.. - vi ..... ) v. L THEORY OF MAXIMA AND MINIMA BY HARRIS HANCOCK, PH.D. (BERLIN), DR. So. (PARIS) PROFESSOR OF MATHEMATICS IX THE UNIVERSITY OF CINCINNATI GINN AND COMPANY BOSTON NEW YORK CHICAGO LONDON ATLANTA DALLAS COLUMBUS SAN FRANCISCO COPYRIGHT, 1917, BY HARRIS HANCOCK ALL RIGHTS RESERVED 217.11 tgfte satbenaum GINN AND COMPANY PRO PRIETORS BOSTON U.S.A. PREFACE Mathematicians have always been occupied with questions of maxima and minima. With Euclid one of the simplest problems of this character was : Find the shortest line which may be drawn from a point to a line, and in the fifth book of the conies of Apollonius of Perga occur such problems as the determination of the shortest line which may be drawn from a point to a given conic section. It is thus seen that a sort of theory of maxima and minima was known long before the discovery of the differential calculus, and it may be shown that the attempts to develop this theory exercised considerable influence upon the discovery of the cal culus. Fermat, for example, after making numerous restorations of two books of Apollonius, often cites this old geometer in. his " method for determining maximum a nd minimum" 1638, a work which in some instances is so closely related to the calculus that Lagrange, Laplace, Fourier, and others wished to consider Fermat as the discoverer of the calculus. This he probably would have been had he started from a somewhat more general point of view, as in fact was done by Newton (Opuscula Newtoni, /, 86-88). Maclaurin (A Treatise of Fluxions, Vol. I, p. 214. 1742), wrote : " There are hardly any speculations in geometry more useful or more entertaining than those which relate to maxima and minima. Amongst the various improvements that began to appear in the higher parts of geometry about a hundred years ago, Mr. de Fermat proposed a method for rinding the maxima and minima. How the methods that were then invented for the mensuration of figures and drawing tangents to curves are comprehended and improved by the method of Fluxions, may be understood from what has already been demonstrated. A general way of iii iv THEOEY OF MAXIMA AND MINIMA resolving questions concerning maxima and minima is also de rived from it, that is so easy and expeditious in the most common cases, and is so successful when the question is of a higher degree, when the difficulty is greater and other methods fail us, that this is justly esteemed one of the most admirable applications of Fluxions." The theory of maxima and minima was rapidly developed along the lines of the calculus after the discovery of the latter. Mathematicians were at first satisfied with finding the necessary conditions for the solution of the problem. These conditions, how ever, are seldom at the same time sufficient. In order to decide this last point, the discovery of further algebraic means was necessary. Descartes had already remarked, in a letter of March 1, 1638, that Fermat s rule for finding maxima and minima was imperfect ; and we shall see that many imperfections still existed for a long time after the invention of the calculus by Newton. As introductory to a course of lectures on the calculus of variations, I have for a number of years given a brief outline of the theory of maxima and minima. This outline is founded on the lectures that were presented by the late Professor Weierstrass in the University of Berlin. It treats the ordinary cases ; that is, where the functions are everywhere regular and where the forms are either definite or indefinite. It was published as a bulletin of the University of Cincinnati in 1903. At that time I expected to publish another bulletin which was to treat the more special cases ; for example, where only one-sided differ entiation enters, the "ambiguous case," where the form is semi- definite, etc. A treatment of these cases, the extraordinary cases, required more study than was anticipated. The bulletin has consequently been delayed so long that I have concluded to give an entirely new exposition of the whole theory. In the preface to the German translation by Bohlmann and Schepp of Peano s Calcolo differenziale e principii di calcolo integrale, Professor A. Mayer writes that this book of Peano not only is a model of precise presentation and rigorous deduction, whose propitious influence has been unmistakably felt upon PREFACE v almost every calculus that has appeared (in Germany) since that time (1884), but by calling attention to old and deeply rooted errors it has given an impulse to new and fruitful development. The important objection contained in this book (Xos. 133-136) showed unquestionably that the entire former theory of maxima and minima needed a thorough renovation ; and in the main Peano s book is the original source of the beautiful and to a great degree fundamental works of Scheeffer, Stolz, Victor v. Dantscher, and others, who have developed new and strenuous theories for extreme values of functions. Speaking for the Germans, Professor A. Mayer, in the introduction to the above- mentioned book, declares that there has been a long-felt need of a work which, for the first time, not only is free from mis takes and inaccuracies that have been so long in vogue but which, besides, so incisively penetrates an important field that hitherto has been considered quite elementary. To a considerable degree these inaccuracies are due to one of the greatest of all mathematicians, Lagrange, and they have been diffused in the French school by Bertrand, Serret, and others. "We further find that these mistakes are ever being repeated by English and American authors in the numerous new works which are constantly appearing on the calculus. It seems, therefore, very desirable in the present state of mathematical science in this country that more attention be given to the theory of maxima and minima ; for it has a high interest as a topic of pure analysis and finds immediate appli cation to almost every branch of mathematics. I have therefore prepared the present book for students who wish to take a more extended course in the calculus as intro ductory to graduate work in mathematics. I do not believe in making university students study abstruse theories in foreign languages, and in this treatise it will be found that the peda gogical side of the presentation is insisted upon; for example, the Taylor development in series is given under at least half a dozen different forms. HARRIS HAXCOCK CONTENTS CHAPTER I FUNCTIONS OF ONE VARIABLE I. ORDINARY MAXIMA AND MINIMA SECTION PAGE 1. Greatest and smallest value of a function in a fixed interval ; maximum and minimum. Several maxima or minima in the same interval. Ordinary and extraordinary, proper and improper maxima and minima. The extremes of functions; upper and lower limits; absolute and relative maxima and minima 1 2. Criterion for maxima or minima values 2 3. Other theorems 4 4. Re surne of the above expressed in a somewhat different form .... 5 II. EXTRAORDINARY MAXIMA AND MINIMA A. Functions which have Derivatives only on Definite Positions 5. Left-hand and right-hand differential quotients 6 6. Criteria as to whether a root of the equation /(x) = offers an extreme of the f unction /(x). Statement and proofs of certain theorems . . 7 B. Functions which have only One-Sided Differential Quotients of a Certain Order for a Value x = X Q 7. Theorems and their proofs 11 C. Upper and Lower Limits of a One-Valued Function which is Continuous for Values of the Argument within a Definite Interval 8. Greatest and least values ; upper and lower limits. Examples 12 9. An interesting example given by Liouville 13 10. Examples in which the functions are discontinuous ; cases in which the upper and lower limits are not reached 14 PROBLEMS 15 vii viii THEORY OF MAXIMA AND MINIMA CHAPTER II FUNCTIONS OF SEVERAL VARIABLES I. ORDINARY MAXIMA AND MINIMA SECTION Preliminary Remarks PAGE 11. Proper and improper extremes 17 12. Criteria for maxima and minima 17 13. Definite, semi-definite, and indefinite forms 19 14. Criterion for a definite quadratic form . 19 II. RELATIVE MAXIMA AND MINIMA 15. Statement of the problem. Derivation of the conditions 21 PROBLEMS 22 CHAPTER III FUNCTIONS OF TWO VARIABLES I. ORDINARY EXTREMES 16. Definitions and derivation of the required conditions. Statement of different cases after Goursat 23 17. The indefinite case 25 18. The definite case 25 19. Summary of the results derived for the two cases above. Example taken from the theory of least squares 26 PROBLEMS 27 Introduction to the Ambiguous Case B 2 A C = 20-23. Exposition of the attending difficulties which are illustrated by means of geometric considerations due to Goursat 27 24. The classic example of Peano showing when we may and may not expect extremes 31 II. INCORRECTNESS OF DEDUCTIONS MADE BY EARLIER AND MANY MODERN WRITERS 25. The Lagrange fallacy followed by Bertrand, Serret, Todhunter, etc. Geometric explanation of this fallacy . 33 III. DIFFERENT ATTEMPTS TO IMPROVE THE THEORY 26. Failure in deriving the required conditions by studying the behavior of the function upon algebraic curves 35 27. Statement of the Scheeffer method 37 28. The method of Von Dantscher outlined 39 29. The Stolzian theorems . . 39 CONTENTS ix CHAPTER IV THE SCHEEFFER THEORY I. GENERAL CRITERIA FOR A GREATEST AND A LEAST VALUE OF A FUNCTION OF Two VARIABLES ; IN PARTICULAR THE EXTRAOR DINARY EXTREMES SECTION PAGE 30. Stolz s analytic proof of the Scheeffer theorem 43 31. Stolz s added theorem by which the nonexistence of extremes may be often ascertained 45 32. Scheeffer s geometric proof of his own theorem 46 33. Failure of this theorem 48 II. HOMOGENEOUS FUNCTIONS 34. Scheeffer s criteria of extremes for such functions 49 III. EXTREMES FOR FUNCTIONS .THAT ARE NOT HOMOGENEOUS 35. In particular the case where the terms of lowest dimension consti tute a semi-definite form. Extreme curves 52 36. The criteria established 54 37-40. A direct and more practical method due to Stolz of deriving the above results 55 41. Exceptional cases . . . . 58 EXAMPLES 59 IV. THE METHOD OF VICTOR v. DANTSCHER 42. The general method outlined and applied to the definite and indefinite cases 62 43-44. Treatment of the semi-definite case . 64 PROBLEMS 69 V. FUNCTIONS OF THREE VARIABLES Treatment in Particular of the Semldefinite Case 45. Extensions of the theorems and proofs given by Stolz and Scheeffer for functions of two variables 70 46. Evident generalizations of 37-41 71 PROBLEMS 72, x THEOKY OF MAXIMA AND MINIMA CHAPTER V MAXIMA AND MINIMA OF FUNCTIONS OF SEVERAL VARIABLES THAT ARE SUBJECTED TO NO SUBSIDIARY CONDITIONS I. ORDINARY EXTREMES SECTION PAGE 47. Nature of the functions under consideration. Definition of regular functions 73 48. Definition of maxima and minima of functions of one and of several variables 74 49. The problem of this chapter proposed. Taylor s theorem for functions of one variable 75 50. Taylor s theorem for functions of several variables 77 51. The usual form of the same theorem 79 52. A condition of maxima and minima of such functions 81 II. THEORY OF THE HOMOGENEOUS QUADRATIC FORMS 63. No maximum or minimum value of the function can enter when the corresponding quadratic form is indefinite. When is a quadratic form a definite form ? 82 54. Some properties of quadratic forms. The condition that the quadratic form <f> (x v x 2 , , x n ) = S^A^XAX^ be expressible as a function of n 1 variables 83 55-61. Every homogeneous function of the second degree (x v x 2 , - -, x n ) may be expressed as an aggregate of squares of linear functions of the variables 85 62. The question of 53 answered 91 III. APPLICATION OF THE THEORY OF QUADRATIC FORMS TO THE PROBLEM OF MAXIMA AND MINIMA STATED IN 47~51 63. Discussion of the restriction that the definite quadratic form must only vanish when all the variables vanish. The problem of this chapter completely solved 92 64. Upper and lower limits 94 CHAPTER VI THEORY OF MAXIMA AND MINIMA OF FUNCTIONS OF SEVERAL VARIABLES THAT ARE SUBJECTED TO SUBSIDIARY CONDITIONS. RELATIVE MAXIMA AND MINIMA 65. The problem stated 96 66. The natural way to solve it 96 67-70. Derivation of a fundamental condition 96 71. Another method of finding; the same condition 98 CONTEXTS xi SECTION PAGE 72. Discussion of the restrictions that have been made 100 73. A geometrical illustration of these restrictions 101 74. Establishment of certain criteria 102 75. Simplifications that may be made - 102 76. More symmetric conditions required 103 I. THEORY OF HOMOGENEOUS QUADRATIC FORMS 77. Addition of a subsidiary condition 103 78, 79. Derivation of the fundamental determinant Ae and the discussion of the roots of the equation Ae = 0, known as the equation of secular variations 104 79. The roots of this equation are all real 107 80-82. Weierstrass s proof of the above theorem 108 83, 84. An important lemma 109 85. A general proof of a general theorem in determinants Ill 86. The theorem proved when the variables are subjected to subsidiary conditions 112 87. Conditions that the quadratic form be invariably positive or invari ably negative 114 II. APPLICATION OF THE CRITERIA JUST FOUND TO THE PROBLEM OF THIS CHAPTER 88. The problem as stated in 65 solved and formulated in a symmetric manner 114 89. The results summarized and the general criteria stated 115 90. Discussion of the geometrical problem : Determine the greatest and the smallest curvature at a regular point of a surface. Derivation of the characteristic differential equation of minimal surfaces . 116 91. Solution of the geometrical problem : From a given point to a given surface draw a straight line whose length is a minimum or a maximum 123 92. Brand s problems. Two problems taken from the theory of light. Keflection. Refraction 126 CHAPTER VII SPECIAL CASES I. THE PRACTICAL APPLICATION OF THE CRITERIA THAT HAVE BEEN HITHERTO GIVEN AND A METHOD FOUNDED UPON THE THEORY OF FUNCTIONS, WHICH OFTEN RENDERS UNNECESSARY THESE CRITERIA 93. Difficulties that are often experienced. Fallacies by which maxima and minima are established when no such maxima or minima exist 134 xii THEOKY OF MAXIMA AND MINIMA SECTION PAGE 94. Definitions : Realm. A position. An n-ple multiplicity. Struc tures. A position defined which lies within the interior of a definite realm, on the boundary, or without this realm . . . 135 95. Statement of two important theorems in the theory of functions 136 96. Upper and lower limits for the values of a function. Asymptotic approach. Geometrical and graphical illustrations 137 97. Digression into the theory of functions 138 II. EXAMPLES OF IMPROPER EXTREMES 98. Cases where there are an infinite number of positions on which a function may have a maximum value 140 99. Reduction of such cases to the theory of maxima and minima proper. The derivatives of the first order must vanish . . . 141 100. The results that occur here are just the condition which made the former criteria impossible 142 101. The previous investigations illustrated by the problem : Among all polygons which have a given number of sides and a given perimeter, find the one which contains the greatest surface-area . 142 102-107. Solution of the above problem 143 102. Cremona s criterion as to whether a polygon has been described in the positive or negative direction 143 PROBLEM 147 108. A problem due to Hadamard 147 III. CASES IN WHICH THE SUBSIDIARY CONDITIONS ARE NOT TO BE REGARDED AS EQUATIONS BUT AS LIMITATIONS 109. Examples illustrating the nature of the problem when the vari ables of a given function cannot exceed definite limits . . . 149 110. Reduction of two inequalities to one 149 111. Inequalities expressed in the form of equations 150 112. Examples taken from mechanics 150 GAUSS S PRINCIPLE 113. Statement of this principle 151 114. Its analytical formulation . 152 115. By means of this principle all problems of mechanics may be reduced to problems of maxima and minima 152 116. Proof of a theorem in the theory of functions 153 THE REVERSION OF SERIK> 117-126. Proof and discussion of this theorem. Determination of upper and lower limits for the quantities that occur 154 123. Unique determination of the system of values of the x s that satisfy the equations 159 CONTENTS xiii SECTION PAGE 124. If the y s become infinitely small with the x s, the x s become infinitely small with the y s. The x s are continuous functions of the y s 160 125. The x s, considered as functions of the y s, have derivatives which are continuous functions of the y s. The existence of the first derivatives 162 126. The x s, expressed in terms of the y s, are of the same form as the given equations expressing the y s in terms of the x s . . 163 127. Conditions which must exist before the ordinary rules of differ entiation are allowable in the most elementary cases .... 163 MISCELLANEOUS PROBLEMS 164 CHAPTER VIII CERTAIN FUNDAMENTAL CONCEPTIONS IN THE THEORY OF ANALYTIC FUNCTIONS I. ANALYTIC DEPENDENCE; ALGEBRAIC DEPENDENCE 128. Rational functions of one or more variables. Functions defined through arithmetical operations. One-value functions. Infi nite series and infinite products. Convergence ...... 166 129. Uniform convergence 167 130. Region of convergence. Differentiation 167 131. Functions of several variables. Functions which behave like an integral rational function . . 168 132. Analytic dependence 169 133. Many-valued functions 169 134. Possibility of expressing many-valued functions through one- valued functions 169 135. An important theorem for the calculus of variations 171 136. The same theorem proved in a more symmetric manner .... 172 137. Application of this theorem and the definition of a structure of the (71 w)th kind in the realm of n quantities. Analytic continuation 174 138. Analytic structures defined in another manner. Structures of the first kind, second kind, etc 175 II. ALGEBRAIC STRUCTURES IN Two VARIABLES 139. A Igebraic structure defined ; regular point ; singular point . . . 