle. (6) Sinusoidal varying magnitudes of like periods may be compounded or resolved graphically by the law of parallelogram of vectors. If two sinusoidal varying magnitudes of like periods are in quad- rature (that is, if their epoch angles differ by 90°), their relation, neglecting their epochs, can be completely expressed by a single com- plex number. Thus let two S.H.M. in quadrature and E„ = 113 sin a{t - h) Ec = 40 cos a{t - ti) (2) (3) Fig. 157. — Composition of two S.H.M. in quadrature by law of addition of vectors. be represented by the circles and by the vectors marked OEo and 0B„, Fig. 157. Call the resultant of these Ei. Then Ei = 11 3 sin fc)(i - «i) + 40 cos u(« - B mum. The first S.H.M. is there- fore said to lag the amount ( (4) or 2/^=-|^ + f- (5) This curve is the parabola y^ = mx, the special location of ■which the student should describe. (4) Construct a graph such that the increase in y varies directly as X. If y varied directly as x, then y would equal kx, where A; is any constant. In the given problem, however, the increase in y (and not y itself) must vary in this manner. Let the initial value of y be represented by z/o. Then the gain or increase of y is repre- sented by 2/ — 2/0. Hence, by the problem, y — yo = kx. (1) Since t/o is a constant, (1) is the equation of the straight line of slope k and F-intercept j/o, which ordinarily would be written in the form y = kx + 2/0. (5) Express the diagonal of a cube as a function of its edge, and graph the function. If the edge of the cube be x, its diagonal is -v/a;" + x^ + x^ or X \/3. If the diagonal be represented by y, we have y =\/3x, which is a straight line. (6) A rectangle whose length is twice its breadth is to be in- scribed in a circle of radius a. Express the area of this rectangle in terms of the radius of the circle. Let the rectangle be drawn in a circle whose equation is x^ + y^ = a'. At a corner of the rectangle we have x = 2y. The area A of the rectangle is 4xy, or 8y^ since x = 2y. From the §222] LOCI 391 equation of the circle we obtain 4y^ + y^ = a^ or y^ = a^/5. Hence A = (8/5)a2. If A and a be graphed as Cartesian variables, the graph is a parabola. (7) A rectangle is inscribed in a circle. Express the area of the rectangle as a function of a half of one side. Here, as above, A = 4x2/ = 4x Va^ — x^- The student should graph this curve, for which purpose a may be put equal to unity. First draw the semicircle y = \/'o^ — x'- For X = 1/6, take one-fifth of the ordinate of this semicircle. For X = 2/5, take two-fifths of the ordinate of the semicircle, and so on. The curv e throug h these points is y = x s/ a^ — x^, from which y = 4x \/a^ — x'' can be had by proper change in the vertical unit of measure. Exercises 1. In polar coordinates, draw the curves: p = 2 cos 8 p = 2 cos 9 + 1 P = 2 cos 9 — 1 p = 2 cos 9 -1- 3. 2. On polar coordinate paper select the point (1, 1). (This means the point whose coordinates are one centimeter, and one radian.) Starting at this point, a point moves so that the radius vector of the moving point is always equal to the vectorial angle. Sketch the curve. Write the polar equation of the curve. 3. A point moves so that one of its polar coordinates, the radius vector, varies directly as the other polar coordinate, the vectorial angle. Write the polar equation of such a curve. Does the curve go through the point (!', 1)? 4. A polar curve is generated by a point Which starts at the point (1, 2) and moves so that the increase in the radius vector always equals the increase in the vectorial angle. Write the equation of the curve. 6. A polar curve is generated by a point which starts at the point (1, 2) and moves so that the increase in the radius vector varies directly as the increase in the vectorial angle. Write the equation of the curve. 6. A ball is thrown from a tower with a horizontal velocity of 10 392 ELEMENTARY MATHEMATICAL ANALYSIS [§222 feet per second. It falls at the same time through a variable distance given by s = 16. 1<^, where t is the elapsed time in seconds and a is in feet. Find the equation of the curve traced by the ball. 7. The point P divides the line AB, of fixed length, externally in the ratio a : 6, that is, so placed that PA/PB = a/b. If the line AB move with its end points always remaining on two fixed lines OX and OK at right angles to each other, then P describes an ellipse of semi- axes a and b. 8. If in the last problem the lines OX and OY are not at right angles to each other, the point P still describes an ellipse. 9. A point moves so as to keep the ratio of its distances from two fixed lines AC and BD constant. Prove that the locus consists of four straight hnes. 10. A sinusoidal wave of amplitude 6 cm. has a node at + 5 cm. and an adjacent crest at + 8 cm. Write the equation of the curve. 11. The velocity of a simple wave is 10 meters per second. The period is two seconds. Find the wave length and the frequency. 12. A polar curve passes through the point (1, 1) and the radius vector varies inversely as the vectorial angle. Plot the curve and write its equation. Consider especially the points where the vectorial angle becomes infinite and where it is zero. Sketch the same func- tion in rectangular coordinates. 13. Rectangles are inscribed in a circle of radius r. Express by means of an equation and plot: (o) the area, and (6) the perimeter of the rectangles as a function of the breadth. 14. Right triangles are constructed on a line of given length h as hypotenuse. Express and plot: (a) the area, and (6) the perimeter as a function of the length of one leg. 16. A conical tent is to be constructed of given volume, V. Express and graph the amount of canvas required as a function of the radius of the base. 16. A closed cylindrical tin can is to be constructed of given volume, V. Plot the amount of tin required as a function of the radius of the can. 17. A rectangular water-tank lined with lead is to be constructed to hold 108 cubic feet. It has a square base and open top. Plot the amount of lead required as a function of the side of the base. 18. An open cylindrical water-tank is to be made of given volume, V. The cost of the sides per square foot is two-thirds the cost of the bottom per square foot. Plot the cost as a function of the diameter. 19. An open box is to be made from a sheet of pasteboard 12 inches square, by cutting equal squares from the four comers and bending up §223] LOCI 393 the sides. Plot the volume as a function of the side of one of the squares out out. 20. The illumination of a plane surface by a luminous point varies directly as the cosine of the angle of incidence, and inversely as the square of the distance from the surface. Plot the illumination of a point on the floor 10 feet from the wall, as a function of the height of a gas burner on the wall. 21. Using the vertical distances between corresponding points on the curves y = sin t and y = — sin t as ordinates and the vertical distances between corresponding points oi y = 2t and j/ = t^ as abscis- sas, find the equation of the resulting curve. 223. Loci Defined by Focal Radii. A number of important curves are defined by imposing conditions upon the distances of any point of the locus from two fixed points, called foci. Pig. 161. — The lepmiscate. (1) A point moves so that the product of its distances from two fixed points is constant. Find the equation of the path of the par- ticle. Let the two fixed points Fi and Fi, Kg. 161, be taken on the X-axis the distance o each side of the origin. Call the distances of P from the fixed points ri and rj. Then the variables ri and rj in terms of ::; and y are ri'^ = y' + (x - a)2 Hence ri' =y' + (x + a)2. nW = [y^ + (a; - o)"] [y^ + {x + a)']. (1) (2) Calling the constant value of tiTi = c', we have, as the Cartesian equation of the locus, y' + (x- ay] [y^ + (x + a)»] = c\ (3) 394 ELEMENTARY MATHEMATICAL ANALYSIS [§223 Fig. 162. — The lemniscate and the Cassinian ovals. which may be written (2/2 + a;2 + a2)2 - 4a^'x^ = c* (4) (x2 + y'y + 2aV + 20^2/2 + a* - iaV = c* (5) V (a;2 + 2/2)2 ^ 2a%x' - y') + c* - a'. (6) If c = a the curve is called the lemniscate, and the Car- tesian equation reduces to (x2+ 2/2)2 = 2a2(a;2 - 2/=^)- (7) For other values of c the curves are known as the Cassinian ovals. The stu- dent will show that when c < u, the curve consists of two separate ovals surround- ing the foci, and for c > a there is but a single oval. The curves are shown in Fig. 162. These curves give the form of the equipotential surfaces in a field around two positively or two negatively charged parallel wires. (2) Construct the curve such that the ratio of the distances of any of the curve from two fixed points is constant. Let the two fixed points be A and B, Fig. 163; let the constant ratio of the distances of any point of the curve from the two fixed points be n/J'2 = mm. To find one point of the locus, draw circles from A and B as centers whose radii are in the ratio m/n. Let these circles intersect at the point P. At P bisect the angle between PA and PB internally and externally by the lines PM and PN respectively. The line AB ia then divided at M internally in the ratio MA/ MB = m/n and externally at N in the ratio NA/NB = m/n, because the bisectors of any angle of a triangle divide the base into segments proportional to the adja- cent sides. Since the external and internal bisectors of any angle Fig. 163. — Construction of 'the curve ri/Ti = m/n, or the circle MPN. §224] LOCI 395 must be at right angles to each other, PM is perpendicular to PiV for any position of P. Hence the locus of P is a circle, since it is the vertex of a right triangle described on the fixed hypotenuse MN. If a large number of circles be drawn for different values of m/n, and if similar circles be described about B, then these circles are known as the dipolar circles. See Fig. 164. In physics it is found that these circles are the equipotential hnes about two parallel wires perpendic- ular to the plane of the paper at A and B and carrying electricity of opposite sign. Fig. 164. — The dipolar circles, or a family of circles made by drawing ri/ra = e for various values of e. Exercises 1. Draw the locus satisfying the condition that the ratio of the distances of any point from two fixed points ten units apart is 2/3. 2. Draw the two circles which divide a line of length 14 internally and externally in the ratio 3/4. 224. The Cycloid. The cycloid is the curve traced by a point on the circumference of a circle, called the generating circle, which rolls without slipping on a fixed line called the base. To find the equation of the cycloid, let OA,'Yig. 165, be the base, P the tracing point of the generating circle in any one position, and Q the angle between the radii SP and SH. Since P was at when the circle began to roll, OH = ad, 396 ELEMENTARY MATHEMATICAL ANALYSIS [§225 if a be the radius of the generating circle. Since »= OD and y = DP, we have x= OH- SP sin e= a(e - sin 6) y= HS- SP cos e= a(i- cos (9). (1) (2) These ar^ the parametric equations of the curve. For most purposes these are more useful than the Cartesian equation. o D H c A - Fig. 165. — Definition of the cycloid. It is readily seen from the definition of the curve, that the locus consists of an unlimited number of loops above the X-axis, with points of contact with the X-axis at intervals of 2xa (the circum- ference of the generating circle) and with maximum points at X - Ta, 3xo, etc. "=*'2Pl6 12 3 4 6 C A Fig. 166. — Construction of the cycloid. 225.* Graphical Construction of the Cycloid. To construct the cycloid, Fig. 166, draw a circle of radius 1.15 inches and divide the circumference into thirty-six equal parts. Draw horizontal lines through each point of division exactly as in the construction of the sinusoid, Fig. 59- Lay off uniform intervals of 1/5 inch each on the X-axis, marked 1, 2, 3, . . . Then from the point of division of the circle pi lay off the distance 01 to the right. §226] LOCI 397 From pi lay off 02 to the right, from pa lay off 03 to the right, etc. The points thus determined lie on the cycloid. The number of divisions of the circumference is of course immaterial except that an even number of division is .convenient. Further the divisions laid off on the base OA must be the same length as the arcs laid off on the circle. Note that by the process of construction above, the vertical distances from OX to points on the curve are proportional to (1 — cos 6) and that the horizontal distances from OY to points on the curve are proportional to (d— sin 6). The analogy of the cycloid to the sine curve is brought out by Fig. 167. A set of horizontal lines are drawn as before and also a sequence of semicircles spaced at horizontal intervals equal to Fig. 167.— Analogy of the cycloid to the sinusoid. the intervals of arc on the circle. The plane is thus divided into a large number of small quadrilaterals having two sides straight and two sides curved. Starting at and sketching the diagonals of successive cornering quadrilaterals the cycloid is traced. If, instead of the sequence of circles, uniformly spaced vertical straight lines had been used, the sinusoid would have been drawn; The sinusoid on that account is frequently called the "companion to the cycloid." 226. Epicycloids and Hj^ocycloids. The curve traced by a point attached to the circumference of a circle which rolls without slipping on the circimiference of a fixed circle is called an epi- cycloid or a hypocycloid according as the rolling circle touches the outside or inside of the fixed circle. If the tracing point is not on the circumference of the rolling circle but on a radius or radius produced, the curve it describes is called a trochoid if 398 ELEMENTARY MATHEMATICAL ANALYSIS [§226 the circle rolls upon a straight line, or an epitrochoid or a hypo- trochoid if the circle rolls upon another circle. Exercises 1. Construct a cycloid by dividing a generating circle of radius 1.15 inches into twenty-four equal arcs and dividing the base into intervals 3/10 inch each. 2. Compare the cycloid of length 2ir and height 1 with a semi- ellipse of length 2ir and height 1. 3. Write the parametric equations of a cycloid for origin C, Fig. 165. 4. Write the parametric equations of a cycloid for origin B, Fig. 165. 6. Show that the top of a rolling wheel travels through space twice as fast as the hub of the wheel. 6. By experiment or otherwise show that the tangent to the cycloid at any point always passes through the highest point of the generating circle in the instantaneous position of the circle pertaining to that point. CHAPTER XIV THE CONIC SECTIONS 227. The Focal Radii of the EUipse. Draw any ellipse with major and minor circles of radii a and 6 respectively, as in Fig. 168. Draw tangents, II' and KK', to the minor circle at the extremities of the minor axes and comJ)lete the rectangle II'KK'. Properties of the elUpse. The points Fi and Fi, in which IK and I'K' cut the major axis, are called the foci of the ellipse. Prom any point on the ellipse draw the focal radii PFj = rj. and PF2 = r^, as shown in the figure. Represent the distance OFi = OF 2 by c. Then it follows from the triangle OIFi that a^ = b« + c^. (1) This is one of the fundamental relations between the constants of the ellipse. > 399 400 ELEMENTARY MATHEMATICAL ANALYSIS [§227 From the triangles PFiD and PFiD there follows: n^ = (c - xY + v' (2) r/ = (c + xy + if. (3) But the equation of the ellipse is b , 2/ = - Vo' - x\ or y^ = ^,(a^ - a;^). (4) Substituting this value of y^ in (2) ri' = c'' - 2cx + x^ + -.(a'' - x^) (5) = c^ -2cx + x^ + ¥ -~i a;^ or by (1) r," = a^ - 2cx + a;^ [l - -^J . (6) Substituting 1 _ ^ - «' - fe' _ ^, we obtain /i2/»2 ri2 = o2 - 2ca; + ^ (7) = [•-"]"" <») Therefore Likewise, from (3), by exactly the same substitutions, there follows r2 = a + "''■ (10) a From (9) and (10), by addition, ri + Tz = 2a. (11) Hence in any ellipse the sum of the focal radii is constant and equal to the major axis, §228] THE CONIC SECTIONS 401 The converse of this theorem, namely, if the sum of the focal radii of any locus is constant, the curve is an ellipse, can readily be proved. It is merely necessary to substitute the values of ri and Ti from (2) and (3) in equation (11), and simplify the resulting equation in x and y; or first square (11) and then substitute ri and r-2 from (2) and (3). There results an equation of the second degree lacking the term xy and having the terms containing x'' and y^ both present and with coefficients of like signs. By §86, such an equation represents an ellipse. Hence the ellipse might have been defined as the locus of a point, the sum of whose distances from two fixed points is constant. An ellipse can be drawn by attaching a string of length 2a by pins at the points F\ and F2 and tracing the curve by a pencil so guided that the string is always kept taut. Or better, take a string of length 2a + 2c and form a loop enclosing the two pins; the entire curve can then be drawn with one sweep of the pencil. The focal radii may also be evaluated in terms of the para- metric or eccentric angle 0. The student may regard the follow- ing demonstration of the truth of equation (11) as simpler than that given above. Since a; = a cos 6, and y = h sm d equation (2) gives ri" =¥ sin^ e -h (c - a cos df (12) = h^aia^e + c^ - 2accos0 + o^ cos^ d. (13) To put the right side in the form of a perfect square, write W = a^ - c^. Then ri' = a^ sin'' — c^ sin'' B + c^ — 2ac cos d + a' cos^ 6 = o^ - 2oc cos e +c'^ cos^ e. (14) Whence Likewise Whence ri = a — c cos 9. (15) rj = a -|- c cos 6. (16) ri -H r2 = 2a. 228. The Eccentricity. The ratio c/a measures, in terms of o as unit, the distance Pf either fopus from the center of the ellipse. 