SYLLABUS OF MATHEMATICS SOCIETY FOK THE PEOMOTION Of ENGINEEEING EDUCATION 1914 QfarttgU Ittiugraitg Siibrarg 3ltl;aia, Nftn lorfc THE ALEXANDER GRAY MEMORIAL LIBRARY ELECTRICAL ENGINEERING THE oirT or Cornell University Library QA 37.A51 1914 Syllabus of mathematics:a symposium comp 3 1924 003 892 829 Cornell University Library The original of tiiis book is in tine Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924003892829 SYLLABUS OF MATHEMATICS A SYMPOSIUM COMPILED BY THE COM- MITTEE ON THE TEACHING OF MATHE- MATICS TO STUDENTS OF ENGINEERING Accepted by the Society foe the Promotion of Engineebing Education AT the Nineteenth Annual Meeting held at Pittsburgh, Pa., June, 1911 Revised to January 1, 1914 Office of the Secretary ITHACA, N. Y. 1914 PRESS OF THE NEW ERA PRINTING COMPANV LANCASTER, PA. TABLE OF CONTENTS. PAGE Eeport of the Committee on the Teaching op Mathematics to Students of Engineering 1 Syllabus op Elbmentakt Algebra 5 Chap. I. Transformation of Algebraic Expressions 6 Chap. II. Solution of Equations 13 Chap. III. Miscellaneous Topics 16 Syllabus op Elementary Geometry and Mensuration 19 Syllabus op Plane Trigonometry 27 Chap. I. Sine, Cosine and Tangent of Acute Angles 28 Chap. II. The Trigonometric Funciions of any Angle .... 33 Chap. III. General Properties of the Trigonometric Punctions 43 Syllabus op Analytic Geometry 47 Chap. I. Eectangular Coordinates 48 Chap. II. The Straight Line 50 Chap. III. The Circle 52 Chap. IV. The Parabola 53 Chap. V. The EUipse 57 Chap. VI. The Hyperbola 61 Chap. VII. Transformation of Coordinates 65 Chap. VIII. General Equations of the Second Degree in x and y 66 Chap. IX. Systems of Conies 70 Chap. X. Polar Coordinates 71 Chap. XI. Coordinates in Space 72 Syllabus op Dipperential and Integral Calculus 75 Chap. I. Functions and Their Graphical Eepresentation. . . 77 Chap. II. Differentiation. Bate of Change of a Function 82 Chap. in. Integration as Anti-Differentiation. Simple Dif- ferential Equations 96 Chap. rV. Integration as the Limit of a Sum. Definite Integrals 102 Chap. V. Applications to Algebra. Expansion in Series; Indeterminate Forms 109 Chap. VI. Applications to Geometry and Mechanics 114 Discussion at the Pittsburgh Meeting 119 Syllabus on Complex Quantities 134 REPORT OF THE COMMITTEE ON THE TEACH- ING OF MATHEMATICS TO STUDENTS OF ENGINEERING. To the Society for the Promotion of Engineering Education: The committee was appointed at a joint meeting of mathe- maticians and engineers held in Chicago, December 30-31, 1907, under the auspices of the Chicago Section of the Ameri- can Mathematical Society, and Sections A and D of the American Association for the Advancement of Science,* and on the suggestion of officers of the Society for the Promotion of Engineering Education who were there present, the com- mittee was instructed to report to this Society. The membership of the committee is as follows : Algee, Philip R., professor of mathematics, U. S. Navy, Annapolis, Md. Campbell, Donald F., professor of mathematics. Armour Institute of Technology, Chicago, lU. Englee, Edmund A., president of the Worcester Polytechnic Institute, Worcester, Mass. Haskins, Charles N., assistant professor of mathematics, Dart- mouth College, Hanover, N. H. Howe, Charles S., president. Case School of Applied Science, Cleveland, Ohio. KuiCHLiNG, Bmil, consulting civil engineer. New York City. Mageudeb, WiUiam T., professor of mechanical engineering, Ohio State University, Columbus, Ohio. Modjeskt, Ralph, civil engineer, Chicago, 111. Osgood, William F., professor of mathematics, Harvard Uni- versity, Cambridge, Mass. Slichtee, Charles S., consulting engineer of the U. S. Recla- mation Service, professor of applied mathematics. Univer- sity of Wisconsin, Madison, Wis. *ror an account of the Chicago meeting, see Science for 1908 (July 12, 24, and 31; August 7 and 28; and September 4). 1 i COMMITTEE ON TEACHING MATHEMATICS. Steinmetz, Cliarles P., consulting engineer of the General Electric Company, professor of electrical engineering, Union University, Schenectady, N. Y. Swain, George F., consulting engineer, professor of civil engineering, Harvard University, Cambridge, Mass. TowNSEND, Edgar J., dean of the College of Science and pro- fessor of mathematics. University of Illinois, Urbana, HI. TuRNBAUEE, Frederick E., dean of the College of Mechanics and Engineering, University of Wisconsin, Madison, Wis. Waldo, Clarence A., head professor of mathematics, Washing- ton University, St. Louis, Mo. Williams, Gardner S., consulting engineer, professor of civil, hydraulic and sanitary engineering. University of Michigan, Ann Arbor, Mich. Woodward, Calvin M., dean of the School of Engineering and Architecture and professor of mathematics and applied mechanics, Washington University, St. Louis, Mo. Woodward, Robert S., president of the Carnegie Institution of Washington, Washington, D. C. ZiwET, Alexander, professor of mathematics. University of Michigan, Ann Arbor, Mich. Huntington, Edward V., chairman, assistant professor of mathematics. Harvard University, Cambridge, Mass. After deliberation, the committee decided that it could best carry out the purpose for which it was appointed by preparing a synopsis of those fundamental principles and methods of mathematics which, in the opinion of the committee, should constitute the minimum mathematical equipment of the stu- dent of engineering. This synopsis, as finally adopted, consists of five parts : 1. A Syllabus of Elementary Algebra ; 2. A Syllabus of Elementary Geometry and Mensuration; 3. A Syllabus of Plane Trigonometry ; 4. A Syllabus of Analytic Geometry; 5. A Syllabus of Differential and Integral Calculus. Two other syllabi, on Numerical Computation and on Ele- mentary Dynamics, were contemplated in the original plan, but were not completed. COMMITTEE ON TEACHING MATHEMATICS. 3 It is hoped that this report may be serviceable in two ways : first, to the teacher, as an indication of where the emphasis should be laid ; and secondly, to the student, as a syllabus of facts and methods which are to be his working tools. It does not include data for which the student would properly refer to an engineers' hand-book; it includes rather just those things for which he ought never to he obliged to refer to any book — the things which he should ha/ve constantly at his fingers' ends. The teacher of mathematics should see to it that at least these facts are perfectly familiar to all his students, so that the professor of engineering may presuppose, with confidence, at least this much mathematical knowledge on the part of his students. On the other hand, if the professor of engineering needs to use, at any point, more advanced mathematical meth- ods than those here mentioned, he should be careful to explain them to his class. The committee has not found it possible to propose a de- tailed course of study. The order in which these topics should be taken up must be left largely to the discretion of the individual teacher. The committee is firmly of the opinion, however, that whatever order is adopted, the principal part of the course should be problems worked by the students, and that all these problems should be solved on the basis of a small number of fundamental principles and methods, such as are here suggested. The defects in the mathematical training of the student of engineering appear to be largely in knowledge and grasp of fundamental principles, and the constant effort of the teacher should be to ground the student thoroughly in these funda- mentals, which are too often lost sight of in a mass of details. A pressing need at the present time is a series of synoptical text-books, which shall present, (1) the fundamental prin- ciples of the science in compact form, and (2) a classified and graded collection of problems (which would naturally be sub- ject to continual change and expansion) . It is the hope of the committee that this report, which is confined to the first part of 4 COMMITTEE ON TEACHING MATHEMATICS. the desired text-book, will stimulate throughout the country practical contributions toward the second. In the early part of its investigation the committee collected a large amount of information in regard to the present status of mathematical instruction for engineering students. Since that time, however, a much more inclusive inquiry has been undertaken by the International Commission on the Teaching of Mathematics, of which the American Commissioners are Professors D. E. Smith, J. W. A. Young and W. F. Osgood, In order to avoid unnecessary duplication, this committee voted to turn over all the results of its own inquiry in this field to the larger commission, to be worked up in accordance with the general scheme adopted by that commission, and to be incorporated in their report. This material is therefore not included in the present report. Respectfully submitted, Edwabd v. Huntington, Chairman. June, 1911. A SYLLABUS OF THE FORMAL PART OF ELEMENTARY ALGEBRA. This syllabus is intended to include tbose facts and methods of ele^ mentary algebra which a student who has completed a course in that subject should be expected to "know by heart" — that is, those funda- mental principles which he ought to have made so completely a perma- nent part of his mental equipment that he will never need to ' ' look them np in a book. ' ' It is not intended as a program of study for beginners, and no at- tempt has been made to arrange the topics in the order in which they should be taught. In reviewing the subject, however, either at the end of the course in algebra, or at the beginning of any later course, such a syllabus will be found serviceable to both teacher and student; and in the hands of a skillful teacher, and supplemented by an adequate collec- tion of problems, it might well be made the basis of a course of study conducted by the "syllabus method." One of the chief defects in the present-day teaching of algebra is the multiplicity of detached rules with which the student's mind is burdened;* and every successful attempt to knit together a number of these detached rules into a single general principle (provided this principle is simple and easily applied) should conduce to economy of mental effort, and di- minish the liability to error. Table or Contents. Chapter I. • Tbansfokmation op Algebkaio Expressions. General laws of addition and multiplication. Type-forms of multiplication (Factoring)'. Fractions. Negatives. Badicals and Imaginaries. Exponents and Logarithms. Chapter II. Solution op Equations. Legitimate operations on equations. To solve a single equation. Quadratic equations. Exponential equations. To solve a set of simultaneous equations. Chapter m. Miscellaneous Topics. Batio and proportion. Variation. Inequalities. Arithmetical, geometric, and harmonic progressions. * For example, in a recent prominent text-book there are no less than ■fifty italicized rules in the part of the book preceding quadratic equations I CHAPTER I. Tbansfoemation op Algebraic Expressions. 1. The ordinary operations of transforming and simplify- ing algebraic expressions should be so familiar to the student that he performs them almost instinctively; at the same time he should be able, whenever called upon, to justify each step of his work by reference to some one or more of a small number of well established principles. For example, if the student is asked iy what authority he replaces by Ti OT Va' + b' by a + b (to mention only two of the eommon- -\- X est bltmders), he will be forced to recognize either that he is making use of methods that he has never proved, and that are in fact erroneous, or else (which is more likely) that he is working altdgether in the dark, without any conscious reason for the steps he has taken. The following list of such principles, while making no pre- tense at logical completeness, will be sufficient for all practical purposes. 2. General laws of addition and multiplication. + 6 = 6 + a. ah = ia. ( Commutative laws. ) (a + 6) +c = a+ (6 + c). (a6)c = a(6c). (Associative laws.) o(6 + c) = o6 + ac. (Distributive law.) a + 0=^o. aXl=o. oX0 = 0. These laws hold when a, b, c are any of the quantities that occur in ordinary algebra, whether "real" or "complex."* The student sh^Ud be constantly encowaged to test general algebraic statements by substi- tuting concrete numerical values. 3. Tjrpe-forms of multiplication (Factoring). The following type-forms of multiplication are the ones that are most important to remember: * This syllabus is confined chiefly to the algebra of real quantities; th« algebra of complex quantities will be treated only incidentally. 6 AIXJEBRA. / a' — b'={a—b){a + h), as_68=(a— 6)(o2 4-a6 + &='), and so on; the general case is best remembered in the form 1 — a;''=(l — a;)(l + a; + x» + a!»H \-x"-^). Note also that in the algebra of real quantities, a" + 6" is divisible by a + 6 when and only when n is odd. Thus : a^ + l»= (a + h) {a' — ab + h'). Further: (x + a) (a; + 6) =x^+ (a + 6)x + a6, and the "binomial theorem": (a + 6 ) 2 = a^ 4- 2a6 + 6S (a + 6) » = o» + 3a^6 + SaJ" + 6», (a_6)2 = a=' — 2aZ> + 6% (a — 6)» = oS — So^^J + Soft^ — 6^ (a + 6)»= o ! -^ • where A! = "ifc factorial "==1 X 2 X 3 X • • • X *. 4. Fractions. o Def. If &x=a, then and only then we write x= T(or <^/b, or a-;- 6). Here o ia called the numerator and 6 the denominator of the fraction. A fraction with a zero denominator, as a/0, does not represent any definite quantity. For, if a is not zero, there is no quantity x such that X s = o ; and if a = 0, then every quantity x will have this property. Hence, tTie denominator of a fraction must always he different from sero. From the definition, a/1 = a ; also a - = 1, - = 0, (a + 0).* * The symbol + means "not equal to." 8 ALGBBBA. To add two fractions with common denominator : a b a + b c c~ ' To multiply two fractions : a X ax b y~ by' To divide by a fraction, "invert the divisor and multiply" : a X a y ay \b ' y~b x~ bx' The value of a fraction is not changed if we multiply {or divide) both the numerator and the denominator by any quantity not zero: a ma , „^ b = ^b ('^ + «)' This is the most important principle concerning fractions. Tor example, to reduce two fractions to a common denominator, we have merely to multiply numerator and denominator of each fraction by a suitable factor. Again, to simplify a complex fraction, we multiply the whole numera- tor and the whole denominator by any quantity which will "absorb" all the mbsidiary denominators. Thus, by multiplying by xyz, we have - + - X y ayz + hxz e -\- A (c + d)a^' z at once, by a single mental process. (The common practice of reducing the numerator and denominator separately, and then inverting the denom- inator and multiplying, is tedious and clumsy.) Def , If bx ==' 1, then x = 1/6, which is called the reciprocal of 6. To divide by 6 (6 =j= 0) is the same as to multiply by the reciprocal of 6. 5. Negatives. Def. If a-\-x=0, then and only then we write 2;= — a. In particular, — ( — a) = a. If a is not zero, — a is always opposite to a ; that is, if a is positive, — a is negative, and if a is negative, — a is positive. AliGEBBA.. i) Thus, if a = — 3, which is a negative quantity, then — a = 3, which is positive. The notation \a\, which is coming into use more and more widely, means the absolute value of a, that is, the numerical value of a regardless of sign ; thus, 1 5 | ^ 5, | — 5 | = 5. The laws of operation with the minus sign are best remem- bered by regarding — a as the product of a and — 1 : (—1) X/a — Vft a — b Vl— X Vl— 33 a/1— X 1—0! X Vl + X Vl + X Vl — X Vl — x^ Def . If a is negative, and n is odd, there will always be one negative value of x such that a;"=o; this value is denoted by -^a, and is called the (principal) mth root of a. Thus |/^^8"= — 2. 7. Imaginaries. If o is negative, and « is euera, then there is no positive or negative nih root of a. Hence, such quantities do not occur in the algebra of positive and negative quantities. They occur only in the more general algehra of complex qwintities; in this algebra every quantity a (except zero) has n distinct nth roots, the notation '\[a being applied, as occasion requires, to any one of these n values. The detailed study of this general algebra is probably too difficult for a first course; for such applications as occur in elementary work, the following working rules are suf&eient : 1) In manipulating a complex quantity of the form V — h. where 6 is positive, write V — & = V — 1 V& =*V&^ and treat i like any other letter ; then simplify the result by the relation t2=— 1. 2) Every complex quantity can be written in the form a-\-i'b, where a and 6 are "real" (that is, positive, negative, or zero) ; and if a-\-ih = a' -^-ib', then a^^a' and b^^V. ALGEBBA. ll In electrical engineering the letter i is used to denote current, and V — 1 is denoted by j. 8. Exponents. The subject of negative and fractional exponents is a part of algebra in which the preparation of the student is apt to be especially unsatis- factory. Definition of negative and fractional exponents. If a is positive, and p and q are any positive integers, then 9. Laws of operation with exponents. If a and b are positive, then : All these laws hold for any values of m and n; the three fundamental ones can readily be recalled to mind through simple special cases, such as a'a', (a')', and (ah)'. The three other laws commonly mentioned, namely am-n=: am/an, o»»/»^ "^O", (^a/'b)m = am/'bm^ are immediate corollaries of those just mentioned. If o is negative, and m not an integer, o"> will, in general, be a complex quantity. In such cases, let a' = — a, so that a' is positive, and write o»»^ ( — l)»»o''», where ( — 1)™ must then be handled according to the rules of operation in the algebra of complex quantities. 10. Logarithms. The subject of logarithms should be taught in logical connection with the subject of exponents. The common practice of separating these subjects, and treating logarithms as a part of trigonometry, is unfortu- nate. Numerous applications of logarithms can be found that have nothing to do with trigonometry; moreover, the training in the use of logarithms which a student gets in trigonometry is usually quite inade- quate as a preparation for the applications of logarithms in any of hia later work outside of surveying. Def. The logarithm of a (positive) number, to any (posi- tive) base, is the exponent of the power to which the base must be raised to produce that number. 12 AXiQEBEA. Thus, the notation x = logiN means Note that negative numbers in general have no logarithms in the algebra of real qnantities. From the laws of exponents we have, whatever the base may be : log {ab) =log o + log b, log (^\ =log o — log b, log (o") =w log a, log >fa= -log a, log 1 = 0, log(6ase)=l. Only two bases are in common use. For purposes of numerical computation, the base chosen is 10, and in this system log(10»)=m. In higher mathematics, the base e = 2.718 •••is used, for the reason that the use of this base simplifies certain formulas in the calculus; in this system log (e") =w. Change of base. To find logeN when logmiV is known, let x^logeiV, that is, e'' = N. Then take the logarithm of both sides of this equation to base 10, and solve for x. The resulting formula, loggW^ (logio^)/(logioe), is so easily obtained in this way that it is not worth while to remember it separately. The approximate values logi„e = .4343, and loge^'= (2.3026) logu^, however, are useful to remember. CHAPTER II. Solution of Equations. 