176 140, 141. Development of an algebraic function y in the neighborhood of a regular point. For such a point there is only one power series in x which when written for y makes identically zero the function which defines the algebraic structure 177 xiv THEORY OF MAXIMA AND MINIMA III. METHOD OF FINDING ALL SERIES FOR y WHICH BELONG TO A k-ply SINGULAR POINT SECTION PAGE 142-143. Practical methods for the derivation of such series under the different conditions that may arise 180 144. Theorem of Stolz which serves as a check on the derivation of the above series 187 145. An example illustrating the general theory ]87 INDEX ...... ]91 THEORY OF MAXIMA AND MINIMA CHAPTER I FUNCTIONS OF ONE VARIABLE I. ORDINARY MAXIMA AND MINIMA 1 . A function f(x) which is uniquely defined for all values of x in the interval (a, I) is said* to have a greatest value or a maxi- mum for the value x = x Qt situated between a and b, if there is a positive quantity 8 such that for all values of h between 8 and + 8 the difference [i] /(*+*) -/(*>=<) exists, which at the same time does not vanish for all these values of h. This function has a smallest value or a minimum if under the same conditions there exists the difference [2] /(* +*)-/(*o)sO, which does not vanish for all values of h between 8 and + B. A function may have several such maxima and minima which may be different from one another ; it may have minima which are greater than maxima. (See the accompanying figure.) When the existence of com plete derivatives in the entire interval under con sideration is presupposed, the maxima and minima which may be derived are called ordinary, but when we have to do with functions whose derivatives exist only on definite points or with functions which have one-sided derivatives and the like, the maxima and minima may be called extraordinary. The discussion will be * See Genocchi-Peano, Calcolo differenziate e principii di calcolo integrate ( 131). 1 2 THEORY OF MAXIMA AND MINIMA restricted at first to ordinary maxima and minima. A maximum of f(x) is called proper by Stolz (G-rundzuge der Differential- und Inte- gralrechnung, Vol. I, p. 199) if in the formula [1] only the sign < stands ; while we have a proper minimum if there stands only the sign > in [2]. The maximum and minimum are improper if in formu las [1] and [2] the sign = also appears, however small 8 may be taken. / 1\ 2 1 For example, y = ( x sin - I has the value + when x = for \ a?/ nir consecutive integral values of n, however large, and that is for intervals as small as we wish. Stolz and others * use the notation extreme or extreme value of a function to denote indifferently either the maximum or minimum of the function. The maximum and minimum of a function defined as above are often denoted as absolute ~\ maximum and minimum, since they depend upon the collectivity of the values of f(x). Opposed to them appear the relative maximum or minimum, which enter if the independent variable x is subjected to a restriction so that h in the formulas [1] and [2] can take only restricted (and not arbitrary) positive and negative values. 2. If the function f(x) has for x = X Q a positive derivative f f (x Q ), the function is becoming larger on this position with in creasing a?, and its values are respectively smaller or greater than those o/(a; ) according as x is smaller or greater than X Q . It is assumed that x lies sufficiently near X Q . In this case the function f(x) has for x = X Q neither a maximum nor a minimum. Similar (mutatis mutandis) conclusions are drawn if / (XQ) is negative. It follows that if the function f(x) has for x = x a finite de rivative that is different from zero, then on this position the function has neither a maximum nor a minimum. If then we exclude from the values of x those to which a defi nite derivative (different from zero) corresponds, there remain either * Extremer Werth was used by R. Baltzer, Elem. d. Math., Bd. I, Aufl. 5, S. 217; Extremum by P. du Bois-Reymond, Math. Ann., Vol. XV, p. 564. t The authors just cited, as also Peano, understand by the absolute maximum and minimum of a function in a given interval the upper and lower limits of the function in this interval, if such limits are reached . See also A. Mayer, Leipz. Ber. (1899) , p. 122, and Lipschitz, Analysis, Vol. II, p. 306, and in particular Voss, Encyklopddie der Math. Wiss., Bd. II, Theil I, Heft I, S. 80, who remarks upon the weak terminology of the subject. FUNCTIONS OF ONE VARIABLE 3 those positions on which the function has no derivative (finite or infinite) or those places on which it has a vanishing derivative. These positions must be further examined if we wish to make ourselves sure of the existence or nonexistence of a maximum or minimum. No rule can be given for the cases where derivatives do not exist. If we assume that the derivative is zero, the following criteria may be used : If f(x) has the derivative / (/-) in the interval (x h X Q H- h), we have, in virtue of the Taylor formula,* where x l lies between # and x. If f (x) becomes zero on the posi tion x = X Q in such a way that it is positive for x < X Q and negative for x > x Q , then (x x^f (x^ is always negative, however x(= X Q ) be taken, and consequently it follows that /(a?) </(# ) for all values of x within the interval X Q li to X Q -f h. The function will there fore be in this case a proper maximum for x = X Q . If, however, f f (x) is negative for X<X Q and positive for x>x Qt the product (x #())/ (#!) is always positive, and the function will therefore be a proper minimum for x = X Q within the interval in question. If f (x) is zero, say, for values of x within the interval X Q x Q +h or within the interval X Q h . x , we have cases of improper ex tremes (maxima or minima). But if f (x) retains a constant sign in the neighborhood of x = X Q , then (x x Q )f (x 1 ) changes its sign according as X>X Q or X<X Q , and the function has neither a maximum nor a minimum. It is thus seen that the function f(x) has on the position x = X Q a maximum or a minimum according to the manner in which its derivative vanishes for x = X Q ; that is, according as we pass from positive to negative values or from negative to positive values with increasing x. It has neither a maximum nor a minimum if the derivative does not change its sign.} * See Pierpont, The Theory of Functions of Real Variables, Vol. I, p. 248. t Leibniz, Vol. V, pp. 220-226, is the first who made a distinction between maximum and minimum. See Maclaurin, A Treatise of Fluxions (1742), Vol. I, chap. Lx, and Vol. II, p. 695; and also Cauchy, Calc. differ., p. 63. With Leibniz, when ^ 0, y is ax a maximum if the curve is concave towards the z-axis, a minimum if the curve is concave away from the x-axis. 4 THEORY OF MAXIMA AND MINIMA 3. Instead of considering the sign of the derivative in the neighborhood of x Qt if we consider the sign of the second deriva tive for x X Q (when this second derivative is different from zero), we have the rule : The function f(x) has on the position X = X Q for which f r (x Q ) = a maximum or a minimum according as f"(x Q ) is negative or positive. Infinite values are always included unless it is stated to the contrary. In fact, if f"(x () )<Q, then f (x) is a decreasing function, and since it is zero for x = X Q , it goes from positive to negative values ; the inverse is the case if/"(x )>0. This rule leaves one in the lurch if /"(aj )= 0. If in general it is assumed that / K)=o, /"W=o, .... /(-() =o, it follows from Taylor s formula that where x l is situated between x and x. As here f (n \x) is assumed to be a continuous function, it retains a constant sign in the neighborhood of X Q . If n is odd, the factor (x x Q ) n changes sign according as x > X Q or x < X Q . Consequently f(x)f(x ) also changes its sign and/(a; ) is neither a maximum nor a minimum. If n is even, the factor (x X O ) H is positive and f(x)f(x ) has always the sign of f (n \x^. It follows that f(x Q ) is a maximum or minimum according as / (w) (^j) is negative or positive. We therefore have the theorem : * If for x = XQ the first and some of the following derivatives vanish, then f(x Q ) is or is not an extreme according as the first nonvanishing derivative is of even or odd order. If it is of even order, there is a maximum or a minimum according as the derivative in question is negative or positive. * See Maclaurin, A Treatise of Fluxions, Vol. I, p. 226 ; Vol. II, p. 695 ; and also Lagrange, (Euvres, Vol. I (1759), p. 4. It was Maclaurin who first gave a correct method of distinguishing maxima from minima. FUNCTIONS OF ONE VAEIABLE 5 4. The following may be regarded as a resume of what has been given above : The function f(x) is supposed to be uniquely denned for all values of x within an interval (a, b), and X Q is a point of this interval. The function f(x) is a proper maximum or a proper minimum for x = X Q if we are able to find a positive number 8 sufficiently small that the difference f(zQ + h)f(x Q ) retains a constant sign when h varies from 8 to + & If this dif ference is positive, the function f(x) is smaller for x = X Q than it is for the values of x neighboring X Q ; it is then a proper minimum. On the contrary, when the difference /(t +&)/(# ) is negative, the function is a proper maximum for x = X Q . If, furthermore, the sign = enters in the cases just mentioned, however small 8 be taken, we have an improper minimum or maximum. When the function f(x) admits a derivative for the value # of the variable, this derivative must be zero. In fact, the two quotients /(s +a)-/(s ) f(x Q -h)-f(x ) h -h which have here by hypothesis the same limit / (#<>) wnen ^ tends towards zero, are of different sign ; it is necessary then that their common limit f f (x Q ) be zero. Inversely, let X Q be a root of the equation f (x)=Q, situated between a and b, and taking the general case suppose that the first derivative which is not zero for x = X Q is the derivative of the nth order and that this derivative is continuous in the neigh borhood of the value X Q . The general formula of Taylor gives here, limiting it to the term of the ?ith degree, /(*+ *) -/W = / <n) K+ Oh) (o < e < i) where e is a quantity that is indefinitely small with h. Let 8 be a positive number such that as x varies from x 8 to X Q + S the absolute value of e is smaller than/ (n) (# ), so that /(# -{-&) /(# ) h n has the same sign as /^(zA If n is odd, we note that this dif- n ! ference changes sign with h ; there is then neither a maximum 6 THEOEY OF MAXIMA AND MINIMA nor a minimum for x = # . If n is even,/(a3 + h) f(x Q ) has the same sign as / (H) (# ) whether & be positive or negative ; the func tion is a minimum if/ (w) (^ ) is positive, and a maximum if/ (?l) (# ) is negative. It follows that for the function to be a maximum or minimum for x = X Q it is necessary and sufficient that the first derivative which vanishes for x = # be of even order.* In geometric language the preceding conditions denote that the tangent to the curve y=f(x) at the point J^ is parallel to OX and is not an inflectional tangent (see Figs. 2-5). o -X FIG. 2 FIG. 3 FIG. 4 X FIG. 5 -X II. EXTRAORDINARY MAXIMA AND MINIMA A. FUNCTIONS WHICH HAVE DERIVATIVES ONLY ON DEFINITE POSITIONS 5. Let the function y=f(x) be uniquely defined for all values of x between X Q 8 and # + 8 and suppose that it is continuous for x = XQ. If the expressions h h h> * See Goursat, Cours D Analyse, Vol. I, pp. 108 et seq. I shall refer hereafter to this work by the name of the author, and by Peano and Stolz I shall designate the works, cited above, of these two mathematicians. FUNCTIONS OF ONE VARIABLE 7 have limiting values when lim h = + 0, each of these expressions is called a one-sided differential quotient,* the first the right-hand, and the second the left-hand, differential quotient of f(x) with regard to x for the value x = x . If the two one-sided differential quotients of /(#) are equal to each other for x = x Qt their common value is called the complete differential quotient of f(x) with respect to x f or x = X Q . If next it is assumed that the one- valued function/"^) is con tinuous for all values of the interval (a, b) and has at least one sided differential quotients, the differential calculus offers a method for the determination of the maxima and minima. For if f(x Q ) is such an extreme of f(x), the quotient h must necessarily either vanish or change sign with k. It may therefore be concluded, as in 2, that the complete differential quotient f f (x) must be zero, and that if the right- hand and left-hand differential quotients of f(x) are different at the point x = X Q , they cannot have the same sign. These require ments are under the existing conditions necessary that f(x Q ) be an extreme of f(x) ; however, as it will be seen in the following, they are not always sufficient. 6. Criteria as to whether a root x = X Q of the equation f f (x)= offers an extreme of the function f(x}.\ THEOREM I. If / (#) vanishes for x = x Qt and if a positive quantity 8 may be so chosen that f(x) has complete differential quotients in the interval (X Q S - X Q + 8), and if f (x) changes sign neither in the interval (# 8 X Q ) nor in the interval (x - - XQ + 8) and also does not remain invariably zero in either of these intervals, then f(x Q ) is or is not a proper extreme of f(x) according as the sign of f(x) hi the first interval is different from or the same as it is in the second interval; and further more, in the first case f(x Q ) is a maximum or a minimum, of * See P. du Bois-Reymond, Math. Ann., Vol. XVI, p. 120 ; see also Pierpont, The Theory of Functions of Real Variables, Vol. I, p. 223. t See Cauchy, Calc. differ., Lesson 7, and see in particular Stolz, pp. 201-210. 8 THEOEY OF MAXIMA AND MINIMA f(x) according as f (x) on its passage through zero, as x with increasing values passes through x Qt changes from the sign -f- to the sign or from the sign to the sign +. This theorem is stated at the end of 3 and there proved ; and as also indicated in that section, the inconvenience arising due to the consideration of the sign f (x) may be obviated if the func tion f(x) has a complete second differential quotient for x = X Q . This leads to the following theorem : THEOREM II. If under the conditions assumed in Theorem I the function f(x) has for x X Q a complete second differential coefficient /"(# ) which is not zero, then f(x Q ) is a proper ex treme of /(#), being a maximum or minimum according as f rt (x Q ) is negative or positive. Due to the definition of a complete second derivative it follows that where RQi) is a quantity that becomes indefinitely small with h. If here the existence of a second derivative of f(x) is assumed only for x x Q9 then, since f (x Q ) = 0, there corresponds to every positive quantity e another quantity 8 such that, if 8 < h < 8, we have , -,, If, say, f"(x Q ) is positive, it follows at once that there must be a positive quantity 8 such that for 8 < h < 8 we have [3] / (+ *)>0. Hence f (xQ+h) must be negative when h is negative and posi tive when h is positive, so that on passing through zero (i.e., when x a; ), f f (x Q + h) passes with increasing x from a negative value to a positive value. Accordingly, in virtue of Theorem I, f(x Q ) is a proper minimum. The above theorem becomes the one given in 3 if it is assumed that there is an interval including the value x = # such that for all points within it a second differential quotient of f(x) FUNCTIONS OF ONE VARIABLE 9 exists, and if it is further assumed that f"(x) is continuous at least at the point x = X Q . THEOREM III. If, furthermore, f"(x ) = and / "(#o) ^ > then f(x Q ) is not an extreme of /(#). For since here it is seen that as f f (x) passes through the value / (# ) = 0, it does not change sign, and consequently /(# ) is not an extreme of f(x). The two preceding theorems are special cases of the two following : THEOREM IV. If for the value x = x we have then /(JJ Q ) is a proper extreme of f(x), being a minimum or maxi mum according a,sfW(x Q ) is positive or negative. For, owing to the supposed existence of the first 2 k differential quotients, there is an interval X Q 8 - X Q + 8 throughout which the differential quotients / (#),/"(), ,/ (2A ~ 1) (^) not only exist but are also continuous functions of x. Accordingly, in virtue of Taylor s formula, we have >2k-l [4] /(** + ft) -/(* (2 _ l which formula is true for all values of h such that 8 < h < 8. Owing to the existence oifW(x Q ), as in the case of formula [3] above, it is seen that for values of h such that 8 < h < 8 we have l/<2t-D(a. + h)>0 or <0 /?/ according as/ (2t) (# ) is positive or negative. If, then, / (2Xr) (# ) is, for example, positive, it is clear that / (2 * ) (a? + /i) is negative for values of h in the interval 8 ... and positive for values of h in the interval <X It follows from [4] that the difference /(# + ^)/(#o) f r a ll values of A- within the interval 8 ... -h 8 (excepting A= 0) is in variably -f or according as/ (2Ar) (# ) is -h or , and correspond ingly we have respectively a proper minimum or a proper maximum. 