26 402 ELEMENTARY MATHEMATICAL ANALYSIS [§229 This ratio is called the eccentricity of the ellipse. In the triangle IFiO, the ratio c/a is the cosine of the angle FiOI, represented in what follows by /3. Calling the eccentricity e, we have e = c/a = cos |8. (1) The ellipse is made from the major circle by contracting its ordi- nates in the ratio m = b/a, or by orthographic projection of the circle through the angle of projection a = cos~i b/a. Hence, as companion to (1) we may write m = b/a = cos a = sin |8. (2) 229. The Ratio Definition of the Ellipse. In Fig. 168, let the tangents to the major circle at I and I' be drawn. Draw a perpendicular to the major axis produced at the points cut by these tangents. These two lines ai'e called the directrices of the ellipse. We shall prove that the ratio PFi/PH (or PFt/PH') is constant for all positions of P. From §227, equation (9) or (15), ri = a — c cos d, (1) /?• (2) (3) From the figure, ON = a sec ION = a But cos |8 = c/a. Hence, ON = a^/c. But PH = 0N - X. Therefore PH = a'^/c - a cos 6. Hence, from (1) and (4), r\ = PI ^ /PH - "'-'' cos 9 PH a'^/c — acos e c a — c cos 6 (4) a a— c cos d or PFi/PH ^ c/a = e = cos |S. . , , . (6) §229] THE CONIC SECTIONS 403 A similar proof holds for the other focus and directrix. Thus, for any point on the ellipse the distance to a focus bears a fixed ratio to the distance to the corresponding directrix. From (5), the ratio is seen to be less than unity. Assuming the converse of the above, the ellipse might have been defined as follows : The ellipse is the locus of a point whose distance from a fixed point (called the focus) is in a constant ratio, less than unity, to its distance from a fixed line (called the directrix). If, in any ellipse, c = 0, it follows that b must equal a and the el- lipse reduces to a circle. If c is nearly equal to a, then from the equation a2 = 62 + c^, it foljows that the semi-minor axis & must be very small. That is, for an eccentricity nearly unity the ellipse is very slender. If the sun be regarded as fixed in space, then the orbits of the planets are ellipses, with the sun at one fobus. (This is "Kepler's First Law.") The eccentricity of the earth's orbit is 0.017. The orbit of Mercury has an eccentricity of about 0.2, which is greater than that of any other planet. Exercises Find the eccentricities and the distance from center to foci of the following elUpses: 1. a;V9 -I- 2/74 = 1. i. 2y = Vl - x". 2. 2/ = (2/3.)-\/36 - xK 5. Qx" + IQy' = 14. 3. 25a;2 + iy' = 100. 6..2a;2 + 3^^ = 1. Find the equation of the ellipse from the following data: 7. e = 1/2, a = 4. Draw this ellipse. 8. c = 4, a = .5. 9. ri = 6 - 2a;/3, ri = 6 + 2a;/3. 10. ri = 5 — 4 cos 0, ri = 5 -j- 4: cos 6. Solve the following exercises: 11. Find the eccentricity of the ellipse made by the orthographic projection of the circle x' + y^ = a' through the angle 60°. , 12. The angle of projection of a circle x' + y' = a' by which an 404 ELEMENTARY MATHEMATICAL ANALYSIS [§230 ellipse is formed is a. Show that the eccentricity of the ellipse is sia a. 13. A circular cylinder of radius 5 is cut by a plane making an angle 30° with the axis. Find the eccentricity of the elliptic section. 14. If the greatest distance of the earth from the sun is 92,500,000 miles, find its least distance. (Eccentricity of earth's orbit = 0.017.) 16. In the ellipse a;*/25 + y'/16 = 1, find the distance between the two directrices. 16. Write the equation of the ellipse whose foci are (2, 0), ( — 2, 0), and whose directrices are x = 5 and a; = — 5. 17. Prove equation 11 §227 by transposing one radical in: V(.x+c)^+y^ + V{x - c)2 + y' = 2o squaring, and reducing to an identity. 18. Obtain the equation of the ellipse from the definition at the top of page 403. 230. The Latus Rectum. The double ordinate through the focus is called the latus rectum of the ellipse. The value of the semi-latus rectum is readily formed from the equation y = (b/a) Va" - a;' by substituting c for x. If I represents the corresponding value oiy, I = (b/a) Va^ - c2 = 6Va (1) (2) since a^ — c'^ = 6^. Hence the entire lnius by 21 = ?^^ a Equation (1) may be also be written l = bVi- c-'/a- = b Vi- e\ (3) In Fig. 168 the distances AF, AN, ON, OB, OF, FN may readily be expressed in terms of a and e as follows in equations (4) to (10). The addition of the formulas (11), (12), (13) brings into a single table all the important formulas of the ellipse. AFi = a - c = a(i - e) (4) §230] THE CONIC SECTIONS 405 e e (5) ON = a sec ^ = 1 (6) e = cos |8 (7) OB = b = a sin /3 = a Vi - e^ (8) OFi = c = ae (9) FiN = OiV - c = a(i - e=)/e (10) 1 = bVa = a(i - e'^) (11) Ti = a — ex = a — X cos |3 (12) ra = a + ex = a + X cos j3 (13) Exercises 1. Find the value in miles of OFi for the case of the earth's orbit. 2. Find the equation of an ellipse whose minor axis is 10 units and in which the distance between the foci is 10. 3. In the ellipse y = (2/3)\/36 - x' find the length of the latus rectum and the value of e. 4. The eccentricity of an ellipse is 3/5 and the latus rectum is 9 units. Find the equation of the ellipse. 6. In (a) X' + 4v2 = 4 and (6) 2x' -|- 32/^ = 6 find the latus rfectum, the eccentricity, and the distances ON and AF. 6. Determine the eccentricities of the ellipses, (o) j/2 = 4s - (l/2)x2 (b) j/2 = 4x - 2x\ . 7. Find the value of /3 for the earth's orbit. (Use the S functions of the logarithmic table.) 8. Find the equation of an ellipse whose latus rectum is 2 units and minor axis is 4. 9. The distance from the focus to the directrix is 16 units. The distance from the vertex to the nearest focus is 6 units. Find the equation of the ellipse. 10. The axes of an ellipse are known. Show how to locate the foci. 11. In an ellipse a = 25 feet, e = 0.96. What are the values of c and 6? 12. For a certain comet (Tempel's) the semi-major axis of the' elliptic orbit is 3.5, and c = 1.4 on a certain scale. For another 406 ELEMENTARY MATHEMATICAL ANALYSIS [§231 comet (Enke's) o = 2.2, e = 0.85. Sketch the curves, taking 3 cm. or 1 inch as unit of measure. 13. If i = 7.2 and e = 0.6, find c, a, 6. 14. An ellipse, with center at the origin and major axis coinciding with the X-axis, passes through the points (8, 3) and (6, 4). Find the axes of the ellipse. 231. Focal Radii of the Hjrperbola. Construct a hyperbola from auxiliary circles of radii o and b, then the transverse axis of the hyperbola is 2a and the conjugate axis is 26. Unlike the case of the ellipse, b may be either greater or less than a. As previously explained, the asymptotes are the extensions of the diagonals of the rectangles BTAO, BT'A'O. From the points /, /', in which the \^ H nX t' Hy^ W^ f i r \m y/ Fo . A' \ V >y A I ^Fi D y^ (2) B' ^y < Fig. 169.— Properties of the hyperbola. asymptotes cut the a-circle, draw tangents to the o-circle. The points Fi, F2 in which the tangents cut the axis of the hyperbola are called the foci. See Fig. 169. The distance OFi or OFa is represented by the letter c. Then, since the triangles FJO and OAT are equal, FJ must equal 6, so that we Jbave the fundamental relation between the constants of "the hyperbola a^ + b^ = c" (1) §232] THE CONIC SECTIONS 407 From any point on either branch of the hyperbola draw the focal radii PFi and PF2, represented by ri and rj respectively. Then from the figure ri^ = (x - c)'+y\ (2) But from the equation of the hyperbola y^ = {V/a?){z^ - a?), (3) hence n^ = (x - cY + Wix" - a?) la"- (4) = {aH^ - 2a^cx + aH^ + Vx^ - a^V") /a" (5) = {cH'' - 2a?ex + a") /a?- (6) = {ex - o?)ya\ (7) Hence r\ = {c/a)x — a. (8) In like manner it may be shown that r2 = {c/a)x + a. (9) Hence from (8) and (9) it follows r2 — Ti = 2a. (10) Thus^ in any hyperbola, the difference between the distances of any point on it from the foci is constant and equal to the transverse axis. The above relation may be derived in terms of the parametric angle 9. Since in any hyperbola x = a sec 8 and y = b tan 6, ri" = V tan'' (9 + (o sec - cf = W tan^ 6 + a' sec" 9 — 2ac sec + c\ To put the right-hand side in the form of a perfect square, write ¥ = c^ - a\ Then ri^ = c' sec'* 9 - 2ac sec 9 + a\ Therefore ri = sec 9 — a. and I2 = c sec 9 + a. (11) (12) 232. The Ratio Property of the Hyperbola. Through the points of intersection of the a-circle with the asymptotes, draw 408 ELEMENTAKY MATHEMATICAL ANALYSIS (§233 IK, and I'K'. These lines are called the diiectrices of the hyperbola. It wiU now be proved that the ratio of the distance of any point of the hyperbola frota a focus to its distance from the corresponding directrix is constant. Adopt the notation c/a = sec ;8 = e. (1) Then, from the figure PFi/PH = ri/{x - ON) = ri/(a sec - o cos ;8) (2) Substituting ri from (11) above: PF^/PH = "^'l^-'* ^ (3) ' a sec — a cos j8 ^ ' = c sec 8 — a c (4) a (^sec e - -j which proves the theorem. The constant ratio e is called the eccentricity of the hyperbola, and, as shown by (1), is always greater than unity. Assuming the converse of the above, it is obvious that the hyper- bola might have been defined as follows: The hyperbola is the locus of a point whose distance from a fixed point (called the focus) is in a constant ratio, greater than unity, to its distance from a fixed line {called the directrix). Proof will be given in §234. 233. The Latus Rectum. The double ordinate through the focus is called the latus rectum of the hyperbola. The value of the semi-latus rectum is readily found from the equation y = (b/a) Vx^ - a^ by snbstituting c f or a;. HI represent the corresponding value of y, I = (b/a) Vc' - a^ = hya. (1) Hence the entire latus rectum is represented by 2l = 2bVa. (2) Equation (1) may also be written i = 6 Ve' - 1. (3) §233] THE CONIC SECTIONS 409 In Fig. 169 the distances AFi, AN, ON, OB, OFi, FiN may readily be expressed in terms of o and e, as follows in equations (4) to (8). Collecting in a single table the other important for- mulas for the hyperbola, we have : AFi = c — a = a(e — i) (4) AN = AF^/e = a(e - i)/e (5) ON = a cos /3 = a/e (6) e = sec /3 (7) OB = b=atan/3 = a Ve^ - i (8) OFi = c = ae FiN = c - OiV = ae - a/e = aCe^ - i)/e (9) 1 = bVa = b Ve^ - 1 = a(e2 - i) (10) ri = ex — a = X sec iS — a (11) Tz = ex + a = X sec |3 + a (12) The important properties of the hyperbola are quite similar to those of the ellipse. It is a good plan to compare them in parallel columns. Ellipse Hyperbola 1. Definition of Foci and Focal Radii 2. a2 = b2 + c2 3. ri + ra = 2a 4. Eccentricity, e = - = cos /3 5. Definition of Directrices PFi 6. The Ratio Property, -pg = e 262 7. The Latus Rectum = — 1. Definition of Foci and Focal Radii 2. a^ + b-' = c2 3. ra — ri = 2a 4. Eccentricity, e = - = sec /3 5. Definition of Directrices PFi 6. The Ratio Property, p^ = e 2b^ 7. The Latus Rectum = — 410 ELEMENTARY MATHEMATICAL ANALYSIS [§234 Exercises 1. Find the eccentricity and axes of sV^ — y'/^^ = 1- 2. Find the eccentricity and latus rectum of the hyperbola con- jugate to the hyperbola of the preceding exercise. 3. A hyperbola has a transverse axis equal to 14 units and its asymptotes make an angle of 30° with the Z-axis. Find the equation of the hyperbola. 4. Find the latus rectum and locate the foci and asymptotes of 4i2 - 362/2 = 144. 6. Locate the directrices of the hyperbola of the preceding exercise. 6. In Fig. 169 show that rz = GK' and ri = GI and hence that rj — ri = IK' or 2a. 7. Find the equation of the hyperbola having latus rectum 4/3 and a = 26. 8. The eccentricity of a hyperbola is 3/2 and its directrices are the lines X = 2 and x = — 2. Write the equation and draw the curve with its asymptotes, a-circle, 6-circle, and foci. 9. Find the eccentricity and axes of 3x^ — 5y' = — 45. 10. Find the eccentricity of the rectangular hyperbola. 11. Describe the shape of a hyperbola whose eccentricity is nearly unity. Describe the form of a hyperbola if the eccentricity is very Large. 12. Describe the hyperbola if b/a = 2, but a very small. 13. Write the equation of the hyperbola if (1) c = 6, o = 3; (2) c = .25, o = 24; (3) c = 17, 6 = 8. / 14. Describe the locus (,x + 1)V7 - iy - 3)V5 = 1. 15. Find the equation of the hyperbola whose center is at the origin and whose transverse axis coincides with the X-axis and which passes through the points (4.5, — 1), (6, 8). 234. The Polar Equation of the Ellipse and Hyperbola. In mechanics and astronomy the polar equations of the ellipse and hyperbola are often required with the pole or origin at the right- hand focus in the case of the ellipse and at the left-hand focus in the case of the hyperbola. In these positions the radius vector of any point on the curve will increase with the vectorial angle when B < 180°. To obtain the polar equation of the ellipse and hyperbola, make use of the ratio property of the curves, namely: That the locus of a point whose distances from a fixed point (called the focus) is in a constant ratio e to its distances from a fixed §234] THE CONIC SECTIONS 411 line (called the directrix), is an ellipse if e < 1 or a hyperbola if e > 1. In Fig. 170 let F be the fixed point, or focus, IK the fixed line, or directrix, P the moving point, and FL = I the semi- latus rectum. Then the problem is to find the polar equation under the condition pg-e (1) If e is understood to be unrestricted in value, the work and the result will apply equally well either to the ellipse or to the hyperbola. Fig. 170. — Polar equation of a conic. When the point P occupies the position L, Fig. 170, we have PF = I and PH = FN, whence from (1) (2) FN = ^-. e Take the origin of polar coordinates at F, and also take FP = p and the angle AFP = 6. Then PH = FN - FD (3) FD = p cos e. (4) 412 ELEMENTARY MATHEMATICAL ANALYSIS [§234 Hence from (2), (3), and (4) PH = - - p cos e. (5) e Substituting these values of FP and PH in (1), clearing of frac- tions and solving for p, we obtain " = I + e cos e ^^^ which is the equation required. When e < 1, (6) is the equation of an ellipse with pole at the right-hand focus. When e > 1, (6) is the equation of a hyperbola with the pole at the left-hand focus. In both cases the origin has been so selected that p increases as d increases. Note: Calling FN (Fig. 170) = n, equation (1) above may be written in rectangular coordinates ^^' + y' =e, (7) n — X x' + y^ = e\n - x)' ^°' which may be reduced to the form / we' \ 2 y' _ e'n' r "•■ 1 - eV "^ 1 - e' (1 -e')^' ^^' By §§86 and 90 this represents an ellipse if e < 1 or a hyperbola if e > 1. Thus starting with the ratio definition (7) we have proved that the curve is an ellipse or a hyperbola; that is, we have proved the statements in italics at end of §§229 and 232. Exercises n 1. Graph on polar paper, form M3, the curve p = j—r-^ i for e = 2, e = 1/2, and e = 1. It will be sufficient in graphing to use 9 = 0°, 30°, 60°, 90°, 120°, 160°, 180°, 210°, . ., 360°. 2. Write the polar equation of an ellipse whose semi-latus rectum is 6 feet and whose eccentricity is 1/3. 3. Write the polar equation of an ellipse whose semi-axes are 5 and 3. 4. Discuss equation (6) for the case e = 0. §235] THE CONIC SECTIONS 413 6. Write the polar equation of a hyperbola if the eccentricity be -\/2 and the distance from focus to vertex be 4. 6. Write the polar equation of the asymptotes of 4 + 5 cos e 9 -and p = 9 e' 4+5 cos I 4 — 5 cos in which a is 1 +e cos (9- «)' 7. Compare the curves p = 8. Discuss the equation p = constant. 235. Ratio Definition of the Parabola. Among the curves of the parabolic type previously discussed, the one whose equation is of Fig. 171. — Properties of the parabola y^ = ipx. the second degree is of paramount importance. On that account when the term parabola is used without qualification, it is under- stood that the curve is the parabola of the second degree, whose equation may be written, y^ = ax or x^ = ay. We shall prove : The locus of a 'point whose distance from a fixed point is always equal to its distance from a fixed line, is a parabola. In Fig. 171, let F be the fixed point and HK the fixed line. Take the origin at A half way between F and HK. Let P be any point satisfying the condition PF == PH. Call OD = x, PD = y, and represent the given distance FK by 2p. Then, from the right triangle PFD, 414 ELEMENTARY MATHEMATICAX ANALYSIS [§230 ppi = 2/2 + FD^ (1) = 2/2 + {x - OFY = 1/2 + (a; - p)2. Since PF by definition equals PH or a; + p, we have {x + vY = 2/2 + {x - v)\ (2) whence y^ = 4px, (3) which is the equation of the parabola in terms of the focal distance, OF or p. The double ordinate through F is called the latus rectum. The semi-latus rectum can be obtained from (3) by placing X = p, whence 1 = 2p, (4) where / is the semi-latus rectum. Hence the entire latus rectum is 4p, or the coefficient of x in equation (3). In Fig. 171, the quadrilateral FLIK is a square since FL and FK are each equal to 2p. 236. Polar Equation of the Parabola. In accordance with the ratio definition of the parabola, its polar equation is found at once from equation (6), §234, by putting e = 1. Hence the polar equation of the parabola is '' = 7Tiosl" (^) For this equation we may make the following table of values: e P 0° 1/2 90° I 180° 00 270° I This shows that the parabola has the position shown in Fig. 171. This is the form in which the polar equation of the parabola is used in mechanics and astronomy. 237. The Conies. It is now obvious that a single definition can be given that will include the ellipse, hyperbola and parabola. §237] THE CONIC SECTIONS 415 These curves taken together are called the conies. The definition fnay be worded: A conic is the locus of a point whose distances from a fixed point (called the focus) and a fixed line (called the directrix) are in a constant ratio. The unity between the three curves was shown by their equation in polar coordinates. Moving the ellipse so that its left vertex passes through the origin, as in §85, and writing the hyperbola with the origin at the right ver- tex (so that both curves pass through the origin in a comparable manner), we may compare each with the parabola as follows: TheeUipse: y^ = 2lx - (b^/a^)x'' (1) The parabola: j/' = 2lx (2) The hyperbola: y'' = 2lx + (b''/a')x'' (3) In these equations I stands for the semi-latus rectum of each of the curves. These equations may also be written 1/2 = 2lx - (l/a)x' (4) 2/^ = 2lx I (5) 2/2 = 2lx + (l/a)x'' (6) whence it is seen that if Z be kept constant while a be increased without limit, the ellipse and hyperbola each approach the parab- ola as near as we please. Only for large values of x, if a be large, is there a material difference in the shapes of the curves. Exercises 1. Write the equation of the circle in the form (1) above. 