CORNELL UNIVERSITY LIBRARIES Mathematics Library White Hall CORNELL UNIVERSITY LIBRARY 3 1924 059 316 103 DATE DUE rETTB \m 1 1 1 GAYLORD PRINTCOIN U.S.A. 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/cu31924059316103 Production Note Cornell University Library produced this volume to replace the irreparably deteriorated original. It was scanned using Xerox software and equipment at 600 dots per inch resolution and compressed prior to storage using CCITT Group 4 compression. The digital data were used to create Cornell's replacement volume on paper that meets the ANSI Standard Z39. 48-1984. The production of this volume was supported in part by the Commission on Preservation and Access and the Xerox Corporation. 1991. BOUGHT WITH TH^J^^COM^ FROM THE' •■•-*» SAGE ENDOWMENT FUND THE GIFT OF Denrg M. Sage 1891 .11' INTRODUCTION INFINITESIMAL ANALYSIS FUNCTIONS OF ONE REAL VARIABLE BY OSWALD VEBLEN Preceptor in Mathematics, Frznceton University AND N. J. LENNES Instructor in Malhemaiia m the WejidelL PktUipa High Schoolt Chicago FIRST EDITION FIRST THOUSAND NEW YORK JOHN WILEY & SONS London: CHAPMAN & HALL, Limited 1907 Copyrigdt, 1907 BI OSWALD VEBLEN awd N. J. LENNES ROBERT DRCHHOHD, PRINTER, NEW TORE PREFACE. A COURSE dealing with the fundamental theorems of infini- tesimal calculus in a rigorous manner is now recognized as an essential part of the training of a mathematician. It appears in the curriculum of nearly every university, and is taken by students as "Advanced Calculus " in their last collegiate year, or as part of "Theory of Functions " in the first year of graduate work. This little volume is designed as a convenient reference book for such courses; the examples which may be considered necessary being supplied from other sources. The book may also be used as a basis for a rather short theoretical course ou real functions, such as is now given from time to time in some of our universities. The general aim has been to obtain rigor of logic with a minimum of elaborate machinery. It is hoped that the system- atic use of the Heine-Borel theorem has helped materially toward this end, since by means of this theorem it is possible to avoid almost entirely the sequential division or " pinching " process so common in discussions of this kind. The definition of a Umit by means of the notion "value approached" has simplified the proofs of theorems, such as those giving necessary and sufficient conditions for the existence of _ limits, and ia general has largely decreased the number of e's and d's. The theory of limits is developed for multiple-valued functions, which gives certain advantages in the treatment of the definite integral. In each chapter the more abstract subjects ?ind those which can be omitted on a first reading are placed in the concluding IV PREFACE. sections. The last chapter of the book is more advanced in character than the other chapters and is intended as an intro- duction to the study of a special subject. The index at the end of the book contains references to the pages where technical terms are first defined. When this work was undertaken there was no convenient source in English containing a rigorous and systematic treat- ment of the body of theorems usually included in even an ele- mentary course on real functions, and it was necessary to refer to the French and German treatises. Since then one treatise, at least, has appeared in English on the Theory of Functions of Real Variables. Nevertheless it is hoped that the present volume, on account of its conciseness, will supply a real want. The authors are much indebted to Professor E. H. Moore of the University of Chicago for many helpful criticisms and suggestions; to Mr. E. B. Morrow of Princeton University for reading the manuscript and helping prepare the cuts; and to Professor G. A. Bliss of Princeton, who has suggested several desirable changes while reading the proof-sheets. CONTENTS. CHAPTER I. The System of Real Numbers. PAoa { 1. Rational and Irrational Numbers 1 J 2. Axiom of Continuity 3 § 3. Addition and Multiplication of Irrationals 7 J 4. General Remarks on the Number System 11 § 5. Axioms for the Real Number System 13 § 6. The Number e 15 § 7. Algebraic and Transcendental Numbers 18 § 8. The Transcendence of e 19 § 9. The Transcendence of ;: 25 CHAPTER II. Sets of Points and of Segments. § 1. Correspondence of Numbers and Points 30 § 2. Segments and Intervals. Theorem of Borel 32 § 3. Limit Points. Theorem of Weierstrass ; 38 § 4. A Second Proof of Theorem 15 42 CHAPTER III. PoNcnoNS IN General. Spbclal Cases of Functions. § 1. Definition of a Function 44 5 2. Bounded Functions 47 { 3. Monotonic Functions. Inverse Functions 49 { 4. Rational, Exponential, and Logarithmic Functions 63 CHAPTER IV. Theory of Limits. § 1. Definitions. Limits of Monotonic Functions 60 § 2. The Existence of Limits 65 vi CONTENTS. PAGS § 3. Application to Infinite Series 70 § 4. Infinitesimals. Computation of Limits 74 § 5. Further Theorems on Limits 81 § 6. Bounds of Indetermination. Oscillation .,.,,., 83 CHAPTER V. CoNTiNuoDs Functions. § 1. Continuity at a Point 87 § 2. Continuity of a Function on an Interval 88 J 3. Functions Continuous on an Everywhere Dense Set 94 § 4. The Exponential Function 97 CHAPTER VI. Infinitesimals and Infinites. § 1. The Order of a Function at a Point 101 § 2. The Limit of a Quotient 105 § 3. Indeterminate Forms 108 § 4. Rank of Infinitesimals and Infinites 114 CHAPTER VII. Derivatives and Differentials. { 1. Definition and Illustration of Derivatives 117 § 2. Formulas of Differentiation itq § 3. Differential Notations 128 § 4. Mean-value Theorems joq % 5. Taylor's Series , , . I 6. Indeterminate Forms , og J 7. General Theorems on Derivatives 144 CHAPTER VIII. Definite Integrals. { 1. Definition of the Definite Integral 151 § 2. Integrability of Functions , c^ 5 3. Computation of Definite Integrals . cq § 4. Elementary Properties of Definite Integrab IfM { 5. The Definite Integral as a Function of the Limits of Integration 171 § 6. Integration by Parts and by Substitution j^ • § 7. General Conditions for Integrability ,-_ CONTENTS. vii CHAPTER IX. Improper Definite Integrals. PAGE 5 1. The Improper Definite Integral on a Finite Interval 191 § 2. The Definite Integral on an Infinite Interval 201 § 3. Properties of the Simple Improper Definite Integral 205 § 4. A More General Improper Definite Integral 210 § 5. Special Theorems on the Criteria of the Existence of the Improper Definite Integral on a Finite Interval 218 } 6. Special Theorems on the Criteria of the Existence of the Improper Definite Integral on an Infinite Interval 223 INFINITESIMAL ANALYSIS. CHAPTER I. THE SYSTEM OF REAL NUMBERS. § 1. Rational and Irrational Numbers. - The real number system may be classified as follows: (1) All integral numbers, both positive and negative, in- cluding zero. (2) All numbers — , where m and n are integers (n^^O). (3) Numbers not included in either of the above classes, such as V2 and w.f Numbers of classes (1) and (2) are called rational or com- mensurable numbers, while the numbers of class (3) are called irrational or incommensurable numbers. As an illustration of an irrational number consider the square root of 2. One ordinarily says that \/2 is 1.4 +, or tit i8 clear that there is no number — such that — 7=2, for if — r = 2. n n' ' n' ' then m'=2n', where m' and 2n' are integral numbers, and 2n' is the square of the integral number m. Since in the square of an integral number every prime factor occurs an even number of times, the factor 2 must occur an even number of times both in n' and 2n', which is impossible because of the theorem that an integral number has only one set of prime factors. 2 INFINITESIMAL ANALYSIS. 1.41 + , or 1.414 + , etc. The exact meaning of these statements is expressed by the following inequalities: t (1.4)2 <2< (1.5)2, (1.41)2 <2< (1.42)2, (1.414)2 <2< (1.415)2, etc. Moreover, by the foot-note above no terminating decimal is equal to the square root of 2. Hence Horner's Method, or the usual algorithm for extracting the square root, leads to an infinite sequence of rational numbers which may be denoted by Oi, a2, as, . . . , a„, . . . (where ai=1.4, 02 = 1.41, etc.), and which has the property that for every positive integral value of n a„b signifies that a is greater than 6. t This involves the assumption that for eveiy nvimber, e, however small there is a positive integrer n such that ttt-K t- This is of course obvious 10** o when e is a rational number. If c is an irrational number, however, the statement will have a definite meaning only after the irrational number has been fully defined. THE SYSTEM OF REAL NUMBERS. 3 other hand, if o^ > 2, let a^ - 2 = e' or 2 + e' =a. Taking n such that r-r- < — , we should have lU" 5 (a« + J^)' < (fln^) + e' < 2 + e' < a; and since Cn + iT^ is greater than a* for all values of k, this would contradict the hypothesis that a is the least number greater than every number of the sequence a\, 02, 03, . . . We also see without difficulty that o is the only nimiber such that «2 = 2. § 2. Axiom of Continuity. The essential step in passing from ordinary raticwial num- bers to the number corresponding to the sjonbol \/2 is thus made to depend upon an assumption of the existence of a number o bearing the unique relation just described to the sequence a^, a2, a„, . . . In order to state this hypothesis in general form we introduce the following definitions: Definition. — ^The notation [x] denotes a set,'f any element of which is denoted by x alone, with or without an index or subscript. A set of numbers [x] is said to have an upjxr bound, M, if there exists a number M such that there is no number of the set greater than M. This may be denoted by M^[x]. A set of numbers [x] is said to have a lower bound, m, if there exists a number m such that no number of the set is less than m. This we denote by m^[x}. Following are examples of sets of numbers: (1) 1, 2, 3. (2) 2, 4, 6, . . . , 2i, . . . (3) 1/2, 1/22, 1/23, . . . , i/2», . . . (4) All rational numbers less than 1. (5) All rational numbers whose squares are less than 2. t Synonyms of set are class, aggregate, collection, assemblage, etc. 4 INFINITESIMAL ANALYSIS. Of the first set 1, or any smaller number, is a lower bound;, and 3, or any larger number, is an upper bound. The second set has no upper bound, but 2, or any smaller nimiber, is a lower bound. The nimiber 3 is the least upper boimd of the first set, that is, the smallest niraiber which is an upper bound. The least upper and the greatest lower bounds of a set of num- bers [x] are called by some writers the upper and lower limits respectively. We shall denote them by B[x] and 5[x] respect- ively. By what precedes, the set (5) would have no least upper bound xmless V2 were coimted as a number. We now state our hjrpothesis of continuity in the following form: Axiom K. If a set [r] of rational numbers having an upper bound has no rational least upper bound, then there exists one and only one number B[r] such that (a) B[r] > /, where / is any number of [r] or any rational rmmr- ber less than some number of [r]. (6) B[r]s', t This axiom implies that the new (irrational) numbers have relations of order with all the rational numbers, but does not explicitly state rela- tions of order among the irrational numbers themselves. Cf. Theorem 2. THE SYSTEM OF REAL NUMBERS. 5 where s' is any rational number not an upper bound of [s]. Moreover, if s" is rational and greater than every s, it is greater than every r. Hence where s" is any rational upper bound of [s]. Then, by the definition of B[s], B[r]=m Definition. — If a number x (in particular an irrational ■number) is the least upper bound of a set of rational numbers [r], then the set [r] is said to determine the number x. Corollary 1. The irrational numbers i and i' determined by the two sets [r] and [/] are equal if and only if there is no num- ber in either set greater than every number in the other set. Corollary 2. Every irrational number is determined by some set of rational numbers. Definition. — If i and i' are two irrational numbers deter- mined respectively by sets of rational numbers [r] and [/] and if some number of [r] is greater than every niunber of [/], then i>i' and i'0, then a+b>b, and that if o<0, then a+b 0, 6 > 0, it follows that ab = B[xy] ; a<0, b<0, " " " ab = B[xy]; a<0, 5>0, " " " a6 = 5[a;j/]; a>0, 6<0, " " " ab = B[.xyl Definition. — If a and 6 are not both rational and [a;] is the set of all rational numbers between and o, and [y] the set of all rationals between and 6,_then if a>0, b>0, ab means B[xy]; if a<0, 6<0, ab means ^xy]■, if o<0, 6>0, ab means B[xy]; if a>0, 6<0, ab means B^xj/]. If a or 6 is zero, then ab=0. THE SYSTEM OF REAL NUMBERS. 9 It is proved, just as in the case of addition, that ab=ha, that a{bc) = idb)c, that if a is rational [ay] is the same set as [xy], that if a>0, b>0, ab>0. Likewise the quotient -r- is defined as a number c such that ac=b, and it is proved that in case a>0, 6>0, then c=b\ —/\, where [y'] is the set of all rationals greater than b. Similarly for the other cases. More- over, the same sort of reasoning as before justifies the usual method of multiplying non-terminated decimals. To complete the rules of operation we have to prove what is known as the distributive law, namely, that o(6+c)=a6+ac. To prove this we consider several cases according as a, b, and c are positive or negative. We shall give in detail only the case where all the numbers are positive, leaving the other cases to be proved by the reader. In the first place we easily see that for positive numbers e and /, if [t] is the set of all the rationals between and e, and [T] the set of all rationals less than e, while [u] and [U] are the corresponding sets for /, then e+f=B[T+U]=B[t+u]. Hence if [x] is the set of all rationals between and a, [y] be- tween and b, [z] between and c, b + c=B[y+z] and hence a{b+c) = B[x{y+z)]. On the other hand ab=B[xy], (ic=^^xz], and therefore db+ac= ^{xy+xz}]. But since the distributive law is true for rationals, x(y+z)=xy+xz. Hence E[x{y+z)]=B[ixy+xz)] and hence aib + c)'=ab+ac. We have now proved that the system of rational, and irra- tional nimabers is not only continuous, but also is such that we may perform with these nimibers all the operations of arith- metic. We have indicated the method, and the reader may 10 INFINITESIMAL ANALYSIS. prove in detail that every rational number may be represented by a terminated decimal, aaO*+Oifc_ilO*-i + . . .+ao+~-+ •■■ +^ = afcfflt_i . . . aoffi_ia_2 . . . ffl_n, or by a circulating decimal, O'kP'k-l • • • ffloffl-lffl-2 ■ . ■ d-i . . . d- id-i ■ . ■ a_j . . . , where i and j are any positive integers such that i<]'; whereas every irrational nimiber may be represented by a non-repeating infinite decimal, aicO-ic-i ■ ■ ■ aoa_ia_2 . . . «_„ . . . The operations of raising to a power or extracting a root on irrational numbers wiU be considered in a later chapter (see page 53). An example of elementary reasoning with the sym- bol jB[x] is to be found on pages 17 and 18. For the present we need only that x", where n is an integer, means the number obtained by multipl3dng x by itself n times. It should be observed that the essential parts of the defini- tions and arguments of this section are based on the assumption of continuity which was made at the outset. A clear under- standing of the irrational number and its relations to the rational number was first reached during the latter half of the last century, and then only after protracted study and much discussion. We have sketched only in brief outline the usual treatment, since it is beUeved that the importance and diffi- culty of a full discussion of such subjects wUl appear more clearly after reading the following chapters. Among the good discussions of the irrational number in the English language are : H. P. Manning, Irrational Numbers and their Representation hy Sequences and Series, Wiley & Sons, New York; H. B. Fine, College Algebra, Part I, Ginn & Co., Boston- THE SYSTEM OF REAL NUMBERS. 11 Dedekind, Essays on the Theory of Number (translated from the German), Open Court Pub. Co., Chicago; J. Piehpont, Theory of Functions of Real Variables, Chapters I and II, Ginn & Co., Boston. § 4. General Remarks on the Number System. Various modes of treatment of the problem of the number system as a whole are possible. Perhaps the most elegant is the following : Assume the existence and defining properties of the positive integers by means of a set of postulates or axioms. From these postulates it is not possible to argue that if p and V q are prime there exists a number a such that ap=q or a = -, i.e., in the field of positive integers the operation of division is not always possible. The set of all pairs of integers jm, n), if \mk, nk\ {k being an integer) is regarded as the same as {Tn,n\, form an example of a set of objects which can be added, subtracted, and multiplied according to the laws holding for positive integers, provided addition, subtraction, and multipli- cation are defined by the equations,t {m,n\®{p,q\ = {mp,Tiq} {m, n!01p, 5! = {mq+np, nq]. The operations with the subset of pairs {m,l} are exactly the same as the operations with the integers. This example shows that no contradiction will be introduced by adding a further axiom to the effect that besides the integers there are numbers, called fractions, such that in the extended system division is possible. Such an axiom is added and the order relations among the fractions are defined as follows: p m .. -<— if pn0 and b>0, then ab>0. These postulates may be regarded as summarizing the prop- erties of the real number system. Every theorem of real analysis is a logical consequence of them. For convenience of reference later on we summarize also the rules of operation with the symbol |a;|, which indicates the "numerical " or "abso- lute" value of x. That is, if a; is positive, la;|=a;, and if x is negative, |x|= —x. \x\ + \ymx + y\ (1) .-. I\xk\^\lxk\, (2) k-l k=l n where Ixk=xi+X2+ . . .+x„. |N-M|^k-2/| = l2/-a;|^W + |j/| (3) i2;-2/| = N-l2/l (4) (5) \X\ X \y\ ~ y If lx-t/|e, then Ek>e--,y Proof .—From the definitions of e and f?„ it follows that '"■^*='^[(ATi)i+(Fr2)i+ •• •+(&+])!]' where [Z] is the set of all positive integers. Hence '^ik+2)...ik+t)]' or e-E,<-g^^^'e. If A>e, this gives Ek>e-^^. THE SYSTEM OF REAL NUMBERS. 17 Theorem 7. e=fil (iH — ) J, where [n] is the set of all positive integers. Proof. — By the binomial theorem for positive integers Hence E„-(l+-) = ^ (^ '-j^, ) « n*-n(n-l) . .. (n-A + 1) ,, k-2 kin'' »-, 7i*-(n-^ + l)fc •^ Tim • ^iZi kin" Hence by factoring _ / 1\" ^(A;-l)(n*-^+n*-'(n-A;+l) 1^ "^ ^(_A; - 1) (n*-i +n*-g(n-A;+l) + ... + (n-A; + l)*-^) "(A;-l);kn*-» £-2 kin" 1 ^ (A;-1)A; (1+-)" d) and from (6) r+^ '^^""n' ^^^ 18 INFINITESIMAL ANALYSIS. whence by the lemma (l + l)">e-V- (3) \ nl n\ n From (1) it follows that e is an upper bound of 1\"- [(^-ri. and from (3) it follows that no smaller number can be an upper bound. Hence (('-y]-' § 7. Algebraic and Transcendental Numbers. The distinction between rational and irrational nimibers, which is a feature of the discussion above, is related to that between algebraic and transcendental numbers. A number is algebraic if it may be the root of an algebraic equation, aox"+0ix"-i + . . .+o„_ia;+a„=0, where n and ao, ai, . . . , a„ are integers and w> 0. A number is transcendental if not algebraic. Thus every rational number — n is algebraic because it is the root of the equation nx—m=0, while every transcendental number is irrational. Examples of transcendental numbers are, e, the base of the system of natural logarithms, and ;:, the ratio of the circumference of a circle to its diameter. The proof that these numbers are transcendental follows on page 19, though it makes use of infinite series which will THE SYSTEM OF REAL NUMBERS. 19 not be defined before page 71, and the function e', which is defined on page 57. The existence of transcendental numbers was first proved by J. LiouviLLE, Comptes Rendus, 1844. There are in fact an infinitude of transcendental numbers between any two num- bers. Cf. H. Weber, Algebra, Vol. 2, p. 822. No particular number was proved transcendental till, in 1873, C. Hermite (Crelle's Journal, Vol. 76, p. 303) proved e to be transcendental. In 1882 E. LiNDEMANN (Mathematische Annalen, Vol. 20, p. 213) showed that k is also transcendental. The latter result has perhaps its most interesting application in geometry, since it shows the impossibihty of solving the classical problem of constructing a square equal in area to a given circle by means of the ruler and compass. This is because any construction by ruler and compass corresponds, according to analytic geometry, to the solution of a special type of alge- braic equation. On this subject, see F. Klein, Famous Prob- lems of Elementary Geometry (Ginn & Co., Boston), and Weber and Wellstein, Encyclop'ddie der Ekmmtarrmiherruitik, Vol. 1, pp. 418-432 (B. G. Teubner, Leipzig). § 8. The Transcendence of e. Theorem 8. // c, Ci, Cz, C3, . . . , c„ are integers (or zero bvi Cj^O), then c+c.c+c,e2+ ..,+c„c»?