10 THE011Y OF MAXIMA AND MINIMA If it is further supposed that f (2k) (x) exists for all values of x in the interval X Q $ X Q + 8 and that fW(x) is a continuous function at least at x = X Q , then, as in 3, due to Taylor s expansion we have [5] /K + A)-/ from which the theorem is obvious. In exactly the same way we may prove THEOREM V. If for x = X Q the 2 k first differential quotients of f(x), viz., / (a ), /"(a ), ., /0*>(a> ), vanish, and if /<** +1 >(a ) *= 0, then f(x Q ) is not an extreme of /(#). REMARK. In the case that a; = x causes every differential quotient of f(x) to vanish, we cannot determine by means of Theorems II, III, and IV whether f(x) is an extreme or not. We must then apply Theorem I. _ j^ For example, it is seen that x = is a minimum of f(x) = e x *. THEOREM V a . If the given function f(x) can be developed in the neighborhood of the point X Q in a series in integral positive powers of x x = h so that f(x) =f(x Q + x - aj ) = c m h m + c m+1 h m +* + ., (c m * 0) then f(x Q ) is not an extreme of f(x) or is an extreme of f(x) according as m is odd or even ; and f(x Q ) is a maximum or mini mum according as c m is negative or positive. For here f(x ) = =f"(x ) . . . . =/<*- D (^ ), while /( w > (a? ) = m ! c w . This theorem may be proved directly by means of the property of series. For under the given assumptions corresponding to every quantity e > 0, we may choose another quantity 8 > 0, such that - e < c m + 1 h + c m+ Ji* H ---- < e. If m is a positive integer and c m < 0, say, and if | X | < 8, then f(x Q + h) - f(x ) < k (c m + e) ; and as e may be taken such that e < c m , the expression on the right-hand side is always negative, so that there is a maximum of f(x) at x = x Q . Similarly, we may prove the remaining part of V a . FUNCTIONS OF ONE VARIABLE 11 B. FUNCTIONS WHICH HAVE ONLY ONE-SIDED DIFFERENTIAL QUOTIENTS OF A CERTAIN ORDER FOR A VALUE x = x * 7. THEOREM VI. If the continuous function/^ 1 ) has for x = x one-sided differential quotients of the first order and of opposite sign (including -f- co and oc), then f(x ) is a proper extreme, being a maximum or minimum according as the right-hand differential quotient of f(x) is negative or positive. For if, say, the left-hand differential quotient is positive, the right-hand one being negative, then there exists a positive quantity B such that according as S < h < or < h < S, we have h It follows that /(a; + h) f(x Q ) < for all values of h that are situated within the interval 8 ... + S. Hence f(x Q ) is a proper maximum. THEOREM VII. If for x = X Q the 2 k first differential quotients of /(<), viz., / (a;), /"(a,-), . . , /<**>(,), vanish, and if /<**>(,) has for X = X Q one-sided differential quotients of contrary sign (+00 and oo included), then the value /(# ) forms a proper extreme of f(x), being a maximum or a minimum according as the right- hand differential quotient is negative or positive. If, for example, the left-hand differential quotient is positive, while the right-hand is negative, that is, h we note, since fW(x Q )= 0, that/^^-h h) < for all values of h within the interval 8 + 8 (the value h = excepted). Hence from formula [5], viz., it is seen at once that f(x Q ) is a proper maximum. THEOREM VIII. If for x = X Q the 2k I first differential quotients vanish, viz., f (x Q )=f"(x () }= . . . =/(2*-D = 0, and if * Stolz, p. 206; see also Pascal, Exercici, etc., pp. 215-222. 1895. 12 THEORY OF MAXIMA AND MINIMA -!)(#) has for x = x one-sided differential quotients of oppo site sign (+ oo and co included), then /(# ) does not form an extreme of /(#). If, however, these differential quotients are both positive or both negative, then /(# ) is a proper minimum or maximum of f(x). This theorem follows from [4] in the same manner as the preceding one did from [5]. Example. If f(x) = xv- (x ^ 0) and /(>) = (x)v(x< 0), show by means of Theorem VI that there is a proper minimum at x = if /x lies be tween and + 1. Verify the same result when /x lies between 2 A: and 2 k + 1 by making use of Theorem VII ; and by using Theorem IV show that f(x) is a proper minimum when /x is situated between 2 k 1 and 2k. C. UPPER AND LOWER LIMITS OF A ONE- VALUED FUNCTION WHICH is CONTINUOUS FOR VALUES OF THE ARGUMENT WITHIN A DEFINITE INTERVAL 8. If the function /(x) is continuous and uniquely defined in the definite interval (a, b), there exist the greatest and the least value* in the interval in question, which are known as the upper and lower limits of the function in this interval ; and, further, the function reaches these limits ; that is, if these limits are denoted by g and Jc, then there is at least one value c of x within the interval a - b for which the function is equal to g t and at least one value d within the same interval for which the same function is equal to Jc. But if the interval within which x varies is indefinitely large, (a, co) or ( oo, b) or ( oo, + co), the function need not have a maximum or a minimum, and also it need not have an upper or a lower limit. This is illustrated in the following examples.! (See also 96.) * Proofs of this and the following statements are found, for example, in Harkness and Morley, Intr. to Analytic Functions, 46, 50; E. B. Wilson, Advanced Cal culus, 19-25. See Peano, Theorem IV, 21, and also Dini, Fundamenti per la teorica delle funzioni di varidbili reali (German translation by Liiroth and Schepp, 36, 47). These proofs are founded upon Weierstrass s lectures, which, in turn, are founded upon the work of Bolzano, Abh. d. Bohmischen Gesellsch. der Wiss., Vol. V, p. 17. t Peano, 132. FUNCTIONS OF ONE VARIABLE 13 Example 1. Divide a number into two parts so that their product is a maximum. (Cf. Ex. 6 at end of 10.) Let a be the given number, x and a x the two summands, and y (a x) x their product. If we consider x as variable, we have y = a2x, which becomes zero for x = ^ We further have / = 2, so that the function y has a maximum for x = ^ ; that is, when both parts are equal, this value being y = - Since, however, the derivative / is positive for x < - and negative for x > -> it follows that the function increases in the interval f x, -J and decreases in the interval (- , + x j. The function has neither an upper nor a lower limit. Example 2. y = a? r . (x > 0) Through differentiation we have / = j^(l + loga:). The first factor is never zero and is always positive. The second factor becomes zero when log x = 1 or x = - The derivative passes therefore from negative (for x < -\ to positive values (for x > V The function has a minimum for x = - = 0.36788 , which is y = 0.676411 . This is also the lower e limit which the function takes in the interval (0, x). The function does not have either a maximum or an upper limit. Example 3. y = art, y = x~ i The derivative is zero for no finite value of x, but is infinite for x = 0. For this value y becomes zero, and the function will have at this point both a minimum and a lower limit with respect to the interval ( x, + oc); for all the values of x cause the function to be greater than zero. The function has neither a maximum nor an upper limit. 9. If we add to the postulates already made in the previous article regarding f(x) that it must have a complete differential quotient for all values of x between a and b, then / (#) vanishes for every value of x between a and b to which one of the values g or k of the function belongs. If, however, /(a) = g, say, then possibly /(a) is only a one-sided maximum of /(), and consequently / (a) is not necessarily zero. This must be borne in mind as we proceed with the problem of determining the numbers g and X\ This is 14 THEORY OF MAXIMA AND MINIMA explained by the simple example (see Liouville s Journal, First Series, Vol. VII, p. 163): In the plane of a circle which is described about the point as center with radius r, let there be given an arbitrary point A which is different from 0. Determine the upper and lower limits of the distances of the point A from a point M of the circumference. Let the positive X-axis be taken passing through 0, and standing perpendicular to it through is erected the Y-axis. The equation of the circle is then * 2 4- y* r 2 ; while AM 2 = (a - x )*+y*=r*+ a 2 - 2 ax. (a) As M passes over all points of the circumference, x takes the values in the interval r - + r. The linear function (a) decreases with increasing values of /:, its differential quotient being a negative constant equal to 2 a. Consequently those values of x to which the upper and lower limits of AM 2 belong, fall on the end-points of the interval x IG. O r H- r. It is seen that r corresponds to the upper limit and + r to the lower limit, giving us as upper and lower limits respectively a + r | and | a r \ . 10. Suppose next that the function f(x) is discontinuous at least on an end-point of the arbitrary interval (a, 5); for example, sup pose that the function is not denned at such a point. If this is the case only for the lim x = a t then in the derivation of the upper and the lower limit we must consider in particular the value of f(x) for the lim x = a + 0. The following examples will make clear the method of procedure (see Stolz, p. 210). Example 1. Consider the function ?/ = for values of x such that log* < x < 1. It is seen that y is negative and decreases with increasing values of*. For when l im *= + (), then limy=-0; and when lim * = 1 0, then lim y QC. Thus the upper limit of y in this interval is zero, while the lower limit is x>, although neither is reached. FUNCTIONS OF ONE VAKIABLE 15 Example 2. y = (1 - x) sin -. (0 < x ^ 1) For these values of .r we have always |#j<l- o If we consider onlv values of x such as x - - - (where n is an , 4 n + 1 integer), we have Hence when n = + cc, the upper limit of y is + 1. Bv writing x = - , it is seen that the lower limit is 1. Neither 4n 1 the upper nor the lower limit of y is reached, although in either case they are finite. PROBLEMS 1. Determine the maxima and minima and the upper and lower limits of (b) y = a cosx + b sin x. e x + 1 (c) y = a + x*. (Pierpont, p. 320.) -1 (g) y = x 2 - ex*. (d) y = l x*. (Maclaurin, Vol. IT, l p. 720.) (h) ? = "*. (e) a- 2 sin_- (The function has a -~i ,,_, x (i) y = xe *. (There is no ex- discontinuous derivative for treme on the position x = 0.) x = 0.) 2. Show that the function //<*> = * Sin f ( " * } l/(0) - has an infinite number of maxima and minima within the interval _ _ 3. When is mp + nq a minimum, where p = V<? 2 + if, q = V 2 + (h y) 2 ? (Leibniz, 1682.) 4. "In venire cvlindrum maximi ambitus in data sphaera." (Fermat, (Eurres, Vol. I, p. 167. 1642.) 5. Find the area of the greatest parabola which may be cut from a given cone. 6. x (x a) has its greatest value when x = - (Euclid, Book VI, Prop. 27.) Cantor (Geschichte der Math., Vol. I, p. 228) says that this is the first example of a maximum in the history of mathematics. 16 THEORY OF MAXIMA AKD MINIMA 7. On a given line AB are two fixed points P l and P 2 . Determine a AP PS third point so that ^^ - is a minimum. (Pappus, Book VII, Prop. 61, *y * ^ 2 and Fermat, CEuvres, Vol. I, p. 140.) 8. Of all sections which pass through the vertex of a cone, determine the one of greatest area. (Severus.) 9. The number a is to be divided into two parts, such that their product multiplied into their difference shall be a maximum. (Tartaglia, General Trattato, Part 5, fol. 88.) 10. A ten-foot pole hangs vertically so that its lower end is four feet from the floor. Find the point on the floor from which the pole subtends the greatest angle. (Regiomontanus. 1471.) 11. The curve y = x 2 has no minimum. (Euler, Differential- rechnung, Vol. Ill, p. 744.) g X 12. Two points P l and P 2 not on the straight line AB are given. Find a point P on AB such that PP + PP^ is a minimum. (Solved by Huygens possibly about 1673. See Huygens, Opera Varia, pp. 490 et seq. Note the letters of De Sluse.) 13. Derive the greatest rectangle that can be described in, and having one of its sides, upon the base of a given triangle. (Simpson, Elements of Plane Geometry (1747), pp. 106 et seq. In this work are also found numerous problems that have to do with areas, volumes, etc.) CHAPTER II FUNCTIONS OF SEVERAL VARIABLES I. ORDINARY MAXIMA AND MINIMA PRELIMINARY EEMARKS 11. We say that the function u =f(x v # 2 , , x n ) becomes a maximum or minimum on the position (a v a 2 , . ., a n ) if for a sufficiently small region about (a lt 2 , -, a n ) we have f(a v a 2 , ., a n )^f(x v x 2 , . . -, x n ) or f(a v a a , -, a n ) ^ f(x v # 2 , - ., x n ). These extremes are proper or improper according as the sign = does not, or does enter. As in 1, it is assumed here that the function has definite partial derivatives which are continuous within the region in ques tion with regard to each of the variables ; and the extremes which may be derived we shall call ordinary. If the partial derivatives do not have such derivatives, the extremes may be called extraor dinary. Such extremes in their generality we shall not attempt to consider. Another class of extraordinary extremes is mentioned in 13, and is later treated in its generality for the case of func tions of two variables ( 20 et seq.). 12. Consider the function of one variable x lt viz., f(x v a v -, a n ). If the function u of the preceding article is a maximum or minimum for x^= a l9 , x n = a n , then f(x lf a 1? -, a n ) will be a maximum or minimum for x 1 = a v Hence (see 2) the derivative f x (x v a 2 , . . ., a n ) must be zero. Similar conclusions may be made for the other variables in u. It follows that if u =f(x v , x n ) has an extreme on the position (a v a 2 , -, a n ), the first partial derivatives of u must be zero. 17 18 THEORY OF MAXIMA AND MINIMA (See Euler, Gale. diff. (1755), p. 645; and Lagrange, Theorie des Fonctions, Vol. II, No. 51.) Write* next x l =a 1 +h 1 t f -, a>, t = a, t + h n t, and put If w =/(#!, , x n ) is an extreme on the position (a v - . ., rc M ), then F(t) is an extreme on the position t = 0. Since by hypothesis the derivatives of u are continuous, it follows also that the same is true of F(t}. We consequently may write It follows from 2 that whatever be the values of h v h 2 , ., h n . We 1 therefore have AK, , )= 0, . . .,/(!, -, a n )= 0, as was just seen. We further have If ^ is to be an extreme for the position under consideration, then F(t) must be an extreme for t = 0, so that for a maximum we must have ( 3) j^ (O) =g 0, and for a minimum ^"(0) i= 0, whatever be the values of h v h z , - ., h n . If for the time being we omit the sign = from the two expressions just written, we have the theorem : In order that the function u be an extreme at the position ( a i> > a n) f or ^hich the first derivatives vanish, it is necessary that the following homogeneous function of the second degree in hi, - -,h n) viz., See also Peano, 134. FUNCTIONS OF SEVERAL VARIABLES 19 assume only positive or only negative values, whatever be the values of h v - , h n , except when these quantities are all simul taneously zero. 13. We distinguish three kinds of integral functions of the second degree, or as they are usually called, quadratic forms* viz., I. Definite forms, which with real values of the variables have always the same sign, that is, only positive values or only negative values, and are only zero when the variables are all zero. II. Semi-definite forms,] which always have the same sign, but which vanish also for other values of the variables that are not all zero. III. Indefinite forms, w^hich with real values of the variables can become both positive and negative, and that too for values of the variables whose absolute values do not exceed an arbitrary small quantity. The theorem of the preceding section may be written as follows : If for x 1 = a v - ., x n = a n the first partial derivatives of the function u=f(x v ., x n ) vanish, and if in the Taylor develop ment t for f(x l -h h !,--, x n 4- h n ) the term which is a homogeneous function of the second degree in h v , h n is an indefinite form, then u on the position (#!,, ) has neither a maximum nor a minimum value. If, on the other hand, that term is a positive defi nite form, then u is a minimum, and if it is a negative definite form, u is a maximum. The case where the form is semi-definite is included under the extraordinary extremes, and we shall consider it later ( 20 et seq.). 14. Next is given a criterion to determine whether a given quad ratic form <t>(h lt -, h n ) is a positive definite quadratic form. If (/> depends only upon one variable h v we shall have <f>=Ahf, and this is positive when and only when A is positive. If <f> depends upon two variables h 1 and 7* 2 , we shall have < = AJil + 2 Bh^ + Chi * See Gauss, Disq. Arithm., p. 271. t So called, for example, by Scheeffer, Math. Ann., Vol. XXXV, p. 555. Gergonne, Gerg. Ann., Vol. XX (1830), p. 331, called attention in particular to this case. J This development is found in full in 50. 20 THEOEY OF MAXIMA AND MINIMA If here </> is a positive definite form, it follows that for h 2 = 0, A! = 0, then (f) = Ahf, and consequently A must be positive. We may also write $ in the form If in this expression we give to \ and h 2 such values that Ah l 4- Bh% = 0, it is seen that <j> takes the form </> = (AC B 2 ) hj. We must therefore have A C B 2 > 0. The conditions A > and AC B 2 >0 are not only necessary, but they are also sufficient that < be a definite quadratic form. In fact, if & 2 =0, we have (AC-JBP)h$>Q and (^ 1 + ^ 2 ) 2 - and consequently the sum of these two expressions, and also </>, is positive. If, in general, (f> depends upon several variables h v A 2 , 7i 3 , ., we may write where A is a constant, B a form of the first degree in h 2 , k s , ., and C a quadratic form in h 2 , A 3 , . If ^ 2 , 7^ 3 , are all zero, but /^ ^ 0, we will have B and C zero and < = Akf. We must therefore have A > 0, if (/> is to be a positive definite form. The form may be written where AC B 2 is a quadratic form of h 2 , h s , . The quantity A x may be determined so that AJi^ + B=Q with the result that Hence the expression AC B 2 must be positive and different from zero. Next write AC & ^(h^ h s , ), where c^ is a quadratic form in 7& 2 , A 3 , - - which is always positive and different from zero except when all the variables vanish. FUNCTIONS OF SEVERAL VARIABLES 21 It follows that the necessary conditions that (/> be a definite positive form are (1) that A be greater than and (2) that AC IP be a positive definite form in the variables 7t 2 , h s , .. These conditions are also sufficient; for if we give to h l an arbitrary value and to A 2 , h 3 , arbitrary values which are not all zero, then of the two summands into which <f> is distributed, the first is positive or zero, while the second is positive. It follows that <f> is positive. On the other hand, if we give to & 2 , & 3 , simultaneously the value zero, then Ji^ must be different from zero, and from (f>=Ahf it is seen that A must be positive. In this way the determination of the question whether a quadratic form is definite and positive is reduced to the determination of the same question in the case of another quadratic form of fewer variables. If then the process is continued, we come to the forms in one or two variables already considered. This subject is further considered in 53 et seq. To determine whether a quadratic form < is definite and nega tive, we have to determine whether < is definite and positive. (See Peano, 137.) II. RELATIVE MAXIMA AND MINIMA 15. To introduce the theory, we shall consider here a simple case involving only three variables. Let it be required to deter mine the extremes of the function u = F(x, y, z), where the variables x y y, z are restricted. Suppose, for example, that they are connected by the equation f(x, y, z) = 0. If from the latter equation z is expressed as a function of x and y, and if this value is substituted in the first equation, we shall have u expressed as a function of x and y. The values #, y which make u a maximum or minimum cause the total derivative du to vanish for all values dx and dy. 22 THEORY OF MAXIMA AND MINIMA dF dF dF We have du = - dx + dy 4- - dz t dx dy dz where dz denotes the differential of z s which is denned through the equation $f y $f dx dy dz If this last equation is multiplied by the indeterminate quan tity X and added to the equation du = 0, we have If in this equation we choose X so that the coefficient of dz vanishes, then corresponding to the maxima and minima values of u the coefficients of dx and dy must also be zero, and we thus have the equations -x=o, g^!