11. Legitimate operations on equations. If a given equa- tion is true, it will still be true if we (a) add any quantity we please to both sides; (6) subtract any quantity we please from both sides; (c) multiply both sides by any quantity we please ; (d) divide both sides by any quantity we please except zero, (e) raise both sides to any positive integral power; (/) *extract any positive integral root of both sides, except that if an even root is extracted, the double sign ± must be used; {g) *take the logarithm of both sides (provided both sides are positive) . In regard to (d), we must never divide both sides by an unknown quantity without first excluding the possibility that that quantity is zero. In (/), the restriction stated means, for example, that from A' -=3 we can infer merely that A=i± y/B; that is, that either A = VB, or A^ — V-B; but we cannot teU which. 12. To solve a single equation in x, means to find aU the values of x that satisfy the equation, or to show that none such exist. Any value of x that satisfies the equation is called a root of the equation. In testing a root, the only safe method is to substitute the given value in each side of the equation separately, and see whether the re- sults, when reduced, are equal. Thus, we should find that x^ — 2 is a root of the equation a; ^ 2 — y/Vi — 2x, and that a; = 4 is not a root. In this connection it should be noticed that if we square both sides of a given equation, the new equation will, in general, have more roots than the given equation. Thus (to use the same example), by squaring X — 2 = — V13 — 2x we have x' — 2a; — 8 = 0. This equation has of course the root — 2, since a; = — 2 satisfies the original equation from * In the algebra of complex quantities (/) and {g) are not applicable. 2 13 14 AliGBBBA. which this was derived; but it has also the root 4, which was not a root of the original equation. The formal process usually called "solving the equation" means merely transforming the equation, by a judicious choice of the legitimate operations, into a form in which the solutions are obvious. If this is not possible, we must have recourse to the method of trial and error which, while often laborious, is always applicable in numerical cases. If an equation is given in the factored form: (a; — a)(a; — j3)(a; — 7) •••=0, then the roots are obviously x = a, a; = /3, x = y, ••• . Thus, the roots of a;(fl; + 2) = are and — 2. 13. Quadratic equations. To solve the quadratic equation ax' + ix + c^O, we may divide through by o: , 6 c x' + -X = a a and then "complete the square": whence, — 6 ± VP — 4 ae or, we may use the general result just obtained as a formula. The quantity which must be added to both sides in "completing the square" is obvious by analogy with a" + 2mx + m?, so that this method requires less effort of the memory than the method of solution by formula. The "method of factoring" is very convenient in certain special cases, when the factors can be obtained by inspection. The method still sometimes used, of first multiplying through by 4a to avoid fractions, is apt to lead to confusion, and should be discouraged. From the formula it is evident that the sum of the roots is AIXJEBEA. 1 5 ^1 + ^2= — 6 A) and the product of the roots is x.^x^ = c/a; also, if the coefficients, a, b, c, are real, the roots will be real- and-distinct, real-and-coineident, or imaginary, according as 6^ — 4ac is positive, zero, or negative. 14. Exponential equations. To solve an equation of the form o'"i=6, when a and 6 are positive, take the logarithm of both sides: x log ac=log b; and then solve for x. 15. To solve a set of simultaneous equation in x, y, z • • • means to find all the sets of values oi. x, y, z, •••, that satisfy all the equations at once, or show that none such exist. Two simultaneous equations of the first degree, as ax-^-'by = c and Ax + By = C, can always be solved in a couple of lines, if the work is arranged as follows: 7z— &y = l — 5 3 — 2 1 (— 35 + 42)1 = — 5 + 9 (12 — 10)j/ = — 2 + 3 whence the values of x and y are obvious, provided aB — bA is not zero. (If aB — bA = 0, there is either no pair of values x, y that satisfies both the equations, or else there are an infinite number of pairs of values that do so; in this latter case, the equations are not independent, that is, either of them can be derived from the other.) The theory of simultaneous equations, and sometimes the numerical computation, is facilitated by the use of determinants. In general, n independent equations will suffice to deter- mine n unknown quantities. CHAPTER III. Miscellaneous Topics. 16. Ratio and Proportion. The simply the fraction a/6y and a "proportion" is simply an equation between two ratios. The notation alb'.'.o'.d should be replaced by the equation a/i^o/d; and all special terminology, such as "taking a proportion by alterna- tion," "by composition," etc., should be dropped in favor of the ordinary language of the equation. 17. Variation. The statement "y varies as x," OT"y varies directly as x," or "y is proportional to x," means y^=^kx, where h is some constant. Similarly, "2/ varies inversely as x," means y = 'k/x; "y varies inversely as the square of a;," means y = Tc/x^. The constant h can always be determined if we know any pair of values of x and y that belong together. The statement "y varies as u and v," means y varies as the product of u and V, that is, y = Tcv/v. 18. Inequalities. The notions of ' ' greater and less ' ' are thoroughly familiar when we are dealing only with positive quantities, but the ex- tension of these terms to the algebra of all real quantities (positive, negative, and zero) is apt to cause some confusion. (a) All real quantities (positive, negative, and zero) may be represented by the points of a directed line (running, say, from left to right) : -> _3 _2 —1 +1 +2 +3 and the notation a<6 (read: "a algebraically less than 6") means simply that a precedes b, or a lies on the left of b, along this line. Similarly, a > 6 (read: "a algebraically greater than 6") means that a comes after b, or lies on the right of 6, along the line. (The idea that a negative quantity is a magnitude whose size is in some way ' ' less than nothing" should be carefully avoided.) 16 AIX3EBRA. 17 Obviously, if a and i are any real quantities, one and only one of the three relations : a = b, a<,i, and a>b, will hold between them ; further, if a < 6 and 6 < c, then a<,c. (6) Complex quantities require for their representation the points of a plane instead of the points of a line, and the symbols < and > are not used in connection with these quantities. Legitimate operations on inequalities. If a given inequality is true, it will stiU be true if we (a) add any quantity we please to both sides; (6) subtract any quantity we please from both sides; (c) multiply both sides by any positive quantity; (d) divide both sides by any positive quantity; (e) raise both sides to any positive power (integral or fractional), provided both sides are positive. (/) take the logarithm of both sides, provided both sides are positive. If we multiply or divide both sides by any negative number, we must reverse the sense of the inequality. The neglect of the rules for handling inequalities is the source of many common errors. 19. Arithmetical Progression. In an arithmetical progression : a, a-\-d, a-\-2d, a-\-3d, •■•, each term is obtained from the preceding by adding a con- stant quantity. The nth term is obviously Z = a+ (w — l)d. a-\-l The sum of n terms is S^ = — „— n. This formula is most easily remembered in the form: S= (average of the first and last terms) X (number of terms). The arithmetic mean between a and b is A=i{a-\-b). 20. Geometric Progression. In a geometric progression : a, ar, ar^, ar', •••, 18 ALGEBEA. each term is obtained from the preceding by multiplying by a constant quantity. The nth term is obviously I = ar^^. a(l — r") The sum of n terms is S= — 5 . 1 — r This fonndla is best remembered in connection with the rule for factoring: 1 — r»=(l — r)(l + r + r2+r»H h**"^)- The geometric mean between a and 6 is G^y/db. The geometric mean is also called the mean proportional. Infinite geometric progression. If | r | < 1, the sum of n terms approaches the limit a as n increases indefinitely (since, in the expression for S, if I r I < 1, r" approaches zero), 21. Harmonic Progression. A harmonic progression is a series of terms whose recip- rocals are in arithmetical progression. (The harmonic pro- gression is not of great importance.) The harmonic mean between a and 6 is H= 5^. a + b A SYLLABUS OF ELEMENTARY GEOMETRY AND MENSURATION. This syllabus is intended to include those facts and methods of ele- mentary geometry which a student should have so firmly fixed in his memory that he will never think of looMng them up in a book. 1. Bight Triangles. In a right triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides (Pythagoras, 580-501 B.C.) ; and the sum of the acute angles is 90°. Examples of right triangles with integral sides: 3, 4, 5; 5, 12, 13. Two right triangles are congruent when they agree with respect to (a) any side and an acute angle; or (6) any two sides. In the "45° triangle" and the "30-60° triangle," the ratios of the sides are as indicated in the figure. 2. Oblique Triangles. In any plane triangle, the sum of the angles is 180°. Hence, an exterior angle of a triangle equals the ^0"^^ sum of the opposite interior angles. X~^ "^^^N Of two unequal sides in a triangle, the greater is opposite the greater angle. A plane triangle is, in general, whoUy determined when any three of its parts (not all angles) are given. 19 20 ELEMBNTABY GBOMETBY AND MENSTJBATION. There are four cases : aC^^f^,^^ (o) two angles (provided their sum is less than 180°) and one side; (&) two sides and the included angle; (c) the three sides (provided the largest is less than the sum of the other ..^itf'^vK. ^ Am^ two); *^" ^ ' *^"l^ (d) two sides and the angle opposite one of them (the "ambiguous case," in which we may have two solutions, or one, or none). Hence the usual rules for testing the equality of two plane triangles. The center of gravity of a plane triangle is the intersection of the three medians, and is two thirds of the way from any vertex to the middle point of the opposite side. 3. Angles in a Circle. An angle inscribed in a semicircle is a right angle. ^ An angle subtended by an arc of a circle at any point of the circumference is equal to half the angle subtended by the same arc at the center. 4. Similar Figures. Proportion. If any two lines are cut by a set of parallels, the corresponding segments are proportional. (Hence the usual rule for dividing a given line into any number of equal parts.) In all problems in proportion, the notation o:6::c:d, and all special terminology, such as "taking a proportion by alternation," "by com- ELEMENTARY GBOMBTBY AND MENSTJBATION. 21 position," etc., should be abandoned in favor of the ordinary language of the equation. For example, if 0/6 = clA, then, by adding 1 to both sides, (a + l>)/6 = (c + d)/dj and by subtracting 1 from both sides, (o — 6)/6=(c — d)/d; etc. If two plane triangles are similar, their corresponding sides are proportional. In a right triangle, the perpendicular from the vertex of the right angle to the hypotenuse is a mean proportional be- tween the segments of the hypotenuse: P" ■ mn. Any two similar fig- ures, in the plane or in space, can be placed in " perspective," that is, so that lines joining corresponding points of the two figures wiU pass through a common point. In other words, of two similar figures, one is merely an enlargement of the other. In two similar figures, if Zc is the factor of proportionality, any length in one = ft X (the corresponding length in the other) ; any area in one = k' X (the corresponding area in the other) ; any volume in one = &' X (the corresponding vol- ume in the other) . 5. Lines and Planes. If a line is perpendicular to a plane, every plane containing that line is perpen- dicular to the plane. J^ U^ 7 22 ELBMENTABY GEOMETBY AND MBNSTTBATION. A dihedral angle is measured by a plane angle formed by two lines, one in each face, perpen- dicular to the edge. 6. Plane Areas. Area of parallelogram = base X altitude. Area of triangle = ^ base X altitude. Area of trapezoid = I sum of II sides X alt. = mid-section X altitude. 7. The Circle, (ir = 3.1416 • • • = 22/7, approximately.) Circumference of circle = 2iir. (Proved by regarding the eirele as the limit of an inscribed or circumscribed polygon; proof rather long.) '^ J/ K J Area of circle = Trr". □ (Proof by regarding circle as limit of sum of triangles radiating out from the center, the altitude of each triangle being the radius of the circle; hence, area of circle ^} circumference X radius.) Area of sector angle of sector area of eirele four right angles ' hence, Area of sector = \r^B, where B is the angle in radians. For area of segment, subtract triangle from sector. ELBMEKTABY GEOMBTEY AND MENSTTEATION. 23 8. The Cylinder. Volume of any cylinder (or prism) = base X altitude. Area of curved surface of any right cylinder (or right prism) = perimeter of base X altitude. (Proof by regarding the area as the limit of a sum of rectangles whose conunon altitude is the altitude of the cylinder; or, by slitting the cylinder along an "element" and rolling the surface out into a rectangle.) 9. The Cone. Volume of any cone (or pyra- mid) = 1/3 base X altitude. (Proof by dissecting a triangular prism; or, more simply, by the in- tegral calculus.) Area of curved surface of a right circular cone (or a regular pyramid) = 1/2 perimeter of base X slant height. (Proof by regarding the area as the limit of a sum of triangles whose common altitude is the slant height of the cone.) Area of frustum of a right circular cone (or of a regular pyramid) =1/2 sum of perimeters of bases X slant height. = perimeter of mid-section X slant height. (Proof by regarding the area as the limit of the trapezoids whose common altitude is the slant height of the frustum.) 24 ELEMENTABY GEOMETBY AND MENStTEATION. 10. The Sphere. Area of a zone = cireumf erence of great circle X altitude of zone. In other words, the area of the sphere cut out by two parallel planes is equal to the area of the portion of the circumscribing cyUnder inter- cepted between the same pair of parallel planes. (Proof by regarding the zone as the limit of a sum of conical frustums.) Hence, Area of sphere = 47rr^ =^ area of four great circles of the sphere. In other words, the area of the spJiere is equal to the area of the curved surface of the circwmscribing cylinder. Volume of sphere = |'r^- (Proof by regarding sphere as limit of a sum of pyramids radi- ating out from the center, the altitude of each pyramid being the radius of the sphere; hence, volume of sphere = J area of sphere X radius.) Area of a lune angle of lune area of sphere four right angles' Area of spherical triangle is proportional to its spherical excess (that is, the excess of the sum of its angles over 180°). (Proof by considering three lunes which have the given triangle in common.) ELEMENTAEY GEOMBTBY AND MBNSUEATION. 25 The following fv/rther theorems, the proof of which involves the inte- gral calculus, are mentioned here also, because they are easy to remerriber and are often serviceable in elementary worlc. 11. Oavalieri's Theorem (1598-1647). Suppose two solids have their bases in the same plane, and let the sections made in each solid by any plane parallel to the base be called "corresponding sections." If then the corre- sponding sections of the two solids are always equal, the vol- umes of the solids will be equal. (Proof by regarding each of the solids as the limit of a pile of thin ) 12. Theorems of Guldin (1577-1643), or of Pappus (about 290 A.D.). 1. Suppose a plane figure revolves about an axis in its plane but not cutting it. Then the volume of the solid thus generated is equal to the area of the given figure times the length of the path traced by its center of gravity. 2. Suppose a plane curve revolves about an axis in its plane but not cutting it. Then the area of the surface thus generated is equal to the length of the given curve times the length of the path traced by its center of gravity. 26 ELBMBNTABT GEOMETEY AND MENSUBATION. 13. The Frismoidal Formula. The prismoidal formula holds for any solid lying between two parallel planes and such that the area of a section at distance x tram the base ia expressible as a polynomial of the second (or third) degree in x. li A, B = areas of the bases, M = area of a plane section midway between the bases, and h = altitude, then Volume of prismoid=^ {A-\-B-\- 4M). D A SYLLABUS OF PLANE TRIGONOMETRY. This syllabus is intended to include those facts and methods of plane trigonometry which a student should have so firmly fixed in his memory that he will never think of looking them up in a book. Table op Contents. Chapter I. Sine, Cosine, and Tangent of Acute Angles. Definitions of sine, cosine, and tangent of an acute angle as ratios between the sides of a right triangle: sin a; = opp/hyp; eos a; = adj/hyp; tan a;=:opp/adj. To trace the changes in these functions, as the angle changes from 0° to 90° (circle of reference). Use of tables. Exact values of functions of 30°, 45°, and 60°. To find remaining functions of an angle when one function is given (draw right triangle). To construct an angle from its tangent. Fundamental relations: sin' x + cos" x^l, tan a; = sin a;/eos x, etc. Solution of right triangles. Problems in orthogonal projection. Problems in composition and resolution of forces, etc. Chapteb II. The Trigonometric IFunctions of ant Angle. Angles in general. Congruent, complementary, and supplementary angles. Units of angular measurement: degree, grade, radian. Definitions of sine, cosine, and tangent of any angle. To trace the changes in these functions, as the angle changes from 0° to 360° (circle of reference). Definitions of cotangent, secant, and cosecant: eot a; = l/tan x, see a; = l/oos x, esc i^l/sin x. Definitions of versed sine and coversed sine: vers a; = 1 — eos x, covers a; = 1 — sin x. Use of the tables: reduction to first quadrant. Solution of oblique triangles. Law of sines: a/Zi^sin A/sin. B. Law of cosines : a'^h' + (^ — 26c cos A. Chapter III. General Properties of the Trigonometric Functions. Belations between the functions of a single angle. Functions of ( — x). Functions of (a;±n90°), etc. Functions of the sum and difference of two angles: sin (a; + y) ^ sin x cos y + eos x sin y, cos (,x + y) = COB X COB y — sin x sin y. Functions of twice an angle, and of half an angle. The inverse functions, sin"'a;, cos-'a;, tan-^ar, etc. Solution of trigonometric equations. 