^0 (1) Proof. — ^The scheme of proof is to find a number such that when it is multipUed into (1) the product becomes equal to a whole number distinct from zero plus a number between +1 and —1, a sum which surely cannot be zero. To find this number N, we study the series t for e*, where A; is an integer {x)=bo+bix+b23^+. . .+6a, the successive derivatives of which are '(x)=bi+2-b2X+. . .+s-b,-x'-\ ,i^\, XT . T (m+1)! , s! ^<">(a;)=b„.-m!+6m+i- ,, ■x+...+b.-7- r-.-a*-" The diagonal in (2) from 6-1! to b,-s \. _^.^ is obviously ^'(A), the next lower diagonal is ^"(A), etc. Therefore by adding equations (2) in this notation we obtain THE SYSTEM OF REAL NUMBERS. 21 eHV.hi+2\b2 + . . .+s\b,) = <}>'ik) + "ik) + . . . + <'Kk) + Jb„,-k'--Rkm, (3) m-l k ^ m + l'^(m + l)(m+2) in which ftjfcm = l + zrTT + 7r-rTT7r— rirr+. • . Remembering that (f>(x) is perfectly arbitrary, we note tnat if it were so chosen that '(.k)=0, "ik)=0,..., .^(''-i)(A:)=0, for every A; (k = l, 2, 3, . . . , n) then equations (2) and (3) could be written in the form m-l +bp-p\ +6p+i-(p+l)!(l+^) +6..«!(l+i|+2-, + ... + (^3^). (4) A choice of (j>{x) satisfying the required conditions is ^{x) = (ao+aiX+as^ + ...+an3^y-j^:Ziy^= (p^iyT"' ^^^ where f{x) = {x-l){x-2)(x-S) . . . (x-n). 22 INFINITESIMAL ANALYSIS Every jfc (*=!, 2, . . . , n) is a p-tuple root of (5). Here p is still perfectly arbitrary, but the degree s of ^(x) is np + p-1. If 4)(x) is expanded and the result compared with '=pWkp+rkp. (6) Before completing our proof we need to show that by choosing the arbitrary prime number p sufficiently large, rtp can be made as small as we please. If a is a number greater than n, \R km] 1 + -+; F < m+1 (m + l)(?ra + 2) 2 1 + -T + W m+1 (m + l)(m + 2) + . + . {x) and since each coefficient of ix) is numerically less than or equal to the correspanding coefficient of XP -1 (p-1)! it follows that (|ao| + |ai|a; + |a2|x2 + . . . + |a„|z»)i', !'-*p|) + Cirip+C2r2p + . . . + c„r„p, =NpC+pW + R, (8) where W is an integer or zero and R is numerically less than unity. Since NpC is not divisible by p and is not zero, while pW is divisible by p, this sum is numerically greater than or €qual to zero. Hence Np(c+Cie+C2^ + . . .+c„e»)?^0. Hence and e is a transcendental number. THE SYSTEM OF REAL NUMBERS. 25 § 9. The Transcendence of n. The definition of the number r. is derived from Euler's formula e^"^" ^ = cos x+\/ — 1 sin x; by replacing x by n, e rN/^=-l (1) If TT is assumed to be an algebraic number, rV - 1 is also an algebraic number and is the root of an irreducible algebraic equation F{x)=0 whose coefficients are integers. If the roots of this equation are denoted by 21, 22, 23, ■ • • , Zn, then, since ^v — 1 is one of the z's, it follows as a consequence of (1) that (e^' + l)(e'= + l)(e^»+l) ...(e^" + l)=0. ... (2) By expanding (2) l + Ie'i + Ie'i+'i + Ie''+'i+h + . . .=0. Among the exponents zero may occur a number of times e.g., (c - 1) t-mes. If then Zi, Zi + Zj, Z, + Zj + Zk, ..., be designated by Xi, X2, X3, ■ ■ . , x„, the equation becomes c + e^'+e^=+.. . + e'"=0, (3) where c is a positive number at least unity and the numbers Xi are algebraic. These numbers, by an argument for which the reader is referred to Weber and Wellstein's Encyclopddie der Elemmiarmathematik, p. 427 et seq., may be shown to be the roots of an algebraic equation /(x)=ao+aiX+a2x2 + . . .+a„x" = 0, . . . (3') 26 INFINITESIMAL ANALYSIS. the coefBcients being integers and Oo 5^0 and a„ j-^O. The rest of the argument consists in showing that equation (3) is impossi- ble when xi, X2, . . . , Xn are roots of (3'). The process is analogous to that in § 8. e^>c.V.b,=bi-V.+hxu{l+^+~ + . . .) , e^..3!63 = 63-3!(l+ff+f)+63X.3(l+^+^; + ...), (4) e^..s!.6.=6..«!(l+ff+... + ^) +,^,.(l+_£L+^^ + ...) The numbers 6i, . . . , 6„ may be regarded as the coefficients of an arbitrary polynomial ^(x) = 6o+M+fe2a;2 + . . .+b^, for which ^'"^(x)^bm-m\+b„+i-^—:rY^-x + ... + b.j r.'X^'". J. »-i The diagonal in equations (4) from 6i-l! to b,-s\ , , is (s-1) obviously (f>'{xk), and the next lower diagonal 4>"{Xk), etc. Therefore, by adding equations (4), e**(l!6i+2!62 + . . .+s!6.) = .^'(xjt) + <^"(a;*) + . . . ■^cj>^'Kxk) + h^-Xk^Rk„„ . (5) i»»=i THE SYSTEM OF REAL NUMBERS. 27 in which 7? - 1 I ■ ^^ I ^fc^ I "*" "^m + l^(m + l)(m+2) + -'- Remembering that (f>{x) is perfectly arbitrary, let it be so chosen that '{xk)=0, ,«(p+l)!(l + j-') A choice of ^(x) satisfying the required conditions is Q^ np—i . ^p—1 4>ix) = "(2? -1)1 {ao+aix+CiX^ + . . . +a„a;")p O np-l.-cp-l - (p-1)! (^("))''- of which every xj is a p-tuple root. If <^(x) is expanded and the result compared with ^(x)=feo+6iX+. . .+b,x', it is plain that feo = 0, &i=0, . . . , 6p_2=0, on account of the factor xP~i ; and Op-i- (p_l)! ' ''p- (p_i)! ^' (p--l)! ' 28 INFINITESIMAL ANALYSIS. where /p, . . . , /, are all integers. The coefficient of e'k in (6) may now be written If the arbitrary number p is chosen as a prime number greater than a^ and a„, Np becomes the sum of aoPa„"P~i, which carnot contain p as a factor, and a number of other integers each of which is divisible by p. Np therefore is not zero and not divisible by p. ip + t)\ Further, since , _^, , — j- is an integer divisible by p when 7^', it follows that all of the coefficients of the last block of terms in (6) contain p as a factor. If then (6) is added by columns, Npe^=pan^p-^[Po+PiXk+P2Xk'' + . . .+Ps-pXk'-p] 8 + Ib„-Xk"'-Rk„,. . (7) m= 1 where Po, Pi, . . . , P^^p are integers. It remains to show tha.tIb,n-Xk"'-Rk„, can be made small at will by a suitable choice of the arbitrary p. As in the proof of the transcendence of e, it follows that ' Op |rifcp| = Ib^-Xk""- Rkm < 7 TT-j • e", m=l \P — 1^! where Q = \arJ'\a(}ao\ + \ai\a + . . . + |a„|a), and a is the largest of the absolute values of Xkik = l, . . . , n). If now p is chosen as a prime number, greater than unity, greater than ao . . . an and greater than c, and so great also that |rip| <-, it follows directly from equation (7) that THE SYSTEM OF REAL NUMBERS. 29 = Njfi+pan^p--^{PoSo+PiSr + . . .+P._pS._p) + iVfcp, (8) where \rkp\ Ihrn-XlT-Jtlcm 2^ 2^^> • ■ ■ covers I— I I— I. . the interval —1 1, because every point of —1 1 is a point of one of the intervals. On the other hand a set of segments -10,-^ 1, . . . , ^ o^^j ^*c-j does not cover the interval because it does not include the points — 1, 1, n, . . . , ^^ , . . . , or 0. In order to obtain a set of segments which does cover the inter- val, it is necessary to adjoin a set of segments, no matter how small, such that one includes -1, one includes 0, one includes 1, 2) 4l • • • The segment including 0, no matter how small it is, must include an infinitude of the points ^, and there are only a finite number of them which do not lie on that segment. It therefore follows that in this enlarged set there is a subset of segments, 34 INFINITESIMAL ANALYSIS. I— I finite in number, which includes all the points of -1 1. This turns out to be a general theorem, namely, that if any set of seg- ments covers an interval, there is a finite subset of it which also covers the interval. The example we have just given shows that such a theorem is not true of the covering of an interval by a set of intervals; furthermore, it is not true of the covering of a seg- ment either by a set of segments or by a set of intervals. I— I Theorem lo.t If an interval a b is covered by any set [a] of segments, it is covered by a finite number of segments ai, . . . , (j„ of [o]. Proof. — It is evident that at least a part of a 6 is covered by a finite number of 0, y=0, y=b>0 determine the boundary of P. Let O^yi^b. Upon the interval i of the line SETS OF POINTS AND OF SEGMENTS. 37 y=yi, cut off by P, those parallelograms of [p] that include points of i as interior points determine a set of segments [;r] such that every point of t is an interior point of one of these seg- ments ;:. There is by Theorem 10 a finite subset of [n], 7:1. . . r.n, including every point of i, and therefore a finite subset pi . . . pn of [p], including as interior points every point of i. Moreover, since the number of pi . . . p„ is finite, they include in their interior all the points of a definite strip, e.g., the points be- tween the lines y=y\—e and y=yi-\-e. y=b 1/1 V=o Fig. 4. Thus for every y\ {O^yi^b) we obtain a strip of the parallel- ogram P such that every point of its interior is interior to one of a finite number of the parallelograms [p]. These strips in- tersect the j/-axis in a set of segments that include every point of the interval b. There is therefore, by Theorem 10, a finite set of strips which mcludes every point in P. Smce each strip is included by a finite number of parallelograms p, the whole parallelogram P is included by a finite subset of [p]. The generalization of Theorems 11 and 12 is left to the reader. § 3. Limit Points. Theorem of Weierstrass. Definition.— A neighborhood or vicinity of a point a in a line (or simply a line neighborhood of a) is a segment of this fine such that a lies within the segment. We denote a line neighborhood 38 INFINITESIMAL ANALYSIS. of a point a by V{a). The symbol V*(a) denotes the set of all points of V{a) except a itself. The symbols F(oo) and 7*(oo) are both used to denote infinite segments a + oo , and F( — co ) and 'F'*( — oo) to denote infinite segments — ooa.f A neighborhood of a point in a plane (or a plane neighbor- hood of a point) is the interior of a parallelogram within which the point lies. A neighborhood of a point (a, b) is denoted by V(a, b) if (a, 6) is included and by V*{a, b) if (a, b) is excluded. Instead of the three linear vicinities V(a), V(), and F( — oo) we have the following nine in the case of the plaile : V(-aj,aj) V(-oo,6) V (o, oo ,) ■ V(o, 6) V (oofoo) V (oo.fc) V(— oo, 05) V(a,- oo) V(oo,— od) Fig. 5, t This notation is taken from Pibhpont's Theory of Functions of Real Variables. It is used here, however, with a meaning slightly different from that of PlERPONT. SETS OF POINTS AND OF SEGMENTS. 39 It follows at once from a consideration of the scheme for setting the points on the Une into correspondence with all numbers that in every neighborhood of a point there is a point whose corresponding number is rational. Definition. — A point a is said to be a limit point of a set if there are points of the set, other than a, in every neighbor- hood of a. In case of a line neighborhood this says that there are points of the set in every V*{a). In the planar case this is equivalent to saying that (a, b) is a limit point of the set [x, y], either if for every F*(a) and V{b) there is an (x, y) of which x is in y*{a) and y in F(6), or if for every V(a) and 7* (6) there is an {x, y) of which x is in 7(a) and y in V*(b). Thus is a limit point of the set I ^ I, where k takes all positive integral values. In this case the limit point is not a point of the set. On the other hand, in the set 1, 1-J, 1 — 2^, . . . , 1 — oA' • • ■ > 1 is a limit point of the set and also a point of the set. In this case 1 is the least upper bound of the set. In case of the set 1, 2, 3, the number 3 is the least upper bound without being a limit point. The fimdamental theorem about limit points is the following (due to Weierstrass) : Theorem 13. Every infinite hounded set [p] of points on a line has at least one limit point. Proof. — Since the set [p] is bounded, every one of its points lies on a certain interval a b. li the set [p] has no limit point, I— I then about every point of the interval a b there is a segment a which contains not more than one point of the set [p]. By Theorem 10 there is a finite set of the segments [a] such that every point of a 6 and hence of [p] belongs to at least one of them, but each a contains at most one point of the set [p], whence [p] is a finite set of points. Since this is contrary to the hypothesis, the assimiption that there is no limit point is not tenable. 40 INFINITESIMAL ANALYSIS. It is customary to say that a set which has no finite upper bound has the upper bound + oo , and that one which has no finite lower bound has the lower bound — oo . In these cases, since the set has a point in every F*( + oo ) or in every V*( — oo ) + 00 and — 00 are also called limit points. With these con- ventions the theorem may be stated as follows: Theorem i6. Every infinite set of points has a limit point, finite or infinite. The theorem also generahzes in space of any number of dimensions. In the planar case we have: Theorem 17. An infinite set of points lying entirely within a parallelogram has at least one limit point. Theorem 17 is a corollary of the stronger theorem that fol- lows: Theorem 18. // [{x, y)] is any set of number pairs and if a is a limit point of the numbers [x], there is a value of b, finite or + 00 or —00, sv£h that for every V*{a) and V{b) there is an (x, y) of which x is in V*{a) and y is in V{b). Proof. — Suppose there is no value b finite or +00 or — 00 such as is required by the theorem. Since neither +00 nor - 00 possesses the property required of b, there is a 7* (a) and a F(oo ) and a V{-0, however small, there is some n, say n«, such that \bn,—an,\ S„, the sum of the first sideration that any relation might be so expressed led Lejedne Dirichlet to state his celebrated definition, which is the one given above. See the Encyclopadie der mathematischen Wissenschaften, II A 1, pp. 3-5; also Ball's History of Mathematics, p. 378. * 46 INFINITESIMAL ANALYSIS. n terms of a series, is a function of n where n takes only integral values. Again, the amount of food consumed in a city is a function of the number of people in the city, where the inde- pendent variable takes on only integral values. Or the inde- pendent variable may take on all values between any two of its values, as in the formula for the distance fallen from rest by a body in time t, s = -n- 'It follows from the correspondence between pairs of num- bers and points in a plane that the functional relation between two variables may be represented by a set of points in a plane. The points are so taken that while one of the two numbers which correspond to a point is a value of the independent variable, the other number is the corresponding value, or one of the corresponding values, of the dependent variable. Such representations are called graphs of the function. Cases in point where the function is single-valued are: the hyperbola referred to its asymptotes as axes \y=-) ; a straight line not parallel to the y axis {y=ax+b); or a broken line such that no line parallel to the y axis contains more than one of its points. In general, the graph of a single-valued function with a single- valued inverse is a set of points [{x, y)] such that no two points have the same x or the same y. Following is a graph of a function where the independent variable does not take all values between any two of its values. Consider Sn, the sum of the first n terms as a function of w in the series „ , 1 1 1 The numbers on the x axis are the values taken by the independent variable, while the functional relation is repre- sented by the points within the small circles. Thus it is seen that the graph of this function consists of a discrete set of points. (Fig. 6.) SPECIAL CLASSES OF FUNCTIONS. 47 The definition of a function here given is very general. It will permit, for instance, a function such that for all rational values of the independent variable the value of the function is 4 5 Fig. 6. unity, and for irrational values of the independent variable the value of the function is zero. § 2. Bounded Functions. Since the definition of function is so general there are few theorems that apply to all functions. If the restriction that fix) shall be bounded is introduced, we have at once a very im- portant theorem. Definition. — ^A function, f(x), has an upper bound for a set of valves [x] of the independent variable if there exists a finite nimaber M such that f{x) m for every value of X in {x\. A function which for a given set of values of X has no finite upper bound is said to be unbounded on that set, or to have an upper bound + oo on that set, and if it has 48 INFINITESIMAL ANALYSIS. no lower bound on the set the function is said to have the lower bound — 00 on the set. I — 1 Theorem 19. If on an interval aba Junction has an upper bound M, then it has a least upper bound B, and there is at least I — I one value of x, x\ on a b such that the least upper bound of the function on every neighborhood of Xi contained in a b is B. Proof. — (1) The set of values of the function f{x) form a bounded set of numbers. By Theorem 4 the set has a least upper bound B. I — I (2) Suppose there were no point xi on a b such that the least upper bound on every neighborhood of x\ contained in I — I — I— I a—bisB. Then for every x of a 6 there would be a segment Ox containing x such that the least upper bound of /(x) for I— I _ values of x common to (Ji and a 6 is less than B. The set [tTj] is infinite, but by Theorem 10 there exists a finite subset [<;„] of I — 1 the set [ctJ covering a b. Therefore, since the upper bound of fix) is less than B on that part of every one of these segments of [on] which lies on a b, it follows that the least upper bound I — I _ of /(x) on a 6 is less than B. Hence the hypothesis that no point Xi exists is not tenable, and there is a point Xi such that the least upper bound of the function on every one of its I — I _ neighborhoods which lies in a 6 is B. This argument applies to multiple-valued as well as to single- valued functions. As an exercise the reader may repeat the above argument to prove the following: I — I Corollary. — If on an interval 06a function has an upper bound + 00 , then there is at least one value of x, xi on a b such that in every neighborhood of xi the upper bound of the func- tion is -I- 00 , SPECIAL CLASSES OF FUNCTIONS. 49 § 3. Monotonic Functions ; Inverse Functions. Definitions.— If a single-valued function f{x) on an interval a b is such that /(x,) /(X2) whenever xi /(xi) and /(x2)>/(x3), while xtsin Fig. 8. which the function is monotonic. Such a function may be called partitively monotonic (Abteilungsweise monoton). The function /(x) =sin -, for a; 5^0, and /(i) =0, for a; =0, is an example of a function with an infinite number of oscillations on SPECIAL CLASSES OF FUNCTIONS. 51 every neighborhood of a point. fix)=x sin -, for Xf^O, /(O) = 0, and f{x)=x^ sin -, for Xf^O, /(0)=0 have the above property -and also are contmuous (see page 61 for meaning of the term continuous function). There exist continuous functions which have an infinite number of oscillations on every neighborhood of every point. y — x sin i Fig. 9- The first function of this type is probably the one discovered by Weierstrass,t which is continuous over an interval and does not possess a derivative at any point on this interval (see page 150). t According to F. Klein, this function was discovered by Weierstrass in 1851. See Klein, Anwendung der Differential- und Integralrechnung auf Geometrie, p. 83 et seq. The function wa^ first published in a paper en- titled Abhandlungen aus der FunctionerUehre, Du Bois Beyuond, CreU^s .Journal, Vol. 79, p. 29 (1874). 52 INFINITESIMAL ANALYSIS. Other functions of this type have been published by Peano, Moore, and others.f These latter investigators have obtained the function in question in connection with space-filling curves. Theorem 20. If y is a monotonic function of x on the interval a b, with bounds A and B, then in turn xisa svngle-valued monotonic I — I function of y on A B, whose upper and lower bounds are b and a. Proof. — It follows from the monotonic character of 1/ as a function of x that for no two values of x does y have the same Fig. 10. I 1 value. Hence for every value of y on A JB there exists one and t G. Peano, Sut une courbe, qm remplit toiUe une aire plane, Mathematische Annalen, Vol. 36, pp. 157-160 (1890). Cesaro, Sur la representation analy- tique des regions et des courbes qui les remplisent, Bulletin des Sciences Mathi- mati/juss, 2d Ser., Vol. 21, pp. 257-267. E. H. Moore, On Certain Crinkly Curves. Transactions of the American Mathematical Society, Vol. 1, pp. 73-90 (1899). See also Steikitz, Mathematische Annalen, Vol. 52, pp. 58-69 (1899). SPECIAL CLASSES OF FUNCTIONS. 53 only one value of x. That is, a; is a single-valued function of y.f Moreover, it is clear that for any three values of y, yi, 2/2, 2/3, such that 2/2 is between ?