=o, g-x|=o. dx dx dy dy oz dz It is evident that we have these expressions which are sym metric with regard to the three variables if we form the three partial derivatives of F X/, where X is an indeterminate quan tity, and then put the resulting expressions equal to zero. These three equations, together with the two equations /= and u=F, determine the unknown quantities X, x, y, z, u, which correspond to the values of u for which there exist maxima and minima values. We may proceed in the same manner with an arbitrary number of variables and equations of condition. (See Lagrange, Theorie des Fonctions, p. 268.) PROBLEMS 1. Find the minimum value of u, where M = a: 2 + y 2 + z 2 + -.., and where x, y, z, are connected by the equation ax + by + cz + = k. 2. If x l + 2 4- + x n = a, show that x? + x.}+ -> +k is a minimum when x^ = x<, = x n . (Maclaurin.) CHAPTER III FUNCTIONS OF TWO VARIABLES I. ORDINARY EXTREMES 16. Let z=f(x, y) be a continuous function of the two vari ables x and y when the point P with coordinates (x, y) remains within the interior of an area H which is limited by a contour C. We say that this function f(x, y) is a minimum for a point (X Q , y Q ) of the area fl when we can find a positive quantity B such that we have A = /(,, + h, y, + *)-/(* y ) & (i) for all systems of values of the increments h and k that are less than 8 in absolute value. The maximum is defined in a similar manner.* If we exclude the sign = in the expressions A ^ or A ^ 0, the extremes are said to be proper (cf. 1); but if the equality A = exists for certain values of h and k that are less than 8 in absolute value, however small 8 be taken, we have improper extremes. For example, in the case of the surface represented by the equation z =/(#, y}, the axis Oz being vertical, a proper maximum corresponds to an isolated summit, but if these sum mits form a line on the surface, this line will be a line of improper maxima. Consider, for example, the lines generated by revolving the extremes of a plane curve about the Ox-axis. If in the expression (/) we regard y as constant and equal to y , then z becomes a function of one variable x and ( 2) the difference /., . 7 x /., N /(*+*> y) /Kyo) can only retain a constant sign for small values of h if the O f derivative ~ is zero for x = x Qt y = y . ex * See also Goursat, loc. cit. 23 24 THEORY OF MAXIMA AND MINIMA In the same way it may be shown that these values must also cause to be zero. It follows that the systems of values dy which cause f(x, y) to become proper extremes are to be found among the solutions of the two simultaneous equations dx dy conditions which are also necessary for improper extremes. As only ordinary extremes are considered here, the partial deriva tives of the second order of f(x, y) are supposed to be continuous ( 11) in the neighborhood of the values X Q , y Q and are not all zero for x , y Q , and, furthermore, the derivatives of the third order are supposed to exist. If, then, x = x and y=y Q are a solution of the two equations (a), the formula for Taylor s theorem gives us For values of h and k in the neighborhood of zero, it is clear that the trinomial gives its sign to the right-hand side of (ii), and it is evident that the discussion of the sign of this trinomial is going to enjoy a preponderant role. To have an extreme for x = X Q , y = y^ it is necessary and sufficient that the difference A retain a constant sign when the point (x Q + h, y Q + k) remains within the interior of a square sufficiently small which has the point (X Q , y Q ) for center. In this case the difference A will also retain a constant sign if the point (x +h, y Q + k) remains within a circle with radius sufficiently small and center (X Q) y Q ), and inversely ; for we may replace the square by the inscribed circle and reciprocally. Suppose, then, that C is a circle of radius r with the point FUNCTIONS OF TWO VAEIABLES 25 (X Q , y ) as center. We have all the interior points of this circle by writing h = p cos <, k = p sin (f> and causing (/> to vary from to 2 TT and by causing p to vary from r to 4- r. Making this substitution in A, it becomes A = f^cos 2 4> + 2.Ssin</> cos c/> + (7sin 2 </>) + ^ A 21 <J : where yl = -~ > ^ = TT ^ C = ^r > and where L is an expres- cx* dx Q 2y dy sion which retains a finite value in the neighborhood of the point (a? , y ). It is evident that several cases are to be distinguished according to the sign of & AC. 17. First case. B*-AOQ. The equation A cos 2 <f> + 2 B sin <f> cos <f> -h (7 sin 2 < = admits two real roots in tan $, and the left-hand side may be written as the difference of two squares in the form A = [a (a cos < + & sin </>)- /(acos </> + sn where >0, yS>0, and (ab -ba )^Q. By takmg the circle sufficiently small we may neglect the terms of the third and higher degrees in p. If next to the angle </> a value is given such that a cos <f> + b sin <f> = 0, it is seen that A will be negative ; while if we give the angle <f> a value such that a cos<^ + ft sin^ = 0, then A will be positive. It is therefore impossible to find a number r such that the dif ference A retains a constant sign when the absolute value of p is inferior to r, while the angle <> is arbitrary. It follows that the function f(x, y) has no extreme for x = x Qt y = y Q . 18. Second case. B 2 -AC<0. It is evident that A and C must have the same sign. The trinomial A cos 2 < + 2 B cos (f> sin $ 4- C sin 2 < = - [(A cos<f>+B sin <#>) 2 + (AC - JS^sin 2 ^] y does not vanish when <$> varies from to 2 TT. 26 THEORY OF MAXIMA AND MINIMA Let w be the lower limit of the absolute value of the trinomial and let H be the upper limit of the absolute value of the function L in a circle of radius R and center (x n , y n ). \ 0> 9W Q Let r be a positive number inferior to R and to -- Within H the circle of radius r the difference A will have the same sign as the coefficient of /a 2 , that is to say, of A or C. The function /(#, y) has therefore an extreme for x = # , y = y . 19. The above results may be summarized as follows: If at the point # , y we have ay ay ay there is wo extreme ; but if / gy Y ay ay ^Vy / az ay* there is a maximum or minimum according to the sign of the two , . .. ay a 2 / derivatives ^- > ^- There is a maximum if these derivatives are negative, a if they are positive, and it is also seen that we have a proper maximum or minimum.* Example. In the theory of least squares it is required to determine x, y so that the expression (A} u (x, y) = V (a k x + b k y + c*) 2 k = l may be as small as possible. In other words, determine the values of x and y for which u (x, y) is equal to its lower limit. Following the methods indicated above we must solve the two equations Ft is seen that the determinant of these equations is equal to the sum of the i n (n 1) squares (a^ rt$fc) a (&, Z = 1, 2, , n ; k < I), and this deter minant does no^ vanish if among the binomials a#r + l k y there are at least * See Lagrange, Misc. Taur., Vol. I. 1759. FUNCTIONS OF TWO VAKIABLES 27 two which do not differ from each other by a constant factor. Under this assumption the two equations (Z?) have one and only one system of solu tions x , y Q . That does in fact reach its lower limit for this pair of values is seen if we write in (A) x = X Q + , y y + r], and expand. We then have k=n u (x + , y + 77) u (JT O , y ) - V (jt + &A-7/) 2 , k = l which difference is a positive quantity for every system of values except = 0, 77 = 0. PROBLEMS 1. Find a point P of a plane such that the sum PA + PB + PC of its distances to three fixed points of the plane is a minimum. In particular consider the case when BA C > 120, and show that here the point A gives the minimum. (Cavalieri, Exercitationes Geometricae, pp. 504-510. 1647.) 2. In a plane triangle all of the angles have been measured with the same precision and found to have values a, (3, y. On account of the unavoidable error in observation, the sum a + )8 + y does not equal 180. Let the difference 180 (a + /3 + y) be equal to S, where 8 is expressed in circular measure. What values u, v, w (in circular measure) must be added to the three results of measurement if we wish (1) that a + /3 + y + u + v + w = 180, and (2) that u 2 + r 2 + ic 2 be as small as possible ? Answer, u = ^ 8 = v = tc. INTRODUCTION TO THE AMBIGUOUS CASE B 2 AC=Q 20. We shall first note the difficulties that attend this special case, and with Goursat* we shall illustrate these difficulties by means of geometric considerations ; we shall then call attention to erroneous deductions which have been made, and later a method will be given, due to Scheeffer, of determining the extremes for this case, when they exist. Let S be the surface represented by the equation z=f(x,y). If the function f(x, y) has an extreme at the point X Q , T/ O , and if the function and its derivatives are continuous, we must have * Goursat, p. 112. 28 THEORY OF MAXIMA AND MINIMA which shows that the tangential plane to the surface S at the point P Q (with coordinates X Q) y , Z Q ) must be parallel to the ^y-plane. In. order that this point shall correspond to an ex treme, it is necessary that in the neighborhood of the point P the surface S be entirely on one side of the tangential plane. We are thus led to the study of a surface with regard to a tangential plane in the neighborhood of the point of contact. Suppose that the origin has been transposed to the point of contact. The tangential plane being taken as the a^-plane, the equation of the surface is of the form z = ax 2 -f- 2 bxy -f- cy 2 + ay? -f- 3 fix^y -f- 3 yxy 2 + ?/ 3 , (i) where &, b, c are constants and #, /3, 7, 8 are functions of x t y which remain finite when x and y tend towards zero. To deter mine whether the surface S is situated entirely on one side of the a?2/-plane in the neighborhood of the origin, it is sufficient to study the intersection of this surface by the ajy-plane. This intersection is a curve C represented by the equation f(x, y) ax 2 + 2 Ixy -f c?/ 2 -f- ax*+ . . . = 0, (ii) and presents a double point at the origin. 21 . If. & 2 ac is positive, the equation ax*+ 2 Ixy + cy*=- [(ax + %) 2 - (5 2 - ac) 7/ 2 ] = Ch represents two real and distinct straight lines which pass through the origin. Suppose that we take these two lines for the axes of coordinates, and note that this is brought about by a linear change of the variables. The equation (ii) then has the form xy+R(x,y)=Q. (Hi) If in this equation we write y = ux, we have R (X, U3$) . . . =-. ^- (") where it is evident that R(x } ux) is divisible by x*. FUNCTIONS OF TWO VARIABLES 29 It follows from 140 (see also Goursat, 34) that equation (iv) has one and only one root, say u = ?(#), which tends towards zero with x. Hence through the origin there passes one branch of the curve C represented by an equation y = #(#), which is tangent at the origin to the axis Ox. If we interchange the role of the two variables x and y t it is seen that there also passes through the origin a second branch of the curve C which is tangent to the axis Oy. The point O is a double-point with distinct tangents. If, then, 6 2 e > 0, the intersection of the surface S by the tangential plane presents two distinct branches of curve C l and <7 2 which pass through the origin, and the tangents to these two branches of curve at the origin are represented through the equation FIG. 7 If we give to each region of the plane in the neighborhood of the origin the sign of the first term in (iii) } as seen in the figure, it is clear that if a point moves along either of the curves C l or <7 2 , the left-hand side of (iii) y and consequently also of (ii), changes sign as the point passes through the origin. It fol lows that f(x, y) does not have an extreme (cf. 17) at the origin. 22. If b 2 ae<0, the origin is a double isolated point ; for within the ulterior of a circle with sufficiently small radius described about the origin as center, the right-hand side of (ii) only vanishes at the origin itself. To show this write x = p cos <f>, y = r sin <, where x and y are the coordinates of a point in the neighborhood of the origin. Equation (ii) becomes f(x t y} = p 2 (a cos 2 ( + 2 b sin <f> cos < + c sin 2 < + pL) where L is a function of p and < which remains finite when p tends towards zero. Let H be the upper limit of \L\ when p is less than a positive number r. 30 THEORY OF MAXIMA AND MINIMA When <f) varies from to 2 TT; the trinomial a cos 2 </> + 2 b sin (f> cos (/> + c sin 2 < retains a constant sign. Let m be the minimum of its absolute value. It is clear that the coefficient of p 2 does not vanish for any point on the interior of a circle C with radius less than r and 777 having the origin as center. Consequently the equation /(a?, y) = admits of no other solution than x = 0, y = (i.e., p = 0) within the circle. It follows that f(x, y) retains a constant sign when the point x, y moves within the interior of this circle. Hence, also, all the points excepting the origin of the surface S which may be projected upon the circle C are situated upon the same side of the ^cy-plane. The function /(a;, y) t therefore, presents an extreme at the origin (cf. 18). 23. When 6 2 ac = 0, the two tangents at the double point coincide, and there are, in general, two branches of curve tangent to the same straight line, which form a cusp. The complete study of this theory will be found to require a somewhat delicate discussion. For example, y 2 =x? presents at the origin a cusp of the first kind ; that is, one which has the two branches of curve tangent to the Oo>axis lying the one above and the other below this tangent. The curve y* 2 x*y -f zt ofi= presents a cusp of the second kind ; the two branches of curve are tangent to the a?-axis and situated on the same side of it. The equation gives us, in fact, y = x 2 x*. The two values of y have the same sign in the neigh borhood of the origin and are only real when x is positive. The curve ^+ x*y* 6 x*y -f y*= presents two branches of curve which offer nothing peculiar, both being tangent at the origin to the #-axis. We have from this equation from which it is seen that the two branches obtained when we take successively the two signs before the radical have no singu larity at the origin. FUNCTIONS OF TWO VARIABLES 31 It may also happen that the curve is composed of two coincident branches, as is the case of the curve represented by the equation f(x t ?/) = y 2 - 2 o% + ^= ; that is, (y - 2 ) 2 = 0. It is evident that here the left-hand side passes through zero without changing sign. It may also occur that the point # , y Q is a double isolated point, as is presented through the curve at the origin. From the above it is seen that if the origin is a double isolated point, or if the intersection of the surface with the tangent plane is composed of two coincident branches, the function f(x, y) will be an extreme (hi the latter case just given an improper extreme); but if the intersection is composed of two distinct branches which pass through the origin, there will, in general, be no extreme, for the surface again cuts its tangential plane. 24. Take, for example,* the surface which cuts its tangential plane along two parabolas of which the one is interior to the other. That the surface may not cross its tangential plane, it is necessary that if we cut this surface by any cylinder having its elements parallel to Oz and passing through Oz, the curve of intersection shall lie on one side of the #?/-plane. Let y = (f) (x) be the trace of such a cylinder upon the xy- plane, the function <j>(x) being zero for x = 0. If /(O, 0) is to be a minimum, the function f(x, <f> (x)) = F(x), say, ought to be a minimum for x 0, whatever the function <j> (x). To simplify the calculation, suppose that we have chosen the axes of coordinates so that the equation of the surface is of the form . where A is a positive quantity. * This is a generalization due to Goursat (p. llo) of the classic example of Peano (loc. cit., Nos. 133-136). 32 THEORY OF MAXIMA AND MINIMA With this system of axes we have for the origin The derivatives of ^(a?) are For x = = y these formulas become If </> (0) = 0, the function J^(aj) evidently has a minimum for x = ; but if (0) = 0, it is seen that and Hence, in order that F(x) be a minimum, it is necessary that 3 be zero, while dx* must be positive, whatever the value of <"(()). These conditions are not satisfied for the surface considered above, while they are satisfied for the function . z = ?