2. Write the equation of the equilateral hyperbola in the form (3) ■ above. 3. Describe the curve ^ I '' 1 -I- cos (e - a)' where a is a constant. 4. In Fig. 172 translate the curve xy = Ihy suitable change in the equation to the position shown by the dotted curve, if the translation of each point is unity. 416 ELEMENTARY MATHEMATICAL ANALYSIS [§237 6. In Pig. 173 translate the curve j/* = 4px by suitable change in the equation to the position shown by the dotted curve, if the distance each point is moved be 3p. Fig. 172. — A hyperbola translated at an angle of 45° to OX. 6. A bridge truss has the form of a circular segment, as shown in Fig. 174. If the total span be 80 yards and the altitude BS be 20 Y 0^ j^ ^-^^ X — —i. A l» Ci 10 Ca Fig. 173. — A parabola translated Fig. 174. — Bridge truss in form at an angle of 60° to OX. of circular segment. yards, fmd the ordinates CiDi, C2D2, etc., erected at uniform intervals of 10 yards along the chord AAi. 7. A bridge truss has the form of a parabolic segment, as shown in Fig. 175. The span AAi is 24 yards and the altitude OB is 10 §238] THE CONIC SECTIONS 417 erected at yards. Find the length of the ordinates CD, CiDi, . untform intervals of 3 yards along the Une AAi- 238.* The Conies are Conic Sections. The curves nowtnown as the conies were originally studied by the Greek geometers as the sec- X 3 6 9 12 •^'^^ OoS \ \ Ai B Y Cs Ci C A Fig. 175. — Bridge truss in the form of a paraboUc segment. tions of a circular cone cut by a plane. It can be shown that the three classes of curves, parabola, eUipse, and hyperbola, can be made respectively by cutting any circular cone : (1) by a plane parallel to an Fig. 176. — Section of a circular cone. element; (2) by a plane cutting opposite elements of the same nappe of the cone; (3) by a plane cutting both nappes of the cone. The two nappes of a conical surface, it will be remembered, are the two portions of the surface separated by the apex. 27 418 ELEMENTARY MATHEMATICAL ANALYSLS [§239 In Fig. 176, let the plane NDN'D', caUed the cutting plane, cut the lower nappe of a right circular cone in the curve VPV. It can be proved by geometry that this curve is an ellipse. The foci F and F are the points at which the two inscribed spheres SFS' and RF'R' are tangent to the plane ND'. The directrices are the two lines ND and N'D' in which the plane ND' cuts the two planes SHS' and RKR' produced. 239. Tangent to the Parabola. Let us investigate the condition that the line y = mx + b shaU be tangent to the parabola y^ = ipx. First find the points of intersection of these loci by solving the two equations y = mx + 6 (1) 2/2 = 4pa; (2) as simultaneous equations. Eliminating y by substituting the value of y from (1) in (2), mV + 2mbx + b' - ipx = 0, (3) or mV + 2{mb - 2p)x + b^ == 0. (4) Solving for x (see formula for quadratic. Appendix §309, (2)). _ _ mb — 2p 2\/p' — pmb /gx m^ ~ m^ Therefore there are in general two values of x or two points of intersection of the straight line and the parabola. By the defini- tion of a tangent to a curve (§146) this line becomes a tan- gent to the parabola when the two points of intersection become a single point; that is, when the expression under the radical in (5) approaches zero. This condition requires that p^ — pmb = 0, or b = p/m. (6) Therefore when 6 of equation (1) has this value, the line is tangent to the parabola. The equation of the tangent line is, therefore y = mx + p/m. (7) This line is tangent to the parabola y^ = ipx for all values of §240] THE CONIC SECTIONS 419 m. Substituting in (5) the value of & = p/m, we may find the abscissa of the point of tangency i xi = p/m'. (8) Substituting this value of x in (7) the corresponding ordinate o' this point is found to be yi = 2p/m. (9) 240. Properties of the Parabola. In Fig. 171, F is the focus, HK is the directrix, PT is a tangent at any point P The perpendicular PN to the tangent at the point of tangency is called the normal to the parabola. The projection DT of the tangent PT on the X-axis is called the subtangent and the pro- jection DN of the normal PN on the X-axis is called the sub- normal. The line through any point parallel to the axis, as PR, is known as a diameter of the parabola. (a) The subtangent to the parabola at any point is bisected by the vertex. It is to be proved that OT = OD for all positions of P Now OD is the abscissa of P, which has been found to be p/m^. From the equation of the tangent y = mx + p/m, the intercept OT on the X-axis is found by putting y = Q and solving for x. This yields X = — p/m}. This is numerically the same as OD, hence the vertex bisects DT. (6) The subnormal to the parabola at any point is constant and eqval to the semi-latus rectum. The angle DPN has its sides mutually perpendicular to the sides of the angle DTP, hence the angles are equal. Since the tangent of the angle DTP = m, therefore tangent DPN = m. From the right triangle PDN, DN = PD tan DPN = PD m = i2p/m)m = 2p. 420 ELEMENTARY MATHEMATICAL ANALYSIS [§241 Since KF also equals 2p, we have DN = KF. (fi) PFTH is a rhombus. To prove the figure PFTH a rhombus it is merely necessary to show that FT = PH, since PF = PH FT = FO + OT PH = DK = DO + OK But OD = OT and OK = FO. Therefore FT = PH, and 1;he figure is a rhombus. It follows that the two diagonals of the rhombus intersect at right angles on the F-axis. (d) The normal to a parabola bisects the angle between the focal radius and the diameter at the point. We are to show that Z NPF = Z NPR. Since FPHT is a rhombus, Z FPT = Z TPH. But Z TPH = Z RPS, being vertical angles. From the two right angles NPT and NPS subtract the equal angles last named. Hence, Z FPN = Z NPR. It is because of this property of the parabola that the reflectors of locomotive or automobile headlights are made parabolic. The rays from" a source of light at F are reflected in lines parallel to the axis, so that, in the theoretical case, a beam of light is sent out in parallel lines, or in a beam of undiminishing strength. 241. To Draw a Parabolic Arc. One of the best ways of de- scribing a parabolic arc is by drawing a large number of tangent lines by the principle of §240 (c). Since in Fig. 171 the tan- gent is for all positions perpendicular to the focal line FH at the point where the latter crosses OY, it is merely necessary to §241] THE CONIC SECTIONS 421 draw a large number of focal lines, as in Fig. 177, and erect perpendiculars to them at the points where they cross the F-axis. The equations of the tangent lines in Fig. 177 are of the form = mx + p/m (1) Fig. 177. — Graphical construction of a parabolic arc "by tangents." in which p is the constant given by the equation of the parabola, and in which m takes on in succession a sequence of values appro- priate to the various tangent lines of the figure. These lines are said to constitute a family of lines and to envelop the curve to 422 ELEMENTARY MATHEMATICAL ANALYSIS [§242 which they are tangent. The curve itself is called the envelope of the family of lines. The curve of the supporting surface of an aeroplane as well as the curve of the propeller blades is a parabolic arc. The curve of the cables of a suspension bridge is also paraboUc. Exercises 1. Write the equation of the parabola which the family y = mx + 7 /2m envelops. 2. Draw an arc of a parabola if p = 3 inches. 3. At what point is ^ = mx + Z/m tangent to the parabola y' = 121? 4. At what point isy = mx + ll/ira tangent to y^ = 44x? 6. Draw the family of lines y = mx + 1/m for m = 0.4, m = 0.6, m = 0.8, m = 1, m = 2, m = 4, m — 8. 242. Tangent to the Circle. An equation of a tangent hne to a circle can be found, as in the case of the parabola above, by finding, the points of intersection of y = mx + b (1) and x^ + y^ = a^ (2) and then imposing the condition that the two points of intersection shall become a single point. The value of 6 that satisfies this condition when substituted in (1) gives the equation of the re- quired tangent. It is easier, however, to obtain this result by the following method. In Fig. 178 let the straight line be drawn tangent to the circle at T. Let the slope of this line be m. Then m = tan ONT = tan a, if a be the direction angle of the tangent line. The intercept b of t*he line on the F-aris can be ex- pressed in terms of a and a, b = a sec a = ay/l -{- m'^. (3) Hence the equation of the tangent to the circle is y = mx + a-\/i + m^. The double sign is written in order to include in a single equation the two tangents of given slope m, as illustrated in the figure. §243] THE CONIC SECTIONS 423 Exercises 1. Find the equations of the tangents to x' + y^ = 16 making an angle of 60° with the X-axis. 2. Find the equations of the tangents to x' + y^ = 25 making an angle of 45° with the X-axis. 3. Find the equation of tangents to x^ + y^ = 25 parallel to y = 3x - 2. 4. Find the equation of tangents to x^ + y' = 16 perpendicular to y = (l/2)x + 3. 5. Find the equations of the tangents to (a; — 3)^ -|- (?/ — 4)^ = 25 whose slope is 3. FiG. 178. — The equation of a line of given slope, tangent to a given circle. 6. Find the equation of the tangent to the circle by the method of §239. 243. Nonnal Equation of Straight Line. The normal equation of the straight line was obtained in polar coordinates in §71. The equation was written p cos {d — a) = a. (1) In this equation (p, d) are the polar coordinates of any point on the line, a is the distance of the line from the origin, and a is the direction angle of a perpendicular to the line from the origin. (See Fig. 71.) Expanding cos {6 — a) in (1) we obtain f> cos 6 cos a -J- p sin 5 sin a = a. (2) 424 ELEMENTARY MATHEMATICAL ANALYSIS [§243 But for any value of p and d, p cos B = x and p sin 9 = y. Hence (2) may be written in rectangular coordinates X cos a + y sin a = a. (3) This also is called the normal equation of the straight line. If an equation of any line be given in the form ax + by = c (4) it can readily be "reduced to the normal form. Dividing this equation through by s/a^ + b^, " b c .,, X + . y = . (5) Va' + b^ Va^ + h^ VaT+h^' Now aly/a?' + 6^ and b/-\/a^ + h^ may be regarded as the cosine and sine, respectively, of the angle formed with the positive Z-axis by the line joining the origin to the point (a, 6). Calling this angle a, equation (5) may be written X cos a + 2/ sin o; = d, (6) which is of the form (3) above. Inasmuch as the right-hand side of the equation in the normal form represents the distance of the line from the origin, it is best to keep the right-hand Side of the equation positive. The value of a and the quadrant in which it lies is then determined by the signs of cos a and sin a on the left- hand side of the equation. The angle a may have any value from 0° to 360°. Illustrations: • (1) Put the equation Zx — ^y = 10 in the normal form. Here a2 + 62 = 3' + ( - 4)2 =25. Dividing by 5 we obtain {Z/5)x - (4/5)2/ = 2. The distance of this line from the origin is 2. The angle a is the angle whose cosine is 3/5 and whose sine is — 4/5. Therefore, from the tables, a = 306° 52'. (2) Put the equation 3a; — 4v 4- 20 = in the normal form. Trans- posing and dividing by — 1 to make the right-hand side of the equa- tion positive, we obtain — 3a; -(- 4i/ = 20. Here cos a = - 3/5, sin a = 4/5, d = 4. Hence a = 126° 52'. §244] THE CONIC SECTIONS 425 (3) What is the distance between the lines (1) and (2)? The lines are parallel and on opposite sides of the origin. Their distance apart is therefore 2 + 4 or 6. (4) Put X + y = I in the normal form. Here VoM- b^ = V'2. The equation becomes i -s/^x + i ■\/2y = i y/2. a = 45°, d = i \/2. Exercises 1. The shortest distance from the origin to a line is 5 and the direc- tion angle of the perpendicular from the origin to the line is 30°- Write the equation of the line. 2. The perpendicular from the origin upon a straight line makes an angle of 135° with OX, and its length is 2v'2. Find the equation of the line. 3. Write the equation of a straight line in the normal form if a = 60° and d = y/s. 4. Put 2\/3a; + 2?/ = 32 in the normal form and find the numerical values of a and d. 6. Put 2a; — 2i/ = 1 in the normal form and find the values of a and d. 6. Find the equation of the straight line, if the perpendicular from the origin on the line, makes an angle of 46° with the X-axis and its length is ■\/2. 7. Put 2 -ho = 1 ii the normal form. 244. To Translate Any Locus a Given Distance in a Given Direc- tion. To move any locus the distance d to the right we sub- stitute (xi — d) for X in the equation of the locus. To move the locus the distance d in the y direction we substitute (2/1 — d) for y. To move any locus the distance d in the direction a we substitute (xi — d cos a) for x, , , (2/1 — d sin a) for y, which must give the desired equation of the new locus. It is not necessary to use the subscript attached to the new coordinates if the distinction between the new and old coordinates can be kept in mind without this device. The circle x^ + y^ = a^ moved the distance d in the direction a becomes (x — d cos a)^ -i- (y — d sin a)" =• a'^ 426 ELEMENTARY MATHEMATICAL ANALYSIS [§245 which may be changed to x^ — 2Ax, cos a + 2/^ — 2^2/ sin a = a?- — cP. 245. Distance of Any Point From Any Line. Let the equation of the line I, Kg. 179, in the normal form be X cos a + 2/ sin a = a, (1) and let (a;i, t/i) be any point P in the plane. (See Fig. 179.) If the point {xi, yi) is on the opposite side of the line froin the origin, the line can be moved so that it will pass through the point by translating it the proper distance in the direction a. Let the unknown amount of the required translation be represented by d. To translate the line the amount d in the direction a, we must substitute for x and y the values X = x' — d cos a . . y = y' — d cos a We obtain (x' — d cos a) COB a + iy' — d sin a) sin a =a. (3) The line represented by this equation passes through the point (xi, 2/i). Substituting these coordinates for x' and y' and solving for d, we have d = Xi COS a + yi sin a — a. (4) This is the distance of (xi, yi) from the given line. If the given point is on the same side of the line as the origin, as the point Pixi, y^ Fig. 179, then the given line must be translated the distance d in the direction (180° + a), and the result is the same as (4) above except all signs are changed. We are usu- ally interested only in the numerical value of d, so that formula (4) may be used for all cases. When the value of d comes out negative it merely means that the given point is on the same side of the line as the origin. Equation (4) may be interpreted as follows : To find the distance of any point from a given line, put the equa- tion of the line in the normal form, transpose all terms to the left- hand member and siibstitvte the coordinates of the given point for x and y. The absolute value of the left-hand member is the distance of P from the line. §245] THE CONIC SECTIONS 427 The above facts may be stated in an interesting form as follows: Let any line be X cos a + 2/ sin a — a = 0. If the coordinates of any point on this line be substituted in this equation, the left-hand member reduces to zero. If the coordi- nates of any point not on the line be substituted for x and y in the equation, the left-hand member of the equation does not reduce to zero, but becomes negative if the given point is on the origin side of the line and positive if the given point is on the non- origin side of the line . The absolute value of the left-hand member \ \'^ Y \ \ \ 2 \ \ \ \ \ cZ^APi( ^ a \ \ P.ik^, !/=) > \ \ <^ \ \ /- / \ \ -A \ \ \ \_ Fig. 179. — Distance of any point from a given Une. in each case gives the distance of the given point from the line. Thus every line may be said to have a "positive side" and a "negative side." The "negative side" is the side toward the origin. Illtjstbation 1. Find the distance of (—1, 4) from the line 3a; — 41/ = 10. Transpose and put left-hand member in normal form 1^ — iV 0. 428 ELEMENTARY MATHEMATICAL ANALYSIS [§246 Subatitute — 1 for a; and 4 for y. The left member is now the value of d, BO that The result is negative, so that ( — 1, 4) is on the same side of the line as the origin. iLLtrsTRATioN 2. Find the distance of (2, — 4) from x-2 y+3 4 7 Clear of fractions and simplify, 7a; - 4?/ - 26 = 0. Put in normal form, iz^ - i\y - u = 0. Substitute 2 for x and — 4 for y, The point is on the non-origin side of the given line, and irV of one imit from it. Exercises 1. Find the distance of the point (4, 5) from the line 3x + iy = 10. 2. Find the distance from the origin to the line x/3 — 2//4 = 1. 3. Find the distance from (-3, - 4) to 12(a; + 6) = 6(2/ - 2). 4. Find the distance from (3, 4) to the line x/3 — y/4 = 1. 5. Find the distance between the parallel lines y = 2x -\- 3, and y = 2x +5. 6. Find the distance between y = 2x — 3, y = 2x + 5. 7. Find the distance from (0, 3) to 4a; — 3y = 12. 8. Find the distance from (0, 1) to a; + 2 — 2?/ =0. 246. Tangent to a Circle at a Given Point. The equation of a line having a given slope m and tangent to a given circle with center at the origin, was given in §242. We shall now find the equation of the line that is tangent to the circle at a given point (xo, 2/o). The line, a ='p cos {6 — a), (1) or its equivalent, a; cos a + 2/ sin a = a, (2) §247] THE CONIC SECTIONS 429 is tangent to the circle of radius a, and the point of tangency is at the end of the diameter whose direction angle is a. The point of tangency is therefore (o cos a, a sin a). Hence, multiplying (2) through by a, we obtain x{a cos a) + y(a sin a) = a^, (3) or xox + yoy = a.K (4) This is the equation of the hne tangent to the circle of radius a at the point (xq, ya). Thus 3a; + 4?/ = 25 is tangent to a;^ + y"^- = 25 at (3, 4). 