/i and j/3, the corresponding values of ^, Xi, X2, X3, are such that x-^ is between Xi and Xa, i.e., x is a monotonic function of y, which completes the proof of the theorem. CcroUary.— If a function /(x) has a finite number ^ of oscil- lations and is constant on no interval, then its inverse is at most (A + l)-valued. For example, the inverse of y = x^ is double- valued. § 4. Rational, Exponential, and Logarithmic Functions. Definitions. — The symbol a"", where wi is a positive integer and a any real number whatever, means the product of m factors a. This definition gives a meaning to the symbol y=a„x'"+am-ix"'-^ + . . .+aix + ao, •where oo . . . a^ are any real numbers and m any positive inte- ger. In this case y is called a rational integral function of x or a polynomial in x.t In case amX"'+am_iX"'~^+ . . . +ai-x+ao ^~ 6„x" + 6„_iX"-»+ . . . +6i-x+6o ' m and n being positive integers and a* {k=0, . . .m) and bi (Z = 0, . . . n) being real numbers, y is called a rational function of X. If yn+yfi-lR^(x)+y^-m2{x)+ ... +yRn.l{x) +Rnix) =0, where Ri{x) . . . R„(x) are rational functions of x, then y is said to t it is clear that the independent variable y of the inverse function may not take on all values of a continuum even if x does take on all such values. % The notion of polynomial finds its natural generalization in that of a power series y=c^ + c-x + C2-x'+ . . . +c„i"+ . . . For conditions under which a series defines y as a. function of x see Chapter IV, § 3. 64 INFINITESIMAL ANALYSIS be an algebraic function of x. Any function which is not algebraic is transcendental. The symbol a', where a; = — , m and n being positive integers and a any positive real number, is defined to be the nth root of the mth power of a. By elementary algebra it is easily shown that o*i.a=^=o*'+*» and {a''^)'"=a''^'". If y=a', then 2/ is an exponential function of x. At present this function is defined only for rational values of x. Fia. 11. Theorem 21. The function a' for x on the set \ — \ is a monotonic increasing function if l— , then «i ni Til Wi SPECIAL CLASSES OF FUNCTIONS. 55 c*'l and a*'>a*» if a =a"i"2 and a"2 =o"«"i, where mi 712 >in2-ni, which reduces case (c) to case (&). This theorem makes it natural to define a", where a > 1 and x is a positive irrational number, as the least upper bound of all r "1 m numbers of the form La" J, where — is the set of all posi- tive rational nimibers less than x, i.e., a'^=Bta^J. It is, however, equally natural to define a" as ^Lo^J, where I I is the set of all rational numbers greater than x. We shall prove that the two definitions are equivalent. Lemma. — If [x] is the set of all positive raiional numbers, then 5[a^] = l ifa>l and B[a='] = l ifal, the argument in the other case being similar. If x is any positive 7?l 1 rational number, — , then the number - is less than or equal i_ ^ fl 1 to x, and since a' is a monotonic function, a'* 1, there is a number of the form 1+e, where e>0, such that l + eH-ne, and the latter expression is clearly greater than a if a n>-. e Since 5La"J cannot be either greater or less than 1, Theorem 22. // x is any real number, and \~\ the set of all rational numbers less than x, and \-\the set 0} all rational numbers greater than x, then 5La"J = BLaO ifa>l, Bla'^j^Bla'^J ifOl, — 2 n] is zero, _ ai-a„J=Blai[l-a«~ vJJ is also zero. Now if B[a'^]^B[af'], SPECIAL CLASSES OF FU^CT10I^S. 57 Since as is always greater than a", Bla^j-Bla^j = £>0. But from this it would follow that p m a9 -a" is at least as great as e, whereas we have proved that _ a«-o"J=0. Hence 5La»J = sLa9j ifa>l. Definition. — In case x is a positive irrational number, and I — I is the set of all rational numbers greater than x, and [— is the set of all rational numbers less than x, then a» = Bla^J - 5L0 » J if a > 1 and a*=5La«] = B[a"] ifOl, and a monotonic decreasing function if 00) to the base a{a>0) is a number y such that a^=x, or a^°^'==x. That is, the func- tion logo X is the inverse of a''- The identity gives at once logo xi+ logo X2= logo (xi ■ X2) , and (a*')^'=a''"^^ gives Xi-logoX2=logaX2'^. By means of Theorem 20, the logarithm logaX, being the inverse of a monotonic function, is also a monotonic function, increasing if 1 < a and decreasing if 0< a < 1. Further, the func- tion has the upper bound + 00 and the lower bound — 00 , and takes on all real values as x varies from to +00 . Thus it follows that f or i< a, 1 < b, B(\ogb x) =log6 a=log6 (Bx). By means of this relation it is easy to show that the function x", (a;>0) is monotonic increasing for all values of a, a>0, that its lower bound is zero and its upper bound is + w , and that it takes on all values between these bounds. The proof of these statements is left to the reader. The general type of the argument required is exemplified in the following, by means of which we infer some of the properties of the function x". If xi < xz, then l0g2 Xi < log2 X2, and Xi • log2 xi < X2 • log2 X2, and log2 xi *' < log2 X2*». .". Xi*>0) is a monotonic increasing function of x. Since the upper bound of x ■ log2 x = log2 x^ is + <» , the upper bound of X* is +00. The lower bound of x* is not negative, since x>0, and must not be greater than the lower bound of 2=^, since if x<2, x''<2*; since the lower bound of 2* is zerot the lower bound of x* must also be zero. Further theorems about these functions are to be found on pages 64, 81, 97, 123, and 160. •f The lower bound of a' is zero by Theorem 23. CHAPTER IV. THEORY OF LIMITS. §1. Definitions. Limits of Monotonic Functions. Definition. — If a point a is a limit point of a set of values taken by a variable x, the variable is said to approach a upon the set; we denote this by the symbol x = a. a may be finite or +00 or — 00 . In particular the variable may approach a from the left or from the right, or in the case where a is finite, the variable may take values on eacl^ side of the limit point. Even when the variable takes all values in some neighborhood on each side of the limit point it may be important to consider it first as taking the values on one side and then those on the other. Definition. — A value b (6 may be + oo or - oo or a finite number) is a value approached by f{x) as x approaches a if for every V*{a) and V(b) there is at least one value of x such that x is in V*{a) and f(x) in V(b). Under these conditions f(x) is also said to approach 6 as x approaches a. Definition. — If b is the only value approached as x ap- proaches a, then b is called the limit of f(x) as x approaches a. This is also indicated by the phrase "f{x) converges to a unique limit b as X approaches a," or "f{x) approaches b as a limit," or by the notation L fix) =b. x—a The function f{x) is sometimes referred to as the limitand. The set of values taken by x is sometimes indicated by the sym- bol for a limit, as, for example, 60 THEORY OF LIMITS. 61 L j(x)=b or L f{x)=b or L f{x)=b. x>a xh; then since Bf{x) =b, there would be no value of fix) between b and b', that is, there would be a V(b') which could contain no value of fix) , whence b'>b 'm not a value approached. Sup- pose b'b". If xia x0 there exists a Vt*(a) such that for every x mV*(a), \f{x)-b\0 there exists a 5, >0 sv^h that if \x—a\0 there shall exist a V*{a) such that if Xi and X2 are any two values of x in V,*{a), then |/fe)-/(x2)|0 there ■exists a V *(o) such that if xi and X2 are in V *{a), then l/(^i)-6|<| and \fix2)-b\<^, from which it follows that |/(Xl)-/fe)|<£. tThe E subscript to ^. or to F.*(o) denotes that d, or V,*{a') is a func- tion of e. It is to be noted tha' inasmuch as any number less than S is effective as dt, dt is a multiple- valued function of e. THEORY OF LIMITS. 67 (2) The condition is sufficient. If the condition is satisfied, there exists a V*{a) upon which the function j{x) is bounded. For let 7 be some fixed number. By hypothesis there exists a 'V*{a) such that if x and Xo are on y*(o), then |/(x)-/(xo)| there exists a y,*(a) such that if xi and X2 are any two valves of x in V*{a)^ |/(xi) — /(X2)| < £. Hence by the definition of value approached there is an x, of 7.* (a) for which W.)-'b\<^ (a) and |/(x,)-/(x)l<£ (6) for every x of F.*(a). Hence, combimng (a) and (6), for every x of 7 *(o) we have l/(x)-6|<2£, and hence by the preceding theorem we have L/(x)=6. x-a In case a as well as 6 is finite. Theorem 27 becomes: A necessary and sufficient condition thai Lfix) x±a shaU exist and be finite is that for every e>0 there exists a d.>0 such that \i{Xx)-KXi)\<^ 68 INFINITESIMAL ANALYSIS. for every Xi and X2 such that XiT^a, XzT^a, \xi—a\ there exists a N.>0 such that \Kxi)-f{x2)\N„ X2>N,. The necessary and sufficient conditions just derived have the following evident corollaries : Corollary 1. The expression Lfix)=b, x~a where b is finite, is equivalent to the expression Lif{x)-b)=0, and whether h is finite or infinite L fix) =6 is equivalent to L (-fix)) = —6. Corollary 2. The expressions L/(a;)=0 and L |/(a;)|=0 are equivalent. Corollary 3. The expression Lfix)=b is equivalent to Lfiy+a)=b, »=o where y+a=x. THEORY OF LIMITS. Corollary 4. The expression L/(i)=6 x0 there exists an integer N. such thai if n>N, and n'>N., then \S„-Sn'\0 there exists an integer N. swh that if n>N„ then for every k |o„+o„+i + . . .+a„+fc|0. In this series „ l_rn+l a which shows that the series is convergent. Moreover, it can easily be seen to have the sum :; — . •^ l-r If rf 1, the geometric series is evidently divergent. This result can be used to prove the "ratio-test " for convergence. Theorem 30. // there exists a number, r, 01 for every n, the series is divergent. Proof. — ^The series (1) may be written 02 02 03 . , az an ,n~. Oi+Oi-+ai + ■•• +°'^:r ■•■n — . ■ • (2) 74 and if On Ctn-l INFINITESIMAL ANALYSIS. 0 there exists an integer N. such that if n>N., then \Rn\ £>0, there exists a Vi*{a) for every x of which and a F2*(o) for every x of which Ux)\<^. Hence in any V*(a) common to Vi*{a) and V2*{a) \hix)+f2ixMfi{x)\ + Ux)\f{x)>m for every x on V*{a). Let k be the larger of |m| and \M\. Also by hypothesis there exists for every e a V,*(a) within V*{a) such that if x is in V*ia), then or A;|£(x)|<£. But for such values of x \f{x)-E{x)\E-M. Since hix)>M, this gives hix)+J2{x)>E, which means that /i(x) +J2{x) approaches the limit + 00 . 78 INFINITESIMAL ANALYSIS. Theorem 36. 7/ L/i(x) = + oo or —00, and if f2(x) is such that for a F*(a) /2(x) has a lower bound greater than zero or an upper boundless than zero, then L {/i(x) 72(2;) } is definitely infinite; x±a i.e., if /2(x) has a lower bound greater than zero and Lfi{x) = + co, z = a then L {fi{x) ■f2{x) ! = +<», etc. x~za Proof. — Suppose /2(x) has a lower bound greater than zero, say M, and that L fi{x) = + qo . Then for every E there exists a x~a - E Te*{o) within V*{a) such that for every Xi of F£*(a), fi{xi) > t^, and therefore fi{xi)-f2{xi)jfi{xi)-M>E. Hence by the defini- tion of Umit of a function Lj/i(x) ■/2(a;)! =+». If we consider the case where /2(x) has an upper bound less than zero, we have in the same manner L {/i(x) •/2(a;) t = — <» ■ Similar state- x±a ments hold for the cases in which L /i(x) = — 00 . Corollary. — If /2(x) is positive and has a finite upper bound .andL/i(x) = +oo, thea iv ■ , ^ = + 00 . X±af2{x) Theorem 37. If L /(x) = + (», then L 77-: = 0, and there is' a x~a x~aJ\X) ■vicinity V*{a) upon which /(x) >0. Conversely, if L /(x) =0 and x=a there is a V*{a) upon which /(x) > 0, then L jr— = + 00 . Proof. — If L f{x) = + 00 , then for every e there exists a x-±a V*{(i) such that if x is in V*{a), then /(^)>7 THEORY OF LIMITS. 79 and 77-T<«- since both f{x) and 7^ are positive. Again, if L /(x) =0, then for every e there is a 7.* (a) such x=.a that for X in V*{a), |/(x)|- (/(x) being positive). Hence L 7^-r = + 00 . Corollary 1. If /i(x) has finite upper and lower bounds on some V*{a) and L /2(x) = + « or - 00 , then Corollary 2. If /2(x) is positive and /i(x) has a positive lower bound on some V*{a) and L f2{x)=0, then x£o £( . , . = +00. x=a/2(a;) Theorem 38 (change of variable). If (1) L/i(j)=6i and L/2(2/)=&2 wfecn y takes all values of /i (x) corresponding to valves of x on some V*{a), and if (2) /i{j) f^bi for X on V*{a), then L /•,(/, (x)) =62. x-a 80 INFINITESIMAL ANALYSIS. Proof.— (a) Since L /zCj/) =62, for every 7(62) there exists a F*(6i) such that if y is in V*(bi), jiiy) is in V{hi). Since L fi{x) =61, for every F(6i) there exists a 7*(a) in F*(a) such xda that if a; is in y*(o), /i(x) is in F(6i). But by (2) if x is in V*{a), fi{x)^bi. Hence (/?) for every V*(bi) there exists a V*{a) such that for every x in V*(a), /i(x) is in V*{bi). Combining statements (a) and (/3) : for every 1^(62) there exists a F*(a) such that for every x in V*{a) fi{x) is in F*(6i), and hence fzifix)) is in F(62). This means, according to Theo- rem 26, that Lf2ih{x))^b2. x~a Theorem 39. If L /i(x) =6 and L f2(y) =hQ>)} where y takes x±a yxb all valves taken by /i(x) for x on some V*ia), then L/2(/i(x))=/,(6). x~a Proof. — The proof of the theorem is similar to that of Theorem 38. In this case the notation /2(6) implies that 6 is a finite number. Thus for every ej there exists a V,*{a) entirely within V*{a) such that if x is in V,*{a), |/i(x)-fe|0 and oo*:=0ifjfc<0. Corollary 3. If OO and /(x)>0 and 6>0 and Lf{x)=b, ^'^ Llog,/(x)=log,6, under the convention that log,. ( + w ) = + oo and log„ = - oo . The conclusions of the last two corollaries may also be ex- pressed by the equations L(/(x))* = (L/(x))* x—a ar^a and log, L/(x)=L log, '(x). x'-a x±a Corollary 4. If L (/(x))* or L log /(x) fails to exist, then L /(x) does not exist. § 5. Further Theorems on Limits. Theorem 40. 7/ f{x):^b for all valves of a set [x] on a certain V*{a), then every value approached by fix) as x approaches a is less than or equal to b. Similarly if fix)^b for all values of a set [x] on a certain F*(o), then every valve approached by /(x) as x approaches a is greater than or equal to b. Proof. — If f{x)%b on V*{a), then if b' is any value greater than b, and V{b') any vicinity of b' which does not include b, there is no value of x on V*(a) for which /(x) is in V{b'). Hence V is not a value approached. A similar argument holds for the case where f{x)^b. 82 INFINITESIMAL ANALYSIS. Corollary 1. If /(x)^0 in the neighborhood of x=c, then if L fix) exist, L /(x)^0. Corollary 2. If fiixj^fzix) in the neighborhood of x=a, then L /i(x)2 L U^) x±a i£o if both these Umits exist. Proof.— Apply Corollary 1 to fi(x)-f2{x). Corollary 3. If t\{x)'^J2{x) in the neighborhood of x=a, then the largest value approached by /i(x) is greater than or equal to the largest value approached by /2(x). Corollary 4. If /i(x) and ^(x) are both positive in the neighborhood of x=a, and if /i(x)^/2(x), then if L /i(x)=0, it follows that L/2(x)=0. Theorem 41. // [ucf] is a subset of [x], a being a limit point of [x'], and if L f{x) exists^ then L /(a/) exists and x-±a x±a L f{x) = L fix').^ x~a x'±a Proof. — By hypothesis there exists for every V{b) a V*{a) such that for every x of the set [x] which is in V*{a), f{x) is in F(6). Since [x'] is a subset of [x], the same V*{a) is evidently efficient for x on [x']. In the statement of necessary and sufficient conditions for the existence of a limit we have made use of a certain positive multiple-valued function of e denoted by 8,. If a given value is effective as a d„ then every positive value smaller than this is also effective. Theorem 42. For every e for which the set of valves of d, has an upper bound there is a greatest d,. t The notation /(i') is used to indicate that x takes the values of the 8et[x']. THEORY OF LIMITS. 83 Proof. — ^Let B[d.] be the least upper bound of the set of values oi d, for a particular e. If x is such that \x—a\ < B[8,], then there is a d, such that |x-a|<^,. But if \x-a\l, then there are two vicinities of k, V]ik) contained x=a in V2ik) and V2ik) not containing I. By Theorem 26 a Vi*ia) exists such that if x is in 7i*(a), bix) is in V^ik). Further- more, by the definition of bix), if Xi is an arbitrary value of x on Fi*(a), then there is a value of x in a Xj such that fix) is in Vik). Hence k would be a value approached by fix) con- trary to the hypothesis k>l. § 6. Bounds of Indetennination. Oscillation. It is a corollary of Theorem 43 that in the approach to any point a from the right or from the left the least upper bound and the greatest lower bounds of the values approached by fix) are themselves values approached by fix). The four num- bers thus indicated may be denoted by fia+0) = L fix) = L fix), 1=0+0 lia 84 INFINITESIMAL ANALYSIS. the least upper bound of the values approached from the right: /(a-0)=L/(x)=L/(x), x=a — x—a the least upper bound of the values approached from the left: j{a+0)=L f(x)=L fix), x'-a+O <— the greatest lower bound of the values approached from the right : /(o-0)=L/(x)=L/(x), lio- — > x±a the greatest lower bound of the values approached from the left. If all four of these values coincide, there is only one value approached and L fix) exists. If /(a+0) and fia+0) coincide, x=a ■ this value is denoted by /(a+0) and is the same as L fix). x>a x±a Similarly if /(o-O) and /(a-0 ) coincide, their common value, L fix), is denoted by /(a - 0) . The larger of /(o+O) and /(a-0) xC2 and fix")C2 and /(x")C2, or no x" such that f{x") C2 and f{x")b2 is the least upper bound of the values approached; 02 may then be so chosen that b2B. Therefore B cannot be the least upper bound. Since the least upper bound may not be either less than 62 or greater than 62, it must be equal to 62- A similar argument will prove bi to be the greatest lower bound of the values approached. CHAPTER V. CONTINUOUS FUNCTIONS. § I. Contintiity at a Point. The notion of continuous functions will in this chapter, as in the definition on page 61, be confined to single-valued func- tions. It has been shown in Theorem 34 that if /i(i) and fzix) are continuous at a point x=a, then /lU)±/2(l), /l(x)-/2(x), /i(x)//2(x), (/2(X)?^0) are also continuous at this point. Corollary 1 of Theorem 39 states that a continuous function of a continuous function is continuous. The definition of continuity at a;=a, namely, Lf{x)=f{a), x-a is by Theorem 26 equivalent to the following proposition : For every £>0 0 such that if \x-a\0 there exists a d,>0 such that if \xi-a\ there exists a V,{o-) such that on V,(o) the oscillation of f{x) is less than e. Theorem 46. // f{x) is continvaus at a point x=a and if f(a) is positive, then there is a neighborhood of x=a upon which the function is positive. Proof. — If there were values of x, [x'] within every neighbor- hood of x=a for which the function is equal to or less than zero, then by Theorem 24 there would be a value approached by /(a/) as a/ approaches a on the set [a/]. That is, by Theorem 40, there would be a negative or zero value approached by f{x), which would contradict the hypothesis. § 2. Continuity of a Function on an Interval. Definition. — ^A function is said to be continuous on an in- terval a 6 if it is continuous at every point on the interval. Theorem 47. If f{x) is cordinujous on a finite interval a b, then for every e>0,a b can be divided into a finite number of equal intervals upon each of which the oscillation of f{x) is less than e.f t The importance of this theorem in proving the properties of continu- ous functions seems first to have been recognized by Goursat. See his Coura d' Analyse, Vol. 1, page 161. CONTINUOUS FUNCTIONS. 89 Proof. — By Theorem 45 there is about every point oi a b a. segment a upon which the oscillation is less than e. This set of segments [o] covers a b, and by Theorem 11 a 6 can be divided into a finite number of equal intervals each of which is interior to a there exists a d,>0 such that for any two values of x, Xi, and X2, on a b where \Xi-X2\S„ Xi and X2 are on the same F(x') and consequently |/(xi)— /(x2)li. Corollary. — Whether /j(x) and /2(x) are continuous or not, if L /i(x)=4-oo and L /2(x)=-oo, there exists a pair of CONTINUOUS FUNCTIONS. 93 sequences [xj and [x/] such that L {/l(Xi)+/2(x/)| »=« is + 00 or — 00 . Theorem 52. // y is a function, f{x), of x, monotonic and con- I — I tinuous on an interval a b, then x=f~^{y) is a function of y which is monotonic and continuous on the intenal f{a) f{b). Proof. — By Theorem 20 the function f~'^{y) is monotonic and has as upper and lower bounds a and h. By Theorems 50 and 51 the function is defined for every value of y between and including /(a) and f{b) and for no other values. We prove the I — I function contmuous on the interval /(a) f{b) by showing that it is continuous at any point y=yi on this interval. As y I- — I approaches j/i on the interval /(a) j/i, f~^(y) approaches a definite limit g by Theorem 25, and by Theorem 40 a0 there exists a 5,>0 such that for any two values of a/, Xi, and X2' an a b, for which \xi' -X2'\0 there exists a d,>0 such that for every x' for which \x' -x^| <5„ \fix')-f{x.,)\| <2£, e being arbi- trary and b a constant different from /(xi"). But this is con- trary to the fact proved above, that L j{x') exists and is equal to /(xi). Hence the function is continuous at every point of the I— I interval a b. The uniqueness of the function follows directly from Theorem 54. , This theorem can be applied, for example, to give an ele- gant definition of the exponential function (see Chap. III). We first show that the function a" is uniformly continuous on the set of all rational values between xi and X2, and then define CONTINUOUS FUNCTIONS. 