/ 2 4- #* It is thus seen that in the ambiguous case, where B* A C = 0, the derivation of the necessary and sufficient conditions for the extremes of functions of only two variables is going to be accompanied by difficulties. It is also evident that in the case of three or more variables these difficulties will be correspond ingly augmented. FUNCTIONS OF TWO VARIABLES 33 II. INCORRECTNESS OF DEDUCTIONS MADE BY EARLIER AND MANY MODERN WRITERS 25. One of the greatest mathematicians of all times, Lagrange (Theorie des Fonctions, p. 290), writes: If all the terms of the first and second dimensions [see formula (7/) of 16] vanish, it is necessary for the existence of a maximum or minimum that all the terms of the third dimension in h v 7* 2 , shall disappear and that the quantity composed of terms where h v h z , (cf. 51) form four dimensions shall be always positive for the minimum and always negative for the maximum when fi v /?, have any values whatever. Following Lagrange, all writers on this subject made the same incorrect deductions until Peano, in the remarks to Nos. 133-136 found in the Appendix to his Calcolo, wrote : " The proofs for the criteria by which the maxima and minima of functions of several variables are to be recognized, and which are given in most books, depend upon the theorem that in the Taylor develop ment for functions of several variables the ratio of the remainder after an arbitrary term to this term has a limit zero when the increments of the variables approach zero. This theorem is in general false when the term in question is not a definite form with respect to the increments of the variables, and when it is a definite form, the theorem needs proof." These fallacious conclusions are found, for example, in Bertrand (Calciil Differentiel, p. 504), and also in Serret (Calc., p. 219), who writes: The maxima or minima exist if for the values h v A 2 , which cause d?f and dPf to vanish the derivative d*f has invariably the minus or plus sign. Here d 2 /, c? 3 /, denote the homogeneous integral forms of the second, third, degrees in h lt A 2 , , when the function / is expanded by Taylor s theorem (cf. 51). Todhunter (pp. 227-229 of the 1864 and 1881 editions of his Calculus), for the semi-definite case where B 2 AC= 0, writes the Taylor expansion for a function of two variables in the form (see () of 16) , 2 , , X2 where R is the remainder term. 34 THEORY OF MAXIMA AND MINIMA The condition which it appears that he considered as suffi cient for an extreme is that A and R must have the same sign, and if the terms of the second dimension are zero for the position or positions in question, then also the terms of the third dimension must be zero. That this is not true is seen at once by observing Peano s classic example ^ y) = (y- P 2 x 2 ) (y - q 2 x 2 ), where the conditions just mentioned exist, although there is no extreme at the origin, as already seen in 24. Professor Pierpont (Bull, of the Am. Math. Soc., Vol. IV, p. 536) says, " Our English and American authors seem to be ignorant of these facts." Write Peano s example in the form It is seen that the function /(#, y) is positive in the neighbor hood of the origin upon every straight line through it; however, upon the parabola y = mx 2 the function in the neighborhood of the origin is positive, zero, or negative according as am 2 + 2 bm + c is positive, zero, or negative. We may further illustrate this as follows : Let the equations denote two curves through the origin. The function Fir 8 will have positive values for values of x, y on the arc BA of a circle with origin at the center and radius sufficiently small and negative values on the arc AD. Hence the function f(x, y) has minimum values on all straight lines through the origin that cut the arcBA and maximum values on the lines through the origin that cut the arc AD. FUNCTIONS OF TWO VARIABLES 35 If, further, the two curves 4> (x, y] = 0, NP (a;, y) = have a com mon tangent at the origin with their curvatures lying in the same direction, it is seen that all possible straight lines through the origin are such that the coordi- Y nates of any points on them cause f(x, y) to have positive values. This is true, for example, of the function already considered, f>o FIG. 9 In the spaces above and below both curves we have/(#, y) > 0, while this function is negative for the spaces between the two curves; so that there is a minimum upon every straight line through the origin, although there is a maxiumm * of f(x, y) for all points on the curve y = III. DIFFERENT ATTEMPTS TO IMPROVE THE THEORY 26. The existence of an extreme of the function f(x, y) at the origin, for example, a minimum, depends upon the condition that there exists an upper limit g such that the function f(x t y) for all values of x y y which satisfy the condition is positive ; or, geometrically speaking ( 16), this condition implies that there exists a circle with center (0, 0) within which the func tion is everywhere positive with the exception of the position (0, 0) itself. Instead of considering the values of such a function for the coordinates of points on straight lines through the origin, which lines may be written in the form x = ak, y = flk, * Note in this connection Scheeffer, Math. Ann.,Vo\. XXVI, p. 197 ; and Vol. XXXV, p. 545. S6 THEORY OF MAXIMA AND MINIMA a, ft being arbitrary constants, it would be natural to raise the question whether we could not determine the sufficient conditions for such extremes by studying the more general curves expressed through the algebraic equations x (k) = a^k + aj -\ ---- and make the requirement that the function f(x(k), y(k}) shall have an extreme for k = 0, whatever values there may be assigned to the positive integers m and n and to the m + n quantities a lt a 2> > a m> A @2> > Pn> & being of course assumed that all the quantities a and /3 are not simultaneously zero. It may, however, be shown that such sufficient conditions cannot be derived in the manner indicated. For if we write 4> (x, y) = y sm 2 x, ip(x, y) = y sin 2 # e x * t we have two curves denned through the equations <E> (x, y) = and M? (x, y) = which have at the origin the a?-axis as a common tangent and a contact of an indefinitely high order. There is consequently no curve of the form (i) which may be laid between these two curves ; for clearly any such curve must have with either of these curves a contact of indefinitely high order which is impossible for an algebraic curve. On the other hand, the function f(x, y) = <S> (x, y) M* (x, y) is positive in the whole plane excepting that part of the plane that is situated between the two transcendental curves, in which it is negative. Hence at the origin there is neither a maximum nor a minimum for the function f(x, y) t although for this function upon every curve (i) there enters a minimum. We may therefore desist from further requirements in this direction, and we shall next call attention to two methods, the one due to Scheeffer and the other to Von Dantscher, which are general in character when the discussion has to do with two variables and which lead to criteria which are of use in practice. FUNCTIONS OF TWO VARIABLES 37 27. Scheeffer s method. We have seen that functions of one variable which have ordinary extremes can be expressed through the Taylor development in the form /( , ;) = /!!(M^ (0 <0<1), (a) 71 I when f(x) and the derivatives f (x), -, f (n ~ l \x) are zero for x = while f (n \x) = for x = 0. For such functions the change in value in the neighborhood of the position x = on either side is faster than that of a given quantity ax n ; that is, positive quantities a, n, and S may be so chosen that for all values of x within the interval S to + S the absolute value of f(x) is greater than the absolute value of ax n , excepting the value x = 0. For since / (n) (0) = 0, we may so determine 8 that for values of x such that 8 ^ x ^ 8 the function / (n) (#) is different from zero. If, then, we choose the quantity ci smaller than the absolute f(n)/ x \ value of - *- in the interval 8 to + 8, then (see formula (a)) n i within this interval the condition |/(<)| > ax n \ is satisfied. Recip rocally, if the last condition exists, the n first derivatives of f(x) cannot all vanish for x = 0. For in the latter case we would have (*+V (n -hi)! and from this it follows that Em 8 = which contradicts the assumption that |/(#)| > \ax n \. There are functions, however, for example e x " (cf. Pierpont, loc. cit., Vol. I, p. 205), for which such quantities n, a, 8, do not _^ exist. In fact, the absolute value of e ** is in the immediate neighborhood of x = smaller than any arbitrary power ax n . We may note that the characteristic property of the above requirement consists in the fact that the behavior of the function in the neighborhood of the origin must be marked with a certain degree of distinctness. 38 THEORY OF MAXIMA AND MINIMA The following consideration leads to the generalization of the above condition for functions of two variables: It is clear that a function f(x, y) which vanishes at the origin, if it is continuous, has upon the circumference of every circle which is described about the origin as center with an arbitrary radius r a greatest and a least value, provided the function does not reduce to a function of one variable r = V y? -f- z/ 2 . The signs of these greatest and least values, which we denote by / x (r) and / 2 (r) respectively, offer for sufficiently small radii r a criterion regarding the appear ance or nonappearance of an extreme at the origin.* For if the two quantities / x (r) and / 2 (r) are positive, there will be a mini mum of f(x, y) at the origin, while if they are both negative, a maximum exists at the origin. The degree of distinctness which marks the behavior of the function at the origin is characterized through the existence of a power ar n with the property that for every value of r within a certain limit g both f 1 (r) and / 2 (r) are in absolute value greater than ar 11 . If this requirement is not satisfied we cannot count upon deriving sure characteristics of extremes through the expansion in series. For in this case the value with which the function f(x, y) in the neighborhood of the position (0, 0) either ap proaches the value zero from the one side, or having passed through zero differs from it on the other side, is so little that this value cannot be expressed through a power ever so high of r. The development in series cannot, therefore, serve to de termine whether the value is a little on the one side or on the other side of zero. As examples of this kind are the function which has a minimum value at the origin, and the function * The behavior of the function f(x, y) at any point JB O , y Q other than the origin may be made by the substitution x = x -f h, y y Q + k, to depend upon the behavior of the function /(K O + h, y Q + k) = F (h, k) for the values h 0, k = 0. FUNCTIONS OF TWO VARIABLES 39 which has neither a maximum nor minimum at the origin. The first function approaches the value zero from the positive direction _! up to the value e ^(for y= 0) while the latter approaches the value zero from the negative direction by the same amount. To this class of functions belong also those functions whose initial terms constitute a semi-definite form and which contain as a factor an even power of a series P(x, y) the terms of which vanish for real pairs of values x, y in every region arbitrarily small where < \x\ < 8, < \y <8 (see 36 and 41). Belonging also to this category of functions are the functions which reach the value zero but do not pass through it for every region arbi trarily small where < x\<8, Q<\y\<8. If on the other hand there exists a power ar n whose value, so long as we remain within a certain limit g, is always smaller than the absolute values of fi(r) and/ 2 (r), then the question whether at the origin an extreme of the function exists may always be answered by a development in series and by a finite num ber of observations. How this is accomplished is found in the next chapter. 28. The method of Von Dantscher. We have seen that by considering the extremes on every line through (0, 0) we are not able to form any conclusions regarding the extremes of the func tion f(x, y) at this point. Von Dantscher s method consists in establishing criteria not only for the extremes on such lines but also for all points in the plane in the neighborhood of the points on these lines and also in the neighborhood and on both sides of the point (0, 0). Although Von Dantscher himself finds that there is "no need of an extension or improvement of the Scheeffer method," I shall give later the method of Von Dantscher, as it is of interest in itself and, besides, it is well to compare the two theories (see 42, 44). 29. The Stolzian theorems.* W r e shall at first assume that the function f(x, y) is continuous with respect to both variables in every point (x, y} of a rectangle that includes the point (0, 0), the * Stolz, p. 213. 40 THEORY OF MAXIMA AND MINIMA sides of the rectangle being parallel to the coordinate axes. We shall state and then prove the following theorems : THEOREM I. A necessary condition that f(0, 0) be a proper extreme of f(x } y} is offered through the existence of an interval 8 + 8, within which x(^=0) lies, and such that the upper limit of f(x } y), when x takes a constant value, the variable y being confined to the interval -\- x x, is had through the value y = < 2 (*)> and the lower limit througli y = ^ (x). This necessary condition in question for a proper maximum is that f(x, 4>%(x)) be invariably less than f(0, 0), and for a proper minimum we must have invariably f(x } ^(x)) greater than f(0, 0). In the first case the upper and lower limits of f(x, y) are both less than /(O, 0) and in the second case they are both greater. -5 F X S y o y FIG. 10 of Note that lim $ 2 (x) = 0, since The same is true The same conditions must be true with regard to the upper and lower limits off(x,y) with constant y such that \y\<&, the vari able x being limited to the interval y -\- y, which limits are reached through the values ^ 2 (y) and / ^r 1 (y) respectively. THEOREM II. The fulfillment of all the conditions made above is sufficient that f (0,0) be a proper extreme of f(x, y). Accordingly f(0, 0) is a proper maximum if there exists a positive quantity 8 such that we have simultaneously for 0< f(x,<t>i(x))<f(Q t 0), [1] and [2] for 0<|y|<8, /(^ 2 (y), y) </(0, 0) with corresponding conditions for a proper minimum. To prove the two theorems just stated we remark first that on account of the continuity of f(x, y) with respect to y the function FUNCTIONS OF TWO VARIABLES 41 f(x, y} with constant x and with the assumption that y takes all values of the interval x - -f x has for all these values a finite upper limit and a finite lower limit, and further that f(x, y) reaches these limits for values y = fa(x) and y = fa (x) (see 8). Hence for values of y such that [3] | y\ = | x\ it is clear that f(x, y) ^f(x, fa(x)). Furthermore, in virtue of the definition of a proper maximum of f(x, y) there must be a positive quantity 8 such that if only \x\ and \y\ are smaller than 8 we must have [4] f(x,y)-f(0,0)<0. It follows, if | x j < 8 and x = and if we substitute y = fa (x) in [4]>that which is in fact the inequality [1]. Reciprocally from [1] and [3] are obtained the inequalities 0<|*|<8 and f(x, y) - /"(O, 0) < 0, where | y | = | x | < 8. If the relation [4] is to be true for all systems of values (x, y) where \x and \y\ are smaller than 8 (excepting x = and y = 0) ? then in addition to [1] we must have the corresponding pair of inequalities [2], which may be derived without trouble. We have corresponding conditions for improper extremes : THEOREM III. In order that f(0, 0) be an improper maximum of f(x, y) it is necessary and sufficient that there exist a positive quantity 8 such that for any x with absolute value less than 8 the value f(x, fa(x)) is not greater than f(0, 0) and for any y with absolute value less than 8 the value /(^^(y), y) is not greater than f(0, ) ; while at the same time corresponding to every positive quantity 8 f which is less than 8 there is at least one value of x or y whose absolute value is less than 8* and for which either f(x, fa(x)) or f(^(y), y} is equal to f(0, 0), The conditions for an improper minimum follow at once. 42 THEORY OF MAXIMA AND MINIMA THEOREM IV. That/(0, 0) may not be a minimum (proper or improper) of f(x, y) it is necessary and sufficient that to every positive quantity 8 there either exists a quantity x r , with absolute value less than S, such that [5] /(* , <,(* )) </(, 0), or that there exist a quantity y , with absolute value less than S, such that [6] /(^i(y ).2/ )</(0, 0); and that /(O, 0) may not be a maximum (proper or improper) f /(# 2/) it i s necessary and sufficient that corresponding to every positive quantity & there may be found either a quantity x" , with absolute value less than , such that [7] /(*", $,(*")) >/(0,0), or a quantity y", with absolute value less than S, such that [8] CHAPTER IV THE SCHEEFFER THEORY I. GENERAL CRITERIA FOR A GREATEST AND A LEAST VALUE OF A FUNCTION OF TWO VARIABLES; IN PARTIC ULAR THE EXTRAORDINARY EXTREMES 30. The theorems of Stolz which were developed in the pre ceding article are closely related to those of Scheeffer, which are of more practical value since the computations required have to do mostly with a few of the initial terms of the expansion of f(x, y) /(O, 0) in ascending positive integral powers of x and y. We shall assume that the function f(x, y) is such that it may be expanded by the Taylor-Lagrange theorem in the form = /to y) + W x to + M>y + ^) + * to + fa>y + 0k)] =/to y) + vito y)+ito y) + WfLx(x + Oh>y + M) + 2 hkfi ,, (x + 0h, y + 9k) + &fy y (x + 0h,y + 0k)], etc., where 0<^<1. If we write x= 0, y and then put h = x, h=y, it is seen that [1] /to y)-/(0, 0) = G n to y)4--R n+1 to y), where G n (x t y) denotes the collectivity of terms of the n first dimensions and R n+l (x, y) is the remainder term (Lagrange, Theorie des Fonctions, Vol. I, p. 40). The Scheeffer theorem. If an index n and positive quantities a and 8 can be determined to satisfy the two postulates (1) that for all values of x such that <\x <& the upper and lower limits of \G H (x, y)\ = a x\ n , with constant values of x and with y limited to the interval -x ---- \- x, and (2} that for all values of y such 43 44 THEORY OF MAXIMA AND MINIMA that 0<|y|<8 the upper and lower limits of \G n (x, y)\^a\y^, where y has constant values and where x lies within the interval y + y> then the two functions f(x, y) and G n (x, y) have simultaneously on the position (0, 0) either a proper maximum or a proper minimum. For, let the lower and upper limits of G n (x, y}, with constant x and with \y\= x\, be G n (x, i(x)) and G n (x, 2 (x)) ( see 29 )5 an( i with constant y and with x \ ^ | y \ let the upper and lower limits of G n (x, y} be n ( 2 (y), y) and ^(^(y), y). Since R n+l (x, y} is a homogeneous integral function of the (n + 1) th dimension in x, y and consists of n + 2 terms, we note that corresponding to any positive quantity e we may always find another positive quantity 8 1 such that if \y\^\x\ and 0<| and also such that if x^ and 0< , then , then \R n+1 (x, y)\ < (n + Hence writing (n -f 2) e x = e and (n + 2) e | y | = e, and denoting the corresponding value of & by 8, it is seen that there is always an interval S - + 8 such that if [2] 0<\x and if [3] 0<|^|<Sand and \y\ =i \x\, then <e |y|, then , y)\ < e |y|- It follows then from [1] and [2] that for values of x, y such that \x\ < 8 and \y\ ^ a;| we have [4] G n (x t 3> l (x))-e\x\<f(x,y)-f(Q, 0) and from [1] and [3] that for values of x y y such that < \y\ < 8 and ^ I y I we have [5] -/(0, 0) THE SCHEEFFER THEORY 45 If next we assume that n (0, 0) is a proper extreme of G n (x,y) and that the two postulates of the theorem have been satisfied, then if 6r n (0, 0) is a minimum it is evident for small values of x and y that G n (x, ^(x)) and G n (^ 1 (y),y) are positive quantities, and from the postulates it follows that for values 0< x\<8 and |y|^ x we have G n (x y 4> 1 (^))^ a\x\ n and for values < \y | < 8 and \x\ ^ \y \ we have G n (V l (y), y) ^ a y \ Accordingly it follows from [4] for values [6] 0<|z|<Sand \y\^\x\ that (a - e)\x\<f(x, y) -/(O, 0); and from [5] for values [7] 0<|*/|<Sand \x =i \y\ that (a - e)\y\<f(x, y)-/(0, 0). Since e may be made smaller than , it follows in both [6] and [7] that/(#, y) /(O, 0) is positive and consequently that/(0, 0) is a proper minimum of f(x, y) (see Stolz s second theorem, 29). If 6r w (0, 0) is a proper maximum of G n (x t y), then with small values of x and y the expressions G n (x, z (x)) and G n (V%(y) t y} must be negative. Hence, due to the postulates for values 0<|^|<3and |y|^ x\, we have G n (x, 4> 2 (a;))^_ a \x\*, and for values and in a similar manner as above it follows that /(O, 0) is a proper maximum of f(x, y). 