247. Tangent to the Ellipse at a Given Point. It is easy to draw the tangent to the ellipse at any desired point. In Fig. 180, Fig. 180. — Tangent to the ellipse at a given point. let Po be the point at which a tangent is desired. Draw the major circle, and let Pj of the circle be a point on the same ordinate asPo. Draw a tangent to the circle at Pi and let it meet the Z-axis at T. Then when the circle is projected to form the ellipse, the straight line PiT is projected to make the tangent to the ellipse. Since T when projected remains the same point and since Po is the projection of Pi, the line through Po and T is the required tangent to the ellipse. 430 ELEMENTARY MATHEMATICAL ANALYSIS [§248 The equation of the tangent PoT is alao readily found. The equation of PiT is xxo + yyo = ffl^ (1) To project his into the line Po^ it is merely necessary to multiply the ordinates y and yo' by b/a; that is, to substitute y = ay/b and 2/o' = ayo/b. Whence (1) becomes xox + a^yoy/b^ = a^ (2) or, dividing by a^, xox/a" + joj/b^ = I (3) which is the tangent to a;2/a2 -I- 2^2/62 = 1 at the point (xo, yo)- Exercises 1. Find the equations of the tangents to the ellipse whose semi-axes are 4 and 3 at the points for which x = 2. 2. Find the equations of the tangents to x'/16 + y^/9 = 1 at the ends of the left-hand latus rectum. 3. Required the tangents to x'/9 + y'/i = 1, making an angle of 45° with the X-axis. [Solve y = x + b and x'/9 + y"/4 = 1 as in §239.] 4. Find the equatioiis of the tangents to sr^/lOO -I- y'/25 = 1 at the points where y = 3. 6. Find the equations of the tangents to x'/S6 + y'/16 = 1 at the ^ints where x = y. 248. The Tangent, Normal, and Focal Radii of the Ellipse. In the right triangle PiOT, Fig. 180, the side PiO is a mean propor- tional between the entire hjrpotenuse OT and the adjacent segment OD. That is a^ = xoX OT, or OT = a^/xo Then FiT = OT - OFi = OT - ae = a'^/xo — ae Wkewis^ FtT = OT + OFi 5= aVaio ■+- ae §249] THE CONIC SECTIONS 431 Therefore FrT/FiT = {a^xa - ae)/{ayxa + ae) = (a — exo)/{a + exo) But by §230 this last ratio is equal to r^/r^. Therefore we may write FiT/F^T = P^F./PoF,. Hence T, which divides the base FiFi of the triangle PoFaFi externally at T in the ratio of the two sides PF^ and PFi of the triangle, lies on the bisector of the external angle FiPoQ of the triangle FJPoFi. This' proves the important theorem: The tangent to the ellipse bisects the external angle between the focal radii at the point. This theorem provides a second method of constructing a tangent at a given point of an elUpse, often more convenient than that of §247, since the method of §247 often runs the construction off of the paper. The normal PoN, being perpendicular to the tangent, must bisect the internal angle F^PoFi between the focal radii F^o and F,Po. Since the angle of reflection equals the angle of incidence for light, sound, and other wave motions, a source of light or sound at Fi is "brought to a focus" again atF^, because of the fact that the normal to the ellipse bisects the angle between the focal radii. 249. Additional Equations of the Straight Liae.^ The equations of the straight line in the slope form y = mx + b (1) and in the normal forms p cos {d— a) = a (2) a; cos a + 2/ sin a = a (3) and the general form ax + by + c = (4) have already been used. Two constants and only two are neces- sary for each of these equations. The constants in the first equation are m and 6; in the second and third, a and a; in the fourth a/c and b/c, or any two of the ratios that result from divid- ing through by one of the coefficients. Equation (4) appears to contain three constants, but it is only the relative size of these that > See §17. 432 ELEMENTARY MATHEMATICAL ANALYSIS [§249 determines the particular line represented by the equation, since the line would remain the same when the equation is multiplied or divided through by any constant (not zero). These facts are usually summarized by the statement that two conditions are necessary and sufficient to determine a straight line. The number of ways in which these conditions may be given is, of course, unhmited. Thus a straight line is determined if we say, for example, that the line passes through the vertex of an angle and bisects that angle, or if we say that the line passes through the center of a circle and is parallel to another line, or if we say that the straight Une is tangent to two given circles, etc. An important case is that in which the line is determined by the requirement that it pass through a given point in a given direc- tion. The equation of the line adapted to this case is readily found. Let the given point be {xi, yi). The line through the origin with the required slope is y = mx. Translate this line so that it passes through (xi, yi) and we have y - yi = m(x - xi). (5) Another way of obtaining the same result is : Substitute the coordinates (.Xi, yi) in (1) 2/1 = mxi + 6. (6) Subtract the members of this from (1) above, so as to eliminate b. There results y - yi = m{x — a;i). (7) This is the required equation. The given point is (a;i, 2/1) and the direction of the line through that point is given by the slope m. Another important case is that in which the straight line is determined by requiring it to pass through two given points. Let the second of the given points be (012, yi). Substitute these coordinates in (5) 2/2 - 2/1 = »»(»2 - a;i). (8) To eliminate ?n, divide the members of (7) by the members of (8) 2/ - 2/1 X — xi 2/2 - 2/1 2:2 - X\ (9) §250] THE CONIC SECTIONS 433 or, as it is usually written L^Zli ^Yl^zll^; (10) X — Xi X2 — Xi This is the equation of a line passing through two given points. Since (10) may be looked upon as a proportion, the equation may be written in a variety of forms. 250. The Circle Through Three Given Points. In general, the equation of a circle can be found (when three pojnts are given. Either of the general equations of the circle {x - hY +{y - kY = a\ (1) or a;" + 2/^ + 2gx + 2/2/ + c = (2) contains three unknown constants, so that in general three con- ditions may be imposed upon them. It is best to illustrate the general method by a particular example. Let the three given points be (— 1, 3), (0, 2), and (5, 0). Then since the coordinates of these points must satisfy the equation of the circle, we obtain from (2) above 1 + 9 - 2ff + 6/ + c = 0, (3) 4 + 4/ + c = 0, (4) 25 +\0g + c = 0. (5) Eliminating c from (3) and (4) and from (4) and (5), we obtain 6 - 2sr^ + 2/ = 0, 21 + 10^ - 4/ = 0. Eliminating / ? = - 5i Whence / = - 8i and a = 30. So the equation of the circle is ■ a;2 4- yi _ lla; _ ny + 30 = 0. 28 434 ELEMENTARY MATHEMATICAL ANALYSIS [§251 Exercises 1. Knd the equation of the line passing through (2, 3) with slope 2/3. 2. Find the equation of the line passing through (2, 3), and (3, 5). 3. Find the line passing through (2, — 1) making an angle whose tangent is 2 with the Z-axis. 4. Find the line through (2, 3) parallel to 2/ = 7a; + 11. 5. A line passes through (—1, — 3) and is perpendicular to y — 2x —' 3. Find its equation. 6. Find the line passing through (— 2, 3), and (—3,-1). 7. Find the equation of the line which passes through (— 1, —3), and (-2,4). 8. Find the slope of the line that passes through ( — 1, 6 ), and (-2,8). 9. Find the equation of the line passing through the left focus and the upper end of the right latus rectum of a;2/2S + y'/9 = 1. 10. Find the equation of the circle passing through (2, 8)„ (5, 7), and (6, 6) . 11. Find the equation of the circle which passes through (1, 2), (- 2, 3), and (- 1, - 1). 12. Find the equation of the parabola in the form y^ = ipx which passes through the point (2, 4). 251. Change from Polar to Rectangular Coordinates. The relations between x, y of the Cartesian system and p, 6 of the polar system have already been explained and use made of them. The relations are here brought together for reference : X = p cos 8 (1) y = p sin 0. (2) By these we may pass from the Cartesian equation of any locus to the equivalent polar equation of that locus. Dividing (2) by (1) and also squaring and adding, we obtain: e = tan-i y/x (3) P = Vx^ + y^ (4) These may be used to convert any polar equation into the Carte- sian equivalent. 262. Rotation of Any Locus. It has already been explained that any locus can be rotated tlirough an angle a by substituting §252] THE CONIC SECTIONS 435 (Ot — a) for d in the polar equation of the locus. It remains to determine the substitutions for x and y which will bring about the rotation of a locus in rectangular coordinates. Let us consider any point P of a locus before and after rotation through the given angle a. Call the coordinates of the point before rotation (x, y) in rectangular coordinates and (p, 6) in polar coordinates.' Then, from (1) and (2), §251, X = p cos 6 (1) t/ = P sin 6. (2) Call the coordinates of the point after rotation (xi, yi) and (Pi, ^i), but note that the value of p is unchanged by the rotation. Then for p (Pi, e,) or the point P', Fig. 181, we may write ] /\ (X1.V1) Xi = p cos 61 (3) 2/1 = p sin 01. (4) Since, however, the rotation requires — that Fig- 181 6 = Oi - ot (5) equations (1) and (2) become X = p cos (di -^ a) = p cos di cos a + p sin 81 sin a (6) 2/ = p sin (di — a) = p sin 61 cos a — p cos 0i sin a. (7) But, from (3) and (4), p cos di and p sin di are the new values of X and y; hence, substituting in (6) and (7) from (3) and (4) we obtain X = xi cos Qi + yi sin a (8) y '= yi cos a — Xi sin a. (9) Hence, if the equatiofi of any locus is given in rectangular co- ordinates, it is rotated through the positive angle a by the sub- stitutions X cos a + y sin a for X y cos Q! — X sin a for y, (10) in which it is permissible to drop the subscripts, if the context shows in each case whether we are^ dealing with the old x and y or with the new x and y. PiP.e) or -Rotation of any locus. 436 ELEMENTARY MATHEMATICAL ANALYSIS [§252 If the required rotation is clockwise, or negative, we must replace a by (— a) in aU of the above equations. Whenever comenient, the eqvation of a curve should he taken in the -polar form if it is required to rotate the locus. Important Facts: The following facts should be remembered by the student: (1) To rotate a curve through 90°, change x to y and y to { — x). (2) Rotation through any angle leaves the expression x^ + y^ (fir any function of it) unchanged. This is obvious since the circle x^ + y^ = a^ is not changed by rotation about (0, 0). Fig. 182. — Effect of rotation on the special forms x^ + y^, 2xy, and x^ - 2/2. (3) Rotation through + 45° changes 2xy to y^ — x^. Rotation through — 45° changes 2xy to x^ ,— y^. (4) Rotation through + 45° changes x^ — y^ to 2xy. Rotation through — 45° changes x'^ — y^ to — 2xy. Statements (3) and (4) follow at once from consideration of the equations i;2 - 2/2 = a^ (1) a' (2) a' <3) a^ (4) 2xy yi _ j;Z - 2xy §253] THE CONIC SECTIONS 437 of the four hyperbolas bearing corresponding numbers (I), (2), (3), (4) in Fig. 182. The proper substitution in any case can be remembered by thinking of the four hyperbolas of this figure. (5) The degree of an equation of a locus cannot he changed by a rotation. This follows at once from the fact that the equations of transformation (8) and (9) are linear. Exercises In order to shorten, the work, use statements (1) to (4) whenever possible. 1. Turn the locus a* — j/' = 4 through 45°. 2. Turn x' + y' = a' through 79°. Turn ixy = 1 through 45°. 3. Turn x cos a -\- y sin a = a through an angle /3. (Since this locus is well known in the polar form, transformation formulas (6) and (7) above need not be used.) 4. Rotate x' - y^ = 1 through 90°. 5. Rotate s" — j/^ = a' through — 45°. 6. Rotate x' - y' = 1 through 30°. 7. Rotate x^ - y" = 4: through 60°. 8. Rotate ixy = 1 through 30°. 9. Rotate x' + 2y' = 1 through 45°. 10. Change the equation (x — a)' + (y — b)' = rHo the polar form. 11. Change p cos 29 = 2a, one of a class of curves known as Cote's spirals, to the Cartesian form. 12. Write the equation of the lemniscate in the polar form. 13. Show that p' — 2p/oicos (9 — 9i) + pi" = a'is the polar equation of a circle with center at (pi, 9i) and of radius a. 14. Write the Cartesian equation of the locus p" = 16 sin 29. 15. Turn p^ = 8 sin 29 through an angle of 45°. 16. Rotate x^ - 2y^ = 1 through 90°. 17. Rotate {x^ + y^)^ + {x' - y')^ = 1 through 45°. 18. Rotate log {x' + y^) = tan {x^ - y^) through 45°. 19. Rotate x^ -&xy -\-y''=\ through 45°. 20. Rotate x^ + y^^ = a}^ through 45°. Show that the result is the parabola y = x^la + o/2, and sketch the curves. 253. Ellipse with Major Axis at 45° to the QX Axis. The ellipse frequently arises in applied science as the resultant of the. projection of the motion of two points moving uniformly on two 438 ELEMENTARY MATHEMATICAL ANALYSIS [§253 circles, as has already been explained in §186. Thus the para- metric equations « X = a cos t (1) 2/ = 6 sin t, (2) define an ellipse which may be considered the resultant of two S.H.M. in quadrature. We shall prove that the equations a; = o cos t y = asm (t + a), (3) (4) define an ellipse, with major axis making an angle of 45° with OX. The graph is readily constructed as in Fig. 183. The Car- tesian equation of the curve is found by eliminating t between / / ^ Y B |\ Ic — ■P' \^ "^ y ¥ \y / ^' A X' / .^^ ■^ 1 y\- \x / , -^ O / A T 3 .■'• / 1 -^ y / c ^ ^ r' Fig. 183. — The ellipse x = a cos t,y — a sin (< + a). (3) and (4). Expanding the sin {t + «) in (4) and substituting from (3) we obtain (5) y = X sin.a + -y/a^ — x"^ cos a. Transposing and squarimg a;^ — 2xy sin a + j/'' = a^ cos' a. (6) By §252 rotate the curve through an angle of (— 46''). We §254] THE CONIC SECTIONS 439 know that (x* + y^) is unchanged and that 2xy is to be replaced by (a;2 — y^). Therefore (6) becomes x^l — sin a) + y^{l + sin a) = a^ cos^a. (7) Replacing cos^ a by 1 — sin^ a, and dividing through by the right-hand member, we obtain a\l + sin a) "^ a^l - sin a) " ^' ^^^ which may be written —-— + -^,= 1, (9) 2a2 cos^ I 20^ sin'' |^ where /8 is the complement of a. Equation (8) or (9) proves that the locus is an ellipse. It is any ellipse, since by properly choosing a and a the denominators in (8) can be given any desired values. Hence the pair of parametric equations (3) and (4), or the Cartesian equation (5) represents an ellipse with its major axis inclined + 45° to the X-axis. 254. She.ar of the Circle. The effect of the addition of the term mx to f(x), in the equation y = f(x), has been shown in §38 to change the shape of the locus by lamellar, or shearing, motion in the Xy-plane. We usually speak of this process as "the shear of the locus y = fix) in the line y — mx." When appUed to the circle ?/ = + V'a^ — x'^ the effect is to move vertically the middle point of each double ordinate of the circle to a position on the line y = mx. The result of the shearing motion is shown in Fig. 184. The area hounded by the curve is unchanged by the shear. The equation after shear is • y = mx + ■\/a'^ — x^. (1) This is the same form as equation (5) of §253, if we put m = and replace y cos ahyyu After the substitution, rotate the curve sma cos a 440 ELEMENTARY MATHEMATICAL ANALYSIS [§255 through 45°, and replace ^i by y cos a. The equation can then be written a^ (1 + sin a) y^ (1 + sin a) (2) Therefore the curve of Fig. 184 is an ellipse. The straight line y = mx passes through the middle points of the parallel vertical chords of the eUipse y = mx + (3) The locus of the middle points of parallel chords of any curve is called a diameter of that curve. We have thus shown that one diameter of the eUipse is a straight line. Since the same reasoning applies to y = mx+ {h/aWa^ - x\ (4) X' / 4 . / A w Fig. 184. — The ellipse looked upon as the shear of the circle OA in a Une M'OM. which may be regarded as any ellipse in any way oriented with respect to the origin, the proof shows that the mid-points of arbi- trarily selected parallel chords of an eUipse is always a straight line. 266. General Equation of the Second Degree. The general equation of the second degree in two variables may be written in the standard form oa;2 + 2hxy + &2/' + "igx + 2/3/ + c = 0. (1) §256] THE CONIC SECTIONS 441 In treatises on Conic Sections it is shown that the general equation of the second degree in two variables represents a conic. Three cases are distinguished as follows: The general equation of the second degree represents an ' eUipse if h^ — ab < a parabola ii h^ — ab = a h3rperboIa if h^ — ab > 0. (2) (3) (4) To render the above classification true in all cases we must classify the "imaginary ellipse," -^ +r^ = — 1, as an ellipse, and other degenerate cases must be similarly treated. The expression h^ — ab is called the quadratic invariant of the equation (1), so Fig. 185. — Confocal ellipses and hyperbolas. Note that the curves of , one set cut the curves of the other set orthogonally. called because its value remains unchanged as the curve is moved about in the coordinate plane. In other words, as the locus (1) is translated or rotated to any new position in the plane, and while of course the coefficients of x^, xy, and y^ change to new values, the function of these coefiicients, h'^ — ab, does not change in value, but remains invariant. The above facts are not proved in this book. 442 ELEMENTARY MATHEMATICAL ANALYSIS [§256 266. Confocal Conies. Fig. 185 shows a number of ellipises and hyperbolas possessing the same foci A and B. This family of curves may be represented by the single equation, in which the parameter k takes on any value less than a^, and in which a > b. If fc satisfies the inequality k < 6^ the curves are ellipses. If k satisfies the inequalities ¥ < k < a\ the curves are hyperbolas. The ellipses of Fig. 185 may be regarded as representing the successive positions of the wave front of sound waves leaving the sounding body AB, or they may be regarded as the equipotential lines around the magnet AB, of which the hyperbolas represent the lines of magnetic force. Exercises 1. Sketch the curve y = 2x + \/4 — as". 2. Draw the curve X = 2 cos B 2/ = 2 sin (9 +7r/6). 3. Find the axes of the elUpse X = 3 cos 6 2/ = 3 sin (9 + 7r/4). 4. Draw the curve y = x + \/6x — x*. 