97 a* on the continuum as that continuous function which coin- — TO cides with a" for the rational values — . The properties of the function then follow very easily. It will be an excellent exer- cise for the reader to carry out this development in detail. § 4. The Exponential Function. Consider the function defined by the infinite series x^ x^ x^ Applying the ratio test for the convergence of infinite series we have .71! ■ (n-1)! n If n' is a fixed integer larger than x, this ratio is always less than — < 1 The series (1) therefore converges absolutely for every n' value of X, and we may denote its sura by e(x). From Chap. I, page 17, we have that Theorem 56. .iMT- where [n] is the set of att positive irUegers, exists and is equal to e(x) for all values of x. 98 INFINITESIMAL ANALYSIS. Proof.— Let E„(x) = 2' ri fc=oA;! (where 0! = 1). Then, since \ n) " ■^(n-l)!'n"^(n-2)!-2!W "^•"'^nlW ' it follows that (x\ "I " / 1 t? ^ \ 1+-) = I (t-.-t ,,,• , J x* < " /I to(w-I) ■■■ (n-A: + l) )m < -^ ri~~fc F • A;!n* Now, since n*-(n-A; + l)* = (A-l){n*-i+n*-2.(n-A + l)+... + (n-A; + l)*-M<(i-l)A;-n*-i, it follows that ^„(x)-(l+3" < 2' .^•e{\x\) ~k=.2ik-2)l-n^ n For a fixed value of x, therefore, we have (l+|)"=E„(x) + ei(»), where ei(n) is an infinitesimal as n= oo. At the same time e{x)=E„{x) + £2{n), where e2{n) is an infinitesimal as n= w. Hence L (l+-Y=e(x). CONTINUOUS FUNCTIONS. 99 Theorem 57. L (iH — ) , where [z] is the set of all real numbers, exists and is equal to e{x). Proof. — If z is any number greater than 1, let nz be the integer such that ni^z0, l+£i l+|>l + j^j ■ . . . . (1) Henc (i+|)"""Mi+j)'>('+S:Ti)""' ' ' <^' « (-3(-3""=(-?)'>(-i:Vi)-*"-V- (3) Since L (l+f ) =1, and L (l+r4l)=l. and L (l+f) =e(x), and L (l+Z-Tj) =e(a;;, the inequaUty (3), together with Corollary 3, Theorem 40, leads to the result: L (l+4)'=e(x). The argument is similar if i<0. Corollary. ^L^(l+-) =e{x), ■where [z] is any set of numbers with limit point + oo. Theorem 58. The function e{x) is the same as e^ where 1 1 , 100 INFINITESIMAL ANALYSIS. Proof. — By the continuity of 3^ as a function of 2 (see Corol- lary 2 of Theorem 39), it follows that, since L (1+-)" =e, L (1+-) =e'. where 2=na;. Hence by Theorem 39 &=L (1+-)' and by the corollary of Theorem 57 the latter expression is equal to e(x) , Hence we have ex = i+a;+|+|^+ (1) (1) is frequently used as the definition of e*, a' being defined as e^logeO. CHAPTER VI. INFINITESIMALS AND INFINITES. § I. The Order of a Function at a Point. An infinitesimal has been defined (page 75) as a function f{x) such that L fix) =0. A function which is unbounded in every vicinity of a; = a is said to have an infinity at a, to be or become infinite at x = a, or to have an infinite singularity at x = a.t The recipro- cal of an infinitesimal at x=a is infinite at this point. A function may be infinite at a point in a variety of ways : (a) It may be monotonic and approach +00 or - 00 as x=a; for example, - as a; approaches zero from the positive side. (5) It may oscillate on every neighborhood of x=a and still approach + 00 or — 00 as a unique limit; for example, • 1 ^ sm--l-2 X as X approaches zero. t It is perfectly compatible with the.se statements to say that while fix) has an infinite singularity at x=a, /{o)=0 or any other finite number. For example, a function which is — for all values of x except x = is left undefined for i=0 and hence at this point the function may be defined as zero or any other number. This function illustrates very well how a function which has a finite value at every point may nevertheless have infinite smgularities. 101 102 INFINITESIMAL ANALYSIS. (c) It may approach any set of real numbers or the set of all real numbers; an example of the latter is . 1 sm- X as X approaches zero. See Fig. 13, page 64. (d) + 00 and — » may both be approached while no other number is approached; for example, - as a; approaches zero from both sides. Definition of Order. — If /(x) and 0(x) are two functions such that in some neighborhood V*{a) neither of them changes sign or is zero, and if where k is finite and not zero, then f{x) and {x) are said to be oi the same order aX- x= a. If x=a<{>ix) ' then f{x) is said to be infinitesimal with respect to 4>{x), and ix) is infinitesimal with respect to f(x), and fix) infinite with respect to {x), and 0(x) is infinite of lower order than /(x).t The independent variable x is usually said to be an infini- tesimal of the first order as x approaches zero, x^ of the second order, etc. Any constant j^O is said to be infinite of zero order, - is of the first order, ^ of the second order, etc. This usage, however, is best confined to analytic functions. In the general case there are no two infinitesimals of consecutive order. Evi- dently there are as many different orders of infinitesimals be- tween X and a;2 as there are numbers between 1 and 2; i.e., xi+* is of higher order than x for every positive value of k. c,. -r /l(x) 1 , ^ foix) oince L j-T-T^T whenever L '-rT~=k, we have x-aj2\X) K x=a hW Theorem 59. If /i(x) is of the same order as fzix), then fzix) is of the same order as f\{x). Theorem 60. The function cf{x) is of the same order as f{x), c being any constant not zero. Proof.— By Theorem 34, L ^^=c. Theorem 61. If fiix) is of the same order as fiix), and /2(x) is of the savfie order as fsix), then fiix) and fsix) are of the same order. t This definition of order is by no means as general as it might possibly be made. The restriction to functions which are not zero and do not change sign may be partly removed. The existence of x=a't>(.x) is dispensed with for some cases in § 4 on Rank of Infinitesimals and In- finites. For an account of still further generalizations (due mainly to Cauchy) see E. Borel, Series a Termes Positifs, Chapters III and IV, Paris, 1902. An excellent treatment of the material of this section together with extensions of the concept of order of infinity is due to E. Borlotti, CaJr colo degli Infinitesimi, Modena, 1905 (62 pages). 104 INFINITESIMAL ANALYSIS. Proof. — By hypothesis L 7^,— - = fci and L r-r-r=k2. By Theorem 34, ^L ^-^ ;£/^=,£ WV (By definition, /2(a;) 5^0 and jz{x) y^O for some neighborhood of x=a.) Hence 7- A^^) 7 7 x=al3W Theorem 62, 7/ /i(x) and /2(x) are infinitesimal (infinite) and neither is zero or changes sign on some V*{a), then f\{x) ■f2{x) is infinitesimal (infinite) of a higher order than either. Proof. L^-^fy^^=L/i(x)=0. (±00.) xia /2W xia Theorem 63. If fi(x), . . . , fn(x) have the same sign on some V*(a) and if f2(x), . . . , fn(x) are infinitesimal (infinite) of the same or higher (lower) order than fi(x), then fl(x)+f2(x)+f3(x)+...+fn(x) is of the same order as fi(x), and if /2(x), faix), . . . , f„(x) are of higher (lower) order than fi(x), then fi(x)±f2(x)±f3(x)±. . .+fn(x) is of the same order as /i (x) . Proof. — ^We are to show that J fl(x)+f2(x)+...+f„(x) i=a fl(x) By hypothesis, xi fAx) -"" xta fl(x) -"" •'•' xia M^^^"' A T A(^) 1 INFINITESIMALS AND INFINITES. 105 Hence, by Theorem 30, r [hix) f2ix) hix) Ux)\ since all the ^'s are positive or zero. Similarly, under the second hypothesis. ^ Mx)±f2(x)±...±Ux) ^ j^ /iJ^ + M^li +/"W ] \h(x) hix) ••• /i(x)i = 1+0 + . ..+0 = 1. Theorem 64. — // /3(x) and fi{x) are infinitesimals with re- spect to fi{x) and /aCx), then J \h{x)+fz{x)\-\h{x)+j^{x)] ^ , J l/i(a:)+/3(x}-l/2(x)+/4(x)| Froot. L ,/,.,, , ^ ^ h(x) -hix) +fx(x) -Uix) +/3(X) ■f2{x) +f3ix) -f.jx ) xj,a h(x)-f2{x) J /l(x)-/2fa) , J /1(X)-/4 (X) J hix)-f2(.x) f3{x)-U(x) xLhix) •/2(X) xl'a hix) ■f2{x) JlafM ■ h{x) \Lh{x) -^ix) ^• § 2. The Limit of a Quotient. Theorem 65. If as x = a, £i(x) is an infinitesimal vnth respect to /i(x) and szix) with respect to fzix), then the valv£s approached by /l(x) + £l(x) ^^^ M£) /2(X)+«2(X) ^^ /2(X) •as X approaches a are identical. 106 INFINITESIMAL ANALYSIS. Proof.— This follows from the identity ^i(x)\ /l(x)+£i(x) _ /i(x) f2{x)+e2{x) fiix)' /i ,i2_(i) 1 + f2{x)' Since tt4 and 4^,-^ are infinitesimal. flix) }2{X) Corollary. — If /i(x) and /2(x) are infinite at a;=a, then fiix)+c /i(x) f,{x)+d ^""^ /2(X) approach the same values. Theorem 66. // L ^,= L ^,^k, and if L^,=l • ^ V ,. i. T hi. x)+h{x) J hix) IS finite, then lc= L -,—, ^ , , , ^ = Li , , . , ' ' x^a^\{x)-V^2{x) xUaA-^^^y provided l^ -1 if k is finite, and provided l>0 if k is infinite. fi{x)+f2{x) f2{x) fi(x)2{x)-f2{x)Mx) Proof. 4>iix)+Mx) Mx) Mx){i{x)+Mx)) ' /l(x)+/2(x) _ Mx) //i(x) /2(X)\ 7 1 \ 4>l{x)+(j>2{x) 4>2{X) \(/)i(x) (f>2{x)l "I ^2{X) ). In case k is finite, the second term of the right-hand member is evidently infinitesimal if Ij^—l and the theorem is proved. In the case where k is infinite we write the above identity in the following form: /1(X)+/2(X) /l(x) 1__.^,/2(X) 1 4>i{x)-\-2(x) 0i(x) ^2(3:) (f>2{x) 9!>i(x) ' «^i.(x) %2(a;) INFINITESIMALS AND INFINITES. 107 Both terms of the second member approach + oo or both - oo if Z>0. Corollary. ~li i{x) and 962(3;) are both positive for some VHa), and if A= L ilM= L ^4^, then L i#tM^ =k whenever k is finite. If k is infinite, the condition must be added that , , - has a finite upper and a non-zero lower bound. Theorem 67. // /i(x) and /2(x) are both infinitesimals asx = a, then a necessary and sufficient condition that L , , - =k {k finite and not zero) x = a /2W is that in the equation fi(x)=k-f2{x) + s(x), e{x) is an infini- tesimal of higher order than /i(x) or fzix). Proof. — {lyThe condition is necessary. — Since L r~-( =/fc, X4a/2(X) }2{X) '' + '^^'^^' or /i(x)=/2(2;)-A;+/2(x)-£'(a;), where L £'(x)=0 (Theorem 31). x~a By Theorems 60 and 61, fi(x) and /2(x) -k are of the same order, since kf^O, while by Theorem 62 s'{x)-f2{x) is of higher order than either /i(x) or f2(x). Hence the function s{x) = s'(x) •/2(x) is infinitesimal. (2) The condition is sufficient. — By hypothesis /i(x) = f2(,x)-k + £(x), where /i(x) and /2(x) are of the same order as £(t) x=a, while e(x) is of higher order than these. Let e'{x) =ryT, /2W / (x) which by hypothesis is an infinitesimal. We then have j-j— t / \ = k + e'{x). Hence, by Theorem 31, L j-j-r = k. 108 INFINITESIMAL ANALYSIS. § 3. Indeterminate Forms.t a Lemma. — // j- and -y are any two fractions such that b and d are both positive or both negative, then the value of a + c b+d I — I ,. , . . a c lies on the interval j- -7. Proof. — Suppose b and d both positive and a ^a+c b^b+d' then db+ad^ab+ be. .'. ad^ be; .'. cd+ad^ cd+bc; a+c ^c • ■ 6+d = d* The other cases follow similarly. Theorem 68. // /(x) and (f>{x), defined on some F(+qo), are 'both infinitesimal as x approaches + 00 , and if for some positive number h, (f>{x+h) is always less than {x) and , f{x+h)-f{x) ^^ then „4>{x+h)—4>{x) exists and is equal to k.X f The theorems of this section are to be used in § 6 of Chap. VII. % This and the following theorem are due to O. Stolz, who generalized them from the special oases (stated in our corollaries) due to Cauchy. See INFINITESIMALS AND INFINITES. 109 Proof.— Let Vi(k) and Vzik) be a pair of vicinities of k such that Vzik) is entirely within T^(^-). By hypothesis there exists an h and an A'o such that if j > A'2, f{x + h)-fix) (l>{x+h)-i>{x) W is in V2{k). Since this is true for every x>X2, f{x + 2h)-fix + h) ix+2h)-{x+2h)-X2, jix+nh)-f{x) (j)ix+nh)—{x+nh)-^{x) {xy Hence for every a; and for every e there exists a value of n, Nx„ such that if n> Nxi, f{x+nh)-fix) fix) {x+nh)—(j)(x) {x) <£. Taking e less than the distance between the nearest end-points fix) cf Viik) and V2ik) it is plain that for every i>A'2, ttt is (pyX) Stolz und Gmeiner, Functionentheorie, Vol. 1, p. 31. See also the referenca to BoBTOLOTn given on page 103. 110 INFINITESIMAL ANALYSIS. on Vi{k), which, according to Theorem 26, proves that Corollary. — If [n] is the set of all positive integers and ^(n + 1) < {x+h)-<}>(x) "' then L ~rl exists and is equal to k. Proof. — By hypothesis, for every pair of vicinities Vi{k) and y2{k), V2(k) entirely within Vi(k), there exists an X2 such that if x>X2, then f{x+h)-f{x) 4{x+h)-{x) is in Viik). From this it follows as in the last theorem that f{ x+nh)-f{x) (x+nh) —{x) is in Vzik). Now make use of the identity fjx+nh) _ f{x+nh)-f{x) fix) 4>{x+nh) (j>{x+nh) ^{x+nh) INFINITESIMALS AND INFINITES. Ill f{x+nh)-fix) / {x) \ fix) '4>{x+nh)-{x)\ ix+nh)/'^{x+nhy ' ' ^^ Let [a/] be the set of all points on the interval X2 X2+h, and for this interval let A2 be an upper bound of \f{x^) | and B2 an upper bound of {x'). Then cl>{x') _ B2 4>{xf +nh) ~ ''^^' "^ ^ {X2+nh) and JL/!./ , An = «2(a/, n) < — - — <^(i' +n;i) ~ ''^ ' '"^ ^ <]>{X2 +nhy Hence for every £ there exists a value of n, N^y, such that if n>N£y^ £1(2/, n)Z2, f{x+nh)-f{x) {x+nh)-ix!+'nh) ix'+nh)-4>ixf) N£y, -'i(^»<2(Ztt;;:) and e2(x',n)<-^ for all values of 2/ on X2 X2 +h. Hence for n> AT J J, f(x'+nh) fix' + nh) -f{x') 4){x'+nh) ^{x'+nh)-4>{x') «7 , «v <^^+^^.)2(:^TT7;+T^'^ and since for a;> Z2 +Neyh there is an n> iVjj, and an a/ between X2 and X2+h such that 3/+nh=x, it follows that if a;> X2 +Ney, ]Kx) f(x'+nh)-fix') l |^(x) 4>(.^+nh)-ix)\'^^^' fix) and therefore, -j-p-!- is on Fi(A). This means, according to Theorem 26, that (2) A=+oo. If the numbers wii and ma are the lower end pomts of 7i(A;) and Y^ih), then <^(x'+nA)-,^(x')>"'2 forx'>Z2. INFINITESIMALS AND INFINITES. 113 If ev is then chosen less than m2-mi, there will exist a value of A'^g^ such that for all values of n>Ngy independently of x' so long as a/ is in I X2 X2 + h. Then, in view of (1), f{x'+nh) ii ^v \ ey ev 1 1 \ Since there is no loss of generality if m2> +1, this proves that for X >X2 +Nsyn, fix) >m2 — £v>mi, {x) (3) A; = — 00 is treated in an analogous manner. Corollary 1. If [n] is the set of all positive integers and if {n-\-\)> ^(n) and L ^(n) = 00 , 11 = 00 . .f r /(n + l)-/(n) then If „f.^(n + l)-^(n)-*' . it follows that L -yt-x exists and is equal to k „-=o?>W I — I Corollary 2. If f{x) is bounded on every interval, x (x + l), and if L f{x + l)-}{x)=k, I— 00 m then L XS3 00 x exists and is equal to k. 114 INFINITESIMAL ANALYSIS. § 4. Rank of Infinitesimals and Infinites. Definition. — If on some V*{a) neither /i(a;) nor /2(x) vanishes, and fi(x) f2ix) and Ihix) are both bounded as x approaches a, f (x) then fi{x) and /2(x) are of the same rank whether L j-i—. exists or not.t The following theorem is obvious. Theorem 70. // /i(x) and /2(x) are of the same order, they are of the same rank, and if fi{x) and f2{x) are of different orders, they are not of the same rank. If fi{x) and ^(a;) are of the same rank, they may or may not be of the same order. Theorem 71. // fi{x) and fiix) are of the same rank as x approaches a, then c-fi{x) and /2(x) are of the same rank, c being any constant not zero. Proof. — By hypothesis for some positive number M, AW hence /2(X) c-fiix) < — If we consider the curve representing the function . 1 " X at the point 2;=0, it is apparent that the Umiting position of A B does not exist, although the function is continuous at the point x=0 if defined as zero for x=0. For at every maxi- mum and minimum of the curve sin—, a; -sin = ±x, and the X X curve touches the lines x=y and x=—y. That is, • — — - — — ^ ' X—Xi approaches every value between 1 and —1 inclusive, as x approaches zero. The notion derivative is fimdamental in physics as well as in geometry. If, for instance, we consider the motion of a body, we may represent its distance from a fixed point as a function of time, /(<). At a certain instant of time h its dis- tance from the fixed point is /(ii), and at another instant ^2 it is f(t2) ; then ti—tz course indicated by the sign of the expression _ 120 INFINITESIMAL ANALYSIS. is the average velocity of the body during the interval of time ti-t2 in a direction from or toward the assumed fixed point. Whether the motion be from or toward the fixed point is of f{t l}-f(t2) . If we consider this ratio as the time interval is taken shorter and shorter, that is, as <2 approaches h, it will in ordinary physical motion approach a perfectly definite limit. This limit is spoken of as the velocity. of the body at the instant ti. Definition. — ^The derivative of a fimction y=f{,x) is denoted df(x) diV by fix) or by Di/(x) or -~^ '^^ T" /'(^) ^ ^^^° referred to as the derived Junction of /(x) . § 2. Formulas of Differentiation. Theorem 74. The derivative of a constant is zero. More precisely: If there exists a neighborhood of Xi such that for every valus of X on this neighborhood fix) =/(ii), then fixi) =0. Proof. — In the neighborhood specified =0 for every value of x. Corollary. — If f(xi) exists and if in every F*(xi) there is a value of X such that fix) =fixi), then fixi) =0. Theorem 75. When for two functions /i(x) and fzix) the derived functions fi'ix) and fz'ix) exist at xi it follows that, except in the indeterminate case 00 — 00 , (a) // fsix) =/i(x) +/2(x), then fzix) has a derivative at Xi and /3'(Xl)=/l'(Xi)+/2'(Xi). (b) If fsix) =/i(x) •f2{x), then fsix) has a derivative at Xi and fz'ix{) =/i'(xi) -/aCxi) +/i(xi) -//(xi). (c) // fzix) =-7-r\, then, provided there is a F(xi) wpon which ./2(x) 7^0, fzix) has a derivative and ,,, . //(Xl)-/2(Xl)-/l(Xi)-//(Xi) DERIVATIVES AND DIFFERENTIALS. 121 Proof. — By definition and the theorems of Chapter IV (which exclude the case oo — oo), (a) //(xO +h'{x.) = L /-iMzM^) + L /?M^MEl) (1) x=Zi X X] x^x\ X — Xi ^ ^ I h(x)-h{x^) ^ Hx)-U{x{) I ^ ^ /i(x)+/2(x)-/ifa)-/2(xO x-^x I X— a;i = L hix)- X- -/3(Xl) -a;i But by definition, /3'(Xl) = __^Hx)- -/3(a;i) -Xi (4) Henoe /3'(a;i) exists, and /a'Cxi) =/i'(xi) +/2'(xi). (&) h{x)=h(x)-Ux). Whenever Xt^xi we have the identity hix) -hjxi) _ /i(x) -Ux) -/i(xi) -/aCxi) X — Xi X — Xi _ /i(x) -/zCx) -/i(xi) -/aCx) +A(xi) -hjx) -/i(xi) -/zCxQ X — Xj =/2(a:) f/ i(x)-/i(xi) X— Xi 1 +,,(,) IteWa^l. "' t X-Xi J But the limit of the last expression exists as x=Xi (except perhaps in the case 00 — 00 ) and is equal to f2{x{)-U\x{)+h{x{)-U'{x{). 122 INFINITESIMAL ANALYSIS. Hence L ^;^ exists and fs'ixi) =h{xi) -h'ixi) +/2'(xi) -fiixi). (0 /^(^)=M^)- The argument is based on the identity /i(x)_/ifa) /2(X) /2(Xl) _ /l(x)-/2fa)-/2(x)-/lfa) , x-ii /2(a;) 72(2:1) -(x-xi) which holds when a; 7^X1 and when f2{x) 7^0. 3^^ fi(x)-f2(x^)-f2{x)-h(xr) f2{x)-f2{Xl)ix-Xi) Jljx) ■J2{x{)-h{x{) ■f2{x{)+h{Xi) ■U(Xl)-f2{x) -hJXi) U{x)-J2{Xi){x-Xi) /zfa) \h{x) -hjxi) I -/i(xi) {/2(x) -/aCzi) i f2{x)-f2iXl)ix-Xl) As before (excluding the case 00 — 00) we have , ,. , /2(Xl)-/l^fa)-/2^fa)'-/lfa) Corollary. — It follows from Theorems 74 and 75 of this chapter that if ]2{x)=a-fi{x) where /i'(x) exists, then /2'(x)=a-/i'(^). Theorem 76. Ifx>0, then -^x^ = A; • x*"i. DERIVATIVES AND DIFFERENTIALS. 123 (o) If A; is a positive integer, we have L — ^ — = L 1 a;*- 1 + x*^2. 3.J _|_ .,. 2.4.3.^4-2 +2.jjr-ij x=ii ■'' ^1 1=1, 171 {b) If A; is a positive rational fraction — , we have x"— Xi" \x"/ — XXi"/ L/ = Li JT ( J-'\"~l / J-^^n^Z ( 1\ ( l\n-l 2 i ~'\X^) + \X"/ • Vli"/ +. . .+ Ui"/ X"-Xi" 1 ( 1\'"-1 Tw:i-m\xi»/ , by the preceding case. 1 / J.\ "•""1 fn ——I But —T-^:^^-m\xi") =^^^1" =A;-Xi*-i. (c) If A; is a negative rational number and equal to — m, then, by the two preceding cases, ^3,^, X-Xi arix, a;"-!!™ X-Xi Xi^" = — Tnxi""*"^ But — TOXi~"'~^ = A;-x*"^ (d) If A; is a positive irrational number, we proceed as follows : 124 INFINITESIMAL ANALYSIS Consider values of x greater than or equal to unity. Let x approach Xi so that x>xi. Since, by Theorem 23, x'' is a monotonic increasing function of k for x>l, it follows that t^^^^^u.}^ y^.u'.SL X — Xi X — Xi X — Xi for all values of k' less than k, and all values of x greater than x^. If k' is a rational number, we have by the preceding cases that xY' KxJ -1 lii, X—Xi Since Xi*~^ is a continuous function of k, it follows that for every number N less than A;xi*~i there exists a rational num- ber ki' loss than k such that jV or — co the convention + 00 — 00 is understood. Cf. Theorem 37. Proof. — To prove this theorem we observe that X = Xi ■^ ^l *=Ii ^ •''1 /l(x)-/i(Xi) By the definition of single-valued inverse (p. 45), x-X'^ _ hiy)-f2{yi) /i(x)-/(xi) y-yi t Theorem 78 gives a sufficient condition for the equality dx dx' dy 126 INFINITESIMAL ANALYSIS. Hence, by Theorems 38 and 34 and 37, x-x. x-xi u=m f2{y)-f2{yi) U'iy)' ]i.x)-f{xx) y-yi Theorem 79. // (1) /i'(x) exists and is finite for x^Xi, and fi(x) is continuous at x=Xi, (2) f2'iy) exists and is. finite for 2/1 =fi{xi), then ^/2{/i(xi)l=/2'(2/i)-/i'(2;i).t Proof. — We prove this theorem first for the case when there is a V*{xi) upon which fi(x) 7^/i(xi). In this case the following is an identity in x : /2{/l(x)i-/2l/lfa)i _ /2i/l(x)}-/2|/l(Xi)} /i(x)-/i(3:i) X-Xi fi(x)-fiixi) ' X-Xi By hypothesis (2) and Theorem 38, ' y^m y-yi xix. h{x)-fi{Xi) By hypothesis (1), Hence, by equation (1) and Theorem 34, we have the existence of ^fffC-rM r /2{/l(x)}-/2J/l(Xi)} f,..,,, . ^2|/i(a;)( = L^ ^3^^^ ■■f2{yi)-fi'(xi). If /i(x)=/i(xi) for values of x on every neighborhood of T=Xi, then, by hypothesis (1) and the corollary of Theorem 74, /'(xi)=0. t Theorem 79 gives a sufficient condition for the equality dz _ dz dy dx dydx' DERIVATIVES AND DIFFERENTIALS. 