31. Stolz s* added theorem. If G n (0, 0) is not an extreme of G n (x, y), the following conditions are sufficient to make it impossible that f(0, 0) should le an extreme of f(x,y): if (1) for all positive values of x and y such that < \ x \ < 8 and < \ y \ < 8, or for all negative values within the same limits, at least one of the two upper limits of G n (x, y) defined above is positive and not less than a\x\ n or * Stolz, p. 218, 46 THEORY OF MAXIMA AND MINIMA a\y\ n respectively, and (2) for all positive values of x and y such that < x | < and < | y \ < 3, or /or all negative values within the same limits, at least one of the two lower limits of G n (x, y) defined above is negative and not greater than a x\ n or a\y\ n respectively; that is, if, under the restrictions just made, G n (x, <j> 2 ( x )) is positive and G n (x, &i(x)) negative, or if G n (SPg (y), y) is positive and G^^y}, y) negative. If we limit x, for example, to the interval ... S, and if we suppose that the following inequalities G n (x, <l> 2 (x)) ^ a x n and G n (x, Q l (x)) ^ a x n exist, it is seen that these two expressions vanish only for x = 0. From [1] and [2] it follows for y = Q^x) and y= <& 2 (x) for values of x within the interval in question f(x, ^(^ and f(x, <S> a (a))-/(0, 0) > (a - e}\x\\ Since we may take e<a, it is seen that in the two expressions just written, the difference on the left-hand side is in the first case negative and in the second case positive, so that /(O, 0) is not an extreme of f(x t y} (see Stolz s fourth theorem, 29). 32. The analytic proof given in 30 of the Scheeffer theorem is essentially due to Stolz. Owing to its importance we shall give Scheeffer s statement of this theorem with his geometric deductions (Math. Ann., Vol. XXXV, p. 553). The Scheeffer theorem otherwise stated. Let f(x, y) be any function as already defined of x, y which vanishes at the origin* and let its behavior in the neighborhood of this point be suffi ciently explicit for the determination regarding the appearance of extremes by means of power series to be possible ; in other words, ive assume that there exists a power ar n such that upon every circle described about the origin as center, whose radius r is not smaller than a definite quantity g, the greatest and the least values of the function f(x, y}, viz., fi(r) and / 2 (r) for all points of the * If /(O, 0) ^ 0, we must write /(x, y) /(O, 0) in the place of f(x, y) in the present discussion. THE SCHEEFFER THEORY 47 circumference of the circle with radius r, are in absolute value greater than ar n . Then in the Taylor-Lagrange development given above ,. . . , x f(x t y) = Gr n (x, y) + M n + 1 (x, y), where R + 1 (#, y) consists of all terms beyond those of the nth dimension, the integral rational function G n (x, y) behaves in the neighborhood of the origin as does the function f(x, y). For, as we shall show, in the first place the greatest and the least values of both functions correspond with respect to sign for every small radius r, and from this it follows that there appear simultan eously at the origin extremes for both functions, if such extremes exist ; and secondly, if a is any quantity situated between and a, then upon the circumference of every circle with radius r (within a certain limit g ) the greatest and the least values of the function G n (x, y) are in absolute value greater than a r r n , and from this it follows also that the degree of distinctness that marks the behavior of G n (x, y) is the same as that of f(x, y}. It is evident that we may replace x and y in the remainder term R n + i(%, y} by r, where r-is the radius of the small circle about the origin within which the point (x, y) is situated ; and at the same time we may replace all coefficients by their absolute values. In this way we have for the absolute value of R n + l (x, y) an upper limit Ar n+ \ We shall take the radius r smaller than -- so that ar n > Ar n + \ Since f^r) and f z (r) are by hypotheses greater in absolute value than ar n , it follows from the equation that those values of x, y on the periphery of the circle with radius r which give f^r) and f 2 (r), cause G n (x, y) and f n (x, y} to have the same sign. If/^r) and/ 2 (r) have the same sign, it follows from the above expression that the greatest and least values of G n (x, y} have this same sign. If the two quantities /! (r) and / 2 (r) have contrary signs, the same is true of G n (x, y) for those values of x, y which produce / x (r) and / 2 (r) ; and 48 THEOKY OF MAXIMA AND MINIMA consequently for a greater reason the greatest and least values of Gr n (x, y) have contrary signs. The second part of the theorem follows in the same way if we take the radius r not only smaller than but also so small A f that ar n Ar n + l >a r n ; that is, if we put g 1 equal to and A take r less than g . It is then evident that the values of x, y which produce fi(r) and / 2 (r) when written in the expression G *( x > y} =f( x > y} - fin+i (^ y} cause the right-hand side to be in absolute value greater than a r n when /! (r) and / 2 (r) have the same sign ; and when these two quantities have contrary signs the corresponding values of G n (x, y) w iU m absolute value be greater than a r n , and the same must a fortiori be true of the greatest and the least values of G n (x, y}. 33. If, however, we cannot find an integer n and a quantity a which satisfy the conditions above, we can make no conclusions regarding the behavior of the function f(x, y) by means of powers series and by using the method indicated. For in this case we shall show by means of simple examples which follow this chap ter that in some cases the function G n (x, y} is invariably positive, while f(x t y) may be also negative ; and in some cases G n (x, y) may be both positive and negative while f(x t y) retains a con stant sign (see Ex. 3, p. 61, and Prob. 2, p. 62). But if the conditions of Scheeffer s theorem exist it is seen that the in vestigation of the function f(x, y) has been reduced to that of the function G n (x t y)\ in other words, the investigation has resolved itself into the question : How can we recognize whether a limit g and a quantity a exist such that ^lpon every circle with radius r<g the greatest and the least values of a given integral function of the nth degree G n (x, y) are in absolute value greater than a r n ? And how can we eventually fix the signs of these greatest and least values and thereby determine the extremes of the function G n (x, y) ? These questions we shall now answer. THE SCHEEFFER THEORY 49 II. HOMOGENEOUS FUNCTIONS 34. In the expansion of f(x, y) /(O, 0) suppose that the first terms that appear form a homogeneous function of the 7ith degree in x and y which is the function G n (x, y). With respect to such a function there are three cases to consider, according as this func tion is a definite form, an indefinite form, or a semi-definite form (see 13). If we write it is seen that G n (x, y) changes upon every straight line through the origin proportionally to the nth power of r. If then G 1 and G 2 are the greatest and the least values of G n (x, y) upon the periphery of the unit circle, then G^r n and G 2 r n are the greatest and least values upon any arbitrary circle r. The signs of G 1 and G 2 may be obtained directly through decomposing G n (x, y) into its linear factors, which may be found by solving an equation of the ?^th degree. For we may write G (\ 1 , ^ \ = x n g (u) , where - = u X/ X and g(iC) is an integral function of the ?tth degree hi u. Owing to the fundamental theorem of algebra, g(u} may be decomposed into factors which are linear and quadratic with negative dis criminants if we restrict all the coefficients to real quantities; or these factors are all linear if we allow imaginary coefficients, the quadratic factors breaking up into two imaginary linear com ponents. If these factors are multiplied by the respective powers of x, we have the corresponding decomposition of G n (x, y) into its linear and quadratic factors. At the outset it is clear that if the degree of G n (x, y) is odd, then G and G 2 must be equal but of opposite sign, since G n (x, y) changes sign when x, y are changed into x, y. Furthermore, note that G n (a)x, (oy) = <o n G n (x, y), where o> is a positive quantity. It follows that if G n (x, y} is positive, negative, or zero, then G n (o>x, o>y) is positive, negative, or zero. 50 THEORY OF MAXIMA AND MINIMA If G n (x, y) is an indefinite form, there are values x, y which give G n (x, y) a positive value, and other values x, y which give it a negative value. Let be a positive quantity however small. It is seen that by a proper choice of co we may find values of x t y where \x < 8 and \y\ < 8 such that G n (x, y} is positive, and other systems of values x, y within the same interval for which G n (x, y) is negative. Accordingly the value G n (Q, 0) is not an extreme of 0,(*|f) : * If, however, n is even, and, first, if the linear factors of G n (x, y) are all imaginary, then G n (x, y) cannot change sign nor vanish. It is a definite form and the quantities G l and G 2 have the same sign. If, secondly, there are real linear factors, and if at least one enters to an odd degree, then G n (x, y) takes both signs. G n (x, y) is then an indefinite form and the sign of G is different from that of 6r 2 . It thirdly, there enter real linear factors, but each only to an even degree, the form G n (x t y} may vanish but it cannot change sign. It is a semi-definite form, and one of the extremes G l and G 2 is zero. In this case by a proper choice of co above it is seen that G n (x, y} vanishes for values of x, y other than zero and situated within the interval | x \ < B and j y \ < 8. In this case G n (Q, 0) is an improper extreme of G n (x, y)\ and the behavior of f(x t y) at the origin cannot be recognized with out further discussion. In all cases t except the last a positive quantity a may be so determined that upon every arbitrary circle r the greatest and least values of the function G n (x t y), viz., G-^r 11 and G%r n , are in absolute value greater than a r 11 ; for we need only take a r smaller than the absolute values of 6^ and 6r 2 . In these cases (again excepting the last) there are found in a sufficiently distinct manner (in the previous precise sense of the word, see 27) either a maximum or a minimum of the function G(x, y), or there does not exist such an extreme. The decomposition of G n (x, y) into its linear factors is not necessary, since we may determine the sign of G^ and G% by * Cf. Stolz, p. 222. t The discussion is for the most part due to Scheeffer, loc. cit. THE SCHEEFFEK THEORY 51 means of elementary algebraic operations. For we may determine the multiple factors of G n (x, y) and write this function in the form where, in general, ^r k is an irreducible factor of the &th degree in x and y with integral coefficients and \ k denotes the number of times this factor occurs. Then by Sturm s theorem we may deter mine for each such function ty k (z, y) the number of real factors and by \ k the number of times such factor is repeated. The theory just outlined of the integral homogeneous functions offers, owing to the Scheeffer theorem for the general theory of maxima and minima of arbitrary functions, the following theorem : If in the development of the function f(x, y) in powers of x, y all terms of the first to the (n V)th dimensions are identically zero, while the terms of the nth dimension constitute a form G n (x, y) homogeneous in x and y, and if, first, G n (x, y) is an indefinite form (which is always the case if n is odd), then on the position (0, 0) there is neither a maximum nor a minimum of the function f(x, y) ; if, secondly, G n (x, y) is a definite form, there enters accord ing to the sign of this form an extreme of f(x, y); if, finally, G n (x, y) is semi-definite, the behavior of the function f(x t y) cannot be recognized from the behavior of G n (x, y}. From this theorem it follows that if /(O, 0) is an extreme of f(x, y}, the terms of the first dimension of the expansion by Taylor s formula of f(x, y} /(O, 0) must be wanting, and conse quently we must have fi (0, 0)=0 and .f,;(0, 0)= 0. If, furthermore, 0) = Ax* + 2 Bxy + O/ 2 + then /(O, 0) is not or is (in fact a proper) extreme of f(x, y) according as A C B 2 is negative or positive. If this discriminant is positive, then /(O, 0) is a maximum or a minimum according as A and C (which necessarily have one and the same sign) are negative or positive (see 14). 52 THEORY OF MAXIMA AND MINIMA But if AC B 2 = 0, a criterion regarding an extreme of f(x, y) with the help only of the terms of the second dimension cannot be had. We must then take in addition terms of the third, fourth, degrees in the above expansion of f(x, y) in order, if possible, to satisfy the postulates of Scheeffer regarding the function G n (x, y), In this case we may write, if A is different from zero, where ^, JJ, denote the collectivity of the terms respectively of the third, fourth, dimensions in x, y. If in this expression we write x = Bt, y = At, it is seen that f(Bt, -At)- and if the constant A 3 is different from zero, it is seen that by giving positive and negative values to t, the above expression may take both positive and negative values, so that there is no extreme of f(x, y) on the position (0, 0). But even if the first term that appears on the right of the expansion in t is of even degree, we cannot conclude that there is an extreme, as is illustrated by the classic example of Peano (see 24), viz., f(x, y} = Ay z +2 Bx*y + Cx*. Further investigation is therefore necessary when the terms of the second degree constitute a semi-definite form, and this case is continued in the following sections. III. EXTREMES OF THE FUNCTION G n (x, ?/), INTEGRAL IN x AND y, WHICH IS NOT HOMOGENEOUS 35. We must next determine whether or not the value G n (Q, 0) is an extreme of G n (x, y) when this function is not homogeneous in x and y and when the terms of the lowest dimension in G n (x, y) constitute a semi-definite form. We must again raise the question regarding the existence of an expression a r n which for all suffi ciently small values of r is to be smaller than the absolute values of the greatest value and of the smallest value of G n (x, y) upon the periphery of a circle of radius r, where r is sufficiently small. THE SCHEEFFER THEORY 53 In order, then, to acquaint ourselves with the .different possi bilities which may enter in the behavior of the function G n (x t y) at the point (0, 0), we take a small circle with radius r and seek upon it the two positions at which the function G n (x, y) takes its greatest and its least value. Call these values the extreme values of G n (x, y). They are found (see 15) by solving the three equations ^ By eliminating X from the first two of these equations we have an equation of the ?ith degree y^-*^ = 0, (*) dx dy an equation which is satisfied by all values of x and y which offer extreme values of G n (x, y} upon any arbitrary circle r. It is known in the theory of algebraic functions that every branch of an algebraic curve of the ?zth order which contains the origin may be expressed in the neighborhood of the origin through an independent variable (, say) in the form x = ak 4- ak 2 -f and this expression for the curve may be made in any number of different ways such that in each of the series for x and y the first coefficient which is different from zero (in case there is one) has an exponent which is ^ n. It follows that both those branches which include the origin of the curve (*), and whose points of intersection with the circles of small radii offer the extreme values of G n (x t y) upon these circles, may be expressed in the form (ii) through an independent parameter k t so long, at least, as we remain in the immediate vicinity of the origin ; that is, so long as very small values are ascribed to k. We shall call these two branches the two extreme curves of the function G n (x, y). 