6. Draw the curve y = x + y/x' — 6i. 6. Show that y = x ± \/6x is a parabola. 7. Sketch the curve y = (I/2)x + Vl6 - x^ 8. Sketch the curve 2/ = 5x sin 60° + cos 60°-\/25 -x«. 9. Discuss the curve x'/a' + y'/b" - 2{xy/ah) sin a = cos^ a. §256] THE CONIC SECTIONS 443 Show that the locus is always tangent to the rectangle x = ± o, y = ± b, and that the points of contact form a parallelogram of constant perimeter 4:\/a' + b^ for all values of a. Hint: Compare with equation (6), §253. 10. Show that x = a cos {8 — a), y = b sin (9 — or) represents an elUpse for all values of a. ■ 11. Prove from equation (8), §253, that the distance from the end of the minor to the end. of the major axis of the resulting ellipse remains the same independently of the magnitude of a. 12. Show that the following construction of the hyperbola xy' = a' is correct. On the — X-axis lay off OC = a. Connect C with any point A on the F-axis. At C construct a perpendicular to AC cut- ting the F-axis in B. At B erect a perpendicular to BC cutting the -|- X-axis at D. Through A draw a parallel to the X-axis and through D draw a parallel to the F-axis. The two lines last drawn meet at P, a point on the desired curve. 13. Explain the following construction of the cubical parabola a^y = x'. Lay off OB on the — F-axis equal to a. From B draw a line to any point C of the X-axis. At C erect a perpendicular to BC cutting the F-axis at D. At D erect a perpendicular to CD cutting the X-axis at E. Lay off OE on the F-axis. Then OE is the ordinate of a point of the curve for which the abscissa is OC. 14. Explain and prove the following construction of the semi- cubical parabola, ay^ = i'. Lay off on the — X-axis, OA = u.. From A draw a parallel to the line y = mx, cutting the F-axis in B. Erect at B a perpendicular to AB cutting the X-a,xis at C, and at C erect a perpendicular to Ou. The point- of intersection with y = mx is a point of the curve. Miscellaneous Exercises 1. Show that sec^ a(l + sec 2a) = 2 sec 2a. sin a + sin 2a 2. Show that ;; — j 1 s — = tan a. 1 "1- cos a -\- cos 2a 3. oL ii. J. COS a -|- sin a cos a — sin a Show that -. i — -. — = 2 tan 2a. cos a — sm. a cos a -|- sm a 4. „, ^, ^ cos (a — ff) 1 + tan a tan fi bnow that 7 — ;— ts — 'i 1 1 ;;' cos (a -f- (3) 1 — tan a tan /3 . 1 -1- tan" 1 Show that . = sec a. 5. 1-tan-^ 444 ELEMENTARY MATHEMATICAL ANALYSIS [§256 6. Solve the equation sin* a — 2 cos a + i =0. 7. Simplify the product (x -2 - v'3)(a; - 2 - iVs){x - 2 + V3)ix -2 + iVS). 8. Express in the form c cos (o — 6) the binomial 30 cos o + 40 sin a. 9. Find tan 6 by means of the formula for tan (A + B), if 8 =tan-i 1/2 + tan-i 1/3. 10. Find sin 9, if 9 = sin-i 1/5 + sin-i 1/7. 11. Find the "equation of a circle whose center is the origin and which passes through the point (14, 17). 12. Discuss the curve X = aS y = a(l — cos 0). • 13. Graph on polar paper p^ = a' cos 29. 14. A fixed point located on one leg of a carpenter's "square" traces a curve as the square is moved, the two arms of the square, however, always passing through two fixed points A and B. Find the equation of the curve. 16. Find the parametric equations of the oval traced by a point attached to the connecting rod of a steam engine. 16. Prove that tan (45° + t) - tan (45° - r) = . ^^f'^! ■ 17. Find the quotient of (6 - 2i) by (3 + 75i). 18. Solye for y by inspection: sin (90° + iy) cos (90° - iy) + cos (90° + ^y) sin (90° - iy) = sin y. 19. Write the parametric equations for the circle, the ellipse, and the hyperbola. 20. The length of the shadow cast by a tower varies inversely as the tangent of the angle of elevation of the sun. Graph the length of the shadow for various elevations of the sun. 21. From your knowledge of the equations of the straight line and circle, graph y = ax + y/a^ — x'^- (See Shearing Motion, §37.) 22. In the same manner, sketch y = a -{■ x -\- y/'a^ — x^- 23. Graph the curve y = 1/x + x'. Has this, curve a minimum point? §256] THE CONIC SECTIONS 445 24. Find by use of logarithmio paper the equations of the curves of Fig. 186. These curves give the amounts in ce^ts per kilowatt- hour that must be added to price of electric power to meet fixed charges of certain given annual amounts for various load factors. 25. The angle of elevation of a mountain top seen from a certain point is 29° 4'. The angle of depression of the image of the mountain top seen in a lake 230 feet below the observer is 31° 20'. Find the height and horizontal distance of the mountain top, and produce a single formula for the solution of the problem. 26. Find the points of intersection of the curves a;2 -I- 2/^ = 4 y^ = 4x. 27. Solve llOx-* + 1 = 2\x-'. 28. Solve 3(a; - 7)(a; - l){x - 2) = (» + 2){x - 7){.x + 3). 29. Solve sin x cos x = 1/4. 30. By means of a progression, show how to find the compound interest on $1000 for 25 years at 5 per cent. 100 90 80 70 I I 50 •s o ^ 40 \ -r \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ s S s ^' ^•> \ s \ s s ^ ■fe, s ■v ^ ^ ^ - .^ !a '■a 1 ] ^rn *-. ~ ^ ^ J is. fSc 1 hi Vr, as ~ ~ - ~ "■ — 0.1 0.2 0.3 0.4 05 0.6 0.7 0.8 0.9 1 Cents per K.W.Hour 11 1.2 1.3 1.4 15 Flo. 186.— Annual fixed charges of $10, $15, and $20 of a certain hydro- electric plant, reduced to cents per kw-hr for various load factors. 446 ELEMENTARY MATHEMATICAL ANALYSIS [§256 31. Find the approximate equations for the following data: (a) Steam pressure : (6) Gas-engine mixture : V = volume, p = pressure. V = volume, -p = pressure. (a) (6) r V V V 2 68.7 3.54 141.3 4 31.3 4.13 115.0 6 19.8 4.73 95.0 8 14.3 5.35 81.4 10 11.3 5.94 6.55 7.14 71.2 63.5 54.6 32. Show that p^ = a^ cos 29 is the polar equation of a lemniscate, 33. When an electric current is cut off, the rate of decrease in the current is proportional to the current. If the current is 36.7 amperes when cut off and decreases to 1 ampere in one-tenth of a second, determine the relation between the current C and the time t.- 34. Write four other equations for the circle p = 2-\/3 sin 9 — 2 cos e. 36. Write four other equations for the sinusoid j/ = sin x — ■\/3 cos X. 36. Find the angle that Zx + iy = 12 makes with ix - Zy = 12. 37. From the equation 9 = 6 sin (2« - 1°) determine the amplitude, period, and frequency of the S.H.M. 38. A simple sinusoidal wave has a height of 3 feet, a length of 29 feet, and a velocity of 7 feet per minute. Another wave with the same height, length, and velocity lags 15 feet behind it. Give the equation of each. 39. Simplfy (3\/3 - 3i) 2( - 1 -I- VZiy (cos 36° + i sin 36°) (cos 20° + i sin 20°) . (2 -I- 2^31) 2(cosll° +isin 11°) §256] THE CONIC SECTIONS 447 40. Calculate (1 - VsifK 41. Plot the amount of tin required to make a tomato can to hold 1 quart as a function of the radius of its base. Deterinine approxi- mately from the graph the dimensions requiring the least tin. 42. Find the axes of the ellipse whose foci are (2, 0) and ( — 2, 0), and whose directrices are x = ± 5. 43. Write the polar equation for the ellipse in problem 42. 44. Find the equation of the hyperbola whose foci are (5, 0) and (— 5, 0), and whose directrices are x = + 2. 46. Write the equation of the hyperbola of problem 44 in polar coordinates. 46. Discuss the curve p(l + cos 9) = 4. Write its equation in rectangular coordinates. 47. Find the foci of the hyperbola 2xy = a". Also its eccentricity. 48. Find the equation of the locus of a point whose distance from the point (3, 4) is always twice its distance from the line 3x + 4y = 12. What is the locus? 49. A point moves so that the quotient of its distance from two fixed points is a constant. Find the equation of the locus of the point. 60. Evaluate log 10 - log2 8 + logy 492. 51. Find the maximum and minimum value of (3 sin j: — 4 cos x). What values of x give these maximum and minimum values? 62. Find the equation of a circle passing through the points (1, 2), (- 1, 3), and (3, - 2). 63. A sinusoidal wave has a wave-length of ?r, a period of tt, and an amplitude of t. Write its equation. 64. Compute the value of each of the following : 1^; 7 ois 47° X 6 cis (- 14°); (7 +61)"; '^Ti+ST. 65. Prove by the addition formulas that: sin (90° -t) = COST, sin (360° - t) = - sinr, sin (90° +t) = COST, tan (r + 270°) = - cotT. 56. Solve x2 + 6x + Vx' -|- 6x -|- 1 = 1- 57. Find the product of 3 - 2i by - 2 + i. 68. Find all the values of (cos e -I- 1 sin e)2; (cos S + i sin 6)^^; -^V, VT. 448 ELEMENTARY MATHEMATICAL ANALYSIS [§250 69. Show that sin (a + 6 + c) = sin o cos b cos c + cos a sin b cos c + cos a cos b sin c — sin o sin 6 sin r 60. Draw upon squared paper, using 2 cm. = 1, the curve y" = x By counting the small squares of the paper find the area bounded by the curve and the ordinates x = 1/2, 1, IJ, 2, 2i, 3, 3i, 4, . .By plotting these points upon some form of coordinate paper, find the functional relation existing between the x coordinate and the area imder the curve. 61. The latitude of two towns is 27° 31'. They are 7 miles apart measured on the parallel of latitude. Find their difference in longitude. 62. Solve 3"'"' = 2'+'. Be very careful to take account of all questionable operations. There are two solutions. 63. Find (two problems) the equation connecting: X y 6.8 19.0 14.2 21.6 21.8 23.2 32.0 26.3 46.5 . 31.5 65.0 39.1 78.0 47.0 X y 1.3 21 2.0 25 2.8 29 3.7 33 4.3 35 5.3 38 64. Find the wave length, period, frequency, ampUtude, and velocity for y = 10 sin (2x - 30. 66. Prove that csc^ A „ . — 5— j ^ = sec 2A. csc^ A — 2 66. Find the equation of the elhpse, center at the origin, axes coin- ciding with coordinate axes, passing through the point ( — 3, 5) and having eccentricity 3/5. , 67. Prove (esc 2x) (1 — cos 2x) = sin x sec x. (esc X) (1 — cos x) = ? 68. A S.H.M. has amplitude 6, period 3. Write its equation if time be measured from the negative end of the oscillation. State the difference between a S.H.M. and a wave. §250] THE CONIC SECTIONS 449 69. Sketch on squared paiper : y = V y = 2' y = logz X y = 3" y = logs X y = 5^ y = logs X y = 10- y = logio X 70. Solve 3» - 2.T = 1. 71. Sketch p = a, p = sec 6, p = a sin e, 1 p = -, p = a cos 9, p = — a cos 9, p = (2 — cos e), p = 2 cos 6/ - 3, p = — o sin 9, P = a — a cos 8, P = cos 9 + sin 6. 72. Simplify the expression sin {^ - rj sec ^1 +j;j - sin Q + r^ sec (| - rj 73. Simplify and represent graphically y«-)\^'^^) (1 +»)a + 2i). 74. Find the coordinates of the center, the eccentricity, and the lengths of the semi-axes of: (o) rc^ + Sx + j/^ = 7, (6) x^ -\- 2x + 42/2 - 32/ = 0, (c) a;2 - a; - 2/2 - 2/ = 0, (d) s^ -f-s + j, + 3 = 0. 75. Knd the amplitude, period, frequency and epoch of the fol- lowing S.H.M. 2/ = 7 sin 6i. 2/ = 6 sin 2irt. y = a sin {id -f- e) . 76. Find cis= e. Show that cos 5x = cos^ a; — 10 cos' x sin^ i + 5 cos x sin'' x. 77. Find graphically (on form MZ) the fifth roots of 2^ cis 35°. 78. Complete the following equations : sin (a + b) = ? tan 2x = t cos (a ± 6) = ? cot 2a; = ? tan {a + B)='i sin^ = ? 20 450 ELEMENTARY MATHEMATICAL ANALYSIS [§256 sin 2a; == ? cos - = ? 2 cos 2x = "> cot - = ? 2 79. Solve a' + 1 = 0. 80. y = — St' + 4i — 5 and x = 5t are the parametric equations of a curve. Discuss the curve. 81. Show that [rfcos e +i sin S)] [(r'(cos B' + i sin B')] = rr'lcos (e + B') +i sin (e + 6')]. 82. Two S.H.M. have amplitude 6 and period two seconds. The point executing the first motion is one-fourth of a second in advance of the point executing the second motion. Write the equations of motion. 83. Show that sin 5x = sin* a; — 10 sin' x cos' x -\- 5 sin x cos^ x. CHAPTER XV APPENDIX A REVIEW OF SECONDARY SCHOOL ALGEBRA 300. Only the most important topics are included in this review Prom five to ten recitations should be given to this work before begin- ning regular work in Chapter I. With the kind permission of Professor Hart, a number of the exer- cises have been taken from the Second Course in Algebra, by Wells and Hart. , 301. Special Products. A few simple muItipUcations may be per- formed mentally. (1) The product of the sum and difference of any two numbers: (a -I- 6)(a - 6) = o2 - 62 From this we have (3a; - 2y)i3x + 2y) = 9x^ - ^y^. Exercises Multiply mentally the following : 1. (3a; - J/) (3a; -|- y). 6. (29)(31), or (30 - 1)(30 + 1) = 900 - 1 = 899. 2. (2a; + 7)(2a; - 7). 7. (51) (49). 3. (5a; - y){5x + y). 8. (52)(48), or (50 + 2)(50 - 2). 4. Ixh/ - 3a){x'y + 3o). 9. (103) (97). 6. (o + 3b) (o - 36). 10. (25) (35). (2) A few products of binomials are: \ (o + by = a' + 2ab + 6'. (o - by = o' - 2o6-|- 6^ (a + 6)3 = a> + 3a'b + 3ab' + b\ (a - 6)» = a' - 3a'b + 3a¥ - bK la + h)* = a* + 4o»6 + 6a'b' + 4o6» + 6*. (a - b)* = a* - 4o»6 + 6a'b' - 4a6' + 6". Thus (3 - o) 3 = 27 - 27o + 9a» + a\ and (a; -t- y^Y = x* + 4a;'v«+ Qx^y* + ^y^ + y\ 451 452 ELEMENTAE/y MATHEMATICAL ANALYSIS [§301 Expand mentally the following: 1. (2o - x)K 4. {x - d)*. 2. (a; + 3yy. 5. (1 - a;)'. 3. (2x - 1)'. • 6. (2 + yY. 7. (52)2, or (50 + 2)', or 2500 + 200 + 4 = 2704. 8. (31)2, or (30 + ly. 9. (29)', or (30 - 1)^. (3) The square of a polynomial is illustrated hy the following: (a + b + c)2 = a" + 62 + c2 + 2ab + lac + 2bc. (o + 6 + c + d)2 = a2 + 62 + c2 + d2 + 2o6 + 2oc + 2ad + 26c + 26d + 2cd. (3 - a; + !/)2 = 9 + x2 + 2/2 - 6a;+ 62/ - 2xy. Expand mentally the following : 1. (o + 6 + 2)2. ■ • 4. (2a - X + 3)2. 2. (a + 6 - 2)2. 5. (x2 - 22/2 + 4)2. 3. (a - 6 - c)2. 6. (x - 2o - 62/2)'. (4) The product of two binomials having a common term: (x + a){x + 6) = x2 + (o + b)x +,ab. Thus (x + 5)(x - 11) = x2 + (5 - ll)x + 5( - 11), = x2 - 6x - 55. (x +7)(x + 2) = x2 +9x + 14. (x - 5)(.x + 3) = x2 - 2x - 15. (x2 - 22/) (x2 - 32/) = x" - 5x22/ + 6j/2. Find mentally the value of each of the following : 1. (x + 2)(x + 3). 6. (3x + 22/)(3x - 7y). 2. (x - 2)(x + 3). 7. (x2 - 3)(x2 - 4). 3. (x - 2)(x - 3). 8. (3x1/ - z)(3x2/+ 7z). 4. (x + 2)(x - 3). 9. (x22/2 - 3)(x22/2 - 10). 5. (x2 + 52/) (x2 - 52/). 10. (x - 2y){2x - 2y). (5) rfte product of two general binomials: {ax + 6) (ex + d) = ocx2 + (6c + ad)x + bd. Thus (3o - 4b) (2a + 76) = (3a) (2a) + (- 8 + 21)ab + (- 4 b) (7b) = 6a2 + 13ab - 28b2. Find mentally the following products: 1. (5x - 22/)2. 4. (2m + 3)(m + 4). 2. (a + llb)(a + 36). 5. (2/2 + 4z)(2/2 + 4z). 3. (a - 2u)(a + 12»). 6. (3x2/ - 7)2. §302] REVIEW OF SECONDARY SCHOOL ALGEBRA 453 7. (Sw^w - 4:)(.3uh; + 4). 29. (2 - 3s«)(5 + 2st). 8. {2x - 5)(a; + 4)i 30. (a^b + 6c) (0^6 - 13c). 9. (2r2 - 7)(3r2 + 5). 31. [Ip + 5)(lp - 4). 10. (p2 - Sq){,p^ + 7q). 32. {a' + 7)(o' - 11). 11. (a + l)(o - i). 33. (So + 5) (7a - 8). 12. iix + 5y){ix - By). 34. (1 + 8n)(l - 9n). 13. (u - |)(w - I). 35. (2a - 6") (2a + 3b*). 14. (2a; + 3)(Js + 1). 36. (12a; - i){9x - J). 16. (3x2 4. 46c) (3x' - 46c). 37. (20 - 16z)(3 + 2z). 16. iy -8)(.y + 5). 38. (r^ + 16s) (r^ - s). 17. (X - i){x - f). 39. (a - 6x2) (a 4. ^2^ 18. (1 - 6s) (3 + 2s). 40. (4r + uv)(ir - 5uv). 19. (2< - '7w^){3t - 4u)2). 41. (6x2 _ 1)2, 20. (|u - i)(f« + J). 42. (1 + 23n)(5 - n). 21. (3r - 7<)(5r + 2t). 43. (x* - 2/*)(x« + y*). 22. (11x2 _ I)(i2x2 + 1). 44. (5a2 - 4b)(6a2 - 56). 23. (z2 - 6) (02 + 12). 46. (x2j/ + yH){xh/ - y^x). 24. (x + 32/2) (x - 22/2). 46. (fa + 10) (2a + 1). 26. (6m» - 6s2)(5m' + s2) 47. (9r + 2s) (3r - 4s). 26. (5x + |)(5x -i). ■ 48. (12x2 4. 5) (43,2 _ 3). 27. (3x + 7)(x - 5). 49. (a26* + 4x2)2. 28. (4o - 363)2. 60. (a^ - 5«)(a« + 6«). 61. (a + 6)(a - 6)(a2 + 62)(a* +6<). 302. Symbols of Aggregation. If a sign of aggregation is preceded by the negative sign, change all signs within when the sign of aggre- gation is removed. If the sign of aggregation is preceded by the posi- tive sign, all signs within remain the same when the sign of aggre- gation is removed. 5x2 _ [syi 4. {2x2 _ (2,2 4. 3^2) 4- 5j/2} _ ^2; = 5X2 _ [■^yl 4. {2x2 - y2 - 3x2 4. 52^2} _ 3;2] = 5X2 _ [3y2 4. {42,2 _ 3.2} - x2] = 6x2 _ [3y2 _|_ 4j,2 _ -j2 - x2] = 5X2 _ [7j,2 _ 2x21 = 5x2 _ 7y2 4. 2x2 = 7X2 _ -Jyl Exercises Simplify the following by removing the signs of aggregation : 1. ab - 46^ - (2a2 - 62) _ { _ 502 4. 206 - 862). 2. X - { 2/ -I- z - [x - ( - X - 2/) -f z]) + [z - (2x - 2/)]. 454 ELEMENTARY MATHEMATICAL ANALYSIS [§303 3. o-{ -o — [-o-(-o- 1)]). 4. Syz - [2yz + (9z - 2yz)]. 6. - { -1 -[-1 -(-1)]1. 6. 5x' - [Sy' + {2x^ - {y' + Zx'') + 5y^} - x']. 7. ab - [46^' - (2o» - b^) - [ - 5o' + 2ab - 3b']]. 8. 33/' - I2y' + (9z - 2yx)]. 303. Factoring. Since (a + b)' = a' ± 2ab + b', any expression of the form of the right-hand side can be factored by inspection. Thus, x' - 6xy + 9y' = {x - 3y)' and 4 + 4(o + 6) + (a + 6)2 = (2 + o + 6)' Exercises 1 Factor the followlag by inspection: 1. Qx^ - ZQxy + 25yK 2. 4 + 16« + 16«». 3. a;*j/* + 10a;'j/2z2 + 25z*. 4. 9 + 6(x» + j/») + (I' + !/>)». . 6. a* + 4o26« + 4b<. Since (o + 6) (o — 6) = a' — 6', any expression of the form of the right-hand side can be factored by inspection. Thus, 4o2 - 9b' = (2a -|- 36) (2o - 36). Exercises 2 j Factor the following by inspection: 1. x'j/' - «'• 4. 25 - 3a;'. 2. (o -I- 5)' - c'. 6. 81 - 625x*. 3. c^ - {a-\- 6)'. Since (a 4- 6)(o + c) = o' + (6 + c)a + 6c, any expression of the form of the right-hand side can be factored by inspection. Thus, a;' -5a; - 14 = (a; - 7)(a; +2) Exercises 3 Factor the following by inspection: 1. a;' + 7s + 10. 4. 9i' - 18s - 27. 2. a' + 4aj/ - 21?/'. 5. 25 + 30o - 27a». 3. 4a;' - 18iy + 18i/'. §303] REVIEW OF SECONDARY SCHOOL ALGEBRA 455 Since (a + &)(«' — ah + 6*) = a' + b', any expression of the form of the right-hand side can be factored by inspection. Thus, 27 + 125a;» = (3 + 5x)(,Q - ISa; + 25a;2). Exercises 4 Factor the following by inspection: 1. x'y' + 1. 4. 125 + x'yK 2. x' + y". 5. x' + 8yK 3. 8 + 27a;'. i Since (o — 6)(a^ + ab + 6') = o' — 6', any expression of the form of the right-hand side can be factored by inspection. Thus, 27 - 125a:' = (3 - 5x)(9 + 15a; + 25ai2). Exercises 5 Factor the following by inspection: 1. x>y' - 1. 4. 125 - x^yK 2. x^ — y^, or (a;* + y^)ix' — y^). 5, z^ — 8yK 3. 8 - 9a;'. 6. 27 - 8a'; The following may be factored by grouping the terms. Thus, a'm + o're — m — n = a'(m + n) — (m + ra) = (o' — l)(m + n) = (a - l)(a2 +a + l)(m + n). Exercises 6 Factor the following: 1. ax — ay + bx — by. 4. x^ — xy* — x^y + y^. 