27 CHAPTER I. Sine, Cosinb, and Tangent op Acute Angles. 1. Definition of sine, cosine, and tangent of an acute angle X. — In any right triangle, if a; is one of the acute angles, the sine, cosine and tangent of x are defined as ratios between the sides of the triangle, as follows: side opp. side adj. j"!^ cosx=^i -7^ HS^ fima;=- hypot. tana;= side opp. hypot. Fig. 1. FiQ. 2. side adj. These ratios are pure numbers, depending only on the size of the angle. 2. To trace the changes in these num- bers when the angle changes from 0° to 90°, draw the figure so that the denomi- nator of the ratio is kept constant, say equal to 1 inch, and trace the changes in the numerator. Thus, from Pig. 2, when X goes from 0° to 90°, sin x goes from to 1, and cos x goes from 1 to ; from Fig 3, when x goes from 0° to 90°, tan x goes from to infinity. 3. Tables. — The ratios thus defined are called "trigonometric functions" of the angle, and their values have been tabulated, to 4, 5, or 6 places of decimals, in the "tables of trigonometric functions." Before using the printed tables, the student should make his own table, for a few angles, by graphical con- struction, with a protractor, to two places of decimals.* * It is clear from the figure that the values of cos x from 0° to 90° are the same as the values of sin x in reverse order; note how this fact is made use of to save space in the tables. 28 Fio. 3. TBIGONOMETBY. 29 4. The functions of 30°, 45°, and 60° can be found exactly, without the use of the table. Thus, in the triangles which occur in Fig. 4, it is readily proved by the Pythagorean theorem that if the hypotenuse is 1 inch, the shortest side is ^ in., the longest side is ^V^ m., and the middle-sized side ^y/2 in. Hence any function of 30°, 45°, or 60° can be read ■off the figure by inspection. For example, sin 30° =1, tan 45° = 1, tan 60° = V3 ; etc, 5. It is frequently required to find the remaining functions of an angle when any one function is given. To do this, draw a right triangle, mark one of the angles, and mark two sides to correspond to the given function. Then compute the remain- ing side by the Pythagorean theorem, and read ■off any desired function from the completed figure. For example, Given, tan a; = f. From the figure, sin a; = 2/V13; etc. Given, sin x = a. From the figure, tan x = a/^/1 — a' ; etc. To construct an angle when any one of its functions is given, first find the tangent of the angle ; when the tangent is known, the construction of the angle is obvious. 6. The notation sin^ x, etc., is used as an abbreviation for (sin x)"; etc. The following fundamental relations are easily proved and remembered from the figure : for any angle x, riG. 4. etc, g fun 3 Fig. 5. Pig. 7. Fig. 8. sin^ a; + cos* a; = 1, tana;= sinx cos a; sin (90° — x) =cosa;. cos (90° — a;) =sinx. 30 TEIGONOMETEY. 7. The student should be thoroughly drilled in the defini- ^^ tions of the sine, cosine and tangent, in right // // triangles in all possible positions in the plane ^ Kr-T regardless of lettering. Thus, the mental proc- ess should be as foUows: pointing at the figure, "the tangent of this angle is this side, divided by this side"; etc. ^1_ The following forms of the original equations are especially useful, and should be emphasized: side opp. = hypot. X sine ; side adj. = hypot. X cosine. Solution of Eight Teiangi/es. 8. We recall that in any right triangle, the sum of the squares on the two legs is equal to the square on the hypotenuse, and the sum of the acute angles is 90°. Hence, when either acute angle is known, the other may be found; and the sine of either acute angle is the cosine of the other : c2 = a2-|-62, sin A = cos B. 9. By the aid of a table of sines, cosines and tangents, when any two parts of a right triangle, besides the right angle, are given, the remaining parts may he found (except in the case where the given parts are the two acute angles, in which case the triangle is not determined). For, we have merely to remember the definitions of the func- tions, selecting the equations so that only one unknown ap- pears in each equation ; then solve for the unknown quantity, and compute by the aid of the tables. The results should be checked by substituting in some relation not used in the direct computation.* * This computation, like many other numerical eompatations, can often be shortened by the use of the slide rule, or by the use of logarithms; in fact, tables are provided which give the logarithms of the trigono- metric functions directly in terms of the angles; but the student should thoroughly understand the use of the functions themselves before he begins to use the logarithmic tables. TEIGONOMBTRY. 31 10. Numerous problems involving right triangles : isosceles triangles, polygons, oblique triangles solved by means of right triangles, heights and distances, surveying problems, etc. A*Ac<»-0 Kg. 11. Orthogonal Projection. Components of Forces, etc. 11. The projection of a length AB on any line is the given length times the cosine of the angle between the lines. (Proof from the definition of cosine.) The projection of a plane area upon any fixed plane is the given area times the cosine of the angle between the planes. (Proof by the theo- rem of limits.) 12. The component of a force along any fixed axis is the magnitude of the force times the cosine of the y^ _ y^ angle between the force and the axis. Since we usually require the components along two rectangular axes, it is important to remember that cos (90° — a;)=sina;. The mental process should be as follows : In Fig. 12, the component of F along the j/-axis is F times the cosine of 0; the component of F along the a;-axis is F times the cosine of the other angle, which is F times the sine of 6; that is, Fy=F cose-, Fx=F sine. Similarly, in Fig. 13, Fx=F cos; F^= — Fsin^ (minus, because it pulls backward along that liae). The components of velocities, accelerations, or any other vector quantities are to be handled in the same way. 13. Every problem should be accompanied by a sketch or diagram, to show that the student understands the meaning of each step of his work. And in many cases, an accurate graphical solution on a drawing board may be used as a valu- able check on the correctness of the numerical computation. Fig. 12. Fig. 13. 32 TEIGONOMETBY. 14. Note. That portion of trigonometry which has been outlined up to this point is so elementary in character, and so readily understood and appreciated by the student, that it may well be introduced much earlier in the course tham^ is usually done — perhaps even as early as the elementary course in plane geometry. CHAPTER II. The Trigonometkic Functions op Ant Anghe, 15. Angles in general. — ^An angle, as the term is used in ap- plied mathematics, is the amount of rotation of a moving radius OP about a fixed point 0, measured from a fixed line Fig. 14. OX. Here OX is called the initial line and OP the terminal line of the angle. Counterclockwise rotation is positive, and angles are added and subtracted as algebraic quantities. The quadrants are numbered as in the figure; an "angle in quad- rant II" for example, means an angle whose terminal line lies in quadrant //. 16. Congruent angles are angles differing by any multiple of 360°. 17. Complementary angles are angles whose sum is 90°; supplementary angles are angles whose sum is 180°. 18. Units of angular measurement are: the degree, sub divided into minutes and seconds, or decimally; the grade, 33 34 TBIGONOMETEY. subdivided decimally; and the radian, subdivided decimally. 1 degree = 1° = l/90th of a right angle; 1 grade = 1/lOOth of a right angle (used in France) ; 1 radian ^ angle subtended by an arc equal to the radius. Via. 15. Since ratio of semi-circumference to radius = 7r (where 7r= 3.1416 •••=3% approximately), we have IT radians = 180°, and hence 1 radian = about 57.3°. 19. The radian is especially important in problems concern- ing the motion of a particle in. a circular path. Thus, if r ft. = radius of the circle, s ft. = length of arc traversed, and 6 radians = angle swept over by the moving radius, then s=re. This important equation is not true unless the angle is meas- ured in radians. Again, if V ft. per sec. = linear velocity of the particle in its path, and (1) radians per sec. = its angular velocity, then V = rca. Further, if the angular velocity = m radians per sec. = N rev. per min., then the relation between the numbers o> and N is given by ttN "=30- In all higher mathematics, when a letter is used for an angle, without designating the unit, it is understood that the letter means the number of radians in the angle. TBIGONOMETEY. 35 20. Definition of sine, cosine, and tangent of any angle. — Let X be any angle, swept over by a moving radius revolving from OX to OL, and suppose for convenience of language that OX extends horizontally to tbe right. Assume, for the moment that OX and OL are not perpendicular. From any point P of the moving radius drop a perpendicular on the initial line (or the initial line produced), thus forming a right tri- FiG. 16. angle, called the triangle of reference for the given angle x. In this triangle, the perpendicular MP is called the side oppo- site 0, and is positive if it runs up, negative if it runs down ; the base OM is called the side adjacent to 0, and is positive if it runs to the right, negative if it runs to the left, and the radius OP is called the hypotenuse of the triangle and may always be taken as positive. The sine, cosine and tangent of the angle x are then defined as follows : tana; = side opp. _ sin x side adj. cos a; side opp. side adj. 8ina;=-v- ^, cosa;= ,^^ . hypot. hypot. These ratios are positive or negative numbers, depending only on the position of the terminal side of the angle x, and 36 TBIGONOMBTEY. are called trigonometric fimctions of x. The functions of any angle congruent to x are the same as the functions of x, so that we need consider only the angles in "the first revolu- tion," that is, angles between 0° and 360°. 21. To trace the changes in each function as the angle changes from 0° to 360°, draw the figure so that the denomi- nator of the ratio is kept constant, say equal to 1 inch, and trace the changes in the numerator (Fig. 17 for the sine and cosine ; Fig. 18 for the tangent) . Obviously, the sine wiU be positive for angles in the upper quadrants; the cosine will be positive for angles in the right hand quadrants; and the tangent will be positive in quadrants I and III. The definitions of the functions of 0°, 90°, 180°, and 270°, which were not included above, can now be readily obtained by noting what becomes of the function of a variable angle X when x approaches one of these values as a limit. In using the "circle of reference" be careful to have every angle start from the initial line that extends horizontally to the right. Other Teiqonometeic Functions. 22. Definition of other trigonometric functions. — ^Besides the sine, cosine, and tangent, other functions in common use are the cotangent, the secant, and the cosecant, which are most conveniently defined thus : 1 1 1 cota;=: , seca;= , csca;=-; — . tan X cos x sin x Less important, but often convenient, are the versed sine and the coversed sine: vers x = l — cos x, covers x = l — sin a;. 23. It is worth remembering that the sine and cosine are always less than (or equal to) 1, in absolute value; their reciprocals, the secant and cosecant, are always greater than (or equal to) 1, in absolute value; the tangent and cotangent may have any value, positive or negative; while the versed sine and coversed sine are always positive, ranging from to 2. TEIGONOMETEY. 37 Fig. 17. ^^M Fio. 18. 38 TKIGONOMETEY. .»-90* Fig. 19. 24. Use of the tables: reduction to the first quadrant. — The tables in common use give the values of the functions only for angles between 0° and 90°, that is, only for angles in the first quadrant. To find the functions of an angle x in one of the other quadrants, find first the "reduced angle" in quadrant I (that is, a; — 90°, or x — 180°, or a;— 270°), and then proceed as in the following examples:* (a) To find cos a;, when x is in quadrant II. Draw any angle in quadrant II to represent the angle x (avoiding, however, lines near the middle of the quadrant) and draw the ' ' reduced angle" x — 90° in quadrant I. Then, pointing at the figure, cos x is this line (VW) [divided by the radius], which is the same in length as this line (g) [divided by the radius], which is the sine ot X — 90° ; but the first line is negative; hence cosa; = — sin {x — 90°), where sin (a; — 90° ) , of course, can be found in the table. (6) To find tanaj, when x is in quadrant //. Pointing at the figure, tana; is this line (<) divided by this line (|||), which is the same as this line (VVV) divided by this line (=), which is the cotangent of (x — 90°); but the signs are unlike; hence tan x = — cot {x — 90°), where cot {x — 90°) can be found from the table. Tig. 20. Similarly for any other case. 25. The converse problem of finding the angle correspond- ing to any given function is complicated by the fact that there will be (in general) two angles between 0° and 360° corre- sponding to any given function. The most satisfactory way • The given angle is supposed to be already reduced to an angle be- tween 0° and 360°. TEIGONOMEXBY. 39 to find these two angles, in any numerical case, is to draw the figure, and proceed as in the examples below, ia which x^ in each case represents an angle in the first quadrant which can be found in the table. Given sin a; =; 0.5; x = Xo or 180° — Xo. Given cos a; = 0.8; x = x„ or 360°— a Given tan a; = 0.8; x = Xo or 180° -\-Xg. Given sina;= — 0.5; a; = 180° +Xo or 360° — x,. ■CO^lC"0-S Fig. 24. Given cosa;= — 0.5; a; = 180°— a;o or 180°+: Fig. 26. Given tana;= — 0.8; a; = 180° — x„ or 360°— a;. 40 TBIGONOMETBY. These results are not formulae to be memorized; it is much safer, and more intelligent, to draw the appropriate figure, or to visualize it in the mind, for each case as it arises. The student should be thoroughly drilled in numerical cases, especially for angles in the second quadrant. Notice that an angle is completely determined when we know the value of any one of its functions, and the sign of any other function (not the reciprocal of the first). It we restrict ourselves to angles between 0° and 180°, as in the case of angles in a triangle, then an angle is wholly determined by either its cosine or its tangent ; but there wiU be two angles, x and 180° — x, corresponding to a given sine. 26. The functions of certain angles in the later quadrants, corresponding to 30°, 45°, and 60° in quadrant J, may be found exactly, without the use of the tables, by inspection of the figure (see § 4) . For example, cos 120° = — ^. 27. If it is required to fimd the remaining functions of an angle when one function is given, draw a right triangle and proceed as in § 5, considering only the absolute values of the quantities, without regard to sign; then adjust the sign of the answer according to the quadrant in which the angle lies. Or, the angle may be drawn at once in the proper quadrant. Solution of Oblique Triangles. 28. In any plane triangle the following theorems are easily proved from a figure : (1) The "Law of Sines." — ^Any side is to any other side as the sine of the angle opposite the first side is to the sine of the angle opposite the other side ; in the usual notation : a sin A h Bin JB ' with two analogous formulae obtained by "advancing the letters." TEIGONOMETKY. 41 (2) The "Law of Cosines." — The square of any side is equal to the sum of the squares of the other two sides, minus twice their product times the cosine of the included angle : o^ = 6" _|_ c2 _ 2hc cos A, with two analogous formulse obtained by "advancing the letters." These two laws, with the fact that the sum of the angles is 180°, suffice to "solve" any plane triangle, and are important in many theoretical considerations. The following formulae which are especially adapted to logarithmic computation, give the tangents of the half-angles in terms of the sides, and are included here for reference : ^ A r ^ B r , G " tan — = , tan — = ^ , tan — = 2 s — a^ 2 s — 6' 2 s — c where a + 6 + c S ^ 7i and -4 (s — a) {s — b) (s — c) _ s From these formulae we have at once, = radius of inscribed circle. Area = rs= '\/s{s — a) (s — b) (s — c). 29. The only case which is likely to give any difficulty, is the "ambiguous case" in which the given parts are two sides and the angle opposite one of them. Here we must remem- ber, at a certain point in the work, that when the sine of an angle is given, there will be, in general, two angles corre- sponding to that sine, one the supplement of the other; so that from that point on, the problem breaks up into two separate problems. But if the sine of an angle is 1, then the only value for the angle is 90° ; and if the sine is greater than 1, there is no corresponding angle, a,nd the problem is impossible. It is advisable to construct a fairly accurate figure. 42 TBIGONOMETKT. 30. Problems in oblique triangles, triangulation, etc. In every case at least a rough sketch should be drawn on which the known parts are clearly marked, arid a "blank form" for the computation should be made out for the entire problem, before any of the quantities are looked up in the table. CHAPTER III. General Peoperties of the Teigonometbic Functions. 31. Relations between the functions of a single angle. — The student should convince himself that the following important relations wiU hold for any angle x: sin' X + cos" x=l, tan x = sec* a; = 1 + tan' x. sin a; C08«' All these relations are easily recalled by the aid of the figures. Somewhat less important is the following : csc^a; = 1 + cot^a;. 32. Functions of ( — x). From the figure, sin ( — x) =^ — sin x, cos ( — x) = cos X, tan ( — x) = — tana;. 33. Functions of {90° -\-x), (a; + iSO°),eic.— Any function of a combination like (a;±m90°) or (n90°±a;) can be expressed in terms of a function of x by the use of the figure. For example, find sec (270° — x). Take as x any small angle in the first quadrant, and draw the angle 270° — x. Then, sec (270° — a;) is 1 over the cosine of (270° — x), which, pointing at the figure, is the radius over this line ( VVV ) , which is the same, in length, as the radius over this line ( 5 ) , which is 1 over the sine of x, or esc x. But the signs are opposite ; therefore, §ec (270° — x) = — CSC a;. 