127 Let [x'] be the set of points upon which /i(a;)7^/i(xi). (There is such a set unless f(x) is constant in the neighborhood of x = xi.) Then, by the same argument as in the first case, we have d ^,/2l/i(a;i)! =/2'(j/i)-/i'(xi)=0 for X on the set [x']. Let [x"] be the set of values of x not included in [x']. Then a^Mh (xi) } - ,,L , —:^r—^ = 0' since the limitand function is zero. Hence both for the set [x'] and for the set [x"] the conclusion of our theorem is that the derivative required is zero. Theorem 80. d ^^a^- a- log a. Proof. — Let 2/ = a^, therefore log2/=x-logo dy and, by Theorem 77, dx . whence dy dx' = j/-log a = a* logo. This method also affords an elegant proof of Theorem 76, VIZ., d ;"=n2;"~'. Let J/=x", logj/=nlogx, dy dx _n y 'x' -r=n — =n-x"~i. dx X 128 INFINITESIMAL ANALYSIS. If § 3. Differential Notations. y = f{x) and L K, x=a ^ •*'! we denote f{x)-f{x{) hy Ay, and x-Xi by dx. Then, by Theorem 31, Mj^Ax-K+Ax- £{x), where Ax- s{x) is an infinitesimal with respect to Ay and Ax for x = a. This fact is expressed by the equation dy=K-dx, where K=j'{x). Here dy and dx are any numbers that satisfy this equation. There is no condition as to their being small, either expressed or implied, and dx and dy may be regarded as variable or Fig. 16. constant, large or small, as may be found convenient. When either dx or dy is once chosen, the other is, of course, determined. The numbers dx and dy are called the differentials of x and y respectively. DERIVATIVES AND DIFFERENTIALS 12& In Fig. 16, f{xi) is the tangent of the angle CAB, dx is the length of any segment AB with one extremity at A and parallel to the j-axis, and dy is the length of the segment BC. If x is regarded as approaching xi, then AB' is the infinitesimal ix, WD' is Jy, while Wc' is e{x) -Ax. Hence, by Theorem 73, 'WC' is an infimtesimal of higher order than Ax or Ay. We thus obtain a complete correspondence between deriva- tives and the ratios of differentials. Accordingly, for any for- mula in derivatives there is a corresponding formula in differ- entials. Thus corresponding to Theorem 75 we have : Theorem 8i. When for two junctions /i(x) and /zCx) dfi{x)=fi'{x)-dx and df2{x)=J2ix)-dx at Xi, it follows that (a) ///3(x)=/i(x)+/2(x), (km dfsixi) = |/i'(xi) +/2'(xi) \dx =d/i(xi)+d/2(xi). ib) ///3(x)=/i(x)-/2(x), then dU{x,) = \h'{x^)-U'ix{)\dx = d/i(xi)-d/2(xi). (c) // h{x)=h{x)-f2{x), then d/3(xi) = {/i(xi)-/2'(xi)+/2(xi)+/i'(xi)! -dx = /i(a:i)-d/2(xi)+/2(xi)-d/i(xi). , ^, , , \f2(x,)-h'{x,)-h(xi)-f2'ixr)\-dx then d/3(xi)= {f2{xi)\^ /2(x,)-d/i(x,)-/i(x,)d/2(xi) i/2(Xl)P The rule obtained on page 123 et seq. that the derivative of x* is Jfc • X*- 1 corresponds to the equation dx* = A; • x*- ^ • dx. If , in the 130 INFINITESIMAL ANALYSIS. equation dy=f{x)dx, dx is regarded as a constant while x varies, then dyisa. function of x. We then obtain a differential d2(dy) = {f'{x)-dx\d2X in precisely the same manner that we obtain dy=f{x)-dx. Since d2X may be chosen arbitrarily, we choose it equal to dx. Hence d{dy) =f"{x)dx^. We write this d'^y=f'(,x)-dx'^. The differential coefficient f"{x) is clearly identical with the de- rivative of fix). In this manner we obtain successively d^y=f^^\x)-dx^, etc. We may write these results, ^^fix) ^=f"(x) ^-n-Hx) dx '^^>' dx' ^ ^^^'•••-■dx"~' ^^''• Evidently the existence of the differential coefficient is coexten- sive with the existence of the derivative. § 4. Mean-value Theorems. Theorem 82. // /(x) has a unique and finite derivative at x = xi, then f(x) is continuous at x\. Proof.— The proof depends upon the evident fact that if f(x)-f{xi) approach anything but zero as x approaches Xi, then one of the values approached by m-f{x,) X — Xi is +00 or — 00 . Definition.— The function f(x) is said to have a maximum at x=xi if there exists a neighborhood V(xi) such that (1) No value of fix) in 7(xi) is greater than /(xi). (2) There is a value of x, X2, in 7(xi) such that X2a;i and /(X3)Xi *<^1 Since f(xi) exists these limits are equal, that is, the derivative is equal to zero. Similarly in case of a minimum. Theorem 84. // /(xi)=/(x2), fix) being corUimwus on th£ 132 INFINITESIMAL ANALYSIS. interval Xi X2, and if the derivative exists t at every point between Xi and X2, then there is a value f between Xi and X2 such that /'(?)= 0. The derivative need not exist at Xi and X2- Proof. — (a) The function may be a constant between Xi and X2, in which case f{x) = for all values of x between Xi and X2 by Theorem 74. (6) There may be values of the function between Xi and X2 which are greater than f{xi) and f{x2). Since the function is continuous on the interval Xi X2, it reaches a least upper boimd on this interval at some point X3 (different from Xi and '0:2). By Theorem 83, /'(X3)=0. (c) In case there are values of the function on the interval I — I Xi X2 less than /(xi), the derivative is zero at the minimum point in precisely the same manner as under case (b). Fig. 18. This theorem is called Rolle's Theorem. The restriction that /(x) shall be continuous is unnecessary if the derivative t Not necessarily finite. DERIVATIVES AND DIFFERENTIALS. 133 exists, but simplifies the argument. The proof without this restriction is suggested as an exercise for the reader. The geometric interpretation is that any curve representing a continuous fimction, }{x), such that f{xi) =f{x2), and having a tangent at every point betweeen xi and X2 has a horizontal tangent at some point between them. An immediate gener- alization of this is that between any two points A and B on a curve which satisfies the hypothesis of this theorem there is a tangent to the curve which is parallel to the Une AB. The following theorem is a corresponding analytical generali- zation : I — I Theorem 85. // /(x) is continuous on the interval. Xi X2, and if the derivative exists at every point between X\ and X2, then there is a value of x, x = $, between xi and X2 such that ' X1-X2 Proof. — Consider a function /i(z) such that fii.x)=f{x)-{x-x2) — ^^_^^ ; then fi{xi)=fix2) and /i(x2)=/(x2). Therefore /i(a;i)=/i(x2). Hence, by Theorem 84, there is an x, x = $ on the segment xi X2 such that /i' (0=0. That is, />'©=/'(« -'-^^^-0. Therefore f(f)-^^^^^. This is the "mean-value theorem." Its content may also be ■expressed by the equation f{X2)=f{Xl) + {X2-Xl)n^). 134 INFINITESIMAL ANALYSIS. Denoting xi -x by dx and f by x + Odx, where 0<5<1, it takes the form /(xi +dx) =/(xi) +f (xi + ddx)dx. Theorem 86. // /i(x) and /2(x) are continiious on an interval a b, and if /i'(x) and /a'Cx) exist between a and b, fz'ix) 5^ ± oo , and /2'(x)7^0, /2(a) 5^/2 (&), ^^len there is a value of x, x=f between a and b such that fi(a)-fi( b)_fi'{?) /2(a) -/2(6) /2'(f)- Proof. — Consider a function Since /3(a) =0 and fsib) =0, we have as before /a'Cf) = 0. Therefore AMzMl^M) inereiore /2(a) -/2(6) /a'Ce)" This is called the second mean-value theorem. The first mean-value theorem has a very important extension to "Taylor's series with a remainder," which follows as Theorem 87. § 5. Taylor's Series. The derivative of f{x) is denoted by /"(x) and is called the second derviative of /(x). In general the nth derivative is the derivative of the n - 1st derivative and is denoted by /^"'(x) . Theorem 87. // the first n derivatives of the function f{x) l-l exist and are finite upon the interval a b, there is a value of x, x„ l-l on the interval a b such that DERIVATIVES AND DIFFERENTIALS. 135 /(&)=/(a)+^/'(a)+^V(a) + . ^ (n-1)! ' ^'^^+ „, ; {Xn). Proof. — Let i?„ be a constant such that Fix) =f{x) -fia) - {x-a)na) J-^^f"{a) -... _ (x-a)"-^ ,„_.) (.T-g)" (n-1)! ' '■'^^ n! "" is equal to zero for x=b. Since F{x)=0 for x = a, there is, by Theorem 84, some value of x, Xi,a (n-2)! ^ "^"^ („_i), «n is equal to zero for x=a;i. Since also F'{a) =0, there is a value of X, X2, a(x„)=p(a;n)-ii;n=0. Therefore /i!„ = f"H2:»), whence the theorem. Corollary— In Theorem 87, f^'^Kx) need be supposed to exist only on ab. Definition. — The expression n! n- *"=o "'• is called the remainder, and the infinite series t-o «• is called Taylor's Series. 136 INFINITESIMAL ANALYSIS. a constant different from zero, » /W.(g)(b-g)n then 2 —^ n=0 "• is convergent but not equal to j(b), i.e., If L^-(&-a)» fails to exist and be finite, then 00 /('')(a) „=o «■! (6 -a)" is a divergent series. Hence an obvious necessary and sufficient condition that for a function /(x) all of whose derivatives exist for the values of I, o(a), then : (1) If n is odd, f{x) has neither a maximum nor a mini- mum at a; (2) // n is even, j{x) has a maximum or a minimum according as f'^aXQ or f''\a)>Q. Proof. — By Taylor's theorem, for every x in the vicinity of a fix) =/(a) + (x-o)«/(")(o) + (a;-a)"+i-/(»+»(f«), where ^x is between x and a. Hence j{x) -f{a) = (x- a) "{/(») (a) + (i-a)/("+i)(f ^) } . But since /^""""^'(fi) is bounded and x—a is infinitesimal, there exists a 'V*{a) such that if x is in 7* (a), Kx)-m is positive or negative according as (x-a)"-/(")(o) is positive or negative. (1) If n is odd,(a;— a)"is of the same sign as x—a, and hence for /("'(a) >0 j{x)-f{a)>0 '\ix>a, f{x)-J{a)<0 iix0 ifa;a. (2) If n is even, (x—a)" is always positive, and hence if f"'(a)>0. /(x)-/(o)>0 ifa;>a, /(x)-/(a)>0 ifx(a)<0. /(x)-/(a)<0 if a;>a, 1 tr \ t/ \ ^n -t ^ f then /(a) is a mimmum. /(x)-/(a)<0 if i{x)=0. (2) — , i.e., to compute L -rr^ if L f{x) = ± oo and L 0(1) = ±00. (3) 00 —CO, i.e., to compute L j/(x)— ^(i)| if L f(x)= ±00 x=a x^a and L {x)= ±00. (4) O-oo, i.e., to compute L /(x)-0(i) if L /(x)=0 and Z/ ix)= ±00. (5) 1°°, i.e., to compute L /(x)*(^> if L /(x) = l and L 4>(x) = ± 00 . (6) 0°, i.e., to compute L /(x)*^^) if L /(x) = and L ^(x) =0. x=a x=a x^a (7) 00 0, i.e., to compute L f{xY^''^ if L/(x)=±oo and L 0(x)=O. These problems may all be reduced to one or the other of the first two. The third may be written (since /(x) t-^O on some V*{a)) /(i)-<^(x)=-^ ^(x) = — ^. W) Jixj which is either determinate or of type (1). 140 INFINITESIMAL ANALYSIS. To the cases (5), (6), and (7) we may apply the corollaries of Theorem 39 of Chapter IV, from which it follows (provided fix) 7^0 on some F*(o)), that exists if and only if log L /(x)*(^>= L log/(a;)*(^>= L 9&(a;) log/(x) exists. x=a x=a x^a The evaluation of L -~ ^) comes under case (1) or case (2). The evaluation of cases (1) and (2) is effected by the follow- ing theorems: Theorem 90. // /(x) and 4>{x) are continuous and differentiable and <^(x) is monotonic and (f>'{x) ^0 and '(x) j^ 00 and (1) i] L f{x)=0 and L ix)=0 or (2) if L ,^(x)=±oo,t XwOO 0(x) exists and is equal to K. Proof .—For every positive h we have, by the second mean- value theorem, fix +h)- fix) _f(e.) ix+h)-<}>ix) 'i$^y where ^^ lies between x and x + h. But since f ,, takes on values which are a subset of the values of x, and since L f = 00 t It is not necessary that Lf{x)=aa ■ cf. Theorem 69. DERIVATIVES AND DIFFERENTIALS. 141 which in turn implies L 77 r{ — -~rT = K x^oo{x) ' and this, according to Theorems 68 and 69, gives Corollary. — If f{x) is continuous and differentiable, then L — L fix). The theorem above can be extended by the substitution 1 z = - x-a to the case where x approaches a finite value o. The approach must of course be one-sided. Theorem 91. // fix) and (f>{x) are continuous and differen- tiable on same V*(a) and f{x) is hounded on every finite interval, while (f>{x) is monotonic and (1) L /(x)=0, L 0(1) =0 or x=a x—a (2) L ^(x) =+ 00 or - 00 : thenxf xt¥(i) ' it follows that L 77-r exists and is equal to K. 142 INFINITESIMAL ANALYSIS. fix) Proof. — If L ,,, . exists, the limit exists when the approach x=a V \-^) is only on values of x>a. Consider only such values of x. Then if ^=^' /(^)=/(«+7)=-P'(2) and 4>{x)='^{a+-)=0{z), by hypothesis and Theorem 79, F'iz) and 0'iz) exist and i^'(3)=/'(x)g. Hence If LVi^r^' \ hen, according to Theorem 38, r F'iz) exists and is equal to K. Hence, by Theorem 90, exists and is equal to K. T ZM .t^iz) fix) Hence, by Theorem 38, L ., . exists and is equal to K. We have now derived conditions under which we can state a general rule for computing an indeterminate form. Provided fix) is not zero on every V*ia), any of the forms <3) to (7) can be reduced to Fix) (Pix) W DERIVATIVES AND DIFFERENTIALS. 143 where this is of type (1) or (2). Provided Fix) and 0{x) satisfy the conditions of Theorem 91, the existence of the limit of (a) depends on the existence of the Umit of F'ix) ¥{xj- (^) If (6) is indeterminate, and F'{x) and 0'{x) satisfy the condi- tions of Theorem 91, the limit of (6) depends on the Umit of F"{x) '(X) 'ft-y\> 'W and so on in general. If at each step the conditions of Theorem 91 are satisfied and the form is still indeterminate, the Umit of i?'(n+l)(x) depends on the limit of ^u+iv^ ('i+l) If (n) is indeterminate for all values of n, this rule leads to no result. If for some value of n then all the preceding Umits exist and are equal to K, and so x=o0, then, by Theorem 23, there exists about the point xi a segment (xi— iJ), (xi-f-0, X — Xi and hence, if x>Xi, /(x)>/(xi) and if x/(xi), while x/(x), while x>Xi, \Yliicli is contrary to the hypothesis that the function is monotonic increasing in the neighborhood of x = ii. In the same manner we prove that if the function is monotonic decreasing, and if the derivative exists, then fix) cannot be positive. The following theorem states necessary and sufficient condi- tions for the existence of the progressive and regressive deriva- tives. Conditions for the existence of a derivative proper are obtained by adding the condition that the progressive and regressive derivatives are equal. Theorem 97. // /(x), x>l + |?r. t For references and remarks see page 51 . CHAPTER VIII. DEFINITE INTEGRALS. § I. Definition of the Definite Integral. The area of a rectangle the lengths of whose sides are exact multiples of the length of the side of a unit square, is the num- ber of squares equal to the unit square contained within the rectangle, and is easily seen to be equal to the product of the lengths of its base and altitude.t In case the sides of the rectangle and the side of the unit square are commensurable, the sides of the rectangle not being exact multiples of the side of the square, the rectangle and the square are divided into a set ..of equal squares. The area of the rectangle is then defined as the ratio of the niunber of squares in the rectangle to be measxared to the number of squares in the unit square. Again, the area is equal to the product of the base and altitude. Any figure so related to the unit square that both figures can be divided into a finite set of equal squares is said to be com- mensurable with the miit. The area of a rectangle incommensurable with the unit is defined as the least upper boimd of the areas of all commensur- able rectangles contained within it. It foUows directly from the definition of the product of irrational nvunbers that this process gives the area as the prod- uct of the base and altitude. J t Of course the units are not necessarily squares; they may be triangles, parallelograms, etc. X For the meaning of the length of a segment mcommensurable with the unit segment, compare Chapter II, page 33. 151 152 INFINITESIMAL ANALYSIS. Turning to the figure bounded by the segment a b (which we take on the x axis in a system of rectangular coordinates) the graph of a function y^fix) and the ordinates a;=a and x=6, Fig. 20. we obtain as follows an approximation to the common notion of the area of such figures. Let xo=a, xi, X2, . . . , Xn = b he a. set of points lying in order from a to 6. Such a set of points is called a partition of a h, and is denoted by n. The intervals xq xi, xi Xz, . . . , a;„_ii„ are intervals of ;:. Let xi— zo=.^ia;, X2—xi=J2X, . . . , x„-Xn-i=^„x, and let fi, $2, . . ■ , $n be a set of points such that $1 is on the interval xq Xi, $2 is on I 1 . I 1 Xi X2 . • • , and $n IS on i„_ix„. Then /(fi), fiU), ..., /(f„) are the altitudes of a set of rectangles whose combined area is a more or less close approximation of the area of our figure. Denote this approximate area by S. Then S=m)Jix+m)J2X + . . .+/(f„)i„x= I fih)^kX. k-l As the greatest Ji^ is taken smaller and smaller, the figure DEFINITE INTEGRALS. 153 composed of the rectangles comes nearer to the figure bounded by the curve. In consequence of these geometrical notions we define the area of the figure as the limit of S as the J^x's decrease in- definitely. The area S is the definite integral of /(x) from a to 6. It has been tacitly assumed that the graph of y=f{x) is continuous, since we do not usually speak of an area being enclosed by a discontinuous curve. The definition of the defi- nite integral when stated in its general form admits, however, of functions which are discontinuous in a great variety of ways. A more general definition of the definite integral is as follows : 1 — I I — I Let a h {or h a) he an interval upon which a function f{x) is defined, single-valued and bounded. Let n, stand for any par- I — I I — I tilicn of a b or b a by the points a = Xo, Xi, X2, . . . , x„ = 6 such that the numbers A\X = Xi—a, A2X = X2 — X\,..., J„x~6— x„_i are each numerically less than or equal to d. fl, f2, ■ • • , fn be a set of points on the intervals I 1 I — I I 1 , ., ^ I 1 I 1 I 1 Xq-Xi, Xi X2, ■ . . , Xn-lXn (pr if b But since |/(fft)-/(f«:') 1=^*3/, the difference between (1) and (2) is less than or equal to Aky-\dk'x^Ak"x + . . .\^Aky-\dkx\ and hence \S:,-SA< I Aky-U,^\=0^. ' k=\ I Theorem 98. Every Junction continuous on a b is integrable I — I on a b. Proof. — ^We have to investigate the existence of the limit LS} of the many-valued function Ss as d~0. Since S, ap- proaches at least one value as d approaches zero (see Theorem 24), we need only to prove that it cannot have more than one value approached. Suppose there were two such values, B R — C and C, B>C. Let £ = — —. By the definition of value approached, for every 8 there must exist an S (which we call Sb) such that \SB-B\0, b>0 for every value of m?^ -1, / 6 ^m+1 (j'n+1 x'^dx = ■ m + 1 X^dx= L a{q-l) I q''{aq'')'" a 9=1 i-0 =a'"+i L (3-l)[l + (r+i) + (r+^)' + - ■ . + ir^')^-n (1) (om+l)n_l = La'"+M(3")"'^'-i!jFrri 9 = 1 ^ 162 INFINITESIMAL ANALYSIS. Hence / x^dx = - m + 1 ' T g-1 1 Theorem 103. / -dx = log b—loga, {0°^ a) but n =— i , hence log 3 £V^= ,il^- ^°S (^)=log (|)=log&-loga, since (§6, Chapter VII) l'Hospital's rule gives 8=1 log g The following theorem is of frequent use in computing both derivatives and integrals. I — I Theorem 104. // on an interval a b two functions /(x) and Fix) have the ■property that for every two valries of x, xi and X2, ■where a i^(X2) -2^(Xi) > /(X2)(X2-Xi), then (1), if f{x) is coniiniums, DEFINITE INTEGRALS. 163 and (2) whether f{x) is continuous or not, I }{x)dx exists and is equal to F{h) —F{a). Proof. — We consider first the case fiXi)iX2-Xi)BS, and /(xi)(xi-a)+. . .+/(b)(&-x„_i)F{x2) -Fixi)>f(x2){x2-xi} is identical with the above when we write > instead of <. § 4. Elementary Properties of Definite Integrals. Theorem 105. // a I Ifi$k)^kX\. Hence for every (Sj|/(x)| there is a smaller or equal Si fix), the d's being the same. Hence by CoroUary 2, Theorem 40, LS,\m\>\ LSem\. Theorem io8. // / f{x)dx exists, then I f{x)dx exists and f f(x)dx=- ffixjdx. Proof. — ^This is a consequence of the theorem (Corollary 1 Theorem 27) that L (-/(x)) = - Lfix), a sum cor- for to every S used in defining / f{x)dx corresponds equal to —S which is used in defining / f{x)dx. Similarly to every S' used in defining / f{x)dx there responds a sum -S' used in defining / fix)dx. Hence the function Sg in the definition of / f{x)dx is the negative of the function Sg used in the definition of / f{x)dx. Hence the theorem follows from the theorem quoted. We adjoin the following two theorems, the first of which is an inmiediate consequence of the definition of an integral, and the second a corollary of Theorems 105, 106, and 108. DEFINITE INTEGRALS. 167 Theorem 109. J ^^^ f{x-h)dx exists and is equal to / f{x)dx, provided the latter integral exists.