2. a;' + 3a2 + 3a; - 1. 5. a;* - x^y - xy^ + y\ 3. ax^ - 2axy + ay' + bx' - 2bxy + by', A trinomial of the form px' + gx + ?■, if the product of two bino- mials, may be factored as outlined below. In the product ax + b ex + d OCX' + {be + ad)x + bd 456 ELEMENTARY MATHEMATICAL ANALYSIS [§303 the terms acx'' and hd are called end prodiicts and bcx and adx are called cross proditcts. This most important case of factoring is best learned from the consideration of actual examples. Factor 21x'i + 5a; - 4. Prom the term 21a;*, consider as possible first terms 7s and 3a;, thus (7a; )(3a; ). For factors of (— 4), try 2 and 2, with unhke signs, and signs so arranged that the cross product with larger absolute value shall be positive; thus (7a; — 2)(3i + 2). This gives middle term Sx; incorrect. For (—4) try 4 and 1, with signs selected as be- fore; thus, (7x — l)(3a; + 4). Middle term 25a;; incorrect. Try (7a; + 4) (3a; — 1). Middle term 5x; correct. Factor 2ix' - 17xy + ZyK Try (6a; — 32/)(4x — y). Incorrect, since first () contains factors and given expression does not. Try (fix — y){4:x — 3y). Middle term - 22; incorrect. Try (8a; - 3y)(3x - y). Middle term - 17; correct. Exercises 1 7 Factor the following: 1. 6x' - 7a; + 2. 8. 35u2 + UV - 6t)2. 2. 3x2 -i- 8x + 4. 9. 9*2 - 14* - 8. 3. 6x2 - a; - 2. 10. 121^^ - 35x2/ - 32/2. 4. 9a2 + 15o + 4. 11. 6 - i - 15*2. 5. 66" - 76 -10. 12. 5 + 9s - 18s2. 6. 14x2 + 13^ _ i2y\ 13. 24m2 - 17mn + 3n2. 7. 8z2 - 2yz- 2lyK 14. 28y" - yz - 2zK An expression of the from o* + 0252 -\- b* may be put in the form of the difference of two squares by adding and subtracting a^b'. Thus, a* + a'b' + b< = a* + 2a2b2 + (,2 _ a^b^ = (a2 + 62)2 _ a'b" = (a2 + ab + 62) (a2 - ab + b'). Exercises 8 Factor the following: 1. X* + x'y' + yK 5. 16x* + 36x'y^ + 81yK 2. X* + 4x2 4. 16. 6. a* + a*V + b*. 3. 2/* + iy'z^ + I62 . 7. aV + a^x'y' + y\ 4. 16 + 4«2 + u\ 8, 625x« + 100x2z< + 162». . §304] REVIEW OF SECONDARY SCHOOL ALGEBRA 457 304. To factor a polynomial completely, first remove any monomial factor present; then factor the resulting expression by any of the type forms which apply, until prime factors have been obtained throughout. 'Thus, (a) 5a« - 5&« = 5{a^ - 6«) = b{cfi - V){a^ + 6=) = 5{a - b)(a2 + a6 + V){a + h){a^ - ah -^ h") (b) 42aa;2 + lOox - 8a = 2a{2\x^ + 5x - 4) = 2a(Jx +4) (3a; - 1) (c) 2Cmnu^ - IWmnu + X2imn = 5mn{^' - 20m + 25) = bmnhu - 5Y. Exercises Factor the following expressions: , 1. xV"° - A"*- 22. a;2 + Qx - 27. 2. 9x» - 43/6. 23. c' -64«3. 3. ,25a;« - 1. 24. Sx' - 1. 4. 81 - ^K 26. 1 - 13< - 68«2. 6. 1 - 6ia''b*c\ 26. a;< - Cx^b - SSb". 6. a;' — y^. 27. au" — 4aM!; — i5av^. 7. 225 - aS. 28. 28a2 - a - 2. 8. 121x2 - 1442/2. 29. Ss^ - 17si + 24{2. 9. 49ot« - SQx'y^zK 30. 15r= - r - 6. 10. 169 -:^ a'lx^. 31. iy^ - 3y - 7. 11. 4x2 _ 20x + 25. 32. 641*6 _ 27x3. 12. 9o2 + 6ob + b2. 33. 6ar - 3as + 4a«. 13. a'b^ - nabc - QOcK 34. a^ +2a - 35. 14. r* - llr' + 30. 35. 9x2 ^ i2xy - 32^". 16. 16b2 + 30b + 9. 36. o" + lOab + 25b2. 16. Slu" + 180ua + lOOs;' 37. 625x22/2 - ^. 17. 36a2 - l32o + 121. 38. 3cdy' - 9cdy - 30cd. 18. x'y* - Axy^ + 4. 39. 4ox2 - 25ay*. 19. o2b2 - 2ab - 35. 40. 3y^ +24. ,20. u' +~u3 _ 110. 41. 4x2 _ 27x + 45. 21. a*b2 - 14o2b + 49. 42. 6x2 + 7^ _ 3, 458 ELEMENTARY MATHEMATICAL ANALYSIS [§305 43. -jV' - 1. 58. 2am« - .50a. 44. 10x>y - 5x^y^ - 5xy\ 69. 72 + 7a; - 49a;». 45. »i»n» + 7mn - 30. 60. 31a;' + 23xy - 8yK 46. x^ - Zxy - 70yK 61. 24o» + 26a - 5. 47. mx" + 7mx - 44r«. 62. 1 - 3xy - IQSx'y^ 48. x' - 3a;» - 108x. 63. x^ - Umx + AOm^. 49. x> - yK 64. 26 + 10a5 - 28o%. 60. a;* - hx-^ - 'iAy\ 65. c» + 27(f». 61. 8n« + 18n - 6. 66. Zx^y - 27xy\ 62. 3i* - 12. 67. -^^^ - 4^^*- 63. Stw" - 42ot« + 49<». 68. 49ji<2/ - 196nV- 64. lOa;' - 39a; + 14. 69. a;« - 16a; + 48. 65. 12x« + 11a; + 2. 70. a;' + 23a; - 50. 66. 363;" + 12i - 35. 71. a'. 7. s»-'+V. 15. u""'^ -^ u""'. 23. (a«6»)'. 8. m^'^mr''. 16. aj'-f+i ^- a;'. 24. (r'»s»)''. - ©'■ Exercises 2 27. (- Write each of the following sixteen expressions, using fractional exponents in place of radical signs: 1. v^. 5. v^ 9. v^iT 13. V'o-5. 2. V^- 6. (v^=. 10. (>^3. 14. (-C^a - 6)'. 3. Vc^. 1.-^. 11. X a 8n 3. x^ X K*. 5. o* X o^. 8. (^ H- a^. S r 7. o» Xai:. a^ ^ at = a^t = a4*-M . = aA. 9. h^ 4- hi. 11. Sa^ftt H- 4o26^. 13. 6at -=- 3a*. 10. nfi -H vi A. 12. 9a* H- a*. 15. (a^)^. 14. a6T -=- o 6"C. (a^) ^ = o^ = oT^. » 16. (o*)A. 18. (a*)^. 20. [(x»)f]f. 17. (h^^. 19. (a*)*. 22. (ata;i-2/t)i 21. (si^)r. (a^x^y^ J)*=(at)W)*(2/¥ = ; a*x*j/i''. 23. (o^fe*)*. 26. (36a*a;22/')*. 27. (32x%4)*. 24. (,adi)i. 26. (a^a;^2/*)». 28. (^a'6'c)*. - (i)*- /aiy_(af)i a* VftV (6*)^ 6i -(?; -(i)' -(^)' "■©• A'\* -m 36. (a* + at -f l)(o* -t- a - cji). §309] REVIEW OF SECONDARY SCHOOL ALGEBRA 469 We arrange the work thus: J + J + 1 a'' + a — gi ai +J + a _ of _ o _ ai a' + 2ai + a* - a^ 37. (x + 22/4 + 32/*).(i - 2yi + 32/*). 38. (X* + yh(.x^ - yh- 39. (,J - Sah^ + 4:ah - ah^-)(ai - 2o*6*). 3 i_ 1 ^ ^ 40. (a» - 20"+ 3o")(2a» - a"). Exercises 4 Find the numerical value of each of the following; 1. 2-1. 4. 10-5. 7. 2-\ 10. 1024-*. 2. 4-2. 5. l-» 8. 16-". 11. 512-4. 3. (-2)-». 6. 2-2. 9. 81-4. 12. 625-*. 1 5 5-2 16-i 13. i- 16. (:r^,- 1,7. —^- 19. ^3r- 9 . 1-8 32-4 7-1 14. J-. 16. I^i- 18. ^^iir- 20. — ^• 3-2 8 1 21 49-1 Write each of the following expressions without using negative exponents: 21. x-K 25. 5a-'. 29. (a; + y)-\ 33. 2o«a;-^-4. 22. x'y-K 26.30-^6-4. 30. (- x)-^ 34. (- a^)-^ „o 1 „„ 2a-2 „, X* „^ a-ibi 23. ^Ti- 27. „,. _, - 31. -^- 35. n"^- „. ■mr' „„ 0^6-* „„ 3(rl6-i „„ 3a26-2c-'' 24- -^- 28. _■ _, ■ 32. ^ — 5 — 36. g^,.,,,^,. - a; " a-3^-5 so-fj, 5o ^b =c * Write each of the following expressions in one line: 37 (1. 39 -^. 41 ^^- 43 ''"y" - 38 1. 40 ^^-^. 42 '^E^yll. 44 J??i 38- a' f • 3a-2r» *2- u>z-^ **" ^=F^ 16(a+ b)-'c4 . ^ , 1 , 1, A-. "• (a - 6)-lc-« x3 + x=i + X + a-i 470 ELEMENTARY MATHEMATICAL ANALYSIS [§309 Exercises 6 Perform the indicated operations in each of the following by means of the laws of exponents. 1. o« X o"". 4. 8a-* X 3a'. 7. m"* X m"*. 2. r" X r-i». 6. m-i X tt*. 8. Sax-' X kbx\ 3. c-> -r c-K 6. a;= -=- a;-". 9. o-»6-» -r- ab"'. 10, (- 7o-»6-»)(-4oi'b-')(a-%''a;-'). 11. (2o*6-*)(a-*6*- |o*b* + ah-^). 12. 7a-»b-V-'-i- 80-26-%- 13. S6a;5J/-'^4 -^ Tai-iy-'a-*, 14. 18o-i&lc-5 -=- 6a*bV«. 16. Cai'j^-^ai -r 2a;-^3/iz-i. 16. (a-')!. 17. (o-s)-". 18 (a«)-«. 19. (7i*)-». 20. (r-iyi. 21. (c-')^ 22. (obc)-*. 33. (bj. 42. (a2a;-i+3a=x-i')(4o-i - 5a;-i + 6ax-^) 4a-i - 5a;-i + &ax-' aH-^ + 3a'a;-' 4ox-i - 5a"a;-2 + 6o»a;-» 12a'a;-' - 15a»a;-° + 18a«z-« 4aa;-i + 7a2a:-2 - 9a'a;-= + 18a*a;-* 43. (2a;-* - 3a; + 4a;*) (a;-? - 2a;-* + 3a;-*). 44. (a;-* - 2a;-*^ + y^){x-^ - y^). 46. (3a;* - |x* + 4) X 2a;-*. 46. (a;-^ + x'h + l)(a;"* - 1). 47. (a;-*+ 3/-=) (a;-* - y-^). 48. {x^y + yh{x^ - y-*). §309] REVIEW OF SECONDARY SCHOOL ALGEBRA 471 49. (2o*- 3axi){3a-i .+ 2a;-*) (4a*a;* + 9o-M). 60. (x-^ - x-iyi + x-iy - y^) -■ (a;-* - j/*). x~^ — y^)x-^ — x-^y' + x-'y — y'(3r^ + y x-^ — x-hji x'^y — y' x-^y —y' Bl. (x-' + 2x-°- - Sx-i) -i- (x-\+ 3s-i). 309. Reduction of Surds or Radicals. 1. // any factor of the number under thi radical sign is an exact ■power of the indicated root, the root of that factor may he extracted and written as the coefficient of the surd, while the other factors are left under the radical sign. (1) Thus, VS = V4 X 2 = VW2 = 2v^ (2) Also, /2^+6 g 1 3V2 - 6 ■ a;2 - Vl + x' Vi + Va ' 1 + Vi - x^ 4 76 ELEMENTARY MATHEMATICAL ANALYSIS LOGARITHUB IO|il3l3l4lSl6l7l8l9lia 3 U 5 617 8 9 1 10 II 12 0000 0043 0086 0128 0170 0212 0253 0294 0334 0374 4 8 13 12 17 16 21 25 20 24 30 34 38 28 32 37 0414 0792 0453 0828 0492 0864 0531 0899 0569 0934 0607 0969 0645 1004 0682 1038 0719 1072 0755 II06 4 8 4 7 3 7 3 7 12 II II 10 15 IS 14 14 19 23 19 22 18 21 17 20 27 31 3S 26 30 33 25 28 32 24 27 31 13 14 15 l6 17 Is 19 20 II39 I46I I173 1492 1206 1523 1239 ISS3 1271 1584 1303 1614 I33S 1644 1367 1673 1399 1703 1430 1732 3 7 3 I 3 6 3 6 10 10 9 9 13 12 12 12 16 20 16 19 IS 18 IS 17 23 26 30 22 25 29 21 24 28 20 23 26 I76I 1790 1818 1847 1875 1903 193 1 1959 1987 2014 3 6 3 5 9 8 II II 14 17 14 16 20 23 26 19 22 25 2041 2304 2068 2330 2095 2355 2122 2380 2148 2405 2175 2430 2201 2455 2227 2480 2253 2504 2279 2529 3 5 3 5 3 5 2 5 8 8 8 7 II 10 10 10 14 16 13 15 13 IS 12 15 19 22 24 18 21 23 18 20 23 17 19 22 2553 2788 2577 2810 2601 2833 2625 2856 2648 2878 2672 2900 2695 2923 2718 2945 2742 2967 2765 2989 2 5 2 5 2 4 2 4 7 7 7 6 9 9 12 14 II 14 II 13 II 13 16 19 21 16 18 21 16 18 20 IS 17 19 3010 3032 3054 3075 3096 3118 3139 3160 3I8I 3201 2 4 6 8 II 13 IS 17 19 21 22 23 24 ^1 11 29 30 31 32 33 34 II 11 39 3222 3424 3617 3243 3263 3464 365s 3284 3483 3674 3304 3502 3692 3324 3522 3711 3345 3541 3729 3365 3560 3747 338s 3579 3766 3404 3598 3784 2 4 2 4 2 4 6 6 6 8 8 7 10 12 10 12 9 II 14 16 18 14 15 17 13 IS 17 3802 3979 41SO 3820 3997 4166 3838 4014 4183 3856 4031 4200 3874 4048 4216 3892 4065 4232 3909 4082 4249 3927 4099 4265 3945 4116 4281 3962 4133 4298 2 4 2 3 2 3 5 5 S 7 7 7 9 II 9 10 8 10 12 14 16 12 14 IS II 13 15 4314 4472 4624 4330 4487 4639 4346 4502 4654 4362 4518 4669 4378 4533 4683 4393 4548 4698 4409 4564 4713 4425 4579 4728 4440 4594 4742 4456 4609 4757 2 3 2 3 I 3 5 5 4 6 6 6 8 9 8 9 7 9 II 13 14 II 12 14 ' 10 12 13 4771 4786 4800 4814 4829 4843 4857 4871 4886 4900 I 3 4 6 7 9 10 II 13 4914 5051 SI8S 4928 5065 5198 4942 S079 5211 4955 5092 5224 4969 5105 5237 4983 5119 5250 4997 5132 5263 SOU 5145 5276 5024 5159 5289 5038 5172 5302 13 I 3 I 3 4 4 4 6 S 5 7 8 6 8 10 II 12 9 II 12 9 10 12 S3IS S44I 5563 5328 5453 5575 5340 5465 5587 5353 5478 5599 5366 5490 561I 537.8 5502 5623 5391 5514 563s 5403 5527 5647 5416 5539 5658 5428 5551 5670 I 3 I 2 I 2 4 4 4 5 5 5 6 8 6 7 9 10 II 9 10 II 8 10 II 5682 5798 591 1 5694 S809 S922 5705 S821 5933 5717 5832 5944 5729 5843 5955 5740 5855 5966 5752 5866 5977 5763 5877 5988 5775 5888 5999 5786 5899 6010 I 2 I 2 I 2 3 3 3 5 5 4 6 7 6 7 S 7 8 9 10 8 9 10 8 9 10 40 41 42 43 6021 6031 6042 6053 6064 6075 608 s 6096 6107 6117 I 2 3 4 S 6 8 9 10 6128 6232 633s 5138 5243 634s 6149 6253 6355 6160 6263 6365 6170 6274 6375 6180 6284 6385 6191 6294 6395 6201 6304 6405 6212 6314 6415 6222 632s 6425 I 2 I 2 I 2 3 3 3 4 4 4 5 6 789 789 789 44 643s 6532 6628 6444 6454 6551 6646 6464 6561 6656 6474 6571 6665 6484 6580 6675 6493 6590 6684 6503 6693 6513 6609 6702 6522 6618 6712 I 2 I 2 I 2 3 3 3 4 4 4 S 6 789 7 8 9 7 7 8 49 6721 6812 6903 6730 6821 6911 6739 6830 6920 6749 6839 6928 6758 6848 69J7 6767 6857 6946 677616785 6866 6875 6955 6964 6794 6884 6972 6803 6893 6981 I 2 1 2 1 2 3 3 3 4 4 4 5 5 4 5 4 5 678 678 678 50 6990 6998 7007 7016 7024 7033 7042 7OS0I7059 7067 I 2 3 3 4 sl 6 7 8 REVIEW OF SECONDARY SCHOOL ALGEBRA 477 LoGAEITHMS lo ii|2 1 3(4ISl6|7\8 I9I123U 5 617 89I 51 52 53 7076:7084 7160 7168 72437251 7093 7177 7259 7101 7110 718s 7193 7267 7275 7118 7202 7284 7126 7210 7292 7135 7218 7300 7143 7226 7308 7152 7235 7316 I 2 3 12 2 12 2 3 4 5 3 4 5 3 4 5 678 677 6 6 54 55 73247332 7404 7412 7340 7419 7348 7356 7427 7435 7364 7443 7372 7451 7380 7459 7388 7466 7396 7474 12 2 12 2 3 4 5 3 4 5 667 5 6 7 56 7482 7490 7SS97S66 7634 7642 7497 7574 7649 7505 7513 7582 7589 7657 7664 7520 7597 7672 7528 7604 7679 7536 7612 7686 7543 7619 7694 7551 7627 7701 12 2 12 2 112 3 4 5 3 4 5 3 4 4 5 6 7 5 6 7 567 61 62 64 7709 7716 7782 7789 7853 7860 7723 7796 7868 7731 7738 7803 7810 7875 7882 7745 7818 7889 7752 782s 7896 7760 7832 7903 7767 7839 7910 7774 7846 7917 112 112 112 3 4 4 3 4 4 3 4 4 5 6 7 566 S 6 6 7924 7993 8062 7931 8000 8069 7938 8007 8075 7945 7952 80I4'802I 8082 8089 7959 8028 8096 7966 7973 8035 8041 8l02;8l09 7980 804S 8116 .7987 8055 8122 I Z 2 112 112 3 3 4 3 3 4 3 3 4 566 556 5 S 6 6s 8129 8136 8142 8149 8156 8162 8169 8176 8182 8189 112 3 3 4 5 56 66 ? 69 70 71 72 73 74 8I9S 8261 832s 8202 8267 8331 8209 8274 8338 8215 8280 8344 8222 8287 8351 8228 8293 8357 8235 8299 8363 8241 8306 8370 8248 8312 8376 82S4 8319 8382 112 112 112 3 3 4 3 3 4 3 3 4 5 5 6 5 5 6 456 8388 84s I 8513 8395 8457 8519 8401 8463 8525 8407 8470 8531 8414 8476 8537 8420 8482 8543 8426 8488 8549 8432 8494 8555 8439 Ssoo 8561 8445 8506 8567 112 112 112 234 234 234 4 5 6 456 4 5 5 5f" 8633 8692 8579 8639 8698 !l*5 8645 8704 8591 8651 8710 till 8716 8603 8663 8722 8609 8669 8727 8615 867s 8733 8621 8681 8739 8627 8686 8745 Z I 2 112 112 234 234 234 4 5 5 4 5 5 4 5 5 75 8751 8756 8762 8768 8774 8779 8785 8791 8797 8802 112 233 4 5 5 76 11 8808 8865 8921 8814 8871 8927 8820 8876 8932 882s 8882 8938 8831 8887 8943 8837 8893 8949 8842 8899 8954 8848 8904 8960 8854 8910 8965 8859 8915 8971 112 112 112 233 233 233 4 5 5 4 4 5 4 4 5 81 83 84 8976 9031 908s 8982 9036 9090 8987 9042 9096 8993 9047 910I 8998 9053 9106 9004 9058 9112 9009 901S 9063 '9069 9117J9122 9020 9074 9128 902s 9079 9133 112 112 112 233 233 233 4 4 5 4 4 5 4 4 5 9138 9I9I 9243 9143 9196 9248 9149 9201 9253 9154 9206 9258 9159 9212 9263 9165 9217 9269 9170 9222 9274 9175 9227 9279 9180 9232 9284 9186 9238 9289 112 112 112 233 233 233 4 4 5 4 4 5 4 4 5 85 86 9294 9299 9304 9309 93IS 9320 9325 9330 9335 9340 112 233 4 4 5 9345 939S 9445 9350 9400 9450 9355 940s 9455 9360 9410 9460 9365 9415 946s 9370 9420 9469 9375 9425 9474 9380 9430 9479 9385 9435 9484 9390 9440 9489 112 Oil 1 1 233 223 223 4 4 5 3 4 4 3 4 4 89 90 91 9494 9542 9590 9499 9547 9595 9504 9552 9600 9509 9SS7 960s 9513 9562 9609 9518 9566 9614 9523 9528 9571 9576 9619 9624 9533 9581 9628 9538 9586 9633 Oil 1 1 Oil 223 223 223 3 4 4 3 4 4 3 4 4 92 93 94 9638 968s 9731 9643 9689 9736 9647 9694 9741 9652 9699 9745 9657 9703 9750 9661 9708 97S4 9666 9671 9713 9717 9759 9763 9675 9722 9768 9680 9727 9773 Oil 1 1 1 I 223 223 223 3 4 4 3 4 4 3 4 4 95 96 9777 9782 9786 9791 9795 9800 980s 9809 9814 9818 Oil 223 3 4 4 9823 9868 9912 9827 9872 9917 9832 9877 9921 9836 9881 9926 9841 9886 9930 984s 9890 9934 9850 9854 9894 9899 9939 9943 9859 9903 9948 9863 9908 9952 1 I oil 1 I 223 223 223 3 4 4 3 4 4 3 4 4 99 9956 9961 9965 9969 9974 9978 9983 9987 9991 9996 oil 223 3 3 4 The copyright of that portion of the above table which gives the logarithms of numbers from 1000 to 2000 is the property of Messrs. Macmillan and Company, limited, who, however, have authorised the use of the form in any reprint pub- lished for educational purposes. 478 ELEMENTARY MATHEMATICAL ANALYSIS Logarithms of Tbiqonometric Fttnctions o / log sin d log tan dc log cot log cos ' S T 0.0000 90 w 10 7.4637 3011 7-4637 3011 2.5363 0.0000 so 10 6.4637 6.4637 20 7.7648 1760 12S0 969 7.7648 1 761 1249 969 2.2352 0.0000 40 20 6.4637 6.4637 30 7.9408 7.9409 2.0591 0.0000 30 30 6.4637 6.4637 40 8.0658 8.0658 1.9342 . 0000 20 40 6.4637 6.4637 SO 8.1627 792 669 580 8.1627 792 670 s8o I . 8373 0.000b 10 50 6.4637 6.4638 I 8.2419 8.2419 1.7581 9.9999 89 60 6.4637 6.4638 10 8.3088 8 . 3089 1.6911 9.9999 SO 70 6.4637 6.4638 20 8.3668 SII 458 413 8.3669 5" 457 41S 1.6331 9.9999 40 80 6.4637 6.4638 30 8.4179 8.41S1 I.S8I9 9.9999 30 90 6.4637 6.4638 40 8.4637 8 . 4638 1.5362 9.9998 20 100 6.4637 6.4638 so 8.S0S0 378 348 321 8.S053 378 348 322 1.4947 1. 4569 9.9998 10 110 6.4637 6.4639 2 8.S428 8.S43I 9.9997 88 120 6 . 4636 6.4639 10 8.S776 8.5779 1.4221 9.9997 SO 130 6.4636 6.4639 20 8 . 6097 300 280 8.6101 300 281 263 1.3899 9.9996 40 140 6 . 4636 6.4640 30 8.5397 8.6401 8.6682 I.3S99 9.9996 30 ISO 6 . 4636 6.4640 40 8.6677 263 1.3318 99995 20 160 6.46J6 6 . 4640 so 8 . 6940 248 235 222 8.694s 249 235 223 1.3OSS 9.9995 10 170 6.463s 6.4641 3 8.7188 8.7194 1.2806 9.9994 87 180 6.463s 6.4641 10 8.7423 8.7429 I.2S71 9.9993 SO 190 6.4635 6.4642 20 8.764s 212 8.7652 213 1 . 2348 9.9993 40 200 6.463s 6.464a 30 8.7857 8.786s I. 213s 9.9992 30 210 6.463s 6.4643 40 8.8059 192 8 . 8067 194 1.1933 9.9991 20 220 6.4634 6.4643 so 8.82SI 18s 177 170 8.8261 178 171 1.1739 9.9990 10 230 6.4634 6.4644 4 8 . 8436 8.8613 8 . 8446 8.8624 I.1SS4 9.9989 86 240 6.4634 6.4644 10 1.1376 9.9989 SO 250 6.4633 6.4645 20 8.8783 IS8 152 8.879s IS8 154 1.1205 9.9988 40 260 6.4633 6 . 4646 30 8.8946 8. 8960 1 . 1040 9.9987 30 270 6.4633 6 . 4646 40 8.9104 8.9118 1.0882 9.9986 20 280 6.4632 6.4647 so 8.9256 147 8.9272 148 1.0728 9.998s 10 290 6.4632 6.4648 5 8.9403 8.9420 1.0580 9.9983 8s 300 6.4631 6 . 4649 1 log COB 1 d 1 log cot^ I dc 1 log tan Hog Sin 1 ' ° 1 1 1 1 113 142 138 137 13S 134 130 129 19 7 12s 1 23 1S2 119 117 lis lU 1 U.3 14.2 13.8 13. 7 13.5 13.4 13. » 12.9 12 .7 12.6 12.3 12.2 11. ) 11.7 11.5 11.4 2 28.6 28.4 27.6 27. 4 27.0 26.8 26. a 25.8 2S .4 25.0 !4.6 24.4 23. i 23.4 23.0 22.8 3 42.9 42.6 41.4 41. 1 40.5 40.2 39. D 38.7 3E .1 37.5 16.9 36.6 35. J 35.1 34.5 34.2 1 57.2 56.8 55.2 54. 8 54.0 53.6 52. 9 51.6 6C .8 50.0 ' 19.2 48.8 47. i 46.8 46.0 45. 6 S 71.5 71.0 69.0 68. 6 67.5 67.0 65. a 64.5 63 .5 62.5 H.5 61.0 59. > 58.6 67.6 57.0 6 85.8 85.2 82.8 82. 2 81.0 80.4 78. D 77.4 7( .2 75.0 r3.8 73.2 71. I 70.2 69.0 68.4 7 100.1 99.4 96.6 95. 9 94.5 93.8 91. D 90.3 8! .9 87.6 36.1 85.4 83. i 81.9 80.5 79.8 8 114.4 113.6 110.4 109. 6 108.0 107.2 104. 103.2 101 .6 100.0 98.4 97,6 95. ! 93.6 92.0 91.2 128.7 127.8 124.2 123. 3 121.5 120.6 117. D 116.1 IM .3 112. 5 1 10.7 109.8 107. 1105.3 103.5 102.6 Formulas for using Table directly % ! log sin * = log I* + S °io log cos X = log (90 - *)' + S log cot * = log (90 - *)' + T log tan X = colog (90 — xy + co T V log tan X = log I* + T " « [ log cot * = colog I* + CO T ^ REVIEW OF SECONDARY SCHOOL ALGEBRA 479 Logarithms of Tbigonometeic Functions log sin log tan dc log cot log COS pp so so SO 8.9403 8.9S4S 8. 9682 8.9816 8. 9945 9 . 0070 9.0192 9.031I 9.. 0426 9.0S39 9 . 0648 9.07SS 9.0859 9 . 0961 9.1060 9.1157 9.1252 9.1345 9.1436 9.1525 9.1612 9.1697 9.1781 9.1863 9-1943 9.2022 9.2100 9.2176 9.2251 9.2324 9.2397 log COS 142 137 134 129 125 122 119 IIS 113 109 107 104 102 99 97 95 93 9f 89 8t 85 84 82 80 79 78 76 75 73 8.9420 8.9563 8.9701 8.9836 8 . 9966 9.0093 9.0216 9.0336 9.0453 9.0567 9.0678 9.0786 9.0891 9.0995 9 . 1096 9.1194 9.1291 9.1385 9.1478 9.1569 0.1658 9. 