43 ZTOiaC 44 TEIGONOMETEY. This method requires the memorizing of no rules or for- mulae, besides the definitions of the functions; a very little practice will develop aU the speed and accuracy that can be desired, and the method is one which is readily recalled to mind after long disuse. The special case of complementaiy angles, however, is worth remembering as a separate formula : Any function of (90° — a;)=the co-named function of x. FOKMULAS FOB THE SuM OF TwO ANGLES, EtC. 34. In simplifying trigonometric expressions which occur in calculus, mechanics, etc., the following formulae are so fre- quently required that they should be thoroughly memorized. The abiUty to recognize those relations readily, regardless of the special lettering employed, is a necessary condition for rapid progress in almost any branch of analysis, but it is highly undesirable to extend the list beyond the limits here given. The fundamental formulae from which aU others are derived are these two, the proof of which is obtained from a figure : (1) sin (a; + 2/) ==sina;cosj^-j-cosa;sinj/, 7 (2) cos (a; -f 2/) = cos a; cos 2/ — sin a; sin i/. These and the following formulae should be memorized in words, not in letters : thus, "the sine of the sum of two angles is the sine of the first times the cosine of the second, plus the cosine of the first times the sine of the second," etc. Dividing (1) by (2) and then dividing numerator and de- nomerator by the product of the cosines, we have ,„. J. ^ , \ tan a; + tan y (3) tan (x + y) = — ■ — — -^. ^ ^ V T^:/y 1— tana; tan y Changing the sign of y in these three formulae, and remem- bering the relations for negative angles, we have the corre- sponding formulte for sin (a; — y), cos (a; — y), tan (a; — y), which will be exactly the same as (1), (2), and (3) with aE the connecting signs reversed: TRIGONOMETEY. 45 (4) sin (x — 2/) = sin a; cos y — cos a; sin y, (5) cos {x — y)^ cos a; cos 3/ + sin a; sin y, /cN i. /■ ^ tana; — tan « (6) tan (« — «) = -— -—^. ^ -^ ^ ''' I + tana! tany Putting x=y in (1), (2), and (3) we have at once (7) sin 2a; = 2 sin a; cos x, (8) cos 2a; ^ cos* a; — sin* a; = 1 — 2 sin' x = 2 COS* x — 1, 2 tan X (9) tan 2x = 1 — tan' X Solving (8) first for sin a; and then for cos a;, and putting 2x = y, or x = y/2, we find (10) «i"|-^/-^^'.* (11) whence, -|=±/-±F^. (12) y _i- /l — cos y tan ^.= ± A /q— ; . 2 \l + cos« This last formula may be transformed, by rationalizing numerator or denominator, into tang-=^~/°^y= "^°y . 2 sin y 1 + cos y Other formulas, useful for special purposes, should not be memorized, but should be derived as needed. 35. In proving the identity of two trigonometric expres- sions, it is best to reduce each expression separately to its simplest form. * The phis sign is to be used when sin ly is positive, the mvtvas sign when sin ^j/ is negative. Similarly in the next two f onnulas. 4 46 TEIGONOMETBY. The fallacy of supposing that because a true relation can be deduced from a given equation, the given equation is there- fore necessarily true, should be carefully explained. For example, from the false equation 3 = — 3 we can obtain the true equation 9 = 9 by squaring both sides; or, from the false equation 30° = 150° we can obtain the true equation i^ = % by taking the sine of both sides; but in each of these cases the step taken is not reversible. 36. The following device for transforming an expression of the form a cos a; + 6 sin a; is often useful : = A cos (a; — J5), where A = i/Ca' + 6') and tan B = -. ^ a 37. The inverse functions. The angle between — 90° and +90° whose sine is x is de- noted by sin-i x* The angle between 0° and 180° whose cosine is x is denoted by cos-* X. The angle between — 90° and +90° whose tangent is x is denoted by tan-* x. In simplifying expressions involving these "inverse func- tions," it is weU to take a single letter to stand for each in- verse function; as, i/ = sin-* x, whence, by definition, sin y^=x; etc. 38. Solution of trigonometric equations. Many trigonomet- ric equations can be solved only by the "method of trial and error." In other eases, however, it is possible, by the use of the formulas given above, to transform the given equation into a form involving only a single function of a single angle; if this equation can be solved for the function in question, then the required value (or values) of the angle can be found from the tables or it can be shown that no solution exists. *The symbol sin-* x (or are sin x) is often defined as simply "the angle whose sine is a;"; but since there are many such angles, it is neces- sary to specify which one is to be taken as ' ' the ' ' angle, if the symbol is to have any definite meaning. A SYLLABUS OF ANALYTIC GEOMETRY. Thia syllabus is intended to include those facts and methods of ana- lytic geometry which a student who has completed an elementary course in that subject should have so firmly fixed in his memory that he will never think of looking them up in a book. A course of study in analytic geometry should consist chiefly of problems solved by the students, each problem being solved on the basis of a small number of fundamental formulas such as are here mentioned. This syllabus is confined mainly to the conic sections; but a satis- factory course in analytic geometry should include also the study of many other curves, both in rectangular and in polar coordinates. The syllabus takes up only those properties of curves which can be readily investigated without the aid of the calculus; but the present tendency to introduce the elements of the calculus before any elaborate study of geometry is attempted is to be much encouraged. Table op Contents. Chapter I. Eectangulae Co-okdinates. Chapter II. The Straight Line: Equations or the Form Ax + By + C = 0. Chapter III. The Circle: Equations op the Form XT' + y' -^ Dx + Ey + F = 0. Chapter rv. The Parabola: y' = 2px. Chapter V. The Ellipse: 6V + oy^o'6^ Chapter VI. The Hyperbola: 6W — a''y' = a'l)''. Chapter VII. Transformation op Co-ordinates. Chapter VIII. General Equation op the Second Degree in X and y. Chapter IX. Systems op Conics. Chapter X. Polar Co-ordinates. Chapter XI. Co-ordinates in Space. 47 CHAPTER I. BECTANGULAK COORDINATES. 1. In many geometrical problems it is convenient to describe the position of a point in a plane by giving its distances from two fixed (perpendicular) lines in the plane.* For example, on a map, the distance of a point to the east or west from a fixed meridian is called the longitude of the point, and its distance north or south from the equator is called its latitude. So in general, in any plane, the distance of a point to the right or left from a fixed vertical axis is called the abscissa, x, of the point, and its distance up or down from a fixed horizontal axis is called its ordinate, y. The x and y together are called the coordinates of the point. „ The value of x{=OM) will be positive to the p right, negative to the left; the value of y (=MP) j will be positive upward, negative downward. The ^ . ' - point for which x = Xi and y = yi is denoted by Pi, 0T(xi,yi). M 2. To express the distance between two points in terms of their coordinates : y D= \l{x^ — xif + {y2 — yxY. 3. To fimd the coordinates of the -point half way between two given points: „ X = i(Xi + Xj), I —^^^^'''v^ y = i(2/i + y^)- * We restrict ourselves here to rectangular axes; oblique axes are, however, occasionally useful. 48 AITAIiYTIO GEOMETET. 49 4. To fivd the coordinates of a point P on the line through two fixed points, and such that its distance from the first point is n times the distance between the two points y (PJ' = nP^,): X =: Xi-\- n(X2 — Xi), y=yi-\- n{yi — y^). Here n may be any real number (positive, negative, or zero). 5. To find the slope of a line through two given points: OT = tan <^ = y^~y\ X2 '~~ x^ The angle is the slope. 6. If two lines are parallel, their slopes are equal : mi = m2. If two Unes are perpendicular, the product of their slopes is minus one: mirrii = — 1 . 7. To express the areas of triangles and polygons in terms of the coordinates of the vertices, consider the trapezoids formed by the ordinates drawn to the vertices. 8. In any problem involving an unknown point, remember that two conditions are necessary to determine the coordinates of the point (simultaneous equations in two unknown quantities). CHAPTER II. THE STRAIGHT LINE: EQUATIONS OF THE FORM Ax -{- By -\- C =Z . 9. We have seen that if two conditions are imposed on x and y, the position of the point (x, y) is wholly determined. If only one condition is imposed, the position of the point is only partially restricted. (Examples: a; = 5, a;^ + 2/^ = 25, etc.) The collection of all points which satisfy a given condition im- posed on X and y is called the locus of that condition; and the con- dition itself, expressed in algebraic language, is called the equation of the locus. Thus, the equation of a straight line is the algebraic expression of the condition which x and y must satisfy in order that the point (a;, y) shall lie on the line ; in other words, the equation of a line is an equaiion which is true when the coordinates of any point on the line are substituted for x and y, and false when the coordinates . of any point off the line are substituted for x and y; and so in general for the equation of any locus. 10. To find the coordinates of the points of intersection of two loci whose equations are given, we have simply to find the pairs of values of x and y (if any) which satisfy both the equations at once (simultaneous equations in x and y). 11. To find the equation of a line (not perpendicular to either axis), when its slope, m, and the coordinates of one of its points {Xi, yO, are given: y — yi= mix — Xi). The equation of a line perpendicular to the a;-axis (or the 2/-axis) is, by inspection, a; = a (or 2/ = b). The equation of any straight line is of the form Ax + By + C = 0, and the locus of every equation of the form Ax + By+C = is a straight line. Hence, to plot the locus of such an equation, it is sufficient to find any two of its points. 12. To find the slope of a line whose equation is given (the line being not perpendicular to an axis), write the equation in the form y = i ) a; + ( ) ; then the coefficient of x will be the slope. 50 ANAIiYTIO GEOMBTE.Y. 51 13. To find the equation of a line parallel or perpendicular to a given line and through a given point, remember that vii := m^ for parallel lines, and minis = — 1 for perpendicular lines (see § 6). Special method: if the given line is Ax + By + C = 0, then the parallel is Ax + By = k and the perpendicular is Bx — Ay = K, where k and K are to be determined. 14. To find the angle 6 between two lines whose slopes are given: , -, »W2 — mi ^ tan = ■—-. .* 1 -\- miVii 15- To find the distance between a given point (xq, y^, and a given line: (a) When the inclination of the line, (^, and the coordinates of one of its points, {xi, yi), are given, we have from the figure : QP„=(a;o —xi) sm4>—(yo—yi) cos (f>, (5) When the equation of the line is given in the form Ax -{- By -\- C = 0, use the following formula rf Ax„ -\.Byo + C \/A^ + B^ Here the vertical bars mean " the absolute value of." D = * Proof: By trigonometry, tan (02 — 0i) = tan02 — tan0i_ 1+ tan i t Proof: Show that the foot of the perpendicular from Po to the line Ax + By + C = has the coordinates x^ = xo — hA, 1/2 = 2/0 — hB, where h = (Aw + Byo + C) / (A» + B"). r CHAPTER in. THE CIRCLE : EQUATIONS OF THE FORM I* + J/* -\-Dx -\- Ey •{• F ^ 0. 16. The equation of a circle is the algebraic expression of the condition which x and y must satisfy in order that the point (,x,y) shall lie on the circle (see § 9 and § 10). 17. To find the equation of a circle when its radius, r, and the coordinates (a, /?) of its centre are given: (a;-a)^+(2/-/?)» = r^. When the centre is at the origin (0, 0), this equation becomes X* + J/* = rl 18. The equation of any circle is of the fonn x'-\-y'+Dx-i-Ey+F=0. Conversely, every equation of the form x' + y'-i-I}x + Ey-\r F =0 can be re- duced to the form (x + -^y + (y + -g )' = ^C-D* + E' — 4F), and therefore rep. resents a circle with centre at ( — D/2, — E/2), or a single point, or no locus, ac- cording as D" -|-^ — if is positive, zero, or negative. When we say, in brief, that the locus of any equation of the form :^ + y^ -\- Dx + Ey + ii* = is a "circle," we must understand that the "circle" may be "real," "null," or "imaginary." 19. To find the centre and radius of a circle whose eqvMion is given, do not use a formula, but "complete the squares" of the terms in X and y in each case, and compare with the standard equation in the manner just indicated. 20. In problems concerning tangents to a circle, use the fact that the tangent is perpendicular to the radius drawn to the point of con- tact. 52 CHAPTER IV. THE parabola: ^ = 2px. 21. Definition : The locus of a point which moves so that its distance from a fixed point its distance from a fixed line is ctiUed a parabola. The fixed point is caUed the focus and the fixed line the directrix^ The line perpendicular to the directrix through the focus is called the principal axis. There is evidently only one point of the principal axis which is also a point of the curve, namely the point half way between the focus and the directrix; this point is called the vertex, 22. If we take the vertex as the ori- gin and the principal axis as the axis of x, the equation of the parabola is y" = 2px, where p = the distance between focus and directrix.* 23. The form of the curve is therefore that shown in the- figure, t By definition PF = PH for every point P on the curve. The b re adth of th e curve at the focus is called the laiV;S rectum^, and is equal to 2p. * Proof: If (x, y) is any point on the curve, then \l{x — ipy + (.y — Oy = x + ^Tp. Many British authors write the equation in the form j/' = 4os, to avoid fractions. Other writers use j/' =4pa; for the same purpose; this latter form, however, is unfortunate, since 2p is a fairly well-established notation for the latua rectum in each of the conies. t Thus, when xisO,y is 0. When x increases, y increases, plus and minus; the curve is symmetrical with respect to the x-axis. When x is negative, y ia imaginary. When x = p/2, y = ± p; when x = 2p,y=s± 2p. 53 54 ANAIiTIIC GEOMETEY. 24. To find the eqvation of a tangent to the parabola y^ = 2px, use one of the following formulas : (a) When the point of contact, (x^, y^), is given:* 2/i2/ =p(x + Xt); (&) When the slope, m, is given :t ~^ y y = mx + 2m' A line perpendicular to a tangent at the point of contact is called a normal. If the tangent and normal at any point P meet the principal axis at T and N, the projections of PT and PN on the principal axis are called the suitangent and sui- normal, respectively. The suhtan- gent is bisected by the vertex. The subnormal is constant, equal to the semi-latus rectum, p. 25. The locus of the middle points of a set of parallel chords in the parabola is a straight line parallel to the principal axis; such a line is called a diameter. In the parabola y^ = 2px, if the slope of the parallel chords is m, then the equation of the diameter is y ^ p/m.t * Proof: Let Pa = (a;i + h.,yi-\- k) be a second point on the curve, near Pi; then the slope of the tangent at Pi will be the limit of k/h as Ps approaches Pi along the curve, namely to = p/3/1. Then use § 11. The slope of the curve may also he found hy the general method of the differential calculus. t Proof : Determine p so that the line y = mx + p shall have only one point in common with the curve. [Remember that a quadratic equation A(a' + Bx + C = will have equal roots if B' — 44C = 0.] J Proof: Let (x^, y^) be any point of the required locus; find the points of intersection of the curve and a line through (a;,,, y^ with slope m; then express the condition that (s^, y„) shall be the middle point between these two points. [Remember that the sum of the roots of a quadratic equation As' 4- Ba; -f- C = is — B/A.] VTTy X ANALYTIC GEOMETET. 55 25a. Among the many properties of the parabola which should be worked out as problems, the following may be mentioned as especially important, and easy to remember: 1. The normal at any point P bisects the angle formed by the line from P to the focus and the line through P parallel to the principal axis (paraboUe mirror). 2. If Pi, Pj, . . . are any points on a parabola, the distances of these points from the principal axis are proportional to the squares of their dis- tances from the tangent at the vertex. 3. If the tangents at P and Q inter- sect at T, and if if is the middle point of the chord PQ, then the line through T and M is a. diameter, and the segment TM is bisected by its point of intersec- tion with the curve. 4. The locus of the foot of the perpendicular from the focus on a moving tangent is the tangent at the vertex. 56 ANALYTIC GEOMETET. 5. The locus of the point of intersection of perpen- dicular tangents is the directrix. Note. The usual methods for constructing a parabola should also be given. CHAPTER V. THE ELLIPSE : 6V + aV = ffl*6'. 26. Definition : The locus of a point which moves so that its distance from a fixed point its distance from a fixed line (where e is a constant less than 1), is called an ellipse. The fixed point is called the focus, the fixed hne the directrix, and the constant, e, the eccentricity. The line perpendicular to the directrix through the focus is called the principal axis. There are evidently two points of the principal axis which are also points of the curve; these two points are called the vertices, and the point half way between them is called the centre. 27. If we let 2a = the distance ^\ Y^ ^ Y between the vertices, then :* I |( ^ J^ ^ .J the distance between the centre and either vertex is a; the distance between the centre and the focus is ae; the distance between the centre and the directrix is a/e. 28. If we take the centre as the origin and the principal axis as the axis of x, the equation of the ellipse is ^ + 1-1 where b is an abbreviation for a \/l — e^.f Note that b < a. * Proof: Since the vertices, V and V, are points of the curve, VF/VD = » and V'F/V'D = e; that is, a—CP , a + CF whence CF = ae and CD = a/e. t Proof: If (x, y) is any point on the curve, then X + - e 67 58 ANALYTIC GEOMETBY. 29. The form of the curve is therefore that shown in the figure.* The right triangle enables us to find any one of the three quantities a, b, and e, when the other two are given. The symmetry of the equation shows that the curve might equally well have been obtained, with the same eccentricity, e, from a second focus and directrix, shown on the right. The breadth of the curve at either focus is called the latus rectum, and is equal to 2a(l — e^), or 2b'/a. ^ J. Z _ 1 30. Let P be any point of the ellipse, F and F' the foci, and PH and PH' the perpendiculars from P to the directrices ; then (a) PF/PH = e and PF'/PH' = e, by definition of the curve. Further- more :t &• (6) PF + PF' = 2a. In fact, the ellipse is often defined as the locus of a point which moves so that the sum of its distances from two fixed points is constant. 31. If a circle be described upon the major axis of an elHpse as diameter, each ordinate in the ellipse is to the corresponding ordinate in the circle as h is to a. J In fact, the ellipse is often defined as the locus of points dividing the ordinates of a circle in a constant ratio. From this property it follows that the area of an ellipse is irdb. * Thus, when y =0,x =± a; -when x ■.= 0,y = ±b. The curve is sym- metrical with respect to both axes. In first quadrant, as x increases, y decreases (slowly when x is small, and rapidly when x approaches o). t Proof: PF = e{PH) and PF' = e(PH'), so that PF + PF' = e(HH>) =s e(2a/e) = 2a. JProof : In the ellipse, y=- ^a' — x'; in the circle, y = \/o' — a?. ANALYTIC GEOMETBT. 