\ Theorem no. // any two of the following integrals exist, so does the third, and £ f{x)dx+ Jjf{x)dx= r f{x)dx. Theorem in. If C is any constant and if f{x) is integrable I — I I — I onab, then Cf{x) is integrable on ah and £'cf{x)dx = cj'''fix)dx. n Proof.— Sa= I fi$k)^kX is an S) of the set which defines i— 1 fb J^ f{x)dx and 5/= I^ Cf{$k)J^ is the corresponding S» of the set which defines / Cf{x)dx. Hence our theorem follows immediately from Theorem 34, a special case of which is L Cf{x) =CLf{x). Theorem 112. // fi{x) and fzix) are any two functions each I — I integrable on the interval a b, then f{x) =f\{x) ±f2{x) is integra- I — I bU on ah and rf(x)dx= rfi{x)dx± Pfzixjdx. Proof. — ^The proof depends directly upon the theorem that if L ^i(x)=6i, and L ^(x) =62; then L ^i(x)±^(x)=6i±62 x=a x=a x=a (Theorem 34). t First stated formally by H. Lebesode, Lemons smt VInUgration, Chapter VU, page 98. 168 INFINITESIMAL ANALYSIS. 1-1 Theorem 113. // /i(x) and foix) are integrable onab and such l-l that for every value of x on a b fi(x)tf2{x), then rf,(,x)dx> rf2{x)dx. Proof. — Since Si is always greater than or equal to S2, then, by Theorem 34, L Sit L S2, which proves the theorem. 1=0 oiO Theorem 114. (Maximum-Minimum Theorem.) If (1) the product fi{x)-f2{x) and the factor fi{x) are inte- i-l grable on a b, I 1 (2) /i (x) is always positive or always negative on a b, (.3)' M and m are the least upper and the greatest lower I — I bounds respectively of /2(x) on a b, then m- / fi{x)dx< / fi{x)f2ix)dx f)i{x) -fz^x >M- P fx{x)dx. Proof. — By Theorem 111, M ■ J fi{x)dx== rM-fi{x)dx and m- / fi{x)dx= I m-fi{x)dx. %/ a */o But in case /i(x) is always positive, m-fi{x)M- Ai(i)dx. As an obvious corollary of this theorem we have the Mean- value Theorem : Theorem 115. Under the hypothesis of Theorem II4 there exists a number K, m'S K^M, such that rh{x)-f2(x)dx=K f''h{x)dx. Corollary 1. In case f^^x) is continuous we have a value I — I f of X on a h such that Ai(:c)-/2(x)dx=/2(f) f U^)dx. J a 'J "■ In case /i(x)=l, / /i(x)dx = 6-a, and the theorem reduces to this : Theorem 116. // /(x) is any integrable function on the inter- val a b, there exists a number M lying between the upper and lower I — I bounds of /(x) on a b such that rfix)dx = M{b-a), I — I and if fix) is continuous, there is a value $ of x on a b swk that fj{x)dx = fmh-a). 170 INFINITESIMAL ANALYSIS. In many applications of the integral calculus the expression Cmdx —^ represents the notion of an average value of the dependent variable y=f{x) as x varies from a to 6. An average of an infinite set of values of /(x) is of course to be described only by means of a limiting process. Consider a set of points \-. — I xj, X2, . . . , Xn-i,Xn=b ou thc interval a b such that Xi—a = X2—Xi = X3—X2 = . ■ . = X„-i—X„^2=b—Xn-l. Then M„=- I f{xk), and we define the mean value of f{x), lM}(x)= L M„ if this n=oo ,. . . T% b — a linut exists. But Xk+i—Xk= =ix. If the definite integral / f{x)dx exists, we may write i/ a / f(x)dx= LS», where Si= I f{xk)Ax= I f{xk)^^ = ^^^ 1 f{xk) = (b-a)M„. k-i k"! n n jc^i Therefore L Sa = ib-a) L M„. } = n=ixi We therefore have the theorem: Theorem 117. In case the integral of f{x) exists on the interval yf{x)dx a o b, iMfix) = b — a We note that \M is the same as the K which occvirs in the mean-value theorem, and that the last theorem suggests a simple DEFIXITE IXTEGRALS. 171 method of approximating 'the value of a definite integral by multiplying the average of a finite nmnber of ordinates by b-a. § 5. The Definite Integral as a Function of the Limits of Integration. I 1 Theorem 118. // /(.r) i\>f integrahle on an intenal a b, and I — I r^ if X i^ any poini of a b, I f{x)dx is a contimtous function of x. Proof. — / /(,x)dr exists, b>- Theorem 105, and by the defini- tion of a continuous function we need only to show that L^(^fj{x)dx-fjKx)dj^ =0. By the theorems of the preceding section, A(j)dr - A(x) rf-r = rf{x)dx<\iB- (/ -x) \<\B- [x' -x)\, %/a U a *J I where {B stands for the least upper bound of f{x) on the inter- I — 1 - I — I val J x! , and B for the least upper bound of f{,x) on a b. Smce 5 is a constant, B\x' -x) approaches zero as x' approaches x. and therefore by Theorem 40, Corollary 4, the conclusion of our theorem follows. Theorem 119. // /(x) Vs conXinuoM^ on an inten'ol a b, /fix)dx (aix). Since Fix) and / fix)dx are both differentiable, By the preceding theorem d, dx. fjix)dx=fix). Hence j-^ix) =0, whence, by Theorem 94, <^(a;) is a constant. Theorem 121. // fix) is a continuous function on an interval a b and Fix) is such that then £fix)dx = Fib)-Fia). /; 174 INFINITESIMAL ANALYSIS. Proof. — By the last theorem, f{x)dx=F{x)+c. But 0= rf{x)dx=F{a)+c. Therefore -F(a)=c. Whence f''f{x)dx=F{b) +c=i?'(6) -F(a). The symbol [F{x)fa or |S F{x) is frequently used for ^(6) -F{a). In these terms the above theorem is expressed by the equation ){x)dx = \lF{x). r By this last theorem the theory of definite and indefinite integrals is united as far as continuous functions are concerned, and a table of derivatives gives a table of integrals. For dis- continuous functions the correspondence does not in general hold. That is, there are on the one hand integrable functions ](x) such that / \{x)dx is not differentiable with respect to x, .and on the other hand differentiable functions ^(x) such that ■^'(x) is not integrable. t § 6. Integration by Parts and by Substitution. The formulas for integration by parts and by substitution are ordinarily written as follows: / udv = uv— / vd.u, f M^y- J Kyy%dx. t For a good discussion of this subject the reader is referred to H. Le- BESGUE, L&;ons sur V Integration. DEFINITE INTEGRALS. 175 The following theorems state sufficient conditions for their validity. Theorem 122. (Integration by parts.) £ hix) ■h'{x)dx = \jM ■f2{x)^- fj2{x) ■h'{x)dx, -provided //(x) and fz'ix) exist and are continuous on the interval I — I a b. Proof. — By Theorem 75, £{hix) -Mx)) =/i(x) -k'ix) +k(.x) ■h'i.x). Therefore X di^f^^""^ •/2W)dx = y^ /i(x) ■f2'{x)dx + £''j2{x) ■fi'ix)dx. (The integral exists since it follows from the existence and continuity of //(x) and ^'(a;) that /i(x) and /zCz) are continuous). By Theorem 121, £ii\h(x) •Hx)\dx=hQ>) ■f2ib) -hia) ./2(a). Therefore fjl(x) ■h'{x)dx = [hix) ■hix)l- fj2{x) ■ll'(x)dx. Theorem 123. (Integration by substitution.) If y = cp(x) has a I— I continuous derivative at every point of ab and f(y) is continuous for all values taken byy = (j)(x) as x varies from a to b, where A = (j>(a), B = (b). 176 INFINITESIMAL ANALYSIS. Proof.— By Theorem 120 and by Theorem 79, C being an arbitrary constant. C is determined by letting X = a. Then if x = 6 we have* Theorem fjmy-£mt-d-- f(x)dx = J^ fi4>iy))-^dy, where x — {A), h = 4>{B); provided that both inte- grals exist, and that {y) is non-oscillating and has a finite derivative. Proof. f f{x)dx= L I f{$k)^kX (1) whenever the least upper bound of J4X for each n approaches „ , B-A zero as n approaches + 00 . Now let jy= , yk=A+k-Jy, 4>(yk)-4>(.yk-i)-.dkX. Hence, by Theorem 85, AkX = ^'(j)]^Ay, where ij* lies between j/i and yk-i- Now if fj: = ^(ijA), it will lie between {yk-i); moreover the Akx's are all of the same sign or zero; and since the hypothesis makes ^(t/) uniformly continuous, their least upper bound approaches zero as n approaches + 00 . Therefore f f{x)dx= L I f{^k)^k£ = L I f{4>irik))-'{r,k)-Jy nioo k=l ^fii>iyW(.y)dy, DEFINITE INTEGRALS. 177 provided the latter integral exists. Hence fji'')^^ = fj('f>iy)) '^dy. Corollary. — The validity of this theorem remains if (y) has a finite number of oscillations. Proof. — Suppose the maximum and minimum values of 0 there is an infinite set of partitions n, for which the largest J^a; is less than d, and for each of these there is a value of 0^. If Os stands for any such 0^, then 0) is a many-valued function of b. Theorem 126. A necessary and sufficient condition that a function f{x), defined, single-valued, and hounded on an interval I — I a b, is integrable is that L O,=0. Proof.T-r^e condition is necessary. By Theorem 125 the integrability of f(x) implies B0„=0. Hence for every e there exists a partition t: such that By Lemma 4 there exists a 1?, such that for every s^ whose greatest Jx is less than d, 0,'<0, + £<2s. Hence L O' = 0. The condition is sufficient. Since L O*=0, andO,>0, B0],=0. Hence the function is integrable by Theorem 125. Theorem 127. A necessary and sufficient condition that a function, defined, single-valued, and bounded on an interval a b, shall be integrable on that interval is that for every pair of positive DEFINITE INTEGRALS. 183 numbers a and X there exists a partition n such that the sum of the lengths of those intervals on which the oscillation of the function is greater than a is less than A. Proof. — The condition is necessary. If for a given pair of positive numbers a and X there exists DO ;r such as is required by the theorem, then 0^> a- A for every n, which is contrary to the conclusion of Theorem 125. that The condition is sufficient. For a given positive e choose a and X so that e £ a(b — a)<-^ and XR<-^, where R is the oscillation of the function on a b. Let s^ be a partition such that the sum of the lengths of those intervals on which the oscillation of the function is greater than a is less than A. Then the sum of the terms of 0^ which occur on these intervals is less than XR, and the sum of the terms of 0^ on the remaining intervals is less than aQ) — a) . Therefore On0 the set of points [x„] at which the oscillation of f{x) is greater than or equal to a shall he of content zero.X Proof. — If at every point of an interval c d the oscillation of /(i) is less than a, then about each point of c d there is a segment upon which the oscillation is less than a, and hence by Theorem 11, Chapter II, there is a partition of c d upon each interval of which the oscillation of f{x) is less than a. Now to prove the condition sufficient we observe that if the content of {x„] is zero, there exists for every X a partition TZi such that the sum of the lengths of the intervals containing points of [x,] is less than X. Moreover we have just seen t For further discussion of the notion content see Pierpont, Real Func- tions, Vol. I, p. 352, and Lebesque, Lemons sur I'lntigration. X Compare the example on page 155. DEFINITE INTEGRALS. 185 that the intervals which do not contain points on [x„] can be repartitioned into intervals on which the oscillation is less than a. Hence, by Theorem 127, the function is integrable. To prove the condition necessary we note that on every interval containing a point, x„ the oscillation of j{x) is greater than or equal to or equal to a. Hence, if C[xj>0, the sum of the intervals upon which the oscillation is greater than or equal to ct is greater than C[x<,]. Definition. — A set of points is said to be numerable if it is capable of being set into one-to-one correspondence with the positive integral numbers. If a set [x] is numerable, it can always be indicated by the notation Xi, X2, X3, . . . , Xn, . . . , or \xn\, but if it is not numerable, the notation \xn\ cannot be apphed with the understanding that n is integral. Theorem 129. A 'perfect set of points is not numerably infinite.^ Proof.— Suppose the theorem not true. Then there exists a sequence of points \xn\ containing every point of a perfect set [x]. Let Pi be any point of [x], and a^ bi a segment containing Pi. Let x„, be the first of lx„! within oi 61. Since x„ is a limit point of points of [x], there are points of the set other than Pi and z„, on the segment ai bi. Let P2 be such a point, and let 02 62 be a segment within ai 61 and containing P2 but neither Pi nor x„,. Let x„^ be the first point of {x„i within 02 62. Proceeding in this maimer we obtain a sequence of segments ja, bi\ such that every segment lies within the preceding and such that every segment ai bi contains no point ^ni-ic of the sequence \xn\. By the lemma on page 42, Chapter II, there is a point P on every segment of this set. Since there are points of [x] on every segment a,- bi, P is a limit point of the set [x]. Since [x] is a perfect set, P is a point of [x]. But if P t For definition of perfect set see page 91. 186 INFINITESIMAL ANALYSIS. were in the sequence jz„}, there would be only a finite number of points of [x] preceding P, whereas by the construction there is an infinitude of such points. Theorem 130. A numerably infinite set of sets of points each of content zero cannot contain every point of any interval. Proof. — Let the set of sets be ordered into a sequence {[x]„] . We show that on every segment a b there is at least one point not of ![x]„). Since [x]i is of content zero, there is a segment oi 61 contained in a 6 which contains no point of [x]i. Let [x]„j be the first set of the sequence which contains a point of Ci bi. Since [x]„, is of content zero, there is a segment 02 62 contained in ai 61 which contains no point of [x]„,. Continuing in this manner we obtain a sequence of segments o b,ai 61, ... , a„ bn . . . such that every segment lies within the preceding, and such that a„ 6„ contains no point of [x]i, . . . , [x]„. By the lemma on page 42 there is at least one point P on all these segments. Hence P is a point of a 6 and is not a point of any set of |[x]„!. Theorem 131. The points of discontinuity of an integrabk function form at most a set consisting of a numerable set of sets, each of content zero. Proof. — Let 0 the points where \fix)\> a form a set of content zero. Proof. — ^At every point X where f{x) is continuous, accord- ing to the corollary of Theorem 119, smce I f{x)dx IS a constant. The points of continuity of f{x) J a are everywhere dense, according to Theorem 132. Hence the zero points of /(i) are everywhere dense. At a point of discon- tinuity the oscillation of f{x) is greater than or equal to |/(x)|. Hence the points where |/(x) | > a form a set of content zero. Theorem 134. // rx nx I f{x)dx= I (f>{x)dx I 1 for every X of a b, then f{x)=0 the points where \f{x) —a forms a set of content zero. Proof. — Apply the theorem above to /(x) -<^(x). Theorem 135. // /(x) is integrable from a to b, then |/(x)| is integrable from a tob.'f tThe converse theorem is not true; cf. example given on page 192. 188 INFINITESIMAL ANALYSIS. Proof.— Since 0<0,\Kx)\{x) are both integrabk on an inter- I— I vol a b, then f{x)-4>{x) (1) I— I ^ - IS integrable on a b; and, provided there is a constant m > such I — I that |^(x)| — m>0 for x on a b, then /(x)H-^(x) (2) I— ! is integrable on a b. Proof. — Since f{x) and ^(x) are both integrable on a 6, it follows that for every pair of positive numbers a and A there is a partition tti for fix) and a partition 712 for ^(x) such that the sums of the lengths of the intervals on which the oscilla- tions of f{x) and (x) respectively are greater than a are less than X. Let 7: be the partition consisting of the points of both TTi and 712. Then the sum of the intervals of tz on which the oscillation of either f{x) or (p{x) is greater than a is less than 2A. Let M be the greater of B\f{x)\ and B\(f>{x)\ on a b. Then on any interval of 71 on which the oscillations of /(x) and (p{x) are both less than a the oscillation of /(x)-^(x) is less than aM. Hence the sum of the intervals on which the oscillation of /(x) ■ 0(x) is greater than aM is less than 2A. Since a and A may be chosen so that 2 A and aM shall be any pair of preassigned numbers, it follows by Theorem 127 that 1—1 fix) ■ix) is integrable on a b. In view of the argument above it is sufficient for the second DEFINITE INTEGRALS. 189 part of the theorem to prove that -t-~. is integrable on a 6 if (a;) is integrable and \(f>{x)\ >m. Consider a partition re such that the sum of the intervals on which the oscillation of ^(x) is greater than a is less than X. Since 1 1 ^(Xi) {X2) .. \(Xi)~(X2)\ mXl)\-\{X2)\ it follows that n is such that the sum of the intervals on which the oscillation of -ri-^ is greater than —5 is less than X, and T7-T is integrable according to Theorem 127. A second proof may be made by comparing the integral oscillations of /(x) and ^(x) with those of the functions (1) and (2) and applying Theorem 125. j Theorem 137. // /(x) is an integrable function on an interval 1—1 a b, and if 0 ad„ such that for \yi-y2\(yi)-(y2)\<\ r){x)dx\, since, by the hypothesis and Theorem 105, f^^f{x)dx exists. Hence, by the sufficient condition of Theorem 138, L f)ix)d2 exists and is finite. Theorem 140 I. // / ]{x) dx exists for every x on the segment a b, and if (x—a)''f(x) is bounded on V*(a) for some valve of k, 0m for every x, then L rf{x)d2 ^aJ X cannot exist and he finite. L IMPROPER DEFINITE INTEGRALS. 195 Proof. — (1) In case /jix)dx fails to exist for some value of x between a and b, L f''fix)dx fails to exist because the limitand function does not exist. (2) If fj^''^'^- exists for every value of x between a and h, we proceed as follows : Let 5< 1 be the length of a V*{a) on which f(x) does not change sign, and on which {x — aYj{x)>m, and let Xi be the extrem- ity of this neighborhood, which is greater than a. Then l/(^)l>(^r^>(^^r^* ^^^ ^^^'■y ^ ""^ *^^ neighborhood. Take xi so that (X2 - a) * = 2(a;2 - Xi) . >fe?^^^^-^^) = ^- Then / f{x)dx Hence, by the necessary condition of Theorem 138, L rf{x)dx cannot exist and be finite. Theorem 142. // L f{x)dx cxiste and is finite and if fix) approaches infinity monotonicaliy as x=a on some V*ia), then L (x-a)-/(x)=0, 196 INFINITESIMAL ANALYSIS. 1 or in other words f{x) has an infinity of order lower than ^3^-t Proof.— By means of Theorem 138 it follows from the hypoth- esis that for every £ there exists a 7* (a) within V*{a) such that for every Xi and X2 on a b, and also on V*ia), / fix)dx <£. Let X2 be any point of such a neighborhood and let Xi be so chosen that Xi — a = X2 — Xi . Since Xi and X2 are on V*{a), /(Xi)>/(X2). It follows from Theorem 116 that But / fix)dx >|/(X2)|-(X2-Xi). /(X2) • (X2-X1) = J/(X2) • (x2-a). f L (i— o)-/(x) =0 is not a sufficient condition for the existence of L I l(x)dx, xia'' X as is shown by the following example. Consider a set of points i,, I2 •Cs, . . . , i„, . . . such that Xn — a = 2(i„-|-i — a), Xi — a being unity. 2" Define /(i,) = l, /(x2)=|, /(i,)=2, . . . , /(=r")=^:fri' Letthefunc- tion be linear from f{xO to /(12), from /(xj) to /(xa), etc. Then 1/ Wx\ /(x)dx >i, /(x)J, etc. Since these integrals are all of the same sign, their sum for any given num- ber of terms is greater than the sum of the corresponding number of terms. 2 in the harmonic series. Also (x„ — o) ■ /(xr.) = — tT' ■'^l»e°ce L (x — a)/(x)=0. IMPROPER DEFINITE INTEGRALS. 197 Hence for x = X2, \j{x) \-{x-a)<2e. Since e is arbitrary, and since X2 is any point in V*{a), it follows that L /(x)-(x-a)=0. Corollary. — If / f(x)dx exists for every x between a and h, and l^ f)ix)dz L x=a\J X exists and is finite, and if /(x) is entirely positive or entirely negative, then zero is a value approached by {x — a)-f(x) as x approaches a. Proof. — Consider the case when the function is entirely positive. Suppose zero is not a value approached. Then there exists a pair of positive numbers £ and 8 such that for every X, x — a £. I— I On the interval, a a + d, consider the function x—a Since / ——zdx Jx x—a is a non-oscillating function of x, it follows from Theorem 25 that L f ——dx xLaJ T x-a exists, and by Theorem 142 this limit must be infinite. 198 INFINITESIMAL ANALYSIS. Since 1/(^)1 >^ on the neighborhood under consideration, it follows from Theo- rem 107 and Corollary 2; Theorem 40, that L f f{x)d2 L x=a^ X exists and is infinite, which is contrary to the hypothesis. Theorem 143.! // (1) fii^) and /2(x) are of the same rank of infinity at x = a, or if /i(x) is of lower order than fiix), (2) / fi{x)dx and I f2ix)dx both exist for every x on the segment a b, (3) There is a neighborhood of x^a on which /2(x) does not change sign, P (4) L I f2(.x)dx is finite, J th£n it follows that L I f\{x)dx exists and is finite. t This is what Professor Moore in his lectures calls the relative con- vergence theorem. Theorems 143, 144, 151, 152 in this form are due to him. t We notice that since under the hypothesis /zCi) does not change sign, L I f2{x)dx L I f2{x)dx J X cannot fail to exist either finite or infinite, for it follows from this hypothesis that / fi{x)dx is a non-oscillating function of x and therefore, by Theorem 25 that the limit exists. IMPROPER DEFINITE INTEGRALS. 199 Proof. — Since from the hypothesis L I f2{x)dx exists and is finite, we have by Theorem 138 that for every t there exists a F*(a) such that for every xi and X2 on segment a b and on V*{a) IX X2 J2{x)dx xi \, then L rf{x)di exists and is finite. t Note on page 192 shows that this hypothesis is not redundant. IMPROPER DEFINITE INTEGRALS. 203 Proof. — ^If in the proof of Theorem 140 we write Z),i-* = ^ j^f instead of d.