54 THEORY OF MAXIMA AND MINIMA 36. We must next distinguish between the cases (1) when (excepting for isolated values of r) the extreme values of G n (x, y) are both different from zero and (2) when one of these extremes is zero. If both extremes are different from zero, then the expression G n (x, y), if we write for x and y the two series (ii) which corre spond to an extreme curve, will begin with a term Ak m , which for small values of k determines both the sign and the order of magni tude of the entire expression. This order is the mth order if we 7?7 consider k a quantity of the first order, and it is of the th order if P we consider & M the first order, where If is the smallest exponent that actually appears in (ii). The number //., as we saw above, can at most be equal to n. We have similar quantities A , m r , /// for the second extreme curve. If the two numbers m and m are not both even, there can be no maximum nor minimum of G H (x, y) at the origin, since this function in this case changes sign with k upon an extreme curve. The same is true if m and m 1 are even numbers while A and A have opposite signs, for then the function G n (x, y) shows different signs upon the two extreme curves. If, finally, m and m are both even while A and A f have the same sign, then we have a maximum or minimum of G n (x, y) according as this sign is negative or positive. In all three cases it is clear that a quantity a and an upper limit g of r may be so determined that for r<g the values of G n (x, y) upon both extreme curves are everywhere in absolute value greater than a f r p , where p is the greater of the two , m , m numbers and - /* F If, however, the value of G n (x, y} is invariably zero upon one of the extreme curves, there cannot be a maximum or minimum at the origin, nor is there an expression a r p of the kind required above. But this can only occur when G n (x, y) contains a squared factor which when put equal to defines a real double curve that passes through the origin ; for otherwise, with the vanish ing of G n (x, y) upon crossing the circumference of any circle with THE SCHEEFFER THEORY 55 radius r, there must be a change of sign in G n (x, y). The squared factor enters as a factor to the first power in (ii), so that points on this curve make G n (x t y) identically zero. In the sequel we shall assume that such factors have been di vided out of G n (x, y), so that the case in question does not enter. Under this assumption, which must be tested in every indi vidual case, there exists, in virtue of the considerations already laid down, always a smallest number p associated with which a constant a and an upper limit g of the radius r may be so de termined that upon every circle of radius r<g f the two extreme values of G n (x, y} are in absolute value greater than a r p ; and, in fact, this number p (if the order of r is taken as unity) expresses the degree of the magnitude of the function G n (x, y} upon that one of the two extreme curves upon which this order is the highest. If p is at most equal to ?i, then a r p for small values of r is not smaller than a r 11 , and the two extreme values of G n (x, y) are there fore certainly greater in absolute value than a r n ; but if p is greater than n, then for small values of r at least one of the extreme values of G n (x t y) is in absolute value smaller than ar n , however the constant a may be chosen. It is thus seen that in virtue of the fundamental theorem the function G n (x y y} may be used as a criterion for determining the existence of a maximum or minimum of the function f(x, y), where G n (x, y) consists of the terms of the first to the nth order of f(x, y] only when the characteristic exponent p is at most equal to n. 37. If in an example we wished to discuss the function G n (x, y} in the manner indicated above, we must calculate the coefficients of (ii), which, in general, is a somewhat complicated operation. The following method leads, however, indirectly to the same result, viz., that of finding the extreme values of G n (x, y}, and thus offers an easy method for the criteria in question. The method in question is first to make use of the Stolzian theorems of 29, and then by applying the Scheefferian theorem we may reach the desired conclusions. Accordingly we must determine the upper and lower limits of G n (x, y) with constant x and |y|^|*| as well 56 THEORY OF MAXIMA AND MINIMA as the upper and lower limits of this function with constant y and | a; | = \y\. For brevity put G = G n (x, y). The values of y, viz., y = ^^(x) and y^&^x), which offer the first-mentioned pair of limits, fall either within the interval x + x or upon one of the end-values y = x or y = + x. When they fall within the interval, since G n (x, y) is a continuous function which has a first derivative with respect to y, it is seen that y = 3> 1 (x) and y = 4> 2 (^) are solutions of the equation - = 0. dy In the second case, when they fall upon the end-points of the inter val, then y = x OT y x may offer the desired limit or limits. It is permissible throughout the whole discussion to fix a posi tive quantity a<\ as the upper limit for \x\, where a is taken so small that y = <E> 2 (x) and y = 4> x (x) are convergent series in x, which when substituted in the equation = identically satisfy dy it. Furthermore (see 29), since lim <J>j (x) and lim 4> 2 (x) = 0, it is x=0 x = seen that no constant term can enter these expressions. The method of determining the different values of y which f)C* satisfy the equation = is found in 139 et seq. Let these values be P^x), P 2 (x) t P 3 (x) 3 . (i) 38. We may next see which of these functions may be neglected from the investigation. If P(x) denotes any of the functions PI(X) (i = 1, 2, . ) and if P(x) has the form (1) P(x) = xi>{a + x*R(x)}, where p > and <r>0, then to any arbitrarily chosen e > there corresponds a quantity S > such that there are values x \ < 8 for which | x*R (x) | < e ; and for such values of x we have (2) \P(x)\>\0\{\a-e}. If p lies within the interval 0</o<l and if \x\ is further so diminished that \a e>|^| 1 ~ p , then from (2) it is seen that |P(#)|> x\ and consequently y = P(x) would fall without the fixed interval x - + a; We see, therefore, that any series which THE SCHEEFFER THEORY 57 begins with a term ax* + - - >, where < p < 1, may be neglected from the number of functions given in (i). If, next, p = 1 and a > 1, we may take e so small in (2) that a|e>l, and consequently |P()|>||, so that such series may also be neglected. Furthermore, if one of the series (i) begins with +1 x or 1 .t, and if the second term has the same sign as the first, then evi dently |P(a?)|>||, and such a series may accordingly be neglected from the investigation. 39. The remaining series in (i), together with the values which correspond to the end-points, viz., y = + x and y = x, give, when substituted in G (x, y}, the following functions : G(x, - x) t G(x, + x), G(x, P,(x)) } G(x, P 2 (x)), - . - ; (ii) and we have to determine which of these functions presents the upper and the lower limits of the function G (x, y) for the interval in question. By taking a(<l) sufficiently small the first term in any of the functions (ii) is as a rule sufficient in determining which will give the required upper and lower limits. Of course, if two of the functions (ii) have their initial terms the same, it may be necessary to introduce their second and higher terms to determine which furnish the required limits. Of those functions whose first terms are negative the one with smallest exponent gives the lowest limit; and if two series have the same negative exponent, the one with greater coefficient offers the lower limit. If there is no function in (ii) whose first term is negative, then in determining G(x, &i(z)) we note that of those functions whose first terms are positive that one with highest exponent offers the lowest limit ; while if two functions have first terms with the same exponent, the one with smaller coefficient offers the lower limit. These observations must be made with both positive and negative values of x, where | x < a. If one of the functions in the series (ii) is zero, while the others all begin with a positive term, then G(x, l (z))=Q, etc. We proceed in the same way in determining G(x t 58 THEORY OF MAXIMA AND MINIMA 40. To determine G(^ r 1 (y) > y) and 6r (^ 2 (y), y), taking y con stant, we limit x to the interval y + y. Denote by those values of x which expressed in power series in terms of y satisfy the equation =0. ex The two limits in question are to be found among the functions the method of procedure being the same as above. . When each of the four limits G (^ 1 (x), x), etc. has been deter mined for values of x within the fixed intervals, the Stolzian theorem is at once applicable. If the expansion, say, of G(x t &i(x)) is a k x k + a k+1 x k + 1 + - - - and if k^n, we may at once find a constant e such that and if the same is true of the three other limits the Scheefferian theorem is at once applicable. 41. Exceptional cases. If the f unction G (x, y) contains factors, say x y, then G (x, -P x) identically vanishes. More generally O/~f the equations G(x,y)=Q and = may be satisfied by the cy same series y = P (x). In this case, considered as an integral func tion in y and with arbitrary x, the function G (x, y) has a repeated factor, say Q(x t y), which vanishes for y = P(x). Next suppose that G(x t y) is decomposed into its irreducible factors H^x, y), HI( X > y}> > an d give to x such a value x l that each of these functions is also irreducible when considered as a function of y. Furthermore, since by hypothesis G (x v y) = contains a repeated root y=P(x l ) t it is seen that two of the functions ffi(x v y), H^(x v y), - -, say ^ and H^, vanish for y=P(x^. And since by hypothesis these functions are both irreducible with regard to y, they are identical except as to a multiplicative factor which is independent of y. But as ff l (x, y) and ff 2 (x, y) are identical in y for an indefinitely large number of values such as x = x v it follows that the coefficients of like powers of y in these two THE SCHEEFFER THEORY 59 functions are identical, so that G(x, y) is divisible at least by the square of an integral function H(x, y). If at least one of the four functions, say G(x, ^^(x)) t vanishes for values of x other than x = within the fixed intervals, while for all other values this function retains the same sign, and if the other three functions are invariably of this same sign, then G (0, 0) is an improper extreme of G(x,y). It follows that as a necessary condition for G (x, y) to have an improper extreme on the position x = 0, y = 0, G (x, y) must contain as factor the even power of an integral function H(x, y) which not only vanishes for x = 0, y = but also for values x, y whose absolute values are arbitrarily small. For if, in accordance with the above remarks, G = H k G, where G (x, y) contains no root y = P (x) which is also contained in H(x 9 y), and if k is odd, then as y passes through the value y = P (x) the function H k changes sign and therefore has values with opposite sign. Example 1. Let f(x, y} = ay* + 2 bx z y + ex* + R 5 (x, y}, where > and R 5 (x, y} denotes any series beginning with terms of the fifth order in x and y. Writing G (.r, y) = ay" + 2 bx*y + ex 4 , it is seen that for x constant and 2\/~* 7i j y | = | x |, = 2 (a?/ + bx 2 ) is zero only for y = x 2 . We thus have / b A ac-b 2 . G [x, x 2 } = F \ a I a and G (x, x) = ax 2 2 bx* + cx\ The first expression offers the lower limit, while either G (x, + x) or G (x, x) offers the upper limit. We have three cases to consider : (a) ac b 2 < 0. Then of the two limits one is positive and the other negative. It follows that G (0, 0) is not an extreme of G (x, a;), and as both limits begin with powers of x not exceeding the fourth, the Scheeffer theorem is applicable, which shows that/(0, 0) is not an extreme of f(x, y). (/?) ac b 2 > 0. It follows since a > that c must also be positive. The two limits just derived are both positive. Continuing we must next deter mine the other two limits. When y is constant and | x \ = | y \ , we have by solving the equation = = 4 x (by + ex 2 ) CX : the two values x = and x = \! y-. \ c 60 THEOKY OF MAXIMA AND MINIMA If b ^ the latter value may be neglected ( 38), since the exponent of y lies between and 1. If & = 7 this value coincides with the first. We observe that each of the functions G (0, y) = ay* and G ( y, y} = af + 2 % 3 + cy* is positive. It follows from Stolz s theorem that G (0, 0) is a proper mini mum of G (x, y} ; and since the power of x or y on the right-hand side of any of the four limits is not greater than 4, the Scheeffer theorem shows that/(0, 0) is a proper minimum of f(x, y). (y) ac b 2 = 0. From above G (x, - \ = 0, while the other three limits are all positive. In this case G (0, 0) is an improper minimum and the Scheeffer theorem is not applicable, so long as we regard R^ (x, y~) as an arbitrary power series with initial term of the fifth or higher dimension. (Stolz, p. 235.) Example 2. f(x, y) = y 2 + (ax 2 + 2 bxy + cy 2 ) y + R (x, y}, (a * 0). We have here G (x, y) = y 2 + (ax 2 + 2 bxy + cy 2 ) y. Taking x constant and \y\ = \x\, we find as a solution of ?>.S~*1 = = 2 y + ax 2 + 2 bxy + cy 2 + 2 y (bx + cy) Forming the functions G(x, $(x))=-^-x* + - and G(x, x) = x 2 + [2 b (a + c)];r 3 it is seen that the first furnishes the lower limit, while one of the last functions offers the upper limit. It is evident that with x taken sufficientlv small these two limits have contrary signs, so that G(0, 0) is not an extreme of G (x, y). Furthermore, since the lower limit begins with a power of x greater than 3, the added theorem of 31 is not applicable. Proceeding further and taking y constant and |a;|5fjyj, we have as a solution of 2>/~~i = = 2 y (ax + by} (since y is taken constant) x = -- ?/, which cannot be considered ( 38) unless 1 6 1 < | a . Forming the functions it is seen that both the upper and lower limits are positive. It follows that the added theorem is not applicable. We cannot, therefore, make a negative assertion regarding the extremes of f(x, y}- (Stolz.) THE SCHEEFFER THEORY 61 Example 3. /(*, y) = f + x*y + a; 4 + R 5 (x, y}. In this example we have G (x, y) = y 2 + x 2 y + x 4 . With .r constant and | y \ s | # , we have as the solution of We thus have the functions With y constant and x =\y\> we have from ? = 2xy + * 3 * = 0, ^ It follows at once that == fj fl and The value G*(0, 0) is consequently a proper minimum of G(x, y), and as none of the above series has an initial term with exponent greater than 4, it follows from Scheeffer s theorem thaty(0, 0) is a proper minimum of f(x, y). Although there is a proper minimum for f(x, y} = y 2 + x^y + a; 4 , it may be shown that G(x, y} = y 2 + x 2 y has neither a maximum nor a minimum. (Scheeffer, loc. cit., p. 573.) Example 4. Peano s classic example : /fcy)0(*j) + JW*y)i where G = y 2 ( p 2 + ^ 2 ) a: 2 ?/ + p 2 q 2 x 4 . With a; constant and |y|=i ar|, we have so that Forming the functions it is seen that the upper limit is positive, while the lower limit is negative. It follows that (7(0, 0) is not an extreme of G(x, y} ; and as the initial terms on the right have exponents that are not greater than 4, it follows from the Scheeffer theorem that/(0, 0) is not an extreme of f(x, y). 62 THEOKY OF MAXIMA AND MINIMA Example 5. /(*, y) = G(x, y} + ^ lg (ar, y), where G(x, y) = * 2 2/ 4 - 3 a; 4 / + (a?V 2 - 3 xy 7 + y 8 ) - 10 x 10 /y + 5 x 12 . With x constant and |y| = |#|, we have from fV* = 4 x*y* - 9 xY + (2 x*y - 21 xy* + 8 y 7 - 10 a: 10 ) - 0, as a solution (see 145), y = 2x 2 + f x 4 + = <(*), say. Forming the functions G(x, <(*)) = - 4 x 10 + and G(x, x) = x f < ., which (see again 145) offer the upper and lower limits of G(x, y}, it fol lows from Stolz s theorem and the Scheeffer theorem that neither G(x, y} nor /(a:, y) has an extreme on the position x = 0, y = 0. (Scheeffer, loc. cit. p. 575.) PROBLEMS 1. Show that/(0, 0) is a minimum of /(* 30 = / + * 6 - 108 x^y -x* + R 9 (x, y}. (Stolz.) 2. Writing (*, y) = f - 2 x*y + a: 4 + ?/ 4 , show that G (0, 0) is a minimum for the first function but that /(O, 0) is not a minimum for the second function. Write in the latter expression y = x 2 . (Scheeffer.) IV. THE METHOD OF VICTOR VON DANTSCHER 42. Instead of considering the extremes upon the straight lines through the point P(x Q9 y ) we may derive the criteria for maxima and minima in the neighborhood of the points on these lines on both sides of the point (# , y ) in the ^y-plane. With Von Dantscher* let the straight lines through (# , y ) be denoted by (1) so that or -= x- where \ and /* are real variables such that X 2 + i^= 1 and where /o is a real variable which may have both positive and negative values. *See Math. Ann., Vol. XLII, p. 89. THE SCHEEFFER THEORY 63 For extremes of f(x, y) at the point P we must have /in case of a maximum\ f(x Q + \p, \proper or improper/ ,. / in case of a minimum\ /(<> ^0) - \proper or improper/ for all values of p of a certain interval -p<p<q, while for values p = p or p = q the above difference not only vanishes but changes sign. The thesis of Yon Dantscher may be stated as follows : " If the lower limit, r say, of p in the region X 2 + /n 2 =l is different from zero, then f(x Q , y ) is a maximum or minimum for the surface- neighborhood of the point (x , y ); but if the lower limit of p is zero, then on the position X Q , y Q there is neither a maximum nor a minimum of the function f(x y y)" The decision as to whether a maximum or minimum exists for a given function f(x, y) on a point # , y Q in whose neighborhood f (x, y) can be developed in integral positive powers of x # = h , y y Q = A-, and on which point the first partial derivatives with respect to x and y both vanish, is consequently reduced to the investigation as to whether the quantity p is different from zero or not. If in the supposed development the ?ith dimension is the first whose terms do not all vanish, we write (2) f(z Q +Ji,y Q +*)-f(x Q ,y ) = g(h, Jc) = (h, k) n + (h, *) n+l +(h, &), I + 2 + , (n ^ 2) where (h, k) n denotes the sum of the terms of the ?ith dimension in h and &, etc. If we write in this expression (3) h = \p, k = np, \*+fjfi=l, we have (4) ^,*)=^(X,M) w H-p(X,/*) w+1 -h---] = ^(p; X, /x). 64 THEORY OF MAXIMA AND MINIMA The factor p n may be omitted, since to the value p = there corresponds the position h = 0, k = itself. The quantity r is accordingly nothing other than the lower limit of the absolute values of the real roots of the equation (5) </>(/>; X, /*) = (\, /*) W + (X, /*) w + 1/ >+...= 0. From this the following is at once evident : CASE I. If (h, Jc) n is a definite form (13), that is, one which takes the value zero for the one and only pair of values h = 0, lc = 0, which case can only enter when n is even, then (X, p) n is different from zero for all values X, /* which are different from zero, and consequently |(X, fi) n \ has a lower limit I which is different from zero. We may, consequently, for the region X 2 -f y? = 1, determine a positive quantity r such that for | p | < T we have (X, p) n si>\(\, /*)+!