1745 9.1S31 9.1915 9.1907 9.2078 9.2158 9.2236 9.2313 S-2389 9.2463 log cot 143 138 13s 130 127 123 120 117 114 111 108 105 104 lOI 98 97 94 93 91 89 87 86 84 82 81 80 78 77 76 1.0580 1.0437 1.0299 1.0164 1.0034 0.9907 0.9784 0.9664 0.9547 0.9433 0.9322 0.9214 0.9109 0.9005 0.8904 9.8806 0.8709 0.8615 0.8522 0.8431 o . 8342 0.8255 0.8169 0.8085 0.8003 0.7922 0.7842 0.7764 0.7687 0.7611 0.7537 9.9983 9.9982 9.9981 g.9980 9.9979 9.9977 9.9976 9.9975 9.9973 9.9972 9.9971 9.9969 9.9968 9.9966 9.9964 9.9963 9.9961 9.9959 9-9958 9.9956 9.9954 9.9952 9.9950 9.9948 9.9946 9.9944 9.9942 9.9940 9.9938 9.9936 10 9.9934 O 80 o 8S 10 84 50 40 30 20 o 83 40 30 20 10 82 SO 40 30 20 81 dc log tan log sin "3 II. 3 22.6 33.9 III II. I 22.2 33.3 45.2 67i8 44.4 55. S 66.6 79.1 90.4 101.7 77.7 88.8 99.9 108 10.8 21.6 32.4 107 10.7 21.4 32.1 43.2 64.8 42.8 53. 5 64.2 75.6 86.4 97-2 74.9 85.6 96.3 104 10.4 20.8 31.2 102 10.2 20.4 30.6 41.6 52.0 62.4 40.8 51.0 61.2 72.8 83.2 93.6 il'.6 91.0 109 10. 9 21.8 32-7 43-6 54-5 65.4 76.3 87.2 98.1 lOS 10.5 21.0 3I-S 42 -0 525 63.0 73. S 84.0 94-S lOI 10. 1 20.2 30.3 40.4 50.5 60.6 70.7 80.8 90.9 94 9.4 18.8 28.2 37.6 47.0 58.4 65.8' 75.2 84.6! 18.6 27.9 37.2 46.9 SS.S 65.1 74.4 83.7 91 9.1 18.2 27.3 36.4 46.5 54. 6 63.7 72.8 81.9 89 8.1 17.8 26.7 35.6 44.5 53.4 62. 71.2 80.1 87 8.7 17.4 26.1 31.8 43.5 92.2 60.9 69.6 78.3 86 8.6 17.2 25.8 34.4 43.0 91.6 77.4 86 8.5 17.0 15.5 34.0 42.9 91.0 59.9 68.0 76.9 84 8.4 16.8 25 33 42.0 50.4 98.8 67.2 79.6 16.4 24.6 32.8 41.0 49.2 67.4 65.6 73.8 81 8.1 16.2 2.34 32.4 40 48.6 56.7 64.8 72.9 79 I 78 7.9 7.8 19.815.6 23.7,23.4 31.631. 39.9;39. 47.446. 8 95. 3^94. 6 63.262.4 71.1170.2 99 9.9 19-8 29-7 39.6 49. S S9.4 69.3 79-2 89 -I 77 7.7 15.4 23.1 30.1 38.! 46.: 98 9.8 19.6 29.4 39.2 49 58.8 68.6 78.4 97 9-7 19.4 29.1 38.8 95 9.5 19. 28.5 38. 48.5 47. S 57. o 58.2 67.9 77-6 87.3 66. s 76.0 85. S 76 1 78 I 74 7.6 7.5 7.4 19.2:19.014.8 1.5 61 78 7.3 14.6 21.9 l.4'30. 1.037, i.649 .4167 0'29. 629.2 537.0:36.5 .0j44.4p.g ffsi.ffgi.i .099.298.4 .9|68.6i65.7 Formulas for usine Table inversely Ilog *'■= log sin X — S log «' = log tan X — T colog a/ = log cot X — CO T log (90 — x)' ' log (90 - x)' • colog (90 — xy ' log ■■ log ■ log COS cot tan X — X — S T CO T 480 ELEMENTARY MATHEMATICAL ANALYSIS Logarithms op Trigonometbic Functions o / log sin d log tan dc log cot log cos d pp 10 9 2397 71 70 9-2463 0-7S37 9.9934 3 2 80 73 71 10 9.246S 9-2536 73 73 0.7464 9.9931 SO I 2 7-3 14.6 7-1 14.2 20 9.2538 68 9-2609 71 70 69 0.7391 9-9929 2 40 3 21.9 21-3 30 40 9 . 2606 9.2674 68 66 9-2680 9.2750 0.7320 O.72SO 9-9927 9.9924 3 2 30 20 4 5 29.2 36. 5 28.4 35. S 50 9.2740 66 9.2819 68 0.7181 9.9922 3 2 10 6 43.8 42.6 II 9.2806 54 64 9.2887 66 67 0.7113 9.9919 79 7 SI.l 4?Z 10 9.2870 9.2953 0.7047 9.9917 3 50 8 9 S8.4 6S.7 S6.8 63.9 20 9.2934 53 61 61 9.3020 65 63 0.6980 9-9914 2 40 70 7.0 14.0 59 6.9 13.8 30 9.2997 9.3085 0.691S 9.9912 3 2 30 40 9-3058 9.3149 0.6851 9.9909 20 2 so 9.3119 60 59 58 9.3212 63 61 61 0.6788 9-9907 3 3 . 2 10 3 21.0 20.7 12 9.3179 9.3275 0.672s 9.9904 78 4 28.0 27.6 10 9.3238 9-3336 . 6664 9-9901 SO 35.0 42.0 34-S 41.4 20 9.3296 57 9.3397 61 . 6603 9.9899 3 3 3 40 7 8 9 49.0 56.0 63.0 48-3 55-2 62.1 30 40 9.3353 9.3410 9.3458 9-3517 59 59 0.6542 0.6483 9.9896 9.9893 30 20 SO 9.3466 55 54 54 9-3576 58 57 57 0.6424 9.9890 3 3 3 10 « 68 67 13 9.3521 9-3634 0.6366 9.9887 77 I 6.8 6.7 10 9.357s 9-3691 0.6309 9.9884 50 2 3 13.6 20.4 13.4 20. I 20 9.3629 53 S2 52 9-3748 56 55 5S 0.6252 9.9881 3 3 3 40 4 5 6 27.2 34-0 40.8 26.8 30 9.3682 9.3804 0.6196 9.9878 30 33.5 40.2 40 9-3734 9.3859 0.6141 9.987s 20 SO 9-3786 SI SO 50 9.3914 54 53 S3 0.6086 9.9872 3 3 3 10 7 47.6 46.9 14 9.3837 9.3968 0.6032 9.9869 76 8 54.4 53-6 ID 9.3887 9.4021 0.5979 9 . 9866 50 9 61.2 60.3 20 9.3937 49 48 9.4074 S3 SI 52 0.5926 9.9863 4 3 3 40 66 6.6 6.5 13.0 19. 5 30 9-3986 9-4127 0.5873 9.9859 30 13.2 19-8 40 9 -403s 9.4178 0.5822 9.9856 20 3 SO 9.4083 47 9.4230 51 0.5770 9. 9853 4 10 4 26..^ 26.0 IS 9-4130 9.4281 0.S7I9 9.9849 75 5 33-0 32.5 6 7 39-6 46.2 39.0 45.5 log COS d log cot dc log tan log sin d ° 8 9 52.8 59.4 SO 52.0 58.5 48 1 47 64 fS 6 I 6 59 S8 57 S6, 55 54 53 52 31 1 I 6.4 6.3 6 .1 6 .0 S.9 5.8 5.7 5.6 S-S 5-4 5-3 s 2 S.I 5.0 4-8! 4.7 2 12.8 12.6 12 .2 12 .0 11.8 II. 6 II-4 II. 2 I.O 10.8 10.6 TO 4 10.2 1 0.0 9.6 9.4 3 19.2 18.9 18 ■ 3 18 .0 17.7 17-4 17-I 16.8 6.S 16.2 15-9 15 6 IS. 3 I 5.0 I 4-4 14.1 4 2S.6 2S.2 24 •4 24 .0 23.6 23-2 22.8 22.4 . >2.0 21.6 21-2 20 8 20.4! 0.0 ) 9-2 18.8 S 32.0 3I-S 30 ■ 5 30 .0:29.5 29.0 28. s 28.0 i 7-S 27.0 26.5 26 25.5: S.o: 4.0 23-5 6 38.4 37.8 36 .6 36 -035-4 34-8 34-2 33.6. i3-0 32.4 31.8 31 2 30.6 : 0.0 i 8.8 28.2 7 44.8 44.1 42 .7 42 .041.3 40-6 39-9 39-2 , i8-5 37-8 37-1 36 i 3S.7: S-o; 3-6 32.9 8 SI. 2 SO. 4 48 .8 48 .047.2 46-4 45-6 44.8- t4-0 43-2 42-4 41 40.8 i io-o ; 8.4 37.6 9 S7.6 S6.7 54 -9 54 .o;s3.i S2.2 SI-3 50.4 ' J9-S 48-6 47.7 46 8 45.9^ tS.Oi J3-2 42.3 KEViEW OF SECONDARY SCHOOL ALGEBRA 481 Logarithms op Thigonomethic Functions log sin log tan dc log cot log COS IS o 20 30 40 i6 i8 20 30 40 19 o 9.4130 9.4177 9.4223 9.4269 9.4314 9.43S9 9.4403 9-4447 9-4491 9-4533 9.4576 9.4618 9.4659 9.4700 9-4741 9.4781 9.4821 9.4861 9.4900 9-4939 9.4977 9-5015 9-S052 9.S090 9.5126 9.S163 9.5199 9-5235 9.5270 9-^306 9-5341 9.4281 9.4331 9.4381 9.4430 9.4479 9.4527 9-4575 9.4622 9.4669 9-4716 9-4762 ,4808 4853 14898 9-4943 9-4987 9-5031 9-5075 9-5118 9.5161 9.5203 9-5245 9-5287 9-5329 9-5370 9-5411 9-5451 9-5491 9-5531 9-5571 9-S611 0-5719 0-5669 0,5619 0-5570 0.5521 0-5473 0-5425 0.5378 0-5331 0.5284 O-S238 0.5192 0-S147 0.5102 0-5057 0.5013 o . 4969 0.492s 0.4882 0.4839 0.4797 0.475s 0.4713 0.4671 0.4630 0.4589 0.4549 0.4509 0.4469 0.4429 0.4389 9-9849 9.9846 9.9843 9.9839 9.9836 9.9832 9.9828 9.9825 9.9821 9.9817 9-9814 9-9S10 9.9806 9.9802 9.9798 9-9794 9.9790 9.9786 9-9782 9.9778 9.9774 9.9770 9-9765 9.9761 9.9757 9.9752 9-9748 9 9743 9.9739 9-9734 9-9730 75 72 50 71 25.0 30.0 35-0 40.0 45 -Q 48 4.8 9.6 14.4 ig.2 24.0 28.8 33.6 38.4 43 49 4-9 9.8 14-7 log cos log cot dc log tan log sin 46 1 4 6 2 13 8 18 4 2,1 27 6 ,12 2 36 8 41 4 47 4-7 9-4 14-1 18.8 23. 5 28-2 32.9 37-6 42-3 45 4-5 9.0 13.5 31.5 36.0 40,5 44 13-2 43 12.9 42 4,2 8-4 12-6 1 2 3 41 12.3 40 12.0 39 7.8 1 1.. 7 3.8 7.6 11.4 1 2 3 37 3-7 7-4 11- 1 17-6 22.0 26.4 17.2 21. 5 2S.8 16.8 21.0 25.2 4 5 6 16.4 20.5 24.6 16.0 20.0 24.0 15.6 19. 5 23-4 15.2 19.0 22.8 4 5 6 14.8 18.5 22.2 30.8 35.2 39.6 30.1 34-4 38.7 29.4 33-6 37-8 7 8 9 28.7 32.8 36.9 28.0 32.0 36.0 27-3 31-2 3S-I 26.6 30.4 34-2 7 8 9 25-9 29.6 33.3 36 3.6 7.2 10.8 14-4 18-O 21.6 25.2 28.8 32.4 35 35 7-0 10-5 14.0 17-5 24- 5 28.0 31-5 31 482 ELEMENTARY MATHEMATICAL ANALYSIS LOQABITHMS OF TbIOONOMETRIC FUNCTIONS ■ 1 log Sin d log tan dc log cot log COS d PP ao 9.S34I 34 34 9.5611 39 39 0.4389 9.9730 S 4 70 10 9-S37S 9.5650 0.4350 9.9725 50 4 0.4 20 9.S409 34 34 33 9.5689 38 38 0.43" 9,9721 5 40 I 30 40 9-S443 9.S477 0.5727 9.5766 0.4273 0.4234 9.9716 9.9711 5 5 30 20 2 3 0.8 1.2 so 9-SSiO 33 33 33 9.5804 38 38 . 4196 9.9706 4 5 5 10 4 1.6 31 9.SS43 9.5842 0.4158 9.9702 69 5 2,0 10 9.SS76 9.5879 0.4121 9.9697 SO 6 2.4 30 9.S609 32 32 31 9.S9I7 37 37 37 0.4083 9.9692 5 5 5 40 7 2.8 30 9.S641 9.5954 0.4046 9.9687 30 8 3.2 40 9.5673 9.5991 0.4009 9.9682 20 9 3.6 so 9.S704 32 31 31 9.6028 35 36 0.3972 9.9677 5 5 6 10 aa 9.5736 9.6064 0.3936 9.9672 68 10 9.5767 9.6100 0.3900 9.9667 SO I 2 5 0.5 1.0 20 9.5798 30 31 30 9.6136 36 36 35 0.3864 9.9661 5 S 5 40 30 9.5828 9.6172 0.3828 9.9656 30 3 1.5 40 9.5859 9.6208 0.3792 9.9651 20 4 5 2.0 2.5 SO 9.5889 30 29 30 9.6243 36 35 34 0.3757 9.9646 6 10 23 o 9.5919 9.6279 0.3721 9.9640 % 67 6 3.0 10 9.5948 9.6314 0.3686 9.9635 50 7 3.5 20 9.5978 29 29 29 9.6348 35 34 35 0.3652 9.9629 1 40 8 4.0 30 9.6007 9.6383 0.3617 9.9624 30 9 4.5 40 9.6036 9.6417 0.3583 9.961:8 5 20 so 9.6065 28 9.6452 34 34 33 0.3548 9.9613 6 10 24 9.6093 28 9.6486 0.3514 9.9607 % 66 6 10 9.6121 28 9.6520 0.3480 9.9602 SO I 0.6 20 9.6149 28 9.6SS3 34 33 34 0.3447 9.9596 6 40 2 3 I . 2 1.8 30 9.6177 28 9.6587 0.3413 9.9S90 5 30 40 9.6205 27 9.6620 0.3380 9.9584 5 20 4 2.4 so 9.6232 27 9.6654 33 0.3346 9.9579 6 10 \ 3.0 3.6 as 9.6259 9.6687 0.3313 9.9573 6s 9 4.8 5.4 log cos d log cot dc log tan log sin d / 39 38„ 37 36 35 34 33 32 31 30 29 38 27 I 3.9 3-8 3-7 3.6 3.5 3.4 3.3 3.2 3. I 3.0 2.9 2.8 2.7 2 7.8 7.6 7.4 7.2 7.0 6.8 6.e 6.4 6. 2 6.0 S.8 5.6 5.4 3 II. 7 II. 4 II. I 10.8 10. 5 10.2 9.S 9.6 9. 3 9.0 8.7 8.4 8.1 4 15.6 IS. 2 14.8 14.4 14.0 13.6 13.! 12.8 12. \ 12.0 11.6 II. 2 10.8 S 19. 5 19.0 18.5 18.0 17.5 17.0 16. s 16.0 IS. 5 iS.o 14. S 14.0 13. 5 16.8 16.2 6 23.4 22.8 22.2 21.6 21.0 20.4 19. 8 19.2 18. 3 18.0 17.4 7 27.3 26.6 25.9 25.2 24.5 23.8 23.1 22.4 21. ^ 21.0 20.3 19.6 18.9 8 31.2 30.4 29.6 28.8 28.0 27.2 26. j{ 25.6 24. 5 24.0 23.2 22.4 21.6 9 35. 1 34-2 33-3 32.4 31. 5 30.6 29.7 28.8 27. J 27. 026. 1 25.2 24.3 REVIEW OF SECONDARY SCHOOL ALGEBRA 483 Logarithms of Thigonometbic Functions log Bin d log tan dc log cot log cos d iS 9.62S9 9.6286 9.6313 30 9.6340 40 9 . 6366 26 o SO so 28 10 30 9.6392 9.641S 9.6444 9.6470 9.649s 9.6S21 9.6546 9.6370 9.6595 9.6620 9.6644 9.6668 9 . 6692 9.6716 9.6740 9.6763 9.6787 9.6810 9.6833 9.6856 9.6878 9.6901 9.6923 9.6946 9.6968 9.6990 log cos 9.6687 9.6720 9.6752 9.6785 9.6817 9.6850 9.6882 9.6914 9.6946 9.6977 9 . 7009 9.7040 9.7072 9.7103 9.7134 9.7165 9.7196 9.7226 9.7257 9.7287 9.7317 9.7348 9.7378 9.7408 9-7438 9.7467 9.7497 9.7S26 9.7356 9.758s 9.7614 log cot dc 0.3313 0.3280 0.3248 0.321S 0.3183 0.3150 0.3118 0.3086 0.3054 0.3023 0.2991 o . 2960 0.292& 0.2897 0.2866 0.2835 0.2804 0.2774 0.2743 0.2713 0.2683 0.2652 0.2622 0.2392 0.2562 0.2533 0.2503 0.2474 0.2444 0.2415 0.2386 log tan 9.9573 9.9567 9.9361 9.9SS3 9-9549 9-9543 9-9S37 9-9530 9-9524 9-9518 9-9512 9-9505 9-9499 9.9492 9.9486 9.9479 0-9473 9 - 9466 9-9459 9-9453 9.9446 9.9439 9.9432 ,9425 .9418 .9411 9-9404 9-9397 9-9390 9-9383 9-937S log sin 6S o 64 63 SO o 62 o 61 o 60 6 0.6 1.2 1.8 2.4 3.0 3.6 7 0.7 1.4 2.1 2.8 3.5 4-2 4-9 3-6 6-3 8 0.8 1.6 2.4 3.2 4-0 4-8 3-6 6-4 7-2 ,13 32 31 30 29 27 26 2S 24 23 T 3-3 3-2 3.1 I 3-0 2.0 2.7 2.6 2-5 I 2.4 2.3 2 6.6 6-4 6.2 2 6.0 S.8 5-4 S-2 5-0 2 4-8 4-6 3 9-9 9-6 9.3 3 9-0 8.7 8-1 7.8 7-5 3 7-2 6-9 4 13-2 12.8 12.4 4 12.0 II. 6 10-8 10.4 10 -0 4 9-6 9-2 S I6-I! 16.0 IS. 5 5 15-0 14-S 13-5 13.0 12-3 S 12.0 ir.s 6 19-8 19.2 18.6 6 18.0 17.4 16.2 IS-6 15.0 6 14-4 13.8 7 23.1 22.4 25.6 21.7 '7 21.0 20.3 18-9 18.2 17-5 7 16-8 16. 1 8 26.4 24.8 8 24.0 23.2 21.6 20.8 20.0 8 19-2 1S.4 9 29-7 28.8 27.9 9 27.0 26.1 24-3 23-4 22.5 9 21-6 20.7 4-4 6.6 8.8 15-4 17.6 19.8 484 ELEMENTARY MATHEMATICAL ANALYSIS LOGABITHMS OP TSiaONOMBTKIC FUNCTIONS log sin log tan dc log cot log COS 30 10 9.6990 9.7012 20 30 40 9.7033 9.70SS 9.7076 31 SO 10 9.7097 9.7II8 9.7139 20 30 40 9.7160 9.7I8I 9.7201 32 so 10 9.7222 9.7242 9.7262 20 30 40 9.7282 9.7302 9.7322 33 so 10 9.7342 9.7361 9.7380 4 20 30 40 9.7400 9.7419 9.7438 34 so 10 9.74S7 9.7476 9.7494 20 30 40 9.7SI3 9.7S3I 9.7SSO 35 so 9.7S68 9.7586 9.7614 9.7644 9.7673 9.7701 9.7730 9.77S9 9.7788 9.7816 9.7845 9.7873 9 . 7902 9.7930 9.7958 9.7986 9 . 8014 9 . 8042 9.S070 9.8097 9.812s 9.8153 9.8180 9.8208 9.823s 9 . 8263 9.8290 9.8317 9.8344 9.8371 9.8398 9.8425 9.8452 0.2386 0.2356 0.2327 0.2299 0.2270 0.2241 0.2212 0.2184 O.21SS 0.2127 0.2098 0.2070 0.2042 0.2014 0.1986 O.I9S8 0.1930 0.1903 0.1875 0.1847 0.1820 0.1792 0.176s 0.1737 0.1710 0.1683 o.i6s6 0.1629 0.1602 0.1575 0.1548 9.937s 9.9368 9.9361 9.9353 9.9346 9.9338 9.9331 9.9323 9.931s 9.9308 9.9300 9.9292 9.9284 9.9276 9.9268 9.9260 9.9252 9.9244 9.9236 9.9228 9.9219 9.9211 9.9203 9.9194 9.9x86 9.9177 9.9169 9.9160 9.9151 9.9142 9.9134 o 60 o 58 56 log COS log cot log tan log sin 30 3.0 6-0 9.0 29 2.9 S.8 8.7 12.0 11.6 15.0 18.0 14-5 17.4 21.0 20.3 24.0 27.0 23.2 26.1 2S 2.8 5.6 8.4 11.2 14.0 16.8 19.6 4 25.2 27 22 I 2.7 2.2 2 3 5.4 8.1 Vi 4 10.8 8.8 i 13. S 16.2 11.0 13.2 7 8 18.9 21.6 IS. 4 17.6 9 24.3 19.8 4.2 6.3 8.4 0.5 14-7 16.8 18.9 20 10 I 2.0 1.0 2 4.0 .1.8 3 6.0 5.7 4 8.0 7.6 S 10. 9.5 6 12.0 11.4 7 14.0 16.0 13.3 8 15.2 9 18.0 17. 1 7 0.7 1.4 2.8 3.5 4.2 4.9 S.6 6.3 8 0.8 1.6 2.4 3.2 4.0 4.8 5.6 6.4 7.2 9 0.9 18 1.8 3.6 5.4 7.2 9.0 10.8 12.6 14.4 16.2 REVIEW OF SECONDARY SCHOOL ALGEBRA 485 LOQAEITHMS OF TeIGONOMBTBIC FUNCTIONS, log sin log tan dc log cot log cos 3S 30 40 so 36 o 30 40 so 37 10 20 30 40 SO 38 o 39 o 9-7585 9 . 7604 9.7622 9.7640 9.7657 9-7675 9.7692 9.7710 \ 9.7727 9.7744 9.7761 9.7778 9-7795 9-7811 9.7S28 9-7844 9.7861 9-7877 9.7893 9.7910 9.7926 9.7941 9.7957 9.7973 9.7989 9.8004 9.8020 9.8035 9.8050 9.8066 9.8081 log COS 9.8452 9.8479 9.8506 9.8533 9.8559 9-8586 9-8613 9-8639 9.8666 9.8692 9.8718 9.8745 9.8771 9.8797 9.8824 9-8850 9-8876 9.8902 9.8928 9.8954 g.8980 9 . 9006 9.9032 9-9058 9.9084 9.9110 9.9135 9.9161 9.9187 9.9212 9.9238 log cot dc 0.1548 O.IS2I 0.1494 o . 1467 O.I44I O.I414 0.1387 O.I36I 0.1334 0.1308 0.1282 O.I2S5 0.1229 0.1203 O.II76 Q.I150 o. 1124 0.1098 0.1072 0.1046 0.1020 . 0994 0.0968 0.0942 0.0916 0.0890 0.086s 0.0839 0.0813 p. 0788 0.0762 9-9134 9-9125 9-9I16 9-9107 9.9098 9.9089 9.9080 9.9070 9.9061 9-9052 9 . 9042 9-9033 9-9023 9-9014 9.9004 9.8995 9-8985 9-8975 9-8965 9-8955 9-8945 9-8935 9-8925 9-891S 9-8905 9-8895 9.8884 9-8874 9 - 8864 9-8853 9.8843 log tan log sin o 55 S4 50 40 30 20 o S3 50 9 0.9 1.8 2-7 3-6 4-5 5-4 6-3 7-2 8.1 4.4 S-5 6-6 7.7 8.8 9-9 27 1 2 7 s 4 8 I 10 8 1.1 5 16 2 t8 9 21 6 24 3 26 2.6 S.2 7.8 :o.4 13 o IS. 6 18.2 20.8 23-4 Z-S 18 2-5 1.8 S-0 3.6 7-5 5.4 10. 7-2 r2.5 9-0 15.0 10.8 17.5 12.6 20.0 14.4 22.5 16.2 17 1-7 3-4 5-1 6.8 8-5 II. 9 13-6 IS-3 16 1.6 3.2 4-8 6-4 8-0 9-6 IS l-S 3-0 4-5 6.0 7-S 9-0 486 ELEMENTARY MATHEMATICAL ANALYSIS LOQARITHMS OF TRIGONOMETRIC FUNCTIONS log Bin log tan dc log cot log COS 42 o 43 o 44 o 50 45 9 . 8081 9 . 8096 9.8111 9.812s 9.8140 9.81SS 9.8169 9.8184 9.8198 9.8213 9.8227 9.8241 9.82SS 9.8269 9.8283 9.8297 9.8311 9.8324 9.8338 9.8351 9.836s 9-8378 9.8391 9.840s 9.8418 9.8431 9.8444 9.84S7 9 . 8469 9.8482 9.849s 9.9238 9.9264 9.9289 9.931S 9.9341 9 . 9366 9.9392 9.94*7 9.9443 9.9468 9.9494 9.9SI9 9. 9544 9.9S70 9.9S9S 9. 9621 9.9646 9.9671 9.9697 9.9722 9.9747 9.9772 9.9798 9.9823 9.9S48 9.9874 9.9899 9.9924 9.9949 9. 9975 O . 0000 0.0762 0.0736 0.07II 0.068s o.o6s9 0.0634 o . 0608 0.0583 0.0SS7 O.OS32 , 0.0506 0.0481 0.0456 . 0430 . 040s 0.0379 0.0354 o . 0329 . 0303 0.0278 0.0253 0.0228 0.0202 0.0177 0.0IS2 0.0126 O.OIOI 0.0076 0.0051 0.002s . 0000 log COS log cot dc log tan 9.8843 9.8832 9.8821 9.88IC 9.8800 9.8789 9.8778 9.8767 9.8756 9.8745 9.8733 g.8722 9.87II 9 . 8699 9.8688 9.8676 9 . 8665 9.8653 9.8641 9.8629 0.8618 9.8606 9.8594 9.8582 9.8569 9.8557 9.8545 0.8532 9.8520 9.8507 9.8495 49 so o 48 SO 40 30 47 50 40 30 20 46 SO 40 30 20 log sin 26 2.6 S.2 7.8 10.4 13.0 IS. 6 18.2 20.8 23.4 25 2.5 so .7.5 10. o 12. S IS.O 17. s 20.0 22. S 15 1. 5 3.0 4.5 6.0 7.5 9.0 10. s 12.0 13. 5 14 1.4 2.8 4.2 5.6 7.0 8.4 9.8 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 13 1.3 2.6 3.9 5.2 6.5 7.8 9.1 10.4 II. 7 4.4 s.s 6.6 7.7 8.8 9.9 12 1.2 1:1 4.8 6.0 7.2 ■8.4 9.6 10.8 REVIEW OF SECONDARY SCHOOL ALGEBRA 487 Natural Trigonometric Functions Deg. Radians n Bin n CSC n tan n cot n sec n cos o . 0000 .000 .000 1. 000 1. 00 1.5708 90 I 2 3 0.0I7S 0.0349 0.0524 .017 .035 .052 57.3 28.7 19. 1 .017 .035 .052 57.3 28.6 19. 1 1. 000 1. 001 1. 001 i.OO • 999 .999 I.SS33 1.5359 I.S184 89 88 87 4 1 . 0698 0.0873 . 1047 .070 .087 .105 14-3 II. 5 9.57 .070 .087 .105 14.3 II-4 9.51 1.002 1.004 1.006 .998 .996 .995 1.5010 86 84 7 8 9 0.1222 0.1396 0.IS7I .122 .139 .156 8.21 7.19 6.39 .123 .141 .158 8.14 7.12 6.31 1.008 1. 010 1. 012 .993 .990 .988 1.4486 I. 4312 1.4137 83 82 81 10 0.1745 .174 5. 76 .176 S.67 1. 015 .98s 1.3963 80 11 12 13 0.1920 . 2094 0.2269 .191 .208 .225 5. 24 4.81 4.45 .194 .213 .231 S.14 4.70 4.33 1. 019 1.022 1.026 .982 • 978 • 974 1.3788 1.3614 1.3439 79 78 77 14 IS i6 0.2443 0.2618 0.2793 .242 .259 .276 3.63 ■.HI .287 4.01 3.73 3.49 1. 031 1.035 1.040 ■.III .961 1.3265 1.3090 I.291S 76 75 74 11 19 0.2967 0.3142 0.3316 .292 .309 .326 3.42 3.24 3.07 .306 .325 .344 3.27 3.08 2.90 1.046 1. 051 1.058 • 956 .946 I. 2741 1.2566 1.2392 73 72 71 20 U.3491 .342 2.92 .364 2.75 1.064 .940 1.2217 70 21 22 23 0.366s 0.3840 0.4014 .358 .375 .391 2.79 2.67 2.56 .384 .404 .424 2.61 2.48 2.36 1. 071 1.079 1.086 .934 .927 .921 I . 2043 I. 1868 I . 1694 69 68 67 t 24 0.4189 0.4363 0.4538 .407 Ml 2.46 2.37 2.28 1^1 .4S8 2.25 2.14 2.05 1.095 1. 103 1. 113 .914 .906 .899 1.1519 I. 1345 1.1170 66 64 11 29 0.4712 0.4887 0.S061 1P .485 9.20 2.13 2.06 .510 .532 .554 1.96 1.88 1.80 1. 122 1 .133 1. 143 .891 .883 .875 I . 0996 1.0S21 1 . 0647 62 61 30 0.5236 .500 2.00 • 577 1.73 1. 155 .866 1.0472 60 31 32 33 0.S4" 0.558s 0.5760 .SIS .530 .S4S 1.94 1.89 1.84 .601 .625 .649 1.66 1.60 1. 54 1.167 1. 179 1. 192 .857 .848 .839 1.0297 1.0123 0.9948 59 58 57 34 35 36 0.5934 0.6109 0.6283 .559 .574 .588 1.79 1. 74 1.70 .675 .700 .727 1.48 1-43 1.38 1.206 1.221 1.236 .829 .819 .809 0.9774 0.9599 0.942s 56 55 54 37 38 39 0.6458 0.6632 0.6807 .602 .616 .629 1.66 1.62 1.59 .754 .781 .810 1-33 1.28 1.23 1.252 1.269 1.287 .799 .788 .777 0.9250 0.9076 0.8901 53 52 SI 40 0.6981 .643 x.s6 .839 X.19 1.305 .766 0.8727 SO 41 42 43 0.7156 0.7330 0.7505 .656 .669 .682 1.52 1.49 1.47 .869 .900 • 933 1. 15 HI 1.07 1.32s 1.346 1.367 .7SS ■ 743 .731 0.8S52 0.8378 0.8203 49 48 47 44 45 0.7679 0.7854 .69s .707 1.44 1. 41 .966 1. 00 1.04 1. 