59 a2. To find the equation of a tangent to the ellipse -^ + p = use one of the following formulas :* (a) "When the point of contact, (aii, j/^), is given : £lf _i_ M — 1. „2 i- J2 - i, (J) When the slope, m, is given: y = mx ±: s/a^ni' + 6^ 33. The locus of the middle points of a set of parallel chords in the ellipse is a straight line through the centre; such a line is called a diameter. In the ellipse — + ^r; = 1. a^ 0^ if the slope of the parallel chords is m, then the slope of the diameter is r— .* Any two lines through the centre, such that the product of their slopes is — b^/a'^, are called a pair of conjugate diameters, because each bisects all chords parallel to the other. ; 34. The circle described in § 31 is called the auxiliary circle. If P is any point on the elhpse, and Q the corresponding point on the auxiliary circle (see figure), then the angle 1^ which CQ makes with the axis is called the eccentric angle of the point P. From the figure, and § 31, X = a cos(f) and y = b sin^, where x, y are the coordinates of P. The eccentric angles of the ends of two conjugate diameters differ by 90°. * Proof as in the case of the parabola. 60 ANAIiTTIO GEOMETET. 4a. Among the many properties of the ellipse that should be worked •OBt as problems, the following are especially easy to remember: 1. The normal at any point P bisects the angle formed by the lines joining P with the focL 2. The locus of the foot of the perpendicular from the focus on a moving tangent is the circle on the major -axis as diameter. 3. The locus of the point of intersection of per- pendicular tangents is a circle with radius V«' + 6'. 4, The area of a parallelogram bounded by tan- gents parallel to conjugate diameters is constant. Note. The usual methods for constructing an ellipse should also be given. CHAPTER VI. THE HYPERBOLA : 6V oV = a^b\ 35. Definition : The locus of a point which moves so that its distance from a fixed point its distance from a fixed line {where e is a constant greater than 1), is called a hyperbola. The fixed point is called the focus, the fixed line the directrix, and the constant, e, the eccentricity. The fine perpendicular to the directrix through the focus is called the principal axis. There are evidently two points of the principal axis which are also points of the curve; these two points are called the vertices, and the point half way between them is called the centre. V use one of the following a"" formulas :* (a) When the point of contact, (x^, j/i), is given: a^ h"" ~ ' (6) When the slope, m, is given: y ^ mx ± ^a^m^ — 6^. 43. The locus of the middle points of a set of parallel chords in the hyperbola is a straight hne through the centre; such a line is called a diameter. In the hyperbola a;2 ■w2 "2 — Si ^^ ^>^^ *^® slope of the parallel chords is m, the slope of the diameter will be -^•* Any two lines through the centre, such that the product of their slopes is 62/^2^ are called a pair of conjugate diameters, because each bisects all chords parallel to the other. • Proof as in the case of the parabola. 64 ANALYTIC GEOMETRY. 43a. Among the properties of the hyperbola, the following are easy to remember : 1. If a line cuts the hyperbola and its asymptotes, the parts of the line intercepted between the curve and the asymptotes are equal. In particular, the portion of any tangent intercepted between the asymptotes is bisected by the point of tangency. 2. The area of the triangle bounded by any tan- gent and the asymptotes is constant. Note. The usual methods of constructing a hyperbola — especially the rectangular hyperbola — should be given. CHAPTER VII. TEANSFOEMATION OF COOEDESTATES.* 44. The equation of a curve can often be simplified by a " change of axes," either changing to a new origin (a;,, 1/0), or turning the axes through an angle 6, or both. If {x, y), {x', y'), {x" , y"), are the coordinates of the same point P, referred to three different sets of axes, as in the figures, then Y: ,y// ' ? • •.1" „ % % % \ y 0, \'8 X '" X X='Xo-\-x' 2/ = J/o + 2/' X = x" cos 6 — y" sin 9 1 y = x" sin -\- y" cos d Suppose now that the point P is allowed to move under cer- tain conditions given by an equation in x and y. The same condition can be expressed in terms of x' and y' or in terms of x" and y" by substituting in the given equation the values of X and y just found. This process is called a transforma- tion of coordinates, from the axes x, y to the axes x', y", or to the axes x", y" ; and the new equation can often be made sim- pler than the given equation by a suitable choice of Xo ^^^ 2/0, or 0. * See also the chapter on polar eo-ordinates. t These last formulas are most easily remembered as follows: X = easterly displacement of P, ^ (easterly component of x") -f- (easterly component of y") :=x" cos fl — j^' sin e', 2/ = northerly displacement of P = (northerly component of x") + (northerly component of y") = x" sin e + y" cos 6. 65 CHAPTER VIII. GENERiJj EQUATION OF THE SECOND DEGREE IN X AND y. 45. The general equation of the second degree in x and y is of the form Ax'' + Bxy -{■Cy''-\-Dx + Ey + F = 0. By a suitable transformation of coordinates this equation can always be brought into one or other of the following forms : A'x'' + CY + F' = 0, C'Y + D"x = 0, C'Y + F" = 0, and hence can be shown to represent a conic section, using this term in a general sense to include (1) an ellipse, which may be real, null, or imaginary; (2) a hyperbola, or a pair of intersecting lines; (3) a parabola, or a pair of parallel lines (distinct, coincident, or imaginary).* The student should be able to plot readily the locus of an equation of the second degree in any of the simple cases men- tioned below — ^these being the cases which occur most often in practice. 46. To plot Ax' + Cy' + F = 0, where A and C have the same sign. Find the intercepts on the axes (by putting x = and i/ = 0) ; if both are real, we have an ellipse in which a = the larger of the two intercepts, and h = the smaller ; or ii A = C, the ellipse becomes a circle. If both intercepts are zero, or imaginary, the locus is a single point, or imaginary. * Proof : It B" — 4AC is not zero, transform to parallel axes with origin at (x„, y„), and choose x^ and yo so that the terms of the first degree in the new equation shall vanish; then turn the axes through an angle 6, and choose 6 so that the term in xy shall vanish. If B' — 4AC = 0, turn the axes through an angle d, and choose 8 so that the term in xy shall vanish; then transform to a new origin (x„, y^), and choose «» and 3/o so that the constant term and one of the terms of the first degree, or so that both the terms of the first degree shall vanish. For special methods of abbreviating the computation in numerical cases see § 55, note. 66 ANALYTIC GEOMBTBY. 67 47. To plot Ax'' + Cy" + F = 0, where A and C have oppo- site signs. Unless ^ = 0, one of the intercepts will be real and the other imaginary, and the curve wiU be a hyperbola whose principal axis is the axis on which the intercepts are real. To find the slopes of the asymptotes, divide by x^ and find the limit of y/x as x increases indefinitely. If ^ = 0, the locus is a pair of intersecting lines. 48. To plot Cy^-{-Dx + F = 0. Write this as This is a parabola with vertex at Xo= — F/D, and running out along the positive or negative axis of x. Plotting one or two points will fix the direction, and comparison with the equation y" = 2px will give the semi-latus rectum, p. 49. ToTplotAx''-\-Cy'' + Dx + Ey + F = 0. Write this in the form and "complete the squares"; then reduce to the form Ax'^ + Cy"' .+ #= by an obvious change of origin. 50. To vlot Cy'' + Dx + Ey + F — 0. Complete the square of the terms in y and reduce to the form Oj/^ -j- Dx + F:= by an obvious change of origin. 51. To plot Bxy + jP = 0. This is a rectangular hyberbola referred to its asymptotes (see §41). The equation Bxy-{-Dx-\-Ey-\-F=0 can be reduced to this form by moving the origin to Xo= — E/B, j/o= — D/B. 52. If the equation to be plotted does not come under any of the forms just considered, a fair idea of the position of the curve may be found by the following very elementary method. Solving the equation for y in terms of x, we have, if C is not zero, Bx + E 1 - where X is an expression involving x alone. Finding the 68 ANALYTIC GEOMETRY. values of — (Bx + E)/2C, and adding and subtracting the values of y/XJIG, for various values of x, we can find as many points (a;, y) on the curve as we please. Or, again, solving for x in terms of y, we have, if A is not zero, where T is an expression involving y alone. From this equa- tion we can find values of x corresponding to as many values of y as we please. This method is very easy to remember, but does not give readily the exact dimensions of the curve. 53. The center of the curve wiU be the point of intersection of the two lines Ba; + 2C2/-fE = 0, except when B^ — 440 = 0, in which case these lines will be parallel, and the curve has no center. 54. The slopes of the lines joining the origin with the infi- nitely distant points of the curve (if any) are given by writ- ing the terms of the second degree equal to zero : Ax" -f Bxy -f 02/" = 0, dividing through by a;' (or j/"), and solving for y/x (or x/y). 55. If a more detailed discussion of the curve is required, it is best to foUow the special methods of reduction given in the text-books (compare foot-note in §45).t 56. The student should be familiar with the geometric proof that aU the " conic sections " can be obtained as plane * The student of the calculus will recognize these equations as dF/Bx=Q and aF/dy=:0, where F(x, y) ^Aa? -)- Bxy -\-Cy' -^-Dx-^-Fy ■\-'E-=^ is the equation of the curve. t The resulting f ormulse are given here for reference, although the problem is not one of conunon occurrence. Required, to plot the equation Ai^ -|- Bxy -|- C'f -f Da; -f- -Ey 4- F = 0. ANALYTIC GBOMETEY. 6& Beetions of a right circular cone. It is a profitable exercise to construct a cone, given the vertex and a hyperbolic section. It should also be made thoroughly clear why an elliptic sec- tion is a symmetrical figure instead of egg-shaped. Case I. Central conic. If B' — 4AC is not zero, tiansfonn to the eenter as a new origin: x,= {ZCD — BE)/(B' — ^C), y,= (2AE — BD)/(,JB' — iAC)', then turn the axes through a positive acute angle B given by tan 2B=:B/{A — C). The transformed equation will be ^V-1-CV + JP" = 0, where J" = Da;„/2 + EyJ2 + F, while A' and C are found by solving the equations A' + C'=A + C, A' — C' = ± \/{A — Cy + B', where the sign before the radical is to be + or — acording as B is positive or negative. The reduced equation can be plotted as in §§46, 47. Case II. Pardbolio type. If B' — 4AC^0, the equation may be written in the form (ax + cy)' + I)x + Ey + F = 0, where a = VA while o^VC or c = — VC according as B is positive or negative. The locus will be of the parabolic type. Take as a new axis of x' the line oa; +' cj/ + m ^ 0, where m^= (,aD + cE)/2(A + C), and choose the positive direction along this line so that it shall make a (positive or negative) acute angle with the axis of x. This line will be the principal axis of the curve. Two subcases may now occur. (a) If a/o is not equal to D/E, take as axis of y' the line ox — ay + n =0, where n=(A + C) (m' — F)/(aE — cD). This line will be the tan- gent at the vertex, and the transformed equation will be y'' = 2px', where 2p= (cD — aE)/V{A + C)'. The locus is a true parabola. (6) If a/e = D/E, the equation referred to the axis of x' will be y'^z=m' — F, which represents a pair of distinct, coincident, or imaginary parallel lines. CHAPTER IX. SYSTEMS OP CONICS. 57. If Z7 and Y are expressions of the second degree in x and y, the equations ?7=0 and y=-0 will represent conies; then (a) the equation Z7 + kY = 0, where h is any constant, win represent another conic passing through all the points of intersection of the first two, and having no other points in common with either of them; and (6) the equation VY = will represent a curve made up of the two conies U=0 and y = taken together. Corresponding theorems hold good if U and Y are any expressions in x and y (not necessarily of the second degree). 58. To find the equation of a conic through five points, let M = and V = be the equations of the lines PiPj and PgPt, and let u' = and v' = be the equations of the lines PiPg and P2P4. Then uv + ku'v' = 0, where k is any constant, wiU be the equation of a conic through these four points. It remains to determine k so that this conic shall pass through Pj. 59. The equation — ^> a' + h ' b^ + k where k is an arbitrary constant, represents a family of con- focal ellipses and hyperbolas, which intersect at right angles. 70 CHAPTER X. POLAR COORDINATES. This chapter, placed here for convenience of reference, may well be introduced, in teaching, much earlier in the course. 60. It is often convenient to represent the position of a point P by giving the angle, i^, which the line through and P makes with the a;-axis, and the distance, r, from to P along this line. The angle <^ is caUed the vectorial angle, or simply the angle, of tixe point P, and is measured from the positive direction of the axis of x to the positive direction of the line through and P. The distance r = OP is called the radius vector of the point P, and is positive or negative according as it runs forward or backward along the line through and P. It is customary to take r positive, and let 4> range from 0° to 360°. 61. From the figure, x = rQ,os4>, y = r sin <^, a;!! -f- j/2 -_ j.2^ yi^ -— tan <^. By the aid of these relations, we can trans- form any equation from rectangular to polar coordinates, and vice versa. 62. The polar equation of a conic, referred to the focus as origin, and the principal axis as axis of x (see figure) is V y ? A.'-^^^ ,-^ffl y <3> X 1 — 6 cos Q ' t where p is the semi-latus rectum, and e the eccentricity. 63. Plotting curves in polar coordinates is an excellent exercise in reviewing the trigonometric functions. The work should be so arranged that no critical value of the function occurs between two successive assigned values of 6. 71 CHAPTEE XI. COOBDINATES IN SpACB. 64. Four methods are in use for representing numerically the position of a point in space. If Ox, Oy, Oz are three mu- tually perpendicular axes, the position of any point P may be determined by : (1) Eeetangular coordinates, x, y, z; (2) Polar coordinates in space, r, a, p, y, where the angles a, p, y are subject to the restriction cos^ a + cos^ yS + cos^ 7 = 1; (3) Spherical coordinates, r, , 0, where ^=the latitude of P, and its longitude : (4) Cylindrical coordiuates p, 9, z. The relations between the various sets of coordinates are as follows : x = r cos a, y = r cos p, z = r cos y. x = rcmcos6 p = r cos (fi, y = r cos sin p" = x" + 2/^j s = r sm «* + 2/^ + 2^ = r'' As there is no well-established uniformity in the use of the letters in spherical coordinates, or in the choice of the positive directions along the axes, it is important, in reading any 72 ANALYTIC GEOMBTET. 73 author, to note, on a figure, the exact meanings of the letters he employs. 65. Distance between two points, in terms of their coordi- nates : P1P2 = V(a;i — x^y + (3/1 — y^y + («i — s^y. X c ^ / ^^^ kr /. / ^, y / / / J 1/ / 66. Angle ^ between two lines whose direction cosines are given : cos ^ = ijZa + TOimj + «iWjj, where \ = cos %, m^ = cos jS^, % = cos y^, etc.* 67. Equation of a plane: te + m2/ + ti3 = p, where p ^ perpendicular distance from the origin, and I, m, « = the direction cosines of the normal to the plane, t Every equation of the form Ax -\- By + Cz -\- D = represents a plane ; for, it can be thrown into the form . lx-\~my-{-nz = p by dividing through by y/A^ + B" + C^. "Proof: let (1) and (2) be lines through the origin, parallel to the given lines; on these lines take points Pi, P, at a distance r from the origin; then PiPj' = r» 4- r* — 2rr cos ^ = (Wj — rk) » + (rWi — rwij)' + (rTH — m,)'. t Proof: The foot of the perpendicular is F= (pi, pm, pn) ; take N' = (2pl, 2pm, 2pn) and express the condition that the point (a, y, e) shall be equidistant from and N'. 74 ANALYTIC GEOMETBT. 68. Equation of sphere with center at the origin : aJ* + y"" +2'' = r^- 69. Equation of ellipsoid, with center at the origin : ajVa" + y V6^ + sVc' = 1. 70. Any equation m. x, y, z will represent a surface (real or imaginary), the form of which can be investigated by the method of plane sections. Thus, putting x = x^, the equation becomes an equation in y and z, which represents a curve in the plane a; = Xi; similarly for y^= y^ and 2 = 2:1. 71. Any equation of the second degree in x, y, z represents a (real or imaginary) surface of the second degree, or coni- coid. The types of real conicoids are as follows : (1) Ellipsoid, with semi-axes a, b, c. Special case: ellip- soid of revolution, generated by rotating an ellipse about its major axis (prolate spheroid) or about its minor axis (oblate spheroid). (2) Eyperboloid of two sheets. Special case: generated by rotating a hyperbola about its principal axis. (3) Eyperboloid of one sheet, or ruled hyperboloid. Spe- cial case: generated by rotating a hyperbola about its conju- gate axis. Two sets of straight lines can be drawn on this surface. (4) Elliptic paraboloid. Special case : generated by rotat- ing a parabola about its principal axis. (5) Hyperbolic paraboloid, or ruled paraboloid. A saddle- shaped figure, on which two sets of straight lines can be drawn. (6) Cone, generated by a straight line always passing through a fixed point called the vertex, and always touching a fixed conic, called the directrix. If the directrix is a circle, the cone is a circular cone (right or oblique). If the vertex recedes to infinity, tHe cone becomes a cylinder. On any cone a single set of straight lines can be drawn. The student should become familiar with at least the shapes of these surfaces, through diagrams or models. Any plane section of any surface of the second degree is a conic. A SYLLABUS OF DIFFERENTIAL AND INTEGRAL CALCULUS. This syllabus is intended to include those facts and methods of the calculus which every student who has completed an elementary course in the subject should have so firmly fixed in his memory that he will never think of looking them up in a book. The topics here mentioned are therefore not by any means the only topics that should be included in a course of study, nor does the arrangement of these topics, as classified in the following table of contents, necessarily indicate the order in which they should be presented to a beginner. Table of Contents. past i. functions op a single variable. Chapter I. Functions and theib Gbaphical Eepbesentation. Function and argument. Tables. Graphs. The elementary mathematical functions. Continuity. To find a mathematical function to represent an empirically given curve. Chapter II. Differentiation. Bate op Change op a Function. A. Definitions and notation. Bate of change of a function, or slope of the curve. Derivatives. Increments and differentials. Higher derivatives. B. To find the derivative when the function is given. Bules for differentiating the elementary functions. Differentiation of implicit functions, and of functions expressed in terms of a param- eter. C. To find the derivative when the fimction itself is not given; set- ting up a differential equation. Useful theorems on infinitesimals. D. Applications of differentiation im, studying the properties of a given function. Slope; concavity; points of inflexion. Maxima and minima. Multiple roots. Small errors. Chapter III. Integration as Anti-Differentiation. Simple Dip- fekential Equations. Definition of an integral. Constant of integration. Formal work in integration. Use of tables of integrals. Method of substitution, and method of integration by parts. Simple differential equations. 