/> + (x, ?)+?+ The equation (5) has therefore no root p whose absolute value is not greater than r ; the quantity r is therefore different from zero, and there is consequently a maximum or minimum according as (h, k) n is a negative or positive form. CASE II. If (h, k) n is an indefinite form ( 13), that is, one which for real pairs of values (h, k) takes both positive and nega tive values, then also (X, ji) n is such a form. It is then easy to show that in this case the equation in any interval as small as we please e < p < e, has roots that are different from zero, and consequently r = and also f(x Q , y ) is neither a maximum nor a minimum. 43. CASE III. We come finally to the semi-definite case ( 13) ; that is, one where (h, Jc) n vanishes for pairs of values h, k which are different from zero, but does not change sign. It contains necessarily real linear factors, and, in fact, each one to an even power. The number n is consequently even, THE SCHEEFFER THEORY 65 and it follows tha,t (X, /z) ?l is necessarily also a semi-definite form, whose factors are, say, (6) kfi - hj, kji - h z k, ...,k m h- h m k, so that (h, k) n is of the form (h, k} n = (kji - ^ where l lf / 2 , . l m are positive integers and (h, *)-2( 1 + !,+ is a definite form or a constant. To each such linear factor kji h a k(<r = 1, 2, - , m) of (h, k) n there corresponds a linear factor fjL \ \ a p of (X, /*), where with arbitrary sign of V ^ 4- ^> si^ce this constant enters only to squared terms in (A, k) n . If X, /* approach a pair of values X ff , /JL0. for which (X, /j,) n vanishes, then of the roots of the equation <(/>; \ /*) = (>., /*) + (^ A*) M +i/ H ---- = o, one or several become indefinitely small. Of course we may exclude the case where all the quantities (X, fi) n + v (v^1) simultaneously vanish; for then <(/>; X, /i) = for every arbitrary small value of p, and consequently /(# , y ) is neither a maximum nor a minimum. We have next to see whether among the roots of cf>(p X, p) 0, which become indefinitely small when (X, p} n becomes indefinitely small, there are real roots or not. If no real roots appear, then r > and f(%Q, y) is a maximum if the semi-definite form (h, k) nf when it does not vanish, is negative, while it is a minimum if (h, k) n is positive. When there appear real roots the investigation may be carried out as follows : In order to consider the function <> (p ; X, /*) in the neighborhood of the point \ ff , fjL ff , we write (7) X = X <7 +i/, fjL = n ff +v, where u and v are variable quantities. 66 THEORY OF MAXIMA AND MINIMA Since X 2 + //< 2 = 1 and \* + p% = 1, we must have ^2+^2+ 2X^ + 2/4^ = 0, where it is certain that one of the quantities \ or p v is different from zero. If /^^ we have at once from the equation just written * - (2 X a w + w 2 ), where the positive sign is taken with the root, since from (i) u and v vanish simultaneously. Further, noting the development it is seen that (8) = _*-_ l^i-A^s ---- ; Pa 2 /* 8 2 /1 6 and if X, = 0, (9) = -?- ^i*"-"" V 2 Ag. Writing these values in </> (p ; X, /x), we have u ---- ^ p+ +! and v n + 1 which for sufficiently small values of \u\ and p or of v| and |/o| are certainly convergent and may be arranged in powers of u and p or of v and /a. Since (X a , /* <r ) w = 0, it is seen that <k(0, 0)= 0; the case that ^(0, p) vanishes identically may be excluded, as has already been remarked. If p p is the lowest power of p in </> ff (0, p), the equation ^(O, p) = has exactly p roots p, which become indefinitely small with u or v. We must next see whether there are real roots among these p roots. THE SCHEEFFER THEORY 67 If the equation <f> ff (u, p)= has no real root p which becomes indefinitely small with u or v, then for any arbitrarily small posi tive quantity e a positive quantity 8 cannot be found so small that in the interval e < p < e there is situated a root p of <j> (u t p} or of (f><r( v > P) which is different from zero and which belongs to a value u or v in the interval 8 < % < 8 or &<v <8. Hence there exist positive quantities 8 and e so small that the function fa which vanishes simultaneously with u and p or with v and p in the region 8 < % < 8 or 8 < v < 8, e < p < e takes values that are different from zero on every position u, p or v, p which is different from 0, 0, and these values have neces sarily the same sign. For if 4> a (u , p )>Q and <t> ff (u", p") < 0, then with a continuous passage from the position u r , p to the position u", p" , which both lie within the interior of the realm in question and which passage does not pass through the position 0, 0, there must be a position u , p at which fa(u, p) vanishes ; but there are no such positions. It follows that <0.(0, 0) is itself a maximum or minimum provided the equa tion $ v (u, p)= has no real root which becomes simultaneously indefinitely small with u or v. Inversely, it is also true that if (^(0, 0) is a maximum or minimum of <j> v (u t p), the equation 4> ff (u 9 p)=0 has no real root which becomes indefinitely small with u or v. If, on the other hand, the equation <f> ff (u f p)= has real roots which become indefinitely small with u or v, then $ ff (0, 0) is neither a maximum nor a minimum ; and vice versa, if $ ff (0, 0) is not a maximum or minimum, then in every region as small as we wish 8<u<8 or 8<v< 8, e<p<e there are posi tions u, p or v, p which are different from zero and for which <f> (u t p) or fa(v, p) are zero. Through the above consideration the criterion whether the equation <f> (T = has or has not real roots which become indefi nitely small with u or v is reduced to the investigation whether ^(0, 0) is a maximum or minimum of $ (u t p) or <f> ff (v, p) or not. 68 THEORY OF MAXIMA AND MINIMA We have, therefore, to apply the criteria of Cases I and II of 42 ; that is, to arrange < a in dimensions of u and p or of v and p and to see whether the terms of lowest dimension form a definite or indefinite form. This same process must be applied to each of the m real linear factors ^ ff \ \IA (a- = 1, 2,.-., ra) that are different from one another (p. 65), it being evidently sufficient, since u and v become simultaneously indefinitely small, for those linear factors in which A^ and /^ are both different from zero to consider only one of the functions $(&, p) or <f) ff (v, p). 44. We have, then, the following rule for Case III : If the developments of the functions ^(w, p), </> 2 (w, p), , <t> m (u, p) all begin with definite forms, then f(x Q , y ) is a max imum, when the semi-definite form (h, k) n ^ 0, while it is a min imum if (h, k) n = 0. If only one of the functions $ a (u, p) begins with an indefinite form, then f(x Q) y Q ) is neither a maximum nor a minimum. The case remains undetermined if among all the functions (j)^ (u, p) none of them begins with an indefinite form, while one or several of them begin with a semi-definite form. In this case, for every such function the above process must be again applied. We do not affirm that by using this method a determination may among all conditions be made; but Von Dantscher says " if the method, which has been developed to see whether a series g(h, k) which begins with a semi-definite form has or has not on the position h = 0, k = a maximum or mini mum, fails, the function g(h, k) contains an even power of a series P (h, k) which vanishes for real pairs of values h, k in every region arbitrarily small 0< h < 8, 0<\k\<8" (see 41), Example 1. Peano s classic example : g (h, k) = k*- (p 2 + We have here The semi-definite form p? has the linear factor /x so that either X x = 1, f ji l = 0, or Xj = - 1, /* x = 0. THE SCHEEFFER THEORY 69 The corresponding values of X and /* are (see [7] and [9]) so that <k (r, p ) - r * - Q? 2 + ^) ly> The terms of the second dimension in v and p form an indefinite quad ratic form, so that #(0, 0) is neither a maximum nor a minimum. Example 2. Let g (h, Jc} = - W (h - kf + 2 lik* - 5 WL* + + h*L* - 7 W + 6 h*k - 10 h s + + 3 7^-4 8+ .... \\ e then have < (p; X, /A) = - XV(X-/*) 2 + (2 V 6 - 5 A 2 /* 6 + 3 *V 4 + A.V-7XV+ 6 Xp + (- 10 A 8 + XV + 3 XV 4 - 4 /x 8 )^ 2 + . ... We have here to consider the three linear factors /x x X Xj/x = X, fj^X Xo/x = fi, /x 3 X X 3 /A = X //.. It follows that X^O, ^=1; X.^-1, ^=0; X 3 = 4= ^ = ^= V2 V2 To these values correspond the expressions : X=, fJL = l-U 2 ---- , fji = v, X = - 1 + r 2 + ., 1 1 X = + w, p. = - u ---- . We thus have ^ 3 ( u p) = - w2 + I W P- ip 2 + It is seen that all three of the functions < begin with definite quad ratic forms. The semi-definite initial form is negative when it is not zero; and accordingly g (0, 0) is a maximum. (Von Dantscher, Math. Ann., Vol. XLII, p. 100.) PROBLEMS 1 . Show that g (0, 0) is neither a maximum nor a minimum of the function g (h, k) = h*P - 3 A** 8 + h*k* - 3 hk 7 + k* - 10 h lo k + 5 A u . (See Ex. 5, p. 62.) 2. Apply this method to Ex. 3, p. 61. 3. If 2 2 = a 2 x- y z + (x cos a + y sin a) 2 , find maximum and mini mum values of z and give the geometric interpretation. 4. If z 2 = 2 a Va: 2 + y 2 x 2 + y 2 , find maximum and minimum values of z ; show that there are improper extremes and give geometric signification. 5. Find minimum value of u, where u = (x 2 + y 2 )^. 70 THEORY OF MAXIMA AND MINIMA V. FUNCTIONS OF THREE VARIABLES TREATMENT IN PARTICULAR OF THE SEMI-DEFINITE CASE 45. The theorems and proofs given by Stolz and Scheeffer for functions of two variables may be extended at once to functions of three or more variables. For example, f(x Qt y , Z Q ) is a proper maximum of f(x, y, z) if a positive quantity can be so deter mined that for all systems of values f, 77, f, whose absolute values are smaller than S (excepting f = = 1; = f), we have If the partial derivatives of f(x, y, z) have definite values at every position of a fixed realm R, the coordinates X Q , y Q , Z Q of those positions (if any) in R which offer extremes of the function f(x, y, z) must satisfy the equations To apply the Stolzian theorem we observe, if we limit ourselves to a position X Q = = y Q = Z Q , that the collectivity of positions x, y, z for which x\, \y\, \z are less than 8 are distributed into three kinds of realms : * (1) always with |a?|<8, x constant, and \y\S\x\, \z\S\x\; " - (2) with y constant and |y|<8, where also \x\S\y\, \zs\y\; (3) with z constant and |z|<8 and |*| S |*|, \y\S\,\. To apply the Scheeffer theorem we must consider the difference f(x, y, z)-/(0, 0, 0)= G n (x, y, z)+R n+l (x, y, z), as in 30. The case where G n (x, y, z) is a definite or indefinite form is treated fully in Chapter V. * Stolz, loc.cit., p. 237. THE SCHEEFFER THEORY 71 46. The case where G n (x y y, z) is a nonhomogeneous fuuctiou in which the terms of the lowest dimension constitute a semi- dennite form may be treated in a manner analogous to that given in 37~41, as follows: "We first determine the upper and lower limits of G(x, y, z) with constant x and |y| = 4 || = |4 Geometrically interpreted, this realm constitutes a square whose center is the origin and whose sides are parallel to the y-axis and the z-axis, the length being 2 14 The positions at which G(x, y, z) reaches one of its limits may lie (1) on the vertices, or (2) on the sides, or (3) within the interior of the square. We have, consequently, to form the expressions corresponding to the four vertices G(x, x, x), G(x t x, x), G(x 9 x, x), G(x, - x, - x}. (i) For points on the sides we have to solve for y the equations cy dy ^ for z the equations dz cz Let the solutions of the equations (a} be y = 3(*)> y = %(*)> and let the solutions of (/8) be z=Q l (x], z = Q 2 (x), .... Those functions P (x) and Q (x) which cause y and z to fall without the given square are to be neglected (cf. 38). With the remaining functions we form the expressions G (x, P l (x), x), . . . ; G (x, x, Q l (x)). (ii) For the points within the square we have to determine y and z in terms of x from the equations = oy cz 72 THEORY OF MAXIMA AND MINIMA If we eliminate z from these two equations, we may express y as power series in x without constant term, say y = fa (x) y y = </> 2 (#), ( 29). To each such power series for y, say y = <(#), there corresponds one for z in terms of x t say z = \(x), which two series written in the two equations (7) cause them to vanish identically. With these values of y and z we form the expressions G(x, ^(x), \(x)), G(x t 2 (ar), X 2 (^)). (iii) Among all the functions that are found in (i), (ii), and (iii) we are now able to determine those two which offer the upper and lower limits of the function G(x t y, z) within the interval in question. These limits may be denoted by G(x, <&%(x), A 2 (#)) and G(x, 4> 1 (^ 1 ), A^)). If, next, we take y constant and |$| s&jy |, z | = | ?/ 1, we may derive in a similar manner the upper and lower limits G(W 2 (y), y, M 2 (y)) and G^V^y), y, M^y)). Finally, with z constant and |#|^||, | y | ^ | z , we derive the upper and lower limits , fi a (2), z) and ^(N^), Q^z), z). The Stolzian and Scheefferian theorems are at once applicable to these six functions in three variables, the method of procedure being an evident generalization of these theorems for the functions in two variables. PROBLEMS 1. Make the extension and generalization of Von Dantscher s method to the treatment of functions in three variables. 2. In the line of intersection of two given planes find the nearest point to the origin of coordinates. CHAPTER V MAXIMA AND MINIMA OF FUNCTIONS OF SEVERAL VARIABLES THAT ARE SUBJECTED TO NO SUBSIDIARY CONDITIONS I. ORDINARY EXTREMES 47. It will be presupposed in the following discussion, unless it is expressly stated to the contrary, that not only the quantities that appear as arguments of the functions but also the functions themselves are real, and that the functions, as soon as the vari ables are limited to a definite continuous region, have within this region everywhere the character of one-valued regular functions. Regular functions are defined in the following manner : A func tion f(x) is regular within certain fixed limits of x if the func tion is defined for all values of x within these limits and if for every value a of x within these limits the development f(a + h) =/() + / <) + ^/" ()+... is possible ; the series must be convergent and must in reality (see 136), represent the values of the function within this neighborhood. In other words : A function f(x) is regular in the neighbor hood of the position x a if the function in this neighborhood has everywhere a definite value which changes in a continuous manner with x. (Cf. Weierstrass, Werke, Vol. II, p. 77.) A one-valued analytic function f(x v x%, , x n ) of several vari ables behaves regularly on a definite position (x 1 = a v x 2 = a 2 , . >, x n = a n ) if in the neighborhood of this position we may express the function through a series of the form ^ A ^ v- > * n ( x i- a i) Vl ( x 2- a zY* ( x n~ )""> where v l} z> 2 , ., v n are positive integers or zero, and where the coefficients A v ^ v> , ., Vn are quantities that are independent of the variables. (Cf. Weierstrass, Werke, Vol. II, p. 164.) 73 74 THEORY OF MAXIMA AND MINIMA The discussion is thus limited to such functions as are analytic structures of the nature described more in detail in 130, 131. Only for such functions can we derive general theorems, since for other functions even the rules of the differential calculus are not applicable ; in other words we shall consider only the ordinary extremes. The problem of finding those values of the argument of a function f(x) for which the function has a maximum or mini mum value is not susceptible of a general solution, for, besides the cases of the extraordinary extremes of 5-7, there are func tions which, in spite of the fact that they may be defined through a simple series or through other algebraic expressions and which vary in a continuous manner, have an infinite number of maxima and minima within an interval which may be taken as small as we wish.* Such functions do not come under the present investigation. 48. We say (see 1) that a function f(x) of one variable lias a proper maximum or a proper minimum at a definite position x = a if the value of the function for x = a is respectively greater or less than it is for all other values of x which are situated in a neighborhood x a < S as near as we wish to a. The analytical condition that f(x) shall have for the position x = a a proper a proper maximum, is expressed by f(x)f(a) < \ , ., minimum, is expressed by f(x) /(a) > J In the same way we say a function f(x v x 2 , , x n ) of n variables has at a definite position x l = a v x 2 = a 2 , - - ., x n = a n a proper maximum or a proper minimum if the value of the function for x 1 = a v x 2 = 2 , ., x n = a n is respectively greater * A function may have in an interval as small as we wish (1) an infinite number of discontinuities, (2) an infinite number of maxima and minima, and still be expressed through a Fourier series. See, for example, H. Hankel, Ueber die unendlich oft oscillirenden und unstetigen Functionen (Tubingen, 1870); Lipschitz, Crelle, Vol. LXIII, p. 296: P. du Bois- Reymond, Abh. der Bayer. Akad., Vol. XII, p. 8, and also same volume, Part II, Math.-Phys. Classe (1876). NO SUBSIDIARY CONDITIONS 75 or less than it is for all other systems of values situated in a neighborhood ^..^ (X== lf 2 , ...,) as ?iear as we i0is& to the first position] and the analytical condition that the function f(x v x 2 , , x n ) shall have at the position #! = a v x 2 = a 2 , , x n = a n a proper maximum, isf(x v x%, , < n )f( a i> a i> > fl n) < 0, a proper minimum, isf(x v x 2 , , x n ) f(a v a 2 , , a n ) > 0, f or |a; A a A |<8 x (X = 1, 2,..-, ?i), where the quantities S A are arbitrarily small. Improper extremes take the place of the proper extremes above when we allow the equality sign to appear with the inequality sign, as in 1. 49. The problem which we have to consider in the theory of maxima and minima is, then, to find those positions at which a maximum or minimum really enters. We shall give a brief resume of this problem for functions of