^-''= ' ~ \ and use Theorem 146 in- stead of 138, the proof of Theorem 140 will apply to Theorem 148. Theorem 149. // f{x) does not change sign for x greater than some fixed number D, and if for some positive number m and some number k^\ \{x-aY-f{x)\ >m for every x greater than D, then L f){x)dx cannot exist and be finite. Proof. — By making suitable changes in the proof of Theorem 141 so as to make xi and X2 approach infinity instead of a, that proof applies to this theorem. Theorem 150. // L / f{x)dx 1=00 »/(i exists and is finite, and if f{x) is monotonic for all values of x greater than some fixed number, then L (x-a)-/(x)=0. Proof. — By making slight modifications of the proof of Theorem 142, that proof applies to this theorem. Corollary. — If / f{x)dx exists for every x greater than a, and L f'f{x)dx « exists and is finite, and if /(x) does not change sign for x greater 204 INFINITESIMAL ANALYSIS. than some fixed number, then zero is a value approached by (x-a)/(x) as X approaches oo. The proof is similar to that of the corollary of Theorem 142. Theorem 151. // (1) fi{x) and /2(x) are infinitesimals of the same rank as x approaches 00, or if fi{x) is of higher order than f2{x), (2) / fi{x)dx and I f2{x)dx both exist for every x, a- isling for every x>a, then a necessary condition that L rh{x)dx shall exist and he finite is that L (x-a)-/i(x) = 0. The proof is like that of Theorem 145. § 3. Properties of the Simple Improper Definite Integral. The following definition of the simple improper definite integral is equivalent in substance to that given on page 192, and in form is partly the definition of the general improper definite integral given on page 210. The definite integral of a function is said to exist properly at a point Xi or in the neighborhood of this point, on the interval l-i I — I . . . . a 6 if there exists an interval on ai bi contaimngxi as anmterior point (or as an end point in case Xi=a or Xi=b) ^vrh that the proper definite integral of fix) exists on this : te val. The integral is said to exist improperly at a point xi on the interval aTb if f(x) has an infinite singularity at Xj and there exists an interval ai bi on a b containing Xi as an interior point (or end noint in case Xi=a or ii=6) such that the improper definite I — I . I— I integral exists on each of the intervals a, xi and xi 61. If on an interval a b the definite integral exists properly at every point except a finite number of points, and ex- ists improperly at each of these points, then the improper definite integral is said to exist simply on the interval iTb, or the simple improper definite integral is said to exist on 206 INFINITESIMAL ANALYSIS. 1-1 . I-I the interval a b. Let Xi, X2, . . . , Xn be the points of a 6 at which the integral exists improperly. The simple improper l-l definite integral on a 6 is the sum of the improper definite l-l I— I 1 1 I I — I integrals on the intervals a Xi, xi X2, ■ . . , x„_i x„, x„ b. We denote the simple improper definite integral of f{x) on l-l the interval o 6 by rmdx This symbol is used generically to include the proper as well as the improper definite integral. Theorem 154. // a/o' Po exists a d, such that for every I' and /'' for which lm(/') -D\b and / _ /(x)dd= / _ j{x)dx- / _/(x)da Hence / f{x)dx \b^ a' Po 2|/i(xi)l>2M, and such that (xg —a) 5 J(xi — a). Let Xi, X2, X3, . . . , x„, . . . be a sequence of points dense only at a such that |/i(a;„)|>2|/i(x„_i)|>2''-i-M, and such that |x„— a| ^ i|x„_i -a\. We define /2(x) as follows: hM = - on the points Xi, X2, . . . , x„, . . . . TV IMPROPER DEFINITE INTEGRALS. 219 avd fzix) is linear between the points of the sequence Xi, X2, . . . , Xn,... Then there are values of x on a;„ x„_i such that l/i(a;)|-/2(x)>|-M, whence hi^)-hi^) is unbounded in the neighborhood of a.f Ubviously ^— ^ IS monotomc increasing as x approaches a. Theorem 170. For every function /i(x) which is unbounded in every neighborhood of x = a there exists a non-oscUlating func- tion f2{x) such that L I f2(x)dx exists and is finite, while (.x-a)-fi{x)-f2{x) is unbounded in the neighborhood of x=a. Proof. — According to the lemma there exists a function /3 (x) such that L fa (x) = 0, while fz{x)-fi(x) is imbounded and the fimction fsix) U{x) = x—a is monotonic increasing as x approaches a. Since ix-a)f4(x)-fiix)^f3ix)-f,ix), t In case L f,(x)=2l(x„.-a)/4(xOI, and another value of n, Ui, such that |(x„,-a)/4(x„J| >2|(x„,-a)/4(x„,)|, etc., rim+i being so chosen that |(2„„-a)/4(x„JI > 2l(x„„^i -a)/4(a;n„+i)|. In this manner we select from the sequence (1) a set of tenns forming the convergent series (xi-a)/4(xi) + (x„, -a)/4(x„,) + . . . + (x„^-a)/4(x„J+. . . (2) We then obtain a function f^ix) as follows: For the set of values of X x„„^i2-fi„+i(n + l)i/i(z'„+i)|. (n = l, 2, 3, . . .) On the X axis lay off a set of segments [a„] such that a„ is of length Bn and a;„ is its middle point. On the segments (t„ as bases construct isosceles triangles on the positive side of the X axis whose altitudes are n-|/i(a;)|. The measures of areas of these triangles form a convergent series. Let fsix) be any continuous, monotonic, unbounded function such that L rf3(x)dx exists and is finite. We then define /2(a;) as the function repre- sented by the following curve : (1) Those parts of the boundaries of the isosceles triangles just described which lie above the curve defined by ]z{x). (2) Those parts of the curve defined by ]z{x) which he out- side the triangles or on their boundary. Obviously the func- tion so "defined has the properties specified in the theorem, the points xx, X2,..., Xn', . . . being the set [a/] specified by (3) of the theorem. Theorem 171 means that from the hypothesis that the improper definite integral of f{x) exists on a 6 it is impossible to obtain any conclusion whatever as to the order of infinity or the rank of infinity of f{x) atx=a. This is what one would IMPROPER DEFINITE INTEGRALS. 223 expect a -priori, since the definite integral is a function of two parameters, while the necessary condition in terms of t ounded- ness would be in terms of only one of these. § 6. Special Theorems on the Criteria of the Existence of the Improper Definite Integral on the Infinite Interval. Theorem 172. For every function fi{x) which is unbounded as X approaches 00 there exists a non-osciUating function /2(x) such that L I f2{x)dz z^oDc/a exists and is finite, while (x — a)fi{x)-f2^x) is unbounded as x approaches . Proof. — Obviously the lemma of Theorem 170 can be stated so as to apply to the case where x approaches « instead of 0. If then in the proof of Theorem 161 the set of points ii . . . x„ . . . is so taken that L Xn=CO instead of a, the proof of Theorem 161 applies with the excep- tion that fzix) is non-oscillating decreasing instead of non-oscil- lating increasing. Theorem 173. For every function f\ {x) defined on the interval a 00 there exists a function /2(x) such that (1) /zCx) is corUinuou^ and does not change sign for x greater than a certain fixed number. (2) L I f2(x)dx exists and is finite. 224 INFINITESIMAL ANALYSIS. (3) For X on a certain set [x^ Proof. — Such a function /aCx) may be defined in a manner analogous to that of the proof of Theorem 171. The remarks as to the meaning of Theorems 170 and 171 apply with obvious modifications to Theorems 172 and 173. INDEX. 31 Absolute convergence series, 72 Absolute value, 14 Algebraic functions, 53 " numbers, 18 Approach to a limit, 60 Axioms of continuity, 4, ' ' of the real number system, 13 Bounds of indetermination, 84 " upper and lower, 3, 47 Broad improper definite integral, 211 Change of variable, 79, 126, 175 Class, 3 Closed set, 41 Constant, 44 Content of a set of points, 183 Contin ity at a point, 61 ' axioms of, 4, 31 " over an interval, 88 uniform, 89 Continuous function, 61 ' ' real number system, 4 Continuum, linear, 4 Convergence of infinite series, 71 to a limit, 60 Covering of interval or segment, 33 Decreasing function, 49 Dedekind cut, 7 Definite integral, 153 Dense, 41 Dense in itself, 41 Dependent variable, 44 Derivative, 117 " progressive and regres- sive, 118 Derived function, 120 Difference of irrational numbers, 8 Differential, 128 Differential coefficient, 130 Discontinuity, 62, 63 ' ' of the first and second kind, 85 The references are to pages, of infinite Discrete set, 41 Divergence, 71 Everywhere dense, 41 Exponential function, 54 Function, 44 " algebraic, 51 continuity of, at a point. 61 " continuity of, over an in- terval, 88 ' ' exponential, 54 " graph of, 46 " mfinite at a point, 101 " inverse, 45 " limit of, 60 " monotonic decreasing, 49 " " increasing, 49 " non-oscillating, 49 " oscillating, 49 " partitively monotonic, 50 " rational, 53 " " integral, 53 ' ' transcendental, 54 " unbounded, 47 " uniform continuity of, 89 ' ' upper and lower liound of, 47 ' ' value approached by, 60 Geometric series, 73 Grapn of a function, 46 Greatest lower bound, 4 Improper definite integral, 191 Improper definite integral, broad, 211 Improper definite integral, narrow, 217 Improper definite integral, simple, 205 Improper existence of the definite integral, 205 225 226 INDEX. Increasing function, 49 Independent variable, 44 Infinite, 101, 102 Infinite segment, 32 " series, 70 " " convergence and di- vergence of, 71 Infinitesimals, 75 Infinity as a limit, 40, 47, 60 Integral, definite, 153 existing properly at a point, 205 Integral oscillation, 155 Interval, 32 Inverse function, 45 Irrational number, 1, 4 ' ' numbers, difference of, 8 " product of, 8 " " quotient of, 9 " sum of, 8 Least upper bound, 4 L'Hospital's rule, 139 Limit, lower, 84 " of a function, 60 " of integration, 153 " point, 39 ' ' upper, 84 Limitand function, 60 Linear continuum, 4 Logarithms, 58 Lower bound of a function, 47 " " ■" " set of numbers, 3 " integral, 181 " segment, 12 Many-valued function, 44 Maximum of a function, 130 Mean-value theorem of the differen- tial calculus, 133 Mean-value theorem of the integral calculus, 169 Minimum of a function, 131 Monotonic function, 49 Narrow improper definite integral, 217 Necessary and sufficient condition, 65 Neighborhood, 38 Non-differentiable function, 150 Non-integrable function, 155 Non-numerably infinite set, 185 Non-oscillating function, 49 Nowhere dense, 41 Number, 1 " algebraic, 18 " irrational, 1, 4 Number, system, 4 Numbers, transcendental, 18 ' ' sequence of, 70 " sets of, 3 Numerably infinite set, 185 One-to-one correspondence, 30 Order of function, 102 Oscillation of a function, 49 " " " function at a point, 85 Partitively monotonic, 50 Perfect set, 41 Polynomial, 53 Product of irrational numbers, 8 Progressive derivative, 118 Proper existence of the definite inte- gral at a point, 205 Quotient of irrational mmibers, 9 Rank of infinitesimals and infinites, 114 Rational functions, 53 . ' ' integral functions, 53 " numbers, 1 Ratio test for convergence of infinite series, 73 Real mmiber system, 4, 13 Regressive derivative, 118 RoUe's theorem, 132 Segment, 32 infinite. 32 " lower, 12 Sequence of numbers, 70 Series, infinite, 70 " convergence and divergence of, 71 " geometric, 73 " Taylor's, 134, 135 Sets of numbers, 3 Simple improper definite integral, 205 Single-valued functions, 44 Singularity, 101 Sum of irrational numbers, 7 Taylor's series, 134 Theorem of uniformity, 35 Transcendental functions, 54 " numbers, 18 Unbounded function, 47 Uniform continuity, 89 Uniformity, 35 Upper bound of a function, 47 " "" set of numbers, 3 INDEX. 227 Upper integral, 181 " limit, 84 Value approached by a function, 60 «' " " the independ- ent variable, 60 Variable, 44 ' ' dependent, 44 " independent, 44 Vicinity, 38 V(a), 38 V*{a), 38 SHORT-TITLE CATALOGUE OF THE PUBLICATIONS or JOHN WILEY & SONS, New York. Lokdon: chapman & HALL, Ldhtbd. ARRANGED TTNDER SUBJECTS. 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Insects Injuring Fruits. (In preparation.) Stockbridge's Rocks and Soils 8vo, 2 so Winton's Microscopy of Vegetable Foods Svo, 7 50 Woll's Handbook for Farmers and Dairymen. i6mo, i 50 ARCHITECTURE. Baldwin's Steam Heating for Buildings ismo, a so Bashore's Sanitation of a Country House lamo. i 00 Berg's Buildings and Structures of American Railroads 4to, 5 00 Birkmire's Planning and Construction of American Theatres. Svo, 3 00 Architectural Iron and Steel Svo, 3 50 Compound Riveted Girders as Applied in Buildings. Svo, 2 00 Planning and Construction of High Office Buildings. Svo, 3 50 Skeleton Construction in Buildings Svo, 3 00 Brigg's Modem American School Buildings. 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Carpenters' and Wood- workers' Edition i6mo, morocco, Sabin's Industrial and Artistic Technology of Paints and Varnish 8vo, Siebert and Biggin's Modem Stone-cutting and Masonry 8vo, Snow's Principal Species of Wood 8vo, Sondericker's Graphic Statics with Applications to Trusses, Beams, and Arches. 8vo, Towne's Locks and Builders' Hardware i8mo, morocco. Wait's Engineering and Architectural Jurisprudence 8vo, Sheep, Law of Operations Preliminary to Construction in Engineering and Archi- tecture 8vo, Sheep, Law of Contracts gvo. Wood's Rustless Coatings: Corrosion and Electrolysis of Iron and Steel. .8vo, Worcester and Atkinson's Small Hospitals, Establishment and Maintenance, Suggestions for Hospital Architecture, with Plans for a Small Hospital iimo. The World's Columbian Exposition of 1893 Large 4to ARMY AND HAVY. Bemadou's Smokeless Powder, Nitro-cellulose, and the Theory of the Cellulose M<>'«»'» umo, 2 so • Bruff's Text-book Ordnance and Gunnery gvo 5 „„ Chase's Screw Propellers and Marine Propulsion gvo' 3 00 Cloke's Gunner's Examiner g^o' Craig's Azimuth ...4to[ 3 50 Crehore and Squier's Polarizing Photo-cbronograph gyo 3 00 • Davis's Elements of Law g^^' ^ • Treatise on the Military Law of United States ' ' gvo,' 7 00 Sheep, 7 50 De Brack's Cavalry Outposts Duties. (Carr. ) 24mo, morocco, 2 00 Dietz's Soldier's First Aid Handbook i6mo, morocco, 1 2s • Dudley's Military Law and the Procedure of Courts-martiaL . . Large i2mo, ., so Durand's Resistance and Propulsion of Ships gvo' _ Xo • Dyer's Handbook of Light ArtiUery ...lamo' 3 Eissler's Modem High Explosives g ' • Fiebeger's Text-book on Field Fortification Small gvo' 2 00 Hamilton's The Gunner's Catechism igmo' • Hofi'B Elementary Naval Tactics gvo' I SO 3 00 I SO 3 SO 2 00 3 00 6 00 6 SO S 00 S SO 3 00 4 00 I as I 00 6 OO 7 50 I 50 2 OO 4 OO 5 OO 5 OO lO I OO 7 50 2 so 4 00 I 50 50 4 00 3 00 2 00 2 so I so 2 00 Ingalls's Handbook of Problems in Direct Fire 8to, * Ballistic Tables 8yo, * Lyons'B Treatise on Electromagnetic Phenomena. Vob. L and II. .8vo, each, * Mahan's Permanent Fortifications. (Mercur.) 8to, ball morocco* Manual for Conrts-martiaL i6mo, morocco, * Mercur's Attack of Fortified Places i2mo, * Elements of the Art of War 8vo, Metcalf's Cost of Manufactures — And the Administration of Workshops. .8vo, * Ordnance and Gunnery. 2 vols i2mo, Murray's Infantry Drill Regulations i8mo, paper, Nixon's Adjutants' ManuaL 24mo, Peabody's Naval Architecture 8vo, * Phelps's Practical Marine Surveying 8to, Powell's Army Officer's Examiner i2mo, Sharpe's Art of Subsisting Armies in War iSmo, morocco, * Tupes and Poole's Manual of Bayonet Exercises and Musketry Fencing. 24mo, leather, * Walke's Lectures on Explosives 8vo, Weaver's Military Explosives 8vo, * Wheeler's Siege Operations and Military Mining 8vo, Winthrop's Abridgment of Mihtary Law i2mo, Woodhull's Notes on Military Hygiene i6mo, Young's Simple Elements of Navigation i6mo, morocco, ASSAYING. Fletcher's Practical Instructions in Quantitative Assaying with the Blowpipe. i2mo, morocco, Furman's Manual of Practical Assaying 8vo, Lodge's Notes on Assaying and Metallurgical Laboratory Experiments. . . .8vo, Low's Technical Methods of Ore Analysis 8vo, Miller's Manual of Assaying i2mo. Cyanide Process i2mo, Minet's Production of Aluminnm and its Industrial Uge. (Waldo.) i2mo, O'DriscoU's Notes on the Treatment of Gold Ores 8vo, Ricketts and Miller's Notes on Assaying 8vo, Robine and Lenglen's Cyanide Industry. (Le Clerc.) 8vo, nike's Modem Electrolytic Copper Refining 8vo, Wilson's Cyanide Processes i2mo, Chlorination Process i2mo, ASTRONOMY. Comstock's Field Astronomy for Engineers 8vo, 2 50 Craig's Azimuth 4to, 3 so Crandall's Text-book on Geodesy and Least Squares 8vo, 3 00 Doolittle's Treatise on Practical Astronomy 8vo, 4 00 Gore's Elements of Geodesy ..8vo, 2 50 Hayford's Text-book of Geodetic Astronomy 8vo, 3 00 Merriman's Elements of Precise Surveying and Geodesy 8vo, 2 50 * Michie and Harlow's Practical Astronomy 8vo, 3 00 * White's Elements of Theoretical and Descriptive Astronomy i2mo 00 BOTANY. Davenport's Statistical Methods, with Special Reference to Biological Variation. i6mo, morocco, i 25 Thom^ and Bennett's Structural and Physiological Botany. i6mo, 2 25 Westermaier's Compendium of General Botany. (Schneider.) 8to, 2 00 3 I so 3 00 3 00 3 00 I 00 I 00 2 50 2 00 3 00 4 00 3 00 I so z so CHEMISTRY. * Abegg's Theory of Electrolytic Dissociatioo. (Von Ende.) i2mo, i 25 Adriance's Laboratory Calculations and Specific Gravity Tables i2mo, i 25 Alezeyeff's General Principles of Organic Synthesis. (Matthews.) 8vo, 3 00 Allen's Tables for Iron Analysis 8vo, 3 00 Arnold's Compendium of Chemistry. (Mandel.) Small 8vo, 3 50 Austen's Notes for Chemical Students lamo, i 50 Bemadou's Smokeless Powder. — Nitre-cellulose, and Theory of the Cellulose Molecule i2mo, 2 50 * Browning's Introduction to the Rarer Elements 8vo, i so Bnuh and Penfield's Manual of Determinative Mineralogy 8vo, 4 00 * Claassen's Beet-sugar Manufacture. (Hall and Rolfe.) 8vo, 3 00 Classen's Quantitative Chemical Analysis by Electrolysis. (Boltwood.). .8vo, 300 Cohn's Indicators and Test-papers Z3mo, 2 00 Tests and Reagents 8vo, 300 Crafts's Short Course in Qualitative Chemical Analysis. (Schaeffer.). . . z3mo, i 50 * Danneel's Electrochemistry. (Merriam.) i2mo, 1 25 Dolezalek's Theory of the Lead Accumulator (Storage Battery). (Von Ende.) i3mo, 2 50 Drechsel's Chemical Reactions. (MerrilL) i2mo, i 25 Duhem's Thermodynamics and Chemistry. (Burgess.) 8vo, 4 00 Eissler's Modem High Explosives 8vo, 4 00 Effront's Enzymes and their Applications. (Prescott.) 8to, 3 00 Erdmann's Introduction to Chemical Preparations. (Dunlap.) i2mo, i 25 Fletcher's Practical Instructions in Quantitative Assaying with the Blowpipe. i2mo, morocco, i 50 Fowler's Sewage Works Analyses i2mo, 2 00 Fresenius's Manual of Qualitative Chemical Analysis. (Wells.) 8vo, 5 00 Manualof (Qualitative Chemical Analysis. Part I. Descriptive. (Wells.) 8vo, 3 00 System of Instruction in Quantitative Chemical Analysis. (Cohn.) 2 vols 8vo, 12 so Fuertes's Water and Public Health i2mo, i so Furman's Manual of Practical Assaying 8vo, 3 00 * Getman's Exercises in Physical Chemistry i2mo, 2 00 Gill's Gas and Fuel Analysis for Engineers izmo, i 25 * Gooch and Browning's Outlines of Qualitative Chemical Analysis. Small 8vo, 1 25 Grotenfelt's Principles of Modem Dairy Practice. (WolL) i2mo, 2 00 Groth's Introduction to Chemical Crystallography (Marshall) i2mo, i 25 Hammarsten's Text-book of Physiological Chemistry. (MandeL) Svo, 4 00 Helm's Principles of Mathematical Chemistry. (Morgan.) iimo, i so Bering's Ready Reference Tables (Conversion Factors) i6mo, morocco, 2 so Hind's Inorganic Chemistry 8vo, 3 00 * Laboratory Manual for Students i2mo, i 00 Holleman's Text-book of Inorganic Chemistry. (Cooper.) 8vo, 2 50 Text-book of Organic Chemistry. (Walker and Mott.) Svo, 2 50 * Laboratory Manual of Organic Chemistry. (Walker.) i2mo, i 00 Hopkins's Oil-chemists' Handbook Svo, 3 00 Iddings's Rock Minerals Svo, 5 00 Jackson's Directions for Laboratory Work in Physiological Chemistry . . Svo, 125 Keep's Cast Iron Svo, 2 50 Ladd's ISanual of Quantitative Chemical Analysis i2mo, z 00 Landauer's Spectrum Analysis. (Tingle.) Svo, 3 00 * Langworthy and Austen. 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(Tingle.). . xamo. Miller's Manual of Assaying lamo. Cyanide Process i2mo, Minet's Production of Aluminum and its Industrial Use. (Waldo.) . . . . i2mo, Mixter's Elementary Text-book of Chemistry x2mo, Morgan's An Outline of the Theory of Solutions and its Results. ...... i2mo, Elements of Physical Chemistry i2mo, * Physical Chemistry for Electrical Engineers i2mo, Morse's Calculations used in Cane-sugar Factories z6mo, morocco, * Muir's History of Chemical Theories and Laws 8vo, Mulliken's General Method for the Identification of Pure Organic Compounds. VoL I Large 8to, O'Brine's Laboratory Guide in Chemical Analysis 8to, O'Driscoll's Notes on the Treatment of Gold Ores 8vo. Ostwald's Conversations on Chemistry. Part One. (Ramsey.) l2mo, " " " " Part Two. (Turnbull.) i2mo, * Pauli's Physical Chemistry in the Service of Medicine. C Fischer.) .... z2mo, * Penfield's Notes on Determinative Mineralogy and Record of Mineral Tests. 8vo, paper, Pictet's The Alkaloids and their Chemical Constitution. (Biddle.) 