00 1.390 1.414 .719 .707 0.8029 0.7854 46 45 n cos n sec n cot n tan n CSC n sin Radians Deg. 488 ELEMENTARY MATHEMATICAL ANALYSIS Antilogakithms ] I Ia|3<4l5|6|7l8 19 I 2 3 14 S 617 8 9l •SO •SI •S2 ■ S3 3162 3170 3177 3184 3192 3199 3206 3214 3221 3228 112 3 4 4 5 6 7 3236 33II 3388 3243 3319 3396 3251 3327 3404 3258 3334 3412 3266 3342 3420 3273 3350 3428 3281 3357 3436 3289 3365 3443 3296 3373 3451 3304 3381 3459 122 12 2 12 2 3 4 5 3 4 5 3 4 5 5 6 7 5 6 7 6 6 7 .S4 J is7 .S8 ■ S9 .6^ .6i .62 .63 .64 it 3467 3548 3631 347S 3SS6 3639 3483 3565 3648 3491 3573 3656 3499 3581 3664 3S08 3589 3673 3516 3597 3681 3524 3606 3690 3532 3f4 3698 3540 3622 3707 12 2 12 2 r 2 3 3 4 5 3, 4 5 3 4 5 6 6 7 678 37IS 3802 3890 3724 3811 3899 3733 3819 3908 3828 3917 3750 3837 3926 3758 3846 3936 3767 38SS 3945 3776 3864 3954 3784 3873 3963 3793 3882 3972 I 2 3 I 2 3 123 3 4 5 4 4 5 4 5 5 i ' I t ' 1 678 3981 3990 3999 4009 4018 4027 4036 4046 40SS 4064 I 2 3 4 5 6 678 4074 4169 4266 4083 4178 4276 4093 4188 4285 4102 4198 4295 4HI 4207 4305 4121 42J7 431s 4130 4227 4325 4140 4236 4335 4IS0 4246 4345 4159 4256 4355 1 2 3 I 2 3 I 2 3 456 456 4 5 6 7 8 9 7 8 9 7 8 9 436s 4467 4S7I 437S 4477 4581 438s 4487 4592 4395 4498 4603 4406 4508 4613 4416 4519 4624 4426 4529 4634 4436 4539 4645 4446 4550 4656 4457 4667 I 2 3 I 2 3 I 2 3 456 456 456 7 8 9 7 8 9 7 9 10 .69 4677 4786 4898 4688 4797 4909 4699 4808 4920 4710 4819 4932 4721 4831 4943 4732 4842 4955 4742 4853 4966 4'|3 4864 4977 4764 4875 4989 4775 4887 5000 I 2 3 I 2 3 I 2 3 4 5 7 467 5 6 7 8 9 10 8 9 lO 8 9 10 .70 5012 S023 S035 S047 S058 5070 S082 SO93 Sios 5117 I 2 4 S 6 7 8 9 II .71 .72 .73 .74 S129 S248 S370 S140 5260 S383 5152 5272 5395 5164 S284 S408 5176 5297 5420 S188 S309 5433 5200 5321 S44S 5212 5333 5458 5224 5346 5470 5236 5358 S483 I 2 4 I 2 -4 I 3 4 5 6 7 5 6 7 568 8 10 II 9 10 II 9 10 II 549S 5623 S7S4 SS08 S636 S768 5S2I 5649 5781 5534 5662 5794 5546 5675 5808 SSS9 S689 5821 5572 5702 5834 5585 S7I5 5848 5598 5728 S86i S6io 5741 587s 1 3 4 I 3 4 134 568 5 7 8 5 7 8 9 10 12 9 10 12 9 II 12 ■77 .78 •79 5888 6026 6166 S902 6039 6180 5916 6053 6194 5929 6067 6209 5943 6081 6223 5957 6095 6237 5970 6109 6252 5984 6124 6266 5998 6138 6281 6012 6152 6295 I 3 4 I 3 4 I 3 4 5 7 8 678 679 10 II 12 10 II 13 10 II 13 .So 6310 6324 6339 6353 6368 6383 6397 6412 6427 6442 I 3 4 679 10 12 13 .81 .82 .83 6457 6607 6761 6471 6622 6776 6486 6637 6792 6501 6653 6808 6si6 6668 6823 6531 6683 6839 6855 6s6l 6714 6871 6577 6730 6887 6592 674s 6902 235 235 235 689 689 689 II 12 14 II 12 14 II 13 14 .85 .86 .i? .88 .89 .90 6918 7079 7244 6934 7096 7261 6950 7112 7278 6966 7129 7295 6»82 7145 7311 6998 7161 7328 70IS 7178 7345 7031 7194 7362 7047 7211 7379 7063 7228 7396 235 235 235 6 8 10 7 8 10 7 8 10 11 13 15 12 13 15 12 13 15 7413 7S86 7762 7430 7603 7780 7447 7621 7798 7816 7482 7656 7834 7499 7674 7852 7516 7691 7870 7534 7709 7889 7SSI 7727 7907 7568 7745 7925 235 245 245 7 9 10 7 9 II 7 9 II 12 14 16 12 14 16 13 14 16 7943 7962 7980 7998 8017 803S 8054 8072 8091 8110 246 7 9 II 13 15 17 •91 .92 •93 8128 8318 8S1I 8147 8337 8S3I 8166 8356 8551 8185 8375 8570 8204 839s 8590 8222 8241 8433 8630 8260 8453 8650 8279 8472 8670 8299 8492 8690 246 246 246 8 9 II 8 10 12 8 10 12 13 IS 17 14 IS 17 14 16 18 ■94 .96 .98 ;99 8710 8913 9120 9333 9SS0 9772 8730 8933 9141 8750 8954 9162 8770 8974 91S3 8790 8995 9204 8810 9016 9226 8831 9036 9247 8851 9057 9268 8872 9078 9290 8892 909? 931 1 2 4 6 246 246 8 10 12 8 10 12 8 II 13 14 16 18 15 17 19 IS 17 19 9354 9572 9376 9594 ^817 9397 9616 9840 9419 9638 986^ 9661 0886 9462 9683 ^08 9484 9705 mi 9506 9727 9954 9528 9750 997.7. 247 247 2 5 7 9 II 13 9 II 13 9 " 14 15 17 20 16 18 20 16 18 20 REVIEW OF SECONDARY SCHOOL ALGEBRA 489 Antilogamthms 10|ll2l3l4IS16|'7l8l9|l2 3l4 S 6' 7 89I ■00 01 ■02 03 04 ■U ■07 ■08 •09 •10 ■II ■12 13 ■14 ■It ■■\l •19 ■20 ■21 ■22 •23 •24 ■25 ■26 :S ■29 1000 1002 1005 1007 1009 1012 1014 1016 1019 IO21 I I I I 222 1023 1047 1072 1026 lOSO 1074 102S 1052 1076 1030 1054 1079 1033 1057 1081 1035 1059 1084 1038 1062 1086 1040 1064 1089 1042 1067 IO91 1045 1069 1094 00 I I 00 I I I I III 222 2 2 2 2 2 2 1096 1122 1 148 1099 II2S IISI 1102 I127 1153 IlO/i II30 II56 I107 1132 I159 1 109 II3S 1161 II12 1138 1 164 II14 I140 1167 III7 1 143 1 169 II19 I146 I172 1 I oil oil 112 112 112 2 2 2 2 2 2 2 2 2 1175 1202 1230 II78 1205 1233 1180 1208 1236 II83 I2II 1239 1 186 1213 1242 I189 1216 1245 I191 1219 1247 1194 1222 1250 II97 1225 1253 I199 1227 1256 I I oil oil 112 112 112 2 2 2 223 2 2 3 I2S9 1262 126s 1268 1271 1274 1276 1279 12S2 128s 1 I 112 223 1288 1318 1349 I29I I32I 1352 1294 1324 1355 1297 1327 1358 1300 1330 1361 1303 1334 1365 1306 1337 1368 1309 1340 1371 13I2 1343 1374 1315 1346 1377 oil oil oil 12 2 12 2 12 2 223 223 233 1380 1413 1445 1384 I4I6 1449 1387 1419 1452 1390 1422 I4SS 1393 1426 1459 1396 1429 1462 1400 1432 1466 1403 1435 1469 1406 1439 1472 1409 1442 1476 1 I I I I I 12 2 12 2 12 2 233 233 233 1479 IS14 IS49 1483 ISI7 ISS2 i486 1521 1556 1489 1524 1560 1493 1528 1563 1496 1531 1567 1500 1535 1570 1503 1538 1574 1507 1542 1578 151O IS4S 1581 oil 1 I 1 I 12 2 12 2 12 2 2 '3 3 233 3 3 3 ISSS IS89 1592 1596 1600 1603 1607 1611 1614 1618 oil 12 2 3 3 3 1622 1660 1698 1626 1663 1702 1629 1667 1706 1633 167 1 1710 1637 1675 1714 1641 1679 1718 1644 1683 1722 1648 1687 1726 1652 1690 1730 1656 1694 1734 I I I I 1 I 2 2 2 2 2 2 2 2 2 3 3 3 3 3^3 3 3 4 1738 1778 1820 1742 1782 1824 1746 1786 1828 1750 1791 1832 1754 1795 1837 1758 1799 1841 1762 1803 1845 1766 1807 1849 1770 1811 1854 1774 1816 1858 oil oil 1 I 2 2 2 2 2 2 223 3 3 4 3 3 4 3 3 4 1862 190S 1950 1866 I9I0 1954 1871 1914 1959 1875 1919 1963 1879 1923 1968 1884 1928 1972 1888 1932 1977 1892 1936 1982 1897 1941 1986 1901 I94S 1991 I I I I 223 223 2 2 3 3 3 4 3 4 4 3 4 4 .30 ■31 •32 ■33 •34 ■35 ■36 1995 2000 2004 2009 2014 2018 2023 2028 2032 2037 1 I 223 3 4 4 2042 2089 2138 2046 2094 2143 2051 2099 2148 2056 2104 2153 2061 2109 2158 2065 2113 2163 2070 2118 2168 2075 2123 2173 2080 2128 2178 2084 2133 2183 01 I 1 I I I 223 223 2 2 3 3 4 4 3 4 4 3 4 4 21S8 2239 2291 2193 2244 2296 2198 2249 2301 2203 2254 2307 2208 2259 2312 2213 2265 2317 2218 2270 2323 2223 2275 2328 2228 2280 2333 2234 2286 2339 112 112 112 233 233 233 4 4 5 4 4 5 4 4 5 ■11 ■39 .40 2344 2399 2455 2350 2404 2460 2355 2410 2466 2360 2415 2472 2366 2421 2477 2371 2427 2483 2377 2432 2489 2382 2438 249s 2388 2443 2500 2393 2449 2506 112 112 l' I 2 233 233 233 4 4 5 4 4 5 4 5 5 2SI2 2518 2523 2529 2535 2541 2547 2553 2559 2564 112 234 4 5 5 •41 .42 .43 •44 it ■.ti ■49 2570 2630 2692 lilt 2698 2582 2642 2704 2588 2649 2710 2594 2655 2716 2600 2661 2723 2606 2667 2729 25l2 2673 2735 2618 2679 2742 2624 268s 2748 112 112 112 234 2 3 4 3 3 4 4 5 5 456 456 2754 2818 2884 2761 2825 2891 2767 2831 2897 2773 2838 2904 2780 2844 2911 2786 28S1 2917 2793 2858 2924 2799 2864 2931 2805 2871 2938 2812 2877 2944 112 I I 2 I I 2 3 3 4 3 3 4 3 3 4 4 5 6 5 5 S 5 5 6 29SI 3020 3090 2958 3027 3097 296s 3034 3105 2972 3041 3112 2979 3048 3119 2985 3055 3126 2992 3062 3133 2999 3069 3141 3006:3013 30763083 314813155 112 112 112 3 3 4 3 4 4 3 4 4 5 5 6 566 566 INDEX (The numbers refer to the pages) Abscissa, 33 Absolute value of complex num- ber, 369 Addition formulas for sine and cosine, 307-309 for tangent, 309 Additive properties of graphs, 142, 295-297 Aggregation, symbols of, 453 Algebraic scale, 3, 357 Alternating current curves, 384 et seq. represented by complex numbers, 384 Amplitude of complex number, 369 of S. H. M., 340 of sinusoid, 117 of uniform circular motion, 102 of wave, 345 Angle, 99 depression, 130 direction, 103 eccentric, 156 elevation, 130 epoch, 340, 345 initial side, 99 phase, 340, 345 that one line makes with an- other, 313 vectorial, 103 Angles, congruent, 100 Angular magnitude, 99 units of measure, 100 velocity, 102 Anti-logarithm, 254 Approximation formulas, 209 Approximations, successive, 196 Argument of function, 12 of complex number, 369 Arithmetical mean, 213 progression, 213-216 triangle, 204 Asymptotes of hyperbola, 60, 165, 167 Auxiliary circles, 155 Axes of ellipse, 154 of hyperbola, 168 Binomial coefficients, graphical representation of, 211, 212 theorem, 204 et seq. Briggs, Henry, 236 system of logarithms, 245 Cartesian coordinates, 33 Cassinian ovals, 394 Catenary, 297 Change of base, 264, 265 of unit, 66, 70, foot note, 77 et seq., 285 Characteristic, 250, 251 Circle and circular functions. Chap. IV, 97 et seq. Circle, dipolar, 395 equation of, 97, 98 sine and cosine, 126-128 tangent to, 422, 428 through three points, 433 491 492 INDEX (The numbers refer to the pages) Circles, auxiliary, 155 Circular functions, 103 et seq. graphical computation of, 106, 115 fundamental relations, 110, 304-318 law of, 132 motion, 102 Cologarithm, 254 Combinations, 199, 202, Chap. VII Common logarithms, 246 Complementary angles, 114, 118 Completing square, 463 Complex numbers, Chap. XII, 357 et seq. defined, 363 laws of, 365 polar form, 369 typical form, 363 Composite angles, functions of, 310-312 Composition of two S. H. M.'s, 343 Compound harmonic motion, 334 interest, 220 law, 277 Computers rules, 328 Conditional equations, 138, 320- 327 Conies, 414, 417 con-focal, 441 sections. Chap. XIV, 399 et seq. Conjugate axis, 168 complex numbers, 367 hyperbolas, 170 Connecting rod motion, 355 Constants and variables, 15 Continuous function, 11 compounding of interest, 278 Coordinate paper, 27, 124, 271, 289 Coordinates, Chap. II, 23 et seq. Cartesian, 34 orthogonal, 124 polar, 123, 434 rectangular, 33 et seq. relation of polar and rectan- gular, 136, 434 Cosecant, 103 Cosine, 103 curve, 117, 126 law, 321 ■ Cotangent, 103 Crest of sinusoid, 116 Cubical parabola, 52 Cubic equation, 192 et seq. "Cut and Try," 149 Cycloid, 395 Damped vibrations, 299 Damping factor, 299 Decreasing function, 63 geometrical series, 221 DeMoivres theorem, 375 Descartes, Ren6, 34 Diameter of any curve, 440 of ellipse, 440 of parabola, 419 Direction angle, 103 Directrix of ellipse, 402, 415 of hyperbola, 408, 415 of parabola, 408, 413 Discontinuous function, 13, 27, 59 Distance of point from line, 426 Distributive law of multiplica- tion, 205, 365 general, 456 INDEX 493 CThe numbers refer to the pages) Double angle, functions of, 315 scale, 4-9, 21, 22, 266-276 of algebraic functions, 21 of logarithmic functions, 266-276 "e," 241, 245, 260, 277 Eccentric angle, 156 Eccentricity of earth's orbit, 403 of ellipse, 401 of hyperbola, 408 of parabola, 413, 415 Ellipse, 52 et seq., 399 et seq., Chaps. V and XIV. axes of, 154 construction, 155, 158, 159 directrices, 402, 415 eccentricity, 402 focal radii, 399, 430 foci, 399 latus recturn, 404 parametric equation, 155 polar equation, 410 symmetrical equation of, 154. tangent to, 429 vertices, 154 Ellipsograph, 158 Elliptic motion, 344, 388 Empirical curves, 46, 283, 291 formulas, 75 Envelope, 422 Epicycloid and epitrochoid, 397 Epoch angle, 340, 345, 348 Equations, conditional, 138 explicit, 154 quadratic, 462 systems, 186-192 simple, 461 single and simultaneous, Chap. VI, 174 with given roots, 178 Even function, 119 Expansion, binomial, 205, 208 Exponential curves, 236-240, 260-264 equation, 240 function, Chap. IX, 234 et seq. compared with power, 286-289 defined, 240, 243, 244 sums of, 295-299 Exponents, definition of, 466 irrational, 243, 244 laws of, 466 Factor theorem, 177 Factorial number, 200 Factoring, 454-460 fundamental theorem in, 459 Family of curves, 78 of lines, 421 Focal radii and foci, 393 of eUipse, 399, 430 of hyperbola, 406 radius of parabola, 414 Fractions, 460 Frequency of S. H. M., 340 of sinusoidal wave, 347 uniform circular motion, 102 Function, periodic, 26, 113 power, 48 et seq., 286 S. H. M., 339 trigononietric, 103 Functions, 10, 11 circular, Chap. IV, 97 et seq., 103 continuous, 11 discontinuous, 13, 27, 59 even and odd, 119 explicit and implicit, 155, 174 exponential, 240,-243, 244, 286 et seq. 494 INDEX (The numbers refer to the pages) Functions, increasing and de- creasing, 63, 147 General equation of second de- gree, 440 Geometrical mean, 217 progression, 217 et seq. Graphical computation, 16 et seq. of circular functions, 106 of integral powers, 19 of logarithms, 237 of product, 16 of quotient, 17 of sq. roots, 18, 21 of squares, 18, 21 solution of cubic, 192 simultaneous equations, 183 et seq. Graph of binomial coefficients, 211, 212 of complex number, 364 of cycloid, 396 of ellipse, 158, 159 of equation, 36 of functions of multiple an- gles, 318, 319 of geometrical series, 236 of hyperbola, 167, 168 of hyperbolic functions, 297 of logarithmic and exponen- tial curves, 236-240, 260 of parabolic arc, 420 of power function, 48-60, 64 of sinusoid, 115 of tangent and secant curves, 143-147 Graphs, suggestions on construc- tion of, 27 nonnstatistical, 35 Half-angle, functions of, 315 . Halley's law, 282 Haridonic analysis, 354 functions, 346 fundamental, 352 motion, Chap. XI, 339 et seq. compound, 352 Hyperbola, Chap. V and XIV. asymptotes, 165, 167 axes, 168 center, 168 conjugate, 170 construction of, 167 eccentricity, 408 foci and focal radii, 406 latus rectum, 408 parametric equations, 165, 167 polar equation, 410 rectangular, 58, 164 symmetrical equation, 166 vertices, 168 Hyperbolic curves, 52, 58 sine and cosine, 296 system of logarithms, 245 Hypocycloid and Hypo-trochoid, 397 i = V^^, 362 Identities, 110, 111, 138, 304r-317 Illustrations from science, 69-76 Image of curve, 57 Increasing function, 63, 147 progression, 214 Increment, logarithmic, 279 Infinite discontinuity, 59 geometrical progression, 221 Infinity, 69 Intercepts, 39, 40 Interest, compound, 220, 277 curve, 237 Interpolation, 252 Intersection of loci, 92, 182 INDEX 495 (The numbers refer to the pages) Inverse of curve, 136 of straight line and circle, 136 trigonometric functions, 137, 360 Irrational numbers, 379 Lamellar motion, 88 Langley's law, 74 Latitude and longitude of a point, 33 Latus rectum of ellipse, 404 of hyperbola, 408 of parabola, 414 Law of circular functions, 132 of complex numbers, 365 of compound interest, 277 of exponential function, 288 of power function, 80-82 of sines, cosines, and tan- gents, 320-327 Lead or lag, 349, 384 Legitimate transformations, 178 Lemniscate, 393 Limit, 221 Limiting lines of ellipse, 161 Loci, Chap. XIII, 387 et seq. defined by focal radii, 393 Theorems on, 61, 62, 65, 85, 88, 135 Locus of points, 35, 36 of equation, 36 Logarithmic and exponential functions, Chap. IX, 234 et seq. coordinate paper, 289-295 curves, 236-240, 260-264 double scale, 266 functions, 240, 244 increment and decrement, 279-282, 299 tables, 252, 253 Logarithm of a number, 236, 244 Logarithms, common, 244 graph, 237-243 properties of, 247-250 systems of, 245 Mantissa, 250 Mean, arithmetical, 213 geometrical, 217 harmonical, 224 Modulus of complex number, 369 of decay, 281, 299 of logarithmic system, 264 Motion, circular, 102 compound harmonic, 352 connecting rod, 355 elliptic, 344, 388 shearing, 87 S. H. M., 339 et seq. Naperian base, 245, 260, 277, 341 system of logs., 245 Napier, John, 234 Natural system of logarithms, 245 Negative angle, 100 functions of, 118, 119 Newton's law, 282 Node, 116 Normal, 136 equation of line, 136, 423 to ellipse, 430 to parabola, 420 Oblique triangles, 320-334 Odd functions, 119 Operators, 359 Ordinate of point, 33 Origin, 34 at vertex, 160, 415 Orthogonal systems, 124 Orthographic projection, 120- 123, 152, 265 496 INDEX (The numbers refer to the pages) Paper, logarithmic, 289 et seq. polar, 124 et seq. rectangular, 33 et seq. semi-log, 271, 283 et seq. Parabola, 52, 413 cubical, 52 polar equation, 414 properties of, 419 semi-cubical, 52 Parabolic curves, 49 et seq., 56, 289 Parameter, 155, 387 Parametric equations, 155 of cycloid, 396 of ellipse, 155 of hyperbola, 165, 166 Pascal's triangle, 204, 205 Periodic functions {see trig.- fcns.), 26, 116 Period of S. H. M., 341 of simple pendulum, 342 of uniform circular mo- tion, 102 of wave, 347 Permutations, 199-202 and combinations, Chap. VII, 198 et seq. Phase angle, 341, 348, 349 Plane triangles, 320-334 Polar coordinates, 123, 434 diagrams of periodic func- tions, 126, 318 equation of ellipse, 410 of hyperbola, 410 of parabola, 414 of straight hne, 135 form of complex number, 369 relation to rectangular, 136, 434 Polynomial, 175 Positive and negative angle, 100, 119 Positive and negative coordi- nates, 33 side of line, 427 Power function. Chap. Ill, 48 et seq. compared with exponen- tial, 286-289 law of, 80-82 practical graph, 76 variation of, 62 Probability curve, 212 Products, special, 451, 452 Progressions, Chap. VIII, 213 et seq. arithmetical, 213-216 decreasing, 214 geometrical, 217-224 harmonical, 224, 225 Projection, orthographic, 120- 123, 152, 265 Proportionality factor, 68 Quadrants, 34 Quadratic equations, 462 systems of equations, 186 Questionable transformations, 178 Radian unit of measure, 101, 102 Radicals, reduction of, 471 Radius vector, 123 Ratio definition of conies, 414, 415 Rationalization, 472 Rational formulas, 75 numbers, 354 Rectangular coords, (see Coordi- nates), Chap. II, 33 et seq. Reflection of curve, 57 Remainder theorem, 175 Reversors, 361 Right angle system, 100 INDEX 497 (The numbers refer to the pages) Root of any complex number, 376 of equation, 91 of function, 91, 177 of utoity, 377 Rotation of locus, 82, 133 polar coordinates, 133-135 rectangular, 434^436 of rigid body, 82 Scalar numbers, 358 Scale, 1, 3 algebraic, 3, 357 functions, 21 arithmetical, 3, 357 double, 4 et seq. logarithmic, 266-276 uniform, 1 Tables, damped vibrations, 301, 302 logarithms, 252, 476 natural trig, functions, 107, 128, 129, 487 powers, 51 of "e," 263 Tangent, 103 graph, 143 law, 323 to circle, 422, 428 to curve, 260 to ellipse, 429 to parabola, 418 Theorems, binomial, 205 et seq. factor, 176 functions of composite an- gles, 310 on loci, 61, 65, 85, 135 remainder, 175 Transformations, legitimate and questionable, 178 Translation, 82, 83 of any locus, 83, 85, 425 32 Translation of rigid body, 82 Transverse axis, 168 Triangle of reference, 103, 108 Triangles, solution of, 129, 320- 338 oblique, 320-338 right, 129-131 Trigonometric curves, 115, 117, 143-147, 319 functions, 103 et seq. Trochoid, 397 Trochoidal waves, 349 Trough of sinusoid, 116 Uniform circular motion, 102 Unit, change of, 66, 70, 77 et seq., 285 of angular measure, 101 Variables and constants, 15 and functions of variables. Chap. I Variation, 67 of power function, 62 Vector, 123 radius, 123 Vectorial angle, 103, 123 Velocity, angular, 102, 339 of wave, 348 Versors, 362 Vertices of ellipse, 154 of hyperbola, 168 Vibrations, damped, 299 Waves, Chap. XI, 339 et seq. compound, 352 length of, 347 progressive, 344 et seq. sinusoidal, 344 et seq. stationary, 350 trochoidal, 349 Zero of function, 91 ■i'li ! I! I! iiiii:i:inii liiiiiiii