75 76 CALCULUS. Ghapteb rv. Integration as the Limit or a Sum. Djctinith Integrals. Definition of the definite integral of f{x)dx from a; = a to ar = 6. Fundamental theorem on the evaluation of a definite integral. Duhamel's theorem. Approximate methods of integration: squared paper; Simpson's rule; the planimeter; expansion in series. Definite integral as a function of its upper limit. Chapter V. Appucations to Algebra: Expansion in Series; In- determinate Forms. Taylor's theorem with remainder. Maclaurin's theorem. Im- portant series. Theorem on indeterminate forms. Chapter VI. Applications to Geometry and Mechanics. Tangent and normal. Subtangent and subnormal. Differential of arc (in rectangular and polar coordinates). Badius of curvature. Velocity and acceleration in a plane curve. PART II. functions OP TWO OR MORE VARIABLES.* * In preparation. CHAPTER I. Functions and Their Graphical Representation. 1. Function and argument. — In many problems in prac- tical life we have to deal with the relation between two variable quantities, one of which depends on the other for its value. For example, the temperature of a fever patient depends on the time; the velocity acquired by a falling body depends on the distance fallen; the weight of an iron ball depends on its diameter, etc. In general, if any quantity y depends on another quantity X, then y is called a function of x, written, for brevity, y = f{x), and the independent variable x is called the argu- ment of the function. More precisely stated, the notation y^f{x) means that to every value of the argument x (within the range considered), there corresponds some definite value of the function, y; the value of y, or f{x), corresponding to any particular value a; = a is denoted by f{a). If several values of y correspond to each value of x, we have what is called a "multiple valued function of x," which is really a collection of several distinct functions. For example, if 2/" = a;, then y = ± Vx, which is a double valued function of x. Any mathematical expression involving a variable a; is a function of x ; but there are many important functional rela- tions which cannot be expressed in any simple mathematical form. 2. A function is said to be tabulated when values of the argument (as many as we please, preferably at regular inter- vals) are set down in one column, and the corresponding values of the function are, set down in another column, op- posite the first. For example, in a table of sines, the angle is the argument, and the sine of the angle is the function. 3. A function may also be exhibited graphically, as follows : Lay off the values of the argument as abscissas along a (hori- zontal) axis. Ox, and at each point of the axis erect an ordi- 6 77 78 CALCULUS. nate, y, whose length shall indicate the value of the function at that point; a curve drawn through the tops of these ordi- nates is called the curve, or the graph, of the function. It should be clearly understood, however, that it is the height of the ordinate up to the curve, rather than the curve itself, that represents the function. In plotting the curve for any function, it is important to indicate on each axis the scale which is used on that axis, and the name of the unit. For example, if we plot distance as a function of the time, the units on the j/-axis may represent feet, and those on the a;-axis, seconds.* The obvious method of obtaining the graph of the sum or difEerence of two functions directly from the graphs of those functions should be noted. 4. The elementary mathematical functions. — In many im- portant cases, the relation between the function and the argu- ment can be expressed by a simple mathematical formula. For example, if s = the distance fallen from rest in the time t, then s = igf- In such cases, the value of the function for any given value of the argument can be found by simple sub- stitution in the formula. The most important elementary mathematical functions are the following: Algeiraic funciions: ex, c/x; x^, x^; ■\/x {x positive). Here V^=tlie positive value of y for which y^ = x. Trigonometric functions: sin x, cos x, tan a; (x in radians) . Exponential function: e" (e = 2.718 . . .). Logarithmic function: loge x (x positive). The student should be thoroughly familiar with the curves of each of these functions, so as to be able to sketch them, or visualize them, at any moment; many of the essential prop- erties of the functions can be obtained by inspection of the curve. • It is not necessary that the lengths representing the units of x and y shall be equal; scales should be so chosen that the completed graph is of convenient size to fit the paper. In applications to geometry, however (see Chapter YI), the scales must be equal. 78a 78b 'C 78c 78d CALCULUS. 79 He should also be familiar with the formulas necessary for handling expressions involving these functions. The better drilled the student is in this formal algebraic work, the more rapid progress can he make in the really vital parts of the sub- ject. (See chapters of this report on algebra and trigo- nometry.) 5. Next in importance are the following: the hyperbolic functions, which are coming more and more into use : sinh x= (e"" — e-'")/2, cosh a;= (e* -}- e-') /2, tanh ic = (e'" — er'')/{e^ -f er") ; the inverse trigonometric functions: sin~^ X = the angle between — 7r/2 and + V2 radians (inclusive) whose sine is x-* cos"^ X = the angle between and ir (inclusive) whose cosine is x; tan-^a;=the angle between — 7r/2 and +ir/2 (inclusive) whose tangent is x ; and the inverse hyperbolic functions: sinh"^ X = the value of y for which sinh y^^x; cosh"^ X = the positive value of y for which cosh y^x; tanh"^ X = the value of y for which tanh y = x. It should be noticed that the curves for the inverse functions can be obtained from the curves for the direct functions by rotating the plane through 180° about the line bisecting the first quadrant. Formulas for the hyperbolic functions resemble those for the trigonometric functions, but the differences are so * The symbol sin"^ x is often defined as simply "the angle whose sine is a; "; but since there are many such angles, it is necessary to specify which one is to be taken aa " the " angle, if the symbol is to have any definite meaning. Thus, if sin x = i, x may equal 7r/6, or Stt/S, etc.; but only one of these values, namely ir/6, is properly denoted by the symbol sin"^ J. Similarly for cos"' x and tan"* x; and also for cosh-* x, which is like Vx in this respect. The conventions adopted to avoid am- biguity may be readily recalled from the figure, if we note that in each case the complete curve consists of two or more "branches," and that that one is taken as the "principal branch" which passes through the origin, or which lies nearest the origin on the positive side of the a-axia 80 CALCULUS. confusing that it is better not to try to memorize any formulas for the hyperbolic functions, but to look them up whenever they are needed. (The list in B. 0. Peirce's Table of Integrals, for example, is entirely adequate.) 6. Continuity. A function y=f{x) is said to be continu- ous at a given point x = a, if a small change in x produces only a small change in y; or more precisely, if /(a;) always ap- proaches /(a) as a limit when x approaches a in any manner. A function may be discontinuous at a given point in three ways: (1) it may become infinite at that point, as y = l/x at a; = 0; or (2) it may make a finite jump, as j/^tan"^ (lA) at a; = 0;* or (3) the limit L/(a;) may fail to exist because of the oscillation of the function in the neighborhood of x = a, as i/ = sin 1/x at a; = 0. In each of these cases, the function is, properly speaking, not defined at the point in question. A good example of a discontinuous function is the velocity of a shadow cast by a moving object on a zig-zag fence. In what follows, we shall confine our attention to functions that are continuous, or that have only isolated points of dis- continuity. 7. To find a mathematical function to represent an em- pirically given curve. — In many cases the form of the func- tion is given only empirically ; that is, the values of the func- tion for certain special values of the argument are given by experiment, and the intermediate values are not accurately known (for example, the temperature of a fever patient, taken every hour). In such cases, the methods of the calculus are not of much assistance, unless some simple mathematical law can be found which represents the function sufficiently accu- * This function approaches v/2 when x approaches from above, and — 7r/2 when x approaches from below. CALCULUS. 81 rately.* This problem of finding a mathematical function whose graph shall pass through a series of empirically given points is a very important one, which is much neglected in the current text-books. The complete discussion of the problem involves, it is true, the theory of least squares, which would undoubtedly be out of place in a first course in the calculus; but an elementary treatment of the problem in simple cases^ would be very desirable.f The curves which are most likely to be worth trying, in any given case, are these : y = a-\-hx (straight line) ; y = a-\-:bx-\- cx^ (parabola) ; yz=a-\- c/{x -f 6) (hyperbola) ; 2/ = a sin (bx-\-c) (sine curve) ; and y = ax"*- In testing this last curve, put y' = log y, a;' = log x, and a' = log a, and see whether y' and x' satisfy the straight line- relation y' = a' -\- mx' ; the use of "logarithmic squared paper " greatly facilitates the process. The student should be familiar with all the possible forms of these curves, for various values of the constants a, b, c, and m. * If no simple law can be found to represent the entire curve, it is sometimes possible to break up the curve into parts, and find a separate law for each part. t Numerous examples may be found in John Perry's "Practical. Mathematics," and in P. M. Saxelby's "Practical Mathematics"' (Longmans, 1905). CHAPTER II. DlFFEEENTIATION. RatE OP CHANGE OF A FtTNCTION. For the sake of clearness, this chapter is divided into four parts, A, B, C, D. A. Detinitions and Notation. 8. Rate of change of function; slope of curve. — Given a function, y = f{x), one of the most important questions we can ask about it is, what is the rate of change of the function at a given instant ? For example, the distance of a railroad train from the starting point is a function of the time elapsed, and we may ask, what is the rate of change of this distance? The answer is, so-and-so many miles per hour. Again, the volume of a metal sphere is a function of the temperature, and we may ask, what is the rate of change of this volume? The answer is, so-and-so many cubic inches per degree. If the graph of the function is a straight line, then clearly the rate of change of the function will be constant; for, at any instant, (change in 2/)/(change in x) =the slope of the line. If the scales along x and y are the same, the slope of the line = tan 0, where is the angle which the line makes with the x axis. If the scales are not the same, the slope of the line may stiU be interpreted as the ratio of the "side opposite" to the "side adjacent" in the triangle of reference for that is — j , and not, as one might expect . (dx)" As an example where the distinction is important, consider x^8 — sin and 3/ = 1 — cos 6, where 6 is the independent variable. 86 CALCULUS. B. To Find the Derivative vfhen the Function is Given. 12. Formal work in differentiation. The student should be thoroughly familiar with the results of differentiating all the elementary functions. A list of the formulas which should be memorized is given below; any other formulas should be worked out as needed, or looked up in a book. To establish these formulas, first prove the following im- portant limits : ,. sin Au , , ,. 1 — cos ^u lim — -. — = 1, and hm = 0, provided u is in radians ; and limfl+ iV = e = 2.718- ••;* and hence prove the formulas for differentiating the sine and the logarithm. The proofs of the other formulas present no difficulty. * These limits having been established, it can then be shown that ,. sin(« + ^") — 8in« ir ... j. j lim ^ — ^ -g- cos u, if u IS meaBnied m degrees, Aub=0 J.OU = 003 u, if u is measared In radians ; lim Iog("+A«)-log^ _ ^^343 . . . ) 1 if the base is 10, = -, if the base is « = 2.718 • • •. u The reason for choosing the radian as the unit angle, and e as the base of the "natural" system of logarithms is the simplification in the formulas for the derivatives of the sine and the logarithm which results from this choice. CALCULUS. 87 Rules fob Differentiating the Elementary Functions OP A Single Variable.* {The first four of these rules are the fundamental ones, pom which all the others can he derived.) The differential of a constant is zero : — dk=0. The differential of the logarithm to the base e of any function is one over that function, times the differential of the function: — d(logea;) = — - dx (e=2.718. . .)■ The differential of the sine of any function (in radians) is tne SDsine of that function, times the differential of the function : d{sm x) = cos X dx. rhe differential of the sum [or difference] of two functions is the differential of the first plus [or minus] the differential of the second: — d{u ■iiv) = du± dv. The differential of a constant times any function is the con- stant times the differential of the function: — d(kx) = k dx.'\ The differential of a function to any constant power is the exponent of the power, times the function to the power one less, times the differential of the function: — d(a;") = nx'^-^dx. Useful special cases of this rule are: — The differential of e with a variable exponent is e with the same exponent, times the differential of the exponent : — d{e=')=e''dx (e =2.718...). • All these rules remain valid when the word ' ' derivative ' ' is put in place of "differential," and the symbol "D" in place of "d." t To prove this and the next five rules, let y = the function, and take the logarithm of both sides before differentiating. •88 CALOtTLUS. The differential of the product of two functions is the first times the differential of the second, plus the second times the differential of the first: — d{uv) =udv + V du. The differential of the quotient of two functions is the denomi- nator times the differential of the numerator, minus the numer- ator times the differential of the denominator, all divided by the denominator squared : — V du—u dv <9- The differential of the cosine of any function is minus the sine of that function, times the differential of the function: — d(cos x)= — sin x dx.* The differential of the tangent of any function is the secant- square of that function, times the differential of the function: — d(tan x) = sec^a; dx.\ The differentials of the inverse sine, the inverse cosine, and ihe inverse tangent, of any function, are given by the following formulas, which the student may put into words for himself: — d(Rirt-^x)= dnr.jX (— iff g sin-'a; g Jjr) l/l — «' d(cos-' x) = ^= dx. (0 g eos-'a; s jt) Vl-x' ' d(tan-' x) = a <^»- ("~ *'^ = *^"'^'^ = *"■) [To find the differential of u to the i^th power, where u and v are any functions, let and take the logarithm to base e of both sides before differ- entiating. — Similarly, to find the differential of the logarithm of u to any base v, let y = log^M, whence ■u^ = u; then differentiate both sides.] * Proof: co3a; = siii (Jtt — x). t Proof: tan a; = sin a;/eos a;. $ Proof: Let y = sm-^a;, that is!i'siny = x; then differentiate both sides. — Similarly for the next two formulas. CALCULUS. 89 The rules on these two pages suffice for the differentiation of any elementary function; they should, he carefully memorized. The differentials of the hyperbolic functions are given by the following formulas, which are also worth remembering: d sinh X = cosh xdx; d cosh x = sinh x dx ; d tanh x = sech^ xdx; hence, d sinh~^ X— , d cosh~' x= , d tanh~' x= z, -,• 13. Differentiation of implicit functions, and of functions expressed in terms of a parameter. (a) Suppose we have an equation connecting x and y, but not giving y explicitly as a function of x\ as, for example, ^x^ + 4^/^ = 36. In finding dy/dx in cases of this kind, in- stead of first solving the equation for y in terms of x, and then differentiating, it is usually better to differentiate the equation through as it stands (remembering that both x and y are variables) ; thus, in the present example we have lSxdx-\-Sydy = 0, whence, dy/dx= — 9x/4i/. This result can then, if desired, be expressed wholly in terms of X, by aid of the original equation. (6) Again, suppose y is given as a function of u and v, where m and v are both functions of x; as, for example, y=u''-\'V sin u. Dififerentiating both sides by the regular iTiles, we have dy = 2udu -\-v cos u dw + sin « dv, whence, collecting the terms in du and dv, and dividing by dx, ^=(2« + ,;co8«)^+(Bm«)^. This result shows how the rate of change of y depends on the rates of change of u and v, which are supposed to be known. (c) Finally, both x and y may be given as functions of a third variable, t; as, x = F(^t), y=f{t). To every value of this auxiliary variable, or "parameter," t, there corresponds a pair of values of x and y, so that here again y is indirectly determined as a function of x. Of course if we can eliminate t 7 90 CALCULUS. we shall have a single equation connecting x and y ; but it is often more convenient to keep the equations in the parameter form. Thus, to find dy/dx, we have merely to differentiate both of the given equations: dx==F'{t)dt, dy=f'{t)dt; and then divide the second result by the first: dy/dx=f'{t)/F'(t). C. To Find the Derivative when the Fitnction Itself is NOT Given ; Setting up a Diffeeential Equation. 