8vo, Pinner's Introductian to Organic Chemistry. (Austen.) i2mo, Poole's Calorific Power of Fuels 8vo, Prescott and Winslow's Elements of Water Bacteriology, with Special Refer- ence to Sanitary Water Analysts i2mo, * Reisig's Guide to Piece-dyeing 8to, Richards and Woodman's Air, Water, and Food from a Sanitary Standpoint. .8vo , Ricketts and Russell's Skeleton Notes upon Inorganic Chemistry. (Part I. Non-metallic Elements.) 8vo, morocco, Ricketts and Miller's Notes on Assaying 8vo, Rideal's Sewage and the Bacterial Purification of Sewage 8vo, Disinfection and the Preservation of Food 8vo, Riggs's Elementary Manual for the Chemical Laboratory 8vo, Robine and Lenglen's Cyanide Industry. 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(Boltwood.) x2mo, z 50 • Wa]ke*s Lectures on Explosives 8vo, 4 00 Ware's Beet-sugar Manufacture and Refining Small 8vo, cloth, 4 00 Washington's Manual of the Chemical Analysis of Rocks 8vo, 2 00 Weaver's Military Explosives 8vo, 3 00 Wehrenfennig*5 Analysis and Softening of Boiler Feed-Water 8vo, 4 00 Wells's Laboratory Guide in Qualitative Chemical Analysis 8vo, t 50 Short Course in Inorganic Qualitative Chemical Analysis for Engineering Students iimo, i 50 Text-book of Chemical Arithmetic i2mo» i 25 Whipple's Microscopy of Drinking-water 8vo, 3 50 Wilson's Cyanide Processes i2mo» i 50 Chlorination Process z2mo, i 50 Winton's Microscopy of Vegetable Foods 8vo, 7 50 Wulling's Elementary Course in Inor^ai^ic, Pharmaceuticai, and Medical Chemistry. z2mo, 3 00 CIVIL ENGINEERING. BRIDGES AND ROOFS. HYDRAULICS. MATERIALS OF ENGINEERING. RAILWAY ENGINEERING. Baker's Engineers' Surveying Instruments z2mo, 3 00 Bixby's Graphical Computing Table Paper 19^X241 inches. 25 Breed and Hosmer's Principles and Practice of Surveying 8vo, 3 00 * Burr's Ancient and Modero Engineering and the Isthmian Canal .... Svo, 3 50 Comstock's Field Astronomy for Engineers 8vo» 2 50 Crandall's Text-book on Geodesy and Least Squares 8vo, 3 00 Davis's Elevation and Stadia Tables 8vo, i 00 Elliott's Engineering for Land Drainage 1200, i 50 Practical Farm Drainage i2mo, z 00 ♦Fiebeger's Treatise on Civil Engineering 8vo, Flemer's Phototopographic Methods and Instruments 8vo, Folwell's Sewerage. (Designing and Maintenance.) 8vo, Freitag's Architectural Engineering. 2d Edition, Rewritten 8vo, French and Ives's Stereotomy 8vo, Goodhue's Municipal Improvements i2mo, Gore's Elements of Geodesy 8vo, Hayford's Text-book of Geodetic Astronomy 8vo, Bering's Ready Reference Tables (Conversion Factors') x6mo, morocco, Howe's Retaining Walls for Earth i2mo, * Ives's Adjustments of the Engineer's Transit and Level x6ino, Bds. Ives and Hilts's Problems in Surveying x6mo, morocco, Johnson's (J. B.) Theory and Practice of Surveying Small 8vo, Johnson's (L. J.) Statics by Algebraic and Graphic Methods 8vo, Laplace's Philosophical Essay on Probabilities. (Truscott and Emory.) . z2mo. Mahan's Treatise on Civil Engineering. (1873.) (Wood.) 8vo, * Descriptive Geometry 8vo, Merriman's Elements of Precise Surveying and Geodesy 8vo, Merriman and Brooks's Handbook for Surveyors z6mo, morocco, Nugent's Plane Surveying 8vo, Ogden's Sewer Design i2mo, Parsons's Disposal of Municipal Refuse 8vo, Patton's Treatise on Civil Engineering 8vo half lealher. Reed's Topographical Drawing and Sketching 4to, Rtdeal's Sewage and the Bacterial Purification of Sewage 8vo, Siebert and Biggin's Modern Stone-cutting and Masonry 8vo, 6 5 00 5 00 3 00 3 so 2 so I 75 2 so 3 00 2 SO 1 25 25 I SO 4 00 2 00 2 00 s 00 z SO 2 SO 2 00 3 50 2 00 2 00 7 SO S 00 4 00 I SO a oo S oo 5 00 2 oo 6 oo 6 50 S 00 S so 3 00 3 so 1 25 3 so Smith's MbihibI of Topographical Drawing. (McMillan,) 8vo, 3 50 Sondericker's Graphic Statics, with Applications to 'irusses, Beams, and Arches. 8vo, Taylor and Thompson's Treatise on Concrete, Plain and Reinforced 8to, * Trautwine's Civil Engineer's Pocket-book i6mo, morocco, Venable's Garbage Crematories in America 8to, Wait's Engineering and Architectural Jurisprudence 8to Sheep, Law of Operations Preliminary to Construction in Engineering and Archi- tecture 8vo, Sheep, Law of Contracts 8vo, Warren's Stereotomy — Problems in Stone-cutting 8to, Webb's Problems in the Use and Adjustment of Engineering Instruments. xomo, morocco, Wilson's Topographic Surveying 8vo, BRIDGES AND ROOFS. Boiler's Practical Treatise on the Construction of Iron Highway Bridges. .8vo, 2 00 • Thames River Bridge 4to, paper, 5 00 Burr's Course on the Stresses in Bridges and Roof Trueses, Arched Ribs, and Suspension Bridges 8vo, 3 50 Burr and Falk's Influence Lines for Bridge and Roof Computations 8to, 3 00 Design and Construction of Metallic Bridges 8to 5 00 Du Bois's Mechanics of Engineering. Vol. II Small 4to, xo 00 Foster's Treatise on Wooden Trestle Bridges 4to, 5 00 Fowler's Ordinary Foundations 8vo, 3 50 Greene's Roof Trusses 8vo, i 25 Bridge Trusses 8to, 2 50 Arches in Wood, Iron, and Stone 8vo 2 50 Howe's Treatise on Arches 8vo, 4 00 Design of Simple Roof-trusses in Wood and Steel , 8vo, 2 00 Symmetrical Masonry Arches 8vo, 2 50 Johnson, Bryan, and Turneaure's Theory and Practice in the Designing of Modem Framed Structures Small 4to, 10 00 Merriman and Jacoby's Teit-book on Roofs and Bridges: Part I. Stresses in Simple Trusses 8vo, 2 so Part n. Graphic Statics 8vo, 2 so Part ni. Bridge Design 8vo, 2 so Part IV. Higher Structures 8vo, 2 50 Morison's Memphis Bridge 4to, 10 00 Waddell's De Pontibus, a Pocket-book for Bridge Engineers . . i6mo, morocco, 2 00 * Specifications for Steel Bridges i2mo, 50 Wright's Designing of Draw-spans. Two parts in one vohime 8vo, 3 SO HYDRAULICS. Barnes's Ice Formation 8vo, 3 00 Bazin's Experiments upon the Contraction of the Liquid Vein Issmng from an Orifice. (Trautwine.) 8vo. 2 00 Bovey's Treatise on Hydraulics ^°' 5 00 Church's Mechanics of Engineering 8vo, 6 co Diagrams of Mean Velocity of Water in Open Channels paper, ' 50 Hydraulic Motors ^vo, 2 00 Coffin's Graphical Solution of Hydrr.uUc Problems i6mo, morocco, 2 50 Flather's Dynamometers, and the Measurement of Power "mo, 3 00 7 4 oo 5 oo 4 oo S oo 6 oo 5 oo 5 oo lO oo 1 00 I so 4 00 3 oo 2 so 3 00 Folwell's Water-supply Engineering 8vo, 4 co Frizell'a Water-power 8vo, s 00 Fuertes's Water and Public Health ■ . izmo, i 50 Water-filtration Works lamo. 2 50 Ganguillet and Kutter's General Formula for tlie Uniform Flow of Water in Rivers and Other Channels. (Hering and Trautwine.) 8vo» 4 00 Hazen's Filtration of Public Water-supply Svo* 3 00 Hazlehurst's Towers and Tanks for Water-works 8vo, 2 50 Herschel's 1x5 Experiments on the Carrying Capacity of Large* Riveted, Metal Conduits 8vo, 2 00 Mason's Water-supply. (Considered Principally from a Sanitary Standpoint.) 8vo» Uerriman's Treatise on Hydraulics. . Svo, * Hichie's Elements of Analytical Mechanics 8vo, Schuyler's Reservoirs for Irrigation, Water-power* and Domestic Water- supply Large 8vo , ■*■ Thomas and Watt's Improvement of Rivers 4to» Turneaure and Russell's Public Water-supplies Svo, Wegmann's Design and Construction of Dams 4to, Water-supply of the City of New York from 1658 to 1895 4to, Whipple's Value of Pure Water Large i2mo, Williams and Hazen's Hydraulic Tables 8vo, Wilson's Irrigation Engineering Smail Svo, Wolff's Windmill as a Prime Mover Svo, Wood's Turbines. Svo, Elements of Analytical Mechanics Svo, MATERIALS OF ENGINEERING. Baker's Treatise on Masonry ConBtruction Svo, 5 00 Roads and Pavements Svo, s 00 Black's United States Public Works Oblong 4to, 5 00 ♦ Bovey's Strength of Materials and Theory of Structures Svo, 7 50 Burr's Elasticity and Resistance of the Materials of Engineering Svo, 7 50 Byrne's Highway Construction Svo, 5 00 Inspection of the Materials and Workmanship Employed in Construction. i6mo, 3 00 Church's Mechanics of Engineering. . Svo, 6 00 Du Bois's Mechanics-of Engineering. Vol. I Small 4to, 7 50 •Eckei's Cements, Limes, and Plasters Svo, 6 00 Johnson's Materials of Construction Large Svo, 6 00 Fowler's Ordinary Foundations Svo, 3 50 Graves's Forest Mensuration. Hvo, 4 co * Greene's Structural Mechanics. . Svo, 2 50 Keep's Cast Iron Svo, 2 50 Lanza's Applied Mechanics Svo, 7 50 Marten's Handbook on Testing Materials. (Henning.) 2 vols. Svo, 7 50 Maurer's Technical Mechanics. , . Svo, 4 00 Merrill's Stones for Building and Decoration Svo, 5 00 Merriman's Mechanics of Materials Svo, 5 00 * Strength of Materials i2mo, i 00 Metcalf 8 Steel. A Manual for Steel-users i2mo, 2 00 Patton's Practical Treatise on Foundations Svo, 5 00 Richardson's Modern Asphalt Pavements Svo, 3 00 Richey's Handbook for Superintendents of Construction i6mo, mor., 4 00 • Ries's Clays: Their Occurrence, Properties, and Uses Svo, 5 00 Rockwell's Roads and Pavements in France i2mo i 25 8 3 oo I 00 3 50 3 oo a oo 5 oo 8 oo 3 oo 3 so 3 so 4 oo Sabin's Industrial and Artistic Technology of Paints acd Varnish. 8vo, Smith's Materials of Machines i3mo, Snow's Principal Species of Wood 8vo, Spalding's Hydraulic Cement i3mo. Text-book on Roads and Pavements xsmo, Taylor and Thompson's Treatise on Concrete, Plain and Reinforced. 8vo, Thurston's Materials of Engineering. 3 Parts 8vo, Part I. Non-metallic Materials of Engineering and Metallurgy SyOp Part n. Iron and Steel gvo, Part m. A Treatise on Brasses, Bronzes, and Other Alloys and their Constituents 8to, Tillson's Street Pavements and Paving Materials 8vo, Waddell's De Pontibus. (A Pocket-book for Bridge Engineers.). .i6mo, mor., 300 • Specifications for Steel Bridges i3mo, 50 Wood's (De V.) Treatise on the Resistance of Materials, and an Appendix on the Preservation of Timber 8vo, 3 00 Wood's (De V.) Elements of Analytical Mechanics 8vo, 3 00 Wood's (H. P.) Rustless Coatings; Corrosion and Electrolysis of Iron and SteeL 8vo, 4 00 RAILWAY ENGIWEERIHG. Andrew's Handbook for Street Railway Engineers 3x5 inches, morocco, x 35 Berg's Buildings and Structures of American Railroads 4to, 5 00 Brook's Handbook of Street Railroad Location. i6mo, morocco, i 50 Butf s Civil Engineer's Field-book. x6mo, morocco, 2 50 Crandall's Transition Curve x6mo, morocco, x 50 Railway and Other Earthwork Tables 8vo, i 50 Dawson's "Engineering" and Electric Traction Pocket-book . . i6mo, morocco, 5 00 Dredge's History of the Pennsylvania Railroad: (1879) Paper, 5 00 Fisher's Table of Cubic Yards Cardboard, 33 Godwin's Railroad Engineers' Field-book and Explorers' Guide. . .i6mo, mor., 3 so Hudson's Tables for Calculating the Cubic Contents of Excavations and Em- bankments 8vo, X 00 Molitor and Beard's Manual for Resident Engineers x6mo, x 00 Nagle's Field M<^n^*^ for Railroad Engineers x6mo, morocco, 3 00 Phitbrick's Field Manual for Engineers i6mo, morocco, 3 00 Searles's Field Engineering x6mo, morocco, 3 00 Railroad SpiraL x6mo, morocco, i 50 Taylor's Prismoidal Formuls and Earthwork 8vo, x 50 * Trautwine's Method of Calculating the Cube Contents of Excavations and Embankments by the Aid of Diagrams 8vo, 2 00 The Field Practice of Laying Out Circular Curves for Railroads. i3mo, morocco, 3 50 Cross-section Sheet Paper, 3S Webb's Railroad Construction x6mo, morocco, s 00 Economics of Railroad Construction Large xsmo, 3 50 Wellington's Economic Theory of the Location of Railways Small 8vo, S 00 DRAWHTG. Barr's Kinematics of Machinery 8vo, 3 50 » Bartlett's Mechanical Drawing 8vo, 3 00 • " " " Abridged Ed 8vo, x so Coolidge's Manual of Drawing 8vo, paper, i 00 9 Coolidge and Freeman's Elements ot General Drafting for Mechanical Engi- neers Oblong 4to, Durley's Kinematics of Hachioes .8to« Emch's Introduction to Projective Geometry and its Applications 8to, Hill's Text-book on Shades and Shadows, and Perspective 8vo, Jamison's Elements of Mechanical Drawing 8vo, Advanced Mechanical Drawing 8vo, Jones's Machine Design : Part I. Kinematics of Machinery 8vo, Part n. Form, Strength, and Proportions of Parts 8vo, MacCord's Elements of Descriptive Geometry 8vo, Kinematics; or. Practical Mechanism 8vo, Mechanical Drawing 4to, Velocity Diagrams 8vo, MacLeod's Descriptive Geometry.. Small Svo, * Mahan's Descriptive Geometry and Stone-cutting 8vo, Industrial Drawing. (Thompson.) 8vo, Moyer's Descriptive Geometry 8vo, Reed's Topographical Drawing and Sketching 4to, Reid's Course in Mechanical Drawing Svo, Text-book of Mechanical Drawing and Elementary Machine Design. Svo, Robinson's Principles of Mechanism Svo, Schwamb and Merrill's Elements of Mechanism Svo, Smith's (R. S.) Manual of Topographical Drawing. (McMillan.) Svo, Smith (A. W.) and Marx's Machine Design Svo, * Titsworth's Elements of Mechanical Drawing Oblong Svo, Warren's Elements of Plane and SoUd Free-hand Geometrical Drawing. i2mo. Drafting Instruments and Operations X2mo, Manual of Elementary Projection Drawing i2mo. Manual of Elementary Problems in the Linear Perspective of Form and Shadow z2mo. Plane Problems in Elementary Geometry i2mo, Primary Geometry. lamo. Elements of Descriptive Geometry, Shadows, and Perspective 8vo, General Problems of Shades and Shadows Svo, Elements of Machine Construction and Drawing Svo, Problems, Theorems, and Examples in Descriptive Geometry Svo, Weisbach's Kinematics and Power of Transmission. (Hermann and Klein.): Svo, Whelpley's Practical Instruction in the Art of Letter Engraving i2mo. Wilsoa's (H. M.) Topographic Surveying 8vo, Wilson's (V. T.) Free-hand Perspective Svo. Wilson's (V. T.) Free-hand Lettering , gvo, Woolf's Elementary Course in Descriptive Geometry Large Svo, ELECTRICITY AND PHYSICS. * Abegg's Theory of Electrolytic Dissociation. (Von Ende.) i2ma, i 25 Anthony and Braclcett's Text-book of Physics. (Magie.) Small Svo 3 00 Anthony's Lecture-notes on the Theory of Electrical Measurements. .. .i3mo, 1 00 Benjamin's History of Electricity 8vo, 3 00 Voltaic Cell Svo, 3 00 Classen's Quantitative Chemical Analysis by Electrolysis. (Boltwood.).Svo, 3 00 * Collins's Manual of Wireless Telegraphy i2mo, i 50 Morocco, 2 00 Crehore and Squier's Polarizing Photo-chronograph Svo, 3 00 * Danneel's Electrochemistry. (Merriam.) i2mo, i 25 Dawson's "Engineering" and Electric Traction Pocket-book. i6mo, morocco, 5 00 10 2 SO 4 00 2 SO 2 00 2 50 2 00 I 50 3 00 3 00 S 00 4 00 I SO I so I SO 3 SO 2 00 S 00 2 00 3 00 3 00 3 00 2 SO 3 00 I 2S I 00 I 2S I SO I 00 1 25 7S 3 SO 3 00 7 so 2 so 5 00 2 00 3 50 2 so 3 OO 3 oo 3 OO 6 oo 4 oo 2 50 12 so Dolezalek's Theory of the Lead Accumulator (Storage Battery). (Von Ende.) i2ino, a jo Duhem's Thermodynamics and Chemistry. (Burgess.) 8vo, 4 00 Flather's Dynamometers, and the Measurement of Power i2mo, 3 00 Gilbert's De Hagnete. (Hottelay.) 8vo, 2 50 Hanchett's Alternating Currents Explained. i2mo, i 00 Bering's Ready Reference Tables (Conversion Factors) i6mo morocco, 2 so Holman's Precision of Measurements 8to, 2 00 Telescopic Hirror-scale Method, Adjustments, and Tests .... 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Svo, 2 E Keep's Cast Iron. ^'O' 11 2 SO 2 50 2 SO 3 00 3 00 6 00 4 00 3 00 I 00 I 00 3 00 2 50 Leach's The Inspection and Aiutlysis of Food with Special Reference to State Control Large 8vo, 7 so * McKay and Larsen's Principles and Practice of Butter-making 8vo, i 50 Uatthewe's The Textile Fibres 8vo, 3 50 Metcalf's SteeL A Uanual for Steel-users: i2mo, 2 00 Hetcalfe'r Cost of Manufacttires — And the Administration of Workshops . 8to, 5 00 Heyer's Modem Locomotive Construction 4to, zo 00 Horse's Calculations used in Cane-sugar Factories i6mo, morocco, z 50 * Reisig's Guide to Piece-dyeing 8vo, 25 00 Rice's Concrete-block Manufacture 8vo, ^ 00 Sabin's Industrial and Artistic Technology of Paints and Vamislu 8vo, 3 00 Smith's PresE-workizig of Metals 8vo, 3 00 Spaldizig's HydrauUc Cement. z2mo, 2 00 Spencer's Hancfbook for Cheznists of Beet-sugar Houses. .... z6mo morocco* 3 00 Handbook for Cane Sugar Manufacturers i6mo morocco* 3 00 Taylor and Thoznpson's Treatise on Concrete, Plain and Reinforced 8vo, 5 00 Thurston's Manual of Steam-boilers, their Designs, Construction and Opera- tiozi. 8to, s do * Walke's Lectures on Explosives 8vo, 4 00 Ware's Beet-sugar Manufacture and Refining Small 8vo, 4 00 Weaver's Military Explosives 8vo, West's American Foundry Practice z2mo, Moulder's Text-book z2mo, Wolff's Windmill as a Prime Mover 8vo, 3 00 SO 50 3 Wood's Rustless Coatings: Corrosion and Electrolysis of Iron and Steel. .8vo, 4 00 MATHEMATICS. I so 50 7S so z2mo, I 7S Baker's Elliptic Functions. 8vo, • Bass's Elements of Differential Calculus Z2mo, 4 Briggs's Elements of Plane Azzalytic Geometry Z2mo i 00 Compton's Moniul of Logarithznic Computations i2ino i 50 Davis's Introduction to the Logic of Algebra. gvo, i 50 • Dickson's CoUege Algebra Large Z2mo! i SO • Introduction to the Theory of Algebraic Equations Large z2mo, z 25 Emch's Introduction to Projective Geometry and its Applications 8vo Halsted's Elements of Geometry. ^vo Elementary Synthetic Geometry gyo' Rational Geometry z2mG • Johnson's (J. B.) Thr:.e-place Logarithmic Tables: Vest-pocket size. paper! is zoo copies for s 00 • Mounted on heavy cardboard, 8X10 inches, 25 zo copies for 2 00 Johnson's (W. W.) Elementary Treatise on Differential Calculus. .Small 8vo, 3 00 Elementary Treatise on the Integral Calculus SmalfSvo, z so Johnson's (W. W.) Curve Tracing in Cartesian Co-ordinates z2mo, z 00 Jolmson's (W. W.) 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Berry's Temperature-entropy Diagram i2mo Carnot's Reflections on the Motive Power of Heat (Thurston.) i2mo, i so Dawson's "Engineering" and Electric Traction Pocket-book i6mo, mor., s 00 Ford's Boiler Making for Boiler Makers i8mo, Goss's Locomotive Sparks gvo Locomotive Performance gvo Hemenway's Indicator Practice and Steam-engine Economy . ismo 14 7 SO 6 00 2 50 6 00 2 SO 7 so 7 SO 4 00 5 00 I 00 2 00 3 00 1 00 8 00 3 50 2 so 2 00 3 00 »5 2 00 S 00 Button's Hechanical Engineering of Power Plants 8to, 5 00 Heat and Heat-engines 8to. s 00 Kent's Steam boiler Economy 8vo, 4 00 Kneass's Practice and Theory of the Injector 8to, i so HacCord's Slide-valves 8vo, 2 00 Meyer's Modem Locomotive Construction 4to, 10 00 Peabody's Manual of the Steam-engine Indicator lamo. i 50 Tables of the Properties of Saturated Steam and Other Vapors 8vo, i oo Thermodynamics of the Steam-engine and Other Heat-engines 8vo, 5 00 Valve-gears for Steam-engines 8vo, i 50 Peabody and Miller's Steam-boilers 8vo, 4 00 Pray's Twenty Years with the Indicator Large 8vo, 2 50 Pupin's Thermodynamics of Reversible Cycles in Gases and Saturated Vapors. 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Svo, 5 00 Wehrenfenning'sAnalysisandSofteningof Boiler Feed-water (Patterson) 8vo, 400 Weisbach's Heat, Steam, and Steam-engines. (Du Bois.) Svo, 5 00 Whitham's Steam-engine Design 8vo, s 00 Wood's Thermodynamics, Heat Motors, and Refrigerating Machines. . .8vo, 4 00 MECHANICS AND MACHINERY. 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Svo, 400 MacCord's Kinematics; or, Practical Mechanism Svo, 5 00 Velocity Diagrams Svo, i 50 * Martin's Text Book on Mechanics, Vol. 1, Statics i2mo, 1 25 Maurer's Technical Mechanics. Svo, 4 00 Herriman's Mechanics of Materials Svo, s 00 * Elements of Mechanics i2mo, i 00 * Michie's Elements of Analytical Mechanics Svo, 4 00 *Parshall and Hobart's Electric Machine Design 4to, half morocco, 12 50 Reagan's Locomotives : Simple, Compound, and Electric. New Edition. Large i2mo, 3 00 Reid's Course in Mechanical Drawing Svo, 2 00 Text-book of Mechanical Drawing and Elementary Machine Design. Svo, 3 00 Richards's Compressed Air i2mo, i 50 Robinson's Principles of Mechanism Svo, 3 00 Ryan, Norris, and Hoxie's Electrical Machinery. VoL I Svo, 2 50 Sanborn's Mechanics: Problems Large 12310, i 50 Schwamb and Merrill's Elements of Mechanism 8to, 3 00 Sinclair's Locomotive-engine Running and Management. i2mo, 2 00 Smith's (O.) Press-working of Metals Svo, 3 00 Smith's (A. W.) 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Large 8vo, i 00 Text-book of Mineralogy 8vo, 4 00 Minerals and How to Study Them "•>«>• ' SO Catalogue of American LocaUties of Minerals Large Bvo, Manual of Mineralogy and Petrography >s™o Douglas's Untechnical Addresses on Technical Subjects "mo, i 00 Eakle's Mineral Tables ™' ' \^ Egleston's Catalogue of Minerals and Synonyms ■• ■ »»<>, so Goesel's Minerals and Metals: A Reference Book i6mo,mor. 300 Groth's Introduction to Chemical Crystallography (Marshall) "mo. i 25 17 7 SO 7 SO 3 00 2 SO 2 so 1 SO 3 00 2 oo I 00 2 so 4 oo I 00 8 00 3 50 z 50 3 00 00 IddingB's Rock Minerals 8vo, s oo HeiTill's Non-metallic Minerals: Their Occurrence and Uses 8to, 4 00 * Penfield's Notes on Determinative Mineralogy and Record of Mineral Tests. 8to, paper, 50 * Richards's Synopsis of Mineral Characters izmo, morocco, i 25 * Rles's Clays: Their Occurrence, Properties, and Uses 8vo, 5 00 Rosenbusch's Microscopical Physiography of the Rock- mak i ng Minerals. 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