14. In many cases it is required to find the rate of change of a function when the function itself is not directly given; in fact it is often easier to find the derivative of a function than it is to find the function itself. For example, a hemispherical bowl of radius r, full of water, is being emptied through a hole in the bottom ; find the rate of change of the volume of water drawn off, regarded as a function of the distance, y, between the level of the water and the center of the bowl. To compute this value directly from the definition, we notice first that the increment AV produced in y by an increment Ay given to y will have a value between TrCr" — y^)Ay and ir[r2 — (y -}-,Ay)'']Ay; dividing either of these values by Ay, and taking the limit of the ratio AV/Ay, we find at once dYJdy^tc (r^ — y^), which gives the re- quired value of dV/dy for any value of y from j/ = to 2/ = r. This process of finding the derivative directly from first principles, as the limit of the ratio of the increments, when the function itself is not given, is called "setting up a dif- ferential equation," since the result of the process is an equation between the differentials of the function and of the argument.* Every problem of this kind is a problem in finding the limit of the ratio of two variable quantities, each of which is approaching zero; and in this connection the following theo- rems on infinitesimals are extremely useful, if not indis- pensable. * The problem of finding the relation between the quantities them- selves when the relation between their differentials is known will be dis- cussed in the next chapter. CALCULUS. 91 15. Theorems on infimtesimals. Def. Any variable quantity that approaches as a limit is called an infinitesimal. For example, Ax, Ay, dx, dy, are infinitesimals. The eTroneons notion that an infinitesimal is a constant quantity which is "smaller than any other quantity, however small, and yet not zero" should be carefully avoided. Notation. The notation lim x^=a, or x-^a (read: "x ap- proaches a as a limit "), means that x = a-\-€, where « is a variable approaching zero. Thus a statement expressed in terms of "lim" or " _» " can always be translated into an equation, which can then be handled by the ordinary rules of algebra. The symbol -^ is preferable to == and seems likely to replace it. Def. If a and p are infinitesimals, and Um (a/p) = 0, then a is said to be an infinitesimal of higher order than p. Tor example, if Am = e . Av, where e itself approaches 0, then Aw is of higher order than Ad. Again, 1 — cos M is of higher order than AO. If the difference between two infimitesimals is of higher order than either, then their ratio approaches 1 as a limit ; and conversely, if the ratio of two infinitesimals approaches 1, then their difference is of higher order than either. Two infinitesimals having this relation may be called " similar " or " equivalent " infinitesimals. Important examples are the following: a convex arc of a curve, and the chord of that arc, are " similar " infinitesimals. Again, sin Ax and tan Ax are both " similar " to Ax, provided Ax is in radians. First Eeplacembnt Theorem for Infinitesimals. In finding the limit of the ratio of two infinitesimals, either of them may he replaced by a "similar" infinitesimal, without affecting the value of the Umit. As explained above, two infinitesimals are " similar " : (1) if the difference between them is of higher order than either; or (2) if the Umit of their ratio is 1. (Sometimes the first test is more convenient, sometimes the second.) 92 CALCULUS. This theorem frequently enables us to replace a complicated infinitesimal, like 7r(r + Ar)''Ax, by a simpler one, as irr^Ax; hut it justifies tMs replacement only in the case expressly stated in the hypothesis of the theorem, namely the case in which we are finding the limit of a ratio.* (The fallacy that "infinitesimals of higher order can always be neglected" should be carefully guarded against.) * A second leplacement theorem for infinitesimals will be given in the chapter on Definite Integrals. OALOULtrS. 93 D. Applications op Differentiation in Studying the Properties op a Given Function, 16. That a knowledge of differentiation is of fundamental importance in studying the variation of a given function is evident from the following theorems. Let the given function be 3/ = /(«). I. The value of the derivative at any point shows the slope of the curve at that point. Hence, if the derivative is positive at any point, the curve is rising at that point (as we move in the positive direction along the axis) ; that is, the function is increasing. And if the derivative is negative at any point, the curve is falling at that point; that is, the function is decreasing. II, If the second derivative is positive at any point, the slope is increasing at that point, and hence the curve is con- cave upward; and if the second derivative is negative at any point, the slope is decreasing at that point, and hence the curve is concave downward. 94 CALCULUS. A point where the concavity changes sign is called a point of inflexion; at every such point, the second derivative is or 00.* 17. Maxima and minima. — The application to problems in maxima and minima is immediate. In seeking the largest or smallest value of a given function in a given interval, we need consider only (1) the points where the slope is zero; (2) the points where the slope is infinite (or otherwise discontinuous) ; and (3) the end-points of the interval; for among these points the desired point will certainly be found. In most practical cases it will be a point where the slope is zero. The conditions of the problem wiH usually show clearly which of these points, if any, is a maximum (or a minimum). 18. Multiple roots. — The roots, or the zeros, of a function, are the values of the argument for which the function becomes y=/(X) zero. An inspection of the figure will show that any value of X for which f(x) and f'{x) are both zero simultaneously, will count as at least a double root. * But the second derivative may be zero at points which are not points of inflexion; for example, y=.3^ at a; = 0. CALOUIiTJS. 95 19. Small errors. — The following theorem is very useful in discussing the effect, on a computed value, of small errors in the data : III. If dx is small, dy and Ay are nearly equal. That is, the difference between dy and Ay can be made as small as we please, in comparison with dx, by making dx sufficiently small (except at points where dy/dx does not have a finite value). Thus, if we wish to find approximately the error Ay pro- duced by a small error in x, it will usually be sufficiently accurate to compute, instead of Ay, the simpler value, dy. In problems concerning the relative error, dy/y, or dx/x, it is often convenient to take the logarithm of both sides of the given equation y = f{x) before differentiating. This class of problems is of great practical value. CHAPTER III. INTEGRATION AS THE INVERSE OF DIFFERENTIATION. SIMPLE DIFFERENTIAL EQUATIONS. 20. In many problems in pure and applied mathematics, we have given the derivative [or differential] of a function, and are required to find the function itself. Suppose f{x) [or f{x)dx] is the given derivative [or differ- ential] ; it is required to find a function F{x) which, when dif- ferentiated, will give /(a;) [or /(a;) da;]. Clearly, if one such function F(x) has been found, then any function of the form F(x) + C, where C is any constant, will have the same property. Definition. — Any function F(x) whose differential is f(x)dx is denoted by Jf(x)dx, read: an integral of f{x)dx. The process of finding an inte- gral of a function is called integration, or the inverse of differentiation. If F{x) is any particular integral of f{x)dx, then every integral of f{x)dx can be expressed in the form F{x) -j-O, where (7 is a constant, called the constant of integration. It can be shown that every continuous function has an inte- gral; but this integral may not (in general, will not) be ex- pressible in terms of the elementary functions.* Most of the functions that occur in practice can, however, be integrated in terms of elementary functions, by the aid of a table of integrals, such as B. 0. Peirce's weU-known table of integrals. The entries in such a table can be verified by differentiation. 21. Formal work in integration. — The time devoted to the formal work of integration should not be longer than is nec- * In such cases, an approximate expression for the integral maj be obtained by infinite series. 96 CALCULUS. 97 essary to give the student a reasonable degree of ezpertness in the use of the tables. The following integration formulas should be memorized; they are derived immediately from the corresponding formulas for differentiation. j cudx = c j udx ; ( (^u + v + • • •)dx= I udx + | vdx + • • • ; / afdx = — — — (provided ji + — 1) ; (in words: an integral of any function raised to a constant power, =f: — 1, times the differential of that function, is equal to the function raised to a power one greater, divided by the new exponent) ; /^=log,a;; j'e'dx=d-; j 8ina;(fo;=— cosa;; j cosa;da;=8ina;;' j sec' x da; = tana;; d^ . , , r dx ■ = tan ' X. The constant of integration must be supplied in each case. A large number of integrals can be brought under the form /x"da; by a simple transformation. For example, /cos* xdx = /cos* xeo8xdx=f(l — sin'' x) cos x dx =/cos xdx — /sin* X cos a; da; = /cos a; da; — /(sin x)*d(sin x) = sin X — (sin x) V3. Similarly for any odd power of the sine or cosine. The following integrals are also important, though it is not worth while to memorize them when a table is at hand : I sin' xdx = ^(a;— sin a; cos x) ; j cos' xdx = ^(a; + sin a; cos x) ; /dx , . (v x\ , , 1 + sin a; r dx x =log, tan I T + H =i^ log. q ; — ; i -. — =log. tan -; cos a; ^" \4: 2/ ^ ^'1— sina;' J sma; ^' 2' j sinha;da;= cosha;; j cosha;da; = sinha!; j sech'a;da;=tanha;. 98 CALOTJLUS. 22. Among the other formulas of integration, the following are perhaps the ones that occur most often in practice; they are inserted here for reference, and especially to illustrate the usefulness of the hyperbolic functions. / / / / dx 1 ^ _, a; , , , = - tan -, dx 1, a + X 1 ^ . _.x a' — 9? 2a 'a — X a a dx 1 , X — a 1 ^, _,x T = St log, — r- = - - coth a^ — a' 2a 'x + a a a' dx . _, « _,x or = — cos ' - / — = log. (x + ■•in? + o') , or = sinh~' - , / , = log, (x + -^3? — a'), or = cosh"'-, j Va' — «" rfa: = ^ a; Va' — »* + a' sin""' - , r •>/«'+ a»da; = ^[a;Va^ + a'+a'log.(a;+ V?+a')], or = ^ a; Va^ + a'+o' sinh~' - , j V^^— ^rfa; = ^ [a; V?— «'— a' log, (a; + Va;'— a')] , or = jr a; -yla^—a'— a' cosh"' - . 2L aj 23. Methods of Integration. Among the methods by which a given integral may be reduced to a form in the tables (or an integral in the table to one of the fundamental forms) , the most important are (1) the method of substitution and (2) the method of integration by parts. In the method of substitution, the given integral, [f{x)dx, is expressed wholly in terms of some new variable y (and dy), in the hope that the new integral may be easier to handle than the old one. The substitutions which are most likely to be useful are the following : CALCULUS. 99 (a) y = any part of the given expression whose differential occurs as a factor; 2/ = a;*; y = l/x; y = smx; y = cosx; j/ = tan(a;/2). (6) x = a sm y, or = a tan y, or = a sec y, in expressions involving y/a^ — a;^, or Va^ + a;*, or Va;" — a^ respectively. But much can be done without formal substitution of a new letter, if one remembers that the "x" in the formulas of integration may stand for any function. The method of integration by parts is an application of the formula iudv = uv — ji vdu. Take as dv a part of the given expression which can be readily integrated ; on applying the formula, the new integral may be simpler than the old one. The student should be practiced in both of these methods. 24. Simple differential equations. In a large number of problems in pure and applied mathematics, it is possible to write down an expression involving the rate of change of a desired function more readily than to write down the expres- sion for the function itself. (Compare Chap. II, B.) In other words, it is often easier to write down a relation between the differentials of two variables than to write down the rela- tion between the variables themselves. Such a relation con- necting the differentials of two or more quantities, is called a differential equation, and any function which satisfies the equation, when substituted therein, is called a solution of the equation. Every such problem, then, breaks up into two parts: (1) setting up the differential equation; (2) solving that equation. The first part of the problem has already been treated in Chap. II, B. This part of the problem is too apt to be neg- lected in elementary courses; there is scarcely anything that develops real appreciation of the power of the calculus more effectively than practice in setting up for one's self the differ- ential equations for various physical phenomena. 100 CALCULUS. As to the second part of the problem, namely, the solu- tion of the differential equation, the general plan is to reduce the given equation, by more or less ingenious devices, to the form dy=f{x)dx, or y=ff{x)dx, and then to complete the solution, if possible, by the aid of a table of integrals. In a technical sense, the differential equation is said to be "solved" when it is thus reduced to a simple "quadrature"; that is, to a single integration. The solution of a differential equation of the mth order, that is, an equation involving the nth derivative, wiU contain n arbitrary constants ; to determine these constants, n conditions connecting x, y, y' . . ., y'-"^ must be known (the " initial " or " auxiliary " conditions of the problem). 25. The general discussion of differential equations is too large and too difficult a topic to find a place in a first course in the calculus, but two, at least, of the simpler equations are so important that their solution should be given, as an exercise in integration. These equations are the following: (1) f + .'y = 0,where,' = |. The solution is ^ = CiSin (wf + C2) or, j/ = Cg sin m* + O^ cos nf , where the C's are arbitrary constants. (2) f-n'3,= 0,where,' = |. The solution is y = C^ sinh {nt + CJ, or, 1/ = C»e»» + C,e-"*, where the C's are arbitrary constants. The method of obtaining these results, rather than the re- sults themselves, should be remembered: namely, multiply through by dy, noting that dy/dt = y", and integrate each term, getting \y' ^ + \n^y^ = 0, let m = y/a' — b''; then y = Cjr^* sin [mt + Oj) , or 2/= [Cg sin {mt) -\- C^ cos (m*) ]e-**. Case 2, If o" — 6=' = 0, y = e-^*{C, + CJ). Case 3. If a' — 'b^< 0, let n = y/b^ — a'; then 3/ = de-" sinh (nf + Cj) , or y = (73e-»*»>* + C^e-»-«». 26a. Another important case is the linear differential equa- tion of the first order : where P and Q are functions of a; (or constants), but do not contain y. The solution is given here for reference : ye' = J Qe'dx + const., where F=JPdx. CHAPTER IV. INTEGEATION AS THE LIMIT OF A SUM, DEPINITE INTEGEALS. 27. The limit of a sum. Many problems in pure and ap- plied mathematics can be brought under the following general form: Given, a continuous function, y=f{x), from x^a to x=b. Divide the interval from x = a to x^b into n equal parts, of length Ax=(6 — a)/n* Let x-^jX^tX^, . . . x„ ie values of x, one in each interval; take the value of the func- tion at each of these points, and multiply by Aa;; then form the sum: f{x^)Ax-{-f{x^)Ax -\ [-f{x„)Ax. Required, the limit of this sum, as n increases indefinitely, and Ax i 0. This problem may be interpreted geometrically as the prob- lem of finding the area under the curve y=f{x), between the ordinates x = a and x = b; each term of the sum represents the area of a rectangle whose base is Ax and whose altitude is the height of the curve at one of the points selected. It is easily seen that the difference between the sum of the rec- tangles and the area of the curve is less than a rectangle • It ia not necessary that the parts be equal, provided the largest of them approaches zero when n is made to increase indefinitely. 102 CALCULUS. 103 whose base is Aa; and whose altitude is constant. This dif- ference approaches zero as Aa; = 0; therefore the sum of the rectangles approaches the area of the curve as a limit. In this way, or by an analytic proof, it is shown that the limit of the sum in question always exists. The problem then is, to find the value of this limit. The value of the limit can always be obtained by the fol- lowing fundamental theorem, whenever an integral of the given function f{x) can be found. Fundamental Thbobem of Summation. If Xj^, x^, ■•• x„ are values of x ranging from x = a to x = 'b, as in the state- ment of the general problem above, then lim if(x,)^x +f(x,)Ax + . . -[-{-/(a^JA*] = FQ>Y- F(a), ; where n^o) = fK'^)dx is any function whose derivative is the given function fix). The proof of this remarkable theorem is best given by show- ing that the right hand side of the equation, as weU as the left, is equal to the area under the curve from x = ato x = b; to do this, consider the area from x = a to a variable point x:=x, and find the rate of change of this area regarded as a function of x; hence find the area itself as a function of x, determine the constant of integration in the usual way, and then put a; ^= 6 in the result. Definition. The limit of a sum of the kind described above is called the definite integral of f{x)dx from x = a to x = b, and is denoted by Ji EM)^^, ov r~''f(x)dx. The function obtained by the inverse of differentiation is called, for distinction, an indefinite integral. By the funda- mental theorem just stated, the definite integral is equal to the difference between two values of the indefinite integral : £2''K'^)dx=[fKx)dx]^^- [fA«>)dx]^^^. 104 CALCULUS. The double use of the term "integration" — ^meaning in one case anti' differentiation, and in the other ease finding the limit of a sum — and the fundamental theoTem connecting these two distinct concepts, should be made thoroughly clear.* The concept of the definite integral is the most useful con- cept in the application of the calculus, and the study of problems which can be formulated as definite integrals may well occupy one third of the time of a first course. For example, problems in areas, volumes, surfaces, length of are, center of gravity, moments of inertia, center of fluid pressure, etc. Many of the^ problems require two applications of the fundamental/ theorem. 28. Properties of definite integrals. From the definition of the definite integral we have at once : rf(ix)dx=- rKx)dx; rfix)dx + f fix)dx = r Kx)dx; and, by the aid of a figure, the Mean Value theorem: jT F(^x)S{x)dx = F{X)j'f{_x-)dx, where X is some (unknown) value of x between a and h, and F{x) and /(a;) are any contiauous functions, provided /(a;) does not change sign from x=a to x = i. We have also the following important theorem on change of variable: In evaluating the integral r f{x)dx, if a; is a function of a new variable t, we may replace f{x)dx by its value in terms of t and dt, and replace x=a and a;:=6 • The use of the term in the sense of summation was historically the earlier, and the symbol f is the old English "long s," the first letter of "sum." CALCULUS. 105 by the corresponding values * = a and t=p, without altering the value of the integral, provided that thoroughout the inter- val considered there is one and only one value of x for every value of t, and one and only one value of t for every value of x. 29. AU problems leading to a definite integral are prob- lems in finding the limit of a sum, each term of which is approaching zero, while the number of terms is increasing indefinitely. Whenever a function f{x) can be found, such that all terms of the sum are obtained by substituting suc- cessively x^, x^, etc., in the expression f{x)dx, then the formu- lation of the problem as a definite integral is immediately obvious. The separate terms of the sum, of which f{X]c)dx is a type, are called elements. Thus, in finding the area under a curve, an obvious element of area is the rectangle ydx; if the curve revolves about the a;-axis, the element of volume of the solid thus generated is the cylinder rry'dx. Here y must be expressed as a function of x before the integration can be completed. Again, in polar coordinates, the element of area is the sector, ^r^dO, where r must be a known function of 6. In many cases, however, the proper function ia not so immediately obvious. In such cases, the following theorem is of great service : Second replacement theorem for infinitesmals (Theo- rem OF Duhamel). In finding the limit of a sum of positive terms, each of which approaches zero while the number of terms increases indefinitely, any term may he replaced by a " similar " term without affecting the value of the limit. Two variables a and p are called " similar " if (1) lini|=l, or ifi2) lim°^=0. For example, let us find the weight of a rod whose density, w, and cross-section. A, are both functions of x. The "true element" of weight, AW, corresponding to a given length Aa;, wiU certainly lie between the values w'A'Ax and w"A"Ax, where 8 106 CALCULUS. w\A! are the smallest values, and w",A" the largest values of w and A within the interval from x=x to x^=x-\-^x; but either of these extreme values may be replaced by the simpler value wAAx, where w,A are the values of w and A at the leginning of the interval, for, . . w'A'Ax ^ . w"A"Ax ^ wAAx wAAx Hence, AW itself, which lies between these extremes, can be replaced by wAAx, which is therefore the required "differ- ential element" of weight.* The total weight of the rod, from a; = a to a; ^6, is then equal to the definite integral I wAdx ; where w and A must of course be expressed as functions of x before the integration can be completed. In justifying replacements of this kind by Duhamel's theorem, sometimes the first test is more convenient, some- times the second. When once the common replacements have been justified, the use of the theorem in practice rapidly becomes almost intuitive. 30. Approximate methods of integration. — If the function f{x) is given only empirically, the theorem on evaluating the definite integral by purely mathematical means cannot be ap- plied. In such cases, an approximate value of the definite integral f{x)dx £ may be found by plotting the curve y=fix) on squared paper, and estimating the area by counting squares (and frac- tions of squares) . Another method of approximation is by Simpson's Rule: * When X is the independent variable, it ia immaterial whether w« Tmte Ax or dx. CALCULUS. 107 Divide the area into n panels, where n is even, and number the ordinates from 1 to « + 1 ; then, if Ax is the width of each panel, Area = |Aaj (first ordinate + last ordinate + twice the sum of the other odd ordinates + four times the sum of the even ordinates) . The instnunent known as a plwnimeter provides a mechan- ical means of integration, used especially in measuring the areas of indicator cards. Another and very important method of approximation is by the use of series; see the next chapter. 31. Definite Integral as a function of its upper limit. — If X is a variable, the definite integral I /(«)dai represents the area under the cuirve y=f{x) from x = a to the variable ordinate x=X, and is therefore a function of X, say ^(X). By applying the definition of derivative to this function, it is easy to see from the figure that c^'(X) =/(X) : Thus 4>{X) is one of the indefinite integrals of /(X). 108 CALCULUS. Any indefinite integral which cannot be expressed in terms of known functions can always be written as a definite integral regarded as a function of its upper limit, and its value, for any given value of the argument, can then be found by one of the methods of approximate integration. The elliptic integrals, the most important of which are J (^9=* d6 , /•»=* , - ,„, . == and I i/l-(fc')sin'«d«, 9=0 i/l — (t') sin' fl J«=o •' '^ ^ are handled in this way, by the method of expansion in series. The student should be made familiar with the construction and use of tables of the elliptic integrals. In such tables, W is usually expressed in the f onn sin' a, which empha- sizes the fact that ^<1. CHAPTER V. APPLICATIONS TO ALGEBRA : EXPANSION IN SERIES ; INDETER- MINATB FORMS. Note. — This chapter may be taken, if preferred, immediately after the chapter on differentiation. It is in reality an exten- sion of the "formal work" of that chapter, since it deals with changes in the form of algebraic expressions. 32. Taylor's theorem. — It is often desirable to obtain an approximate expression for a given function, in the neighbor- hood of a given point a; ^ a, in the form of a series arranged according to ascending powers of x — a, with constant coeffi- cients. For values of x near to a, the higher powers of a; — a will then become negligible. The most convenient theorem for this purpose is the fol- lowing: Taylor's Theorem. If f{x) is continuous, and has deriva- tives through the {n-\-l)st, in the neighborhood of a given point x = a, then, for any value of x in this neighborhood, f{x)=S{a-) + t^{x-a)+^-^{x-ay + - n\ ^^ "^ +(n + l)I .^■4^ (.-«)- +^-^ (:«-«)-, where X is some unknown quantity between a and x. The last term, is the error committed if we stop the series with the term in (x — a)» and the formula is useful only when this error be- comes smaller and smaller as we increase the number of terms. 109 110 CALCULUS. This form for the "remainder" B is easUy remembered since it differs from the general term of the series only by the fact that the derivative in the coef&cient of the power of (a; — o) is taken for x^X instead of for x = a.'' (There are also other forms of the remainder which are sometimes useful.) 33. The special case where = is called Maclaurin's Theorem: where X is some unknown quantity between and x. 34. Another special case, obtained by putting m = 0, gives fix)-na)=nX){x-a), where again X is some unknown quantity between a and x. This theorem is called the Law of the Mean, and is of great importance in the theoretical development of the subject. 35. If the error-term in Taylor's Theorem approaches zero as n increases, the formula becomes a convergent infinite series, called the Taylor's series for the given function, about the given point x=a. The series with which the student should be especially fa- miliar are the following : • The simplest proof of this theorem is by means of integration. For example, for the ease n = 2, we have £f"{t)di=f"'(^X)(x-a), where X is some (unknown) constant between a and x (as is evident from a figure) ; but also fj"'{t}dt=f"{x)-f"{a), by the fundamental theorem; so that /"(a,) _/"(a) =f"'(X) ^x-a). Integrating this equation twice between the limits x = a and x^=x, remembering that f"(a) and t"'{X) are constants, we have at once: fix) —na) —f"(^a){x — a) =zf"{X)i{x — ay, f{x)—f{a)—f'{a){x — a)—f"{a)Kx — aY^f"'{X)il{x — ay. CALCULUS. Ill Binomial series: provided \x\ < 1. Sine series : *jj3 ^5 „t sin « = « — g-j + — — -- + •• . (« in radians). Cosine series : s^ X* ^ cos a; = 1 — —^ + — — — + •• • (a; in radians). Exponential series : Next in importance are the series for log (1 + a;), tan"' x, sinh~5', and cosh x. From these series we have the following important approxi- mations, when X is small : vr+^=i+^.;— ., ^=1-.+ ..., _L^=i_i^+..., sin x = x , cos x=\ , etc. An important special case of the binomial series is the geometric series : = l + a; + a;* + ar'+---, provided \x\ < 1. \ — x 36. The student should also understand the comparison test, and the test-ratio test, for the convergence of an infinite series, and the following theorem on alternating series : If the terms of a series are altematoly positive and negative, each being numerically less than or equal to the preceding, and if the nth term approaches zero as n increases, then the series is conver- gent, and the error made by breaking off the series at any given term does not exceed numerically the value of the last term retained. 112 CALCULUS. Further, a power series can he differentiated or integrated term ty term, within the interval of convergence. 37. Indeterminate forms. — The evaluation of indeterminate forms can often be facilitated by the use of the following theorem, in which f{x) and F{x) are functions which possess derivatives at a given point x = a. Theorem of indeterminate forms. If f{x) and F{x) ioth approach zero, or ioth become infinite, when x approaches a, then iimr^]=iimrt^l. The second limit may often be easier to evaluate than the first. The student should thoroughly understand the meaning of indeterminate forms, for which the common symbols^, I'jetc, are merely a suggestive short-hand notation. Thus, "0/0" means that we are asked to find the limit of a function 2/= /(a;) /JF (a;), when f{x) and F{x) bothapproach zero. Now the change in f{x) alone would tend to decrease y numerically, while the change in ^(a;) alone would tend to increase y ; hence we cannot tell, without further investigation, what the combined effect of both changes, taking place simul- taneously, wiU be. Again, the symbol 1°° means that we are asked to find the limit of a function y=f{x)^ '■"'>, when f{x) approaches 1 and F{x) becomes infinite. Now the change in f{x) alone would tend to make y approach 1, while the change in F{x) alone would tend to make y recede from 1 ; hence we cannot tell, with- out further investigation, what the combined effect will be. The student should thoroughly master in this way the meaning of all the seven types of indeterminate forms, namely, 5, ^,0-00, 0°, 1", oo», 00-00. The cases involving exponents are best treated by first find- ing the limit of the logarithm of y, from which the limit of y CALCULUS. 113 can then be obtained. The form 0- oo, or y^=f{x) -F^x), can be written as t/ = , ,„. . , or m= ^ ,},. , which then comes 1/F{x)' " l/f{x) under one of the first two forms. The last form, oo — oo , is usually best handled by the method of series. Before applying the theorem of indeterminate forms, one should, of course, try first to find the required limit by a simple algebraic transformation, if possible. CHAPTER VI. APPUCATIONS TO GEOMETRY AND MECHANICS. In all applications to geometry, in which a curve is repre- sented by an equation connecting x and y, the scales on the x and y axes must he equal (compare §3, footnote). 38. Tangent and normal. — The equation of the tangent at any point (and hence the equation of the normal) can be written down at once when we know the slope and the coordi- nates of the point of contact. Again, to fnd the subtangent or subnormal at any point, we have simply to find the ordinate and the slope at that point, and then solve a right triangle. 39. Differential of arc. If s= length of arc of the curve y=f(x), measured from some fixed point A of the curve, then s, like y, is a function of x, and we may ask what is the x,+/^x rate of change of s with respect to x, that is, what is the value of ds/dx. Now ds/dx=^]im (As/ Ax), and in finding this limit we may replace the arc As by its chord, V {Ax)^-\-{Ay)'; hence ds/da; = lim V1+ (Ay/Ax)" = y/l-\- {dy/dx)", or ds:=^/(dxy + {dyr, 114 CALCULUS. 115 as indicated in the figure. This formula, and the correspond- ing relations dx = ds cos , dy = ds sin , are important, and are readily recalled to mind by the figure. In the case of a circle of radius r, if d0=the angle at the center, subtended by the are ds, then ds=rd9, provided the angle is measured in radians. 40. Again, in ease of a curve whose equation is given in polar coordinates, r=/(^), we see at once from the figure, by the aid of the replacement theorem, that ds = si (dry + {rdey and tan i^= -^, where \ji is the angle which the tangent makes with the radius vector produced. 41. Radius of Curvature, — Consider the normal to a given curve at a given point, P, and also the normal at a neighbor- ing point, Q. These two normals wOl intersect at some point C" on the concave side of the curve ; and as Q approaches P, along the curve, this point C will (in general) approach a definite position as a limit. The circle described with a center at this point C and radius equal to CP wiU fit the given curve more closely, in the neighborhood of the point P, than does any other circle. This circle is called the osculating circle, or the circle of curvature, at the point P ; its center G is called the center of curvature, and its radius CP is called the radius of curvature, at the point P. 116 CALCULUS. The radius of curvature may thus be taken as a measure of the flatness or sharpness of the curve ; the smaller the radius of curvature, the sharper the curve. The length of the radius of curvature, B, at any point P is most readily found as foUows : In the triangle PC'Q, we have G'P/PQ = sin Q/siD. A^, where A^ is the angle between the normals (or between the tangents) at P and Q. Therefore ii = lim C'P = lim (chord PQ/sin A.^) sin Q; or, replacing the chord by the arc As, and sin A^ by A<^, and noticing that Q is approaching 90°, so that lim sin Q = l, we have B = lim (As/A<^), or, ds ^ = d4.- This important formula is readily recalled to mind from the figure, if one thinks of the arc As as approximately a circular arc. To express B in terms of x and y, we have only to remember that ds = y{dxy+ {dyy=\Jl + y^dx, and tan 4> = dy/dx ^ y', whence dip = y"dx/(,l + y''); then j;_ (l+/)^ y" CALCULUS. 117 Def, The curvature of a curve at a point is defined as the rate at which the angle is changing with respect to the length of arc s ; that is, d6 1 curvature = t- = -?. . as E If the slope of the curve is small, the curvature is approxi- mately equal to y". Def. The locus of the center of curvature is called the evolute of the curve. The normals to the given curve are tangent to the evolute, and the given curve may be traced by unwinding a string from the evolute. 42. Velocity and acceleration. — Consider a particle moving along a straight line. Its distance from the origin is a func- tion of the time : x = F{t). The velocity of the particle is the rate of change of its distance: v = dx/dt=F'{t)=x'. The velocity wiU be positive or negative, according as the particle is moving forward or backward along the line. The acceleration of the particle is the rate of change of its velocity: A = dv/dt=F"(t)=x". The acceleration wiU be positive or negative according as the velocity is increasing or decreasing (algebraically). If a particle is moving along a plame curve, we must consider the components of its motion along two fixed axes. The components of acceleration along the x- and j/-axes are x" and y" ; the components of acceleration along the tangent and normal are dv/dt and v'^/B, respectively, where V = yjx'^ -f- y''^ = the path velocity, and i2=the radius of curvature. It should be carefully noticed tliat dv/dt is not the whole acceleration, but only that component of the acceleration which lies along the tangent. 118 CALCULtrS. The importance of this application in problems in mechanics is obvious. Note. — ^As explained in the preface of this report, these pages are intended merely to give a resume of the working principles of the calculus with which the student should be perfectly familiar after having taken a course in this subject. The main part of the work of such a course should he proi- lems done hy the students — each problem being solved on the basis of the small number of fundamental theorems here mentioned. discussion on mathematics eepoet, 119 Discussion. Professor Chas. 0. Gunther: It seems to me that in this report some mention should be made of imaginary and com- plex quantities. A little knowledge of these quantities can, for instance, be utilized to good advantage by applying it to that part of the calculus known as integration. In fact, in- tegration can be simplified to the extent of eliminating the usual " reduction formulae " and rendering the use of tables of integrals unnecessary. As found in text-books in general, there are three cases for which the expression dy = cos^e wH'Ode (1) can be easily integrated. Two of these cases include frac- tional values for h and Tc. AU other cases in which h and h are integers can either directly, or by means of a single imaginary trigonometric substitution (tan fl=i sin a, in which a is an imaginary quantity) , be reduced to one or more of the three cases just referred to. The general binomial differential expression dj/ = a;'»(a + 6a;")P/ada; (2) is only another form of (1) since V« + &^" can always be represented by one of the three sides of a right triangle and therefore expressed as a trigonometric function of one of the acute angles of the triangle. To make this transformation the student must know the relation between the hypotenuse and the two sides of a right triangle, the values of the trigonometric functions of an angle in terms of the sides of a right triangle, and the rules for differentiation. Differential expressions involving trinomial surds may be rationalized in a similar manner. The expressions 2=e-cosH (3) ^^=e'Bmbx, (4) 120 DISCUSSION ON MATHEMATICS EEPOBT. may be integrated with great facility if complex quantities are employed, because e"* cos 6a; and e"" sin 6a; are the rec- tangular components of a vector whose modulus is e" and whose argument is bx. The integrals of (3) and (4) are found from the integral of g=eC«+«)^ (5) in which 2 is a complex variable of the form y + iy. The integral of (5) is readily found to be in which C„_.^, ••• Cj, Co, are constants of the form C = C + iC. Equation (6) may be written " = (^T6y^'*'"""'"'"'"'" + ^-1^""' + • • • + Cia; + q,. (7) The integral of (3) is the real part of (7) and the integral of (4) is the imaginary part of (7) divided by i. Again in differential equations we find the linear equations -^ + ay =b cos nx, (8) dv -^ -|- ay = 6 sin inx, (9) and their solutions can be obtained from the solution of the equation ^ + az = 6e"^/ (10) in which z = y-{- iy. The foregoing illustrates a few of the applications of com- plex and imaginary quantities, and includes a first treatment of hyperbolic functions as trigonometric functions of imagi- nary quantities. Some little consideration should also be given to the com- plex and imaginary branches of certain curves, as for example, DISCUSSION ON MATHEMATICS EEPOBT. 121 the circle, the ellipse, and the hyperbola. It should be noted that the equation of the circle x' -\-y'' = a^ is also the equation of an imaginary hyperbola for values of a; > a and < — a. This is important, since of the three forms of binomial surds Va^ — a5^ Vft^ + ^^'j V^ — ffl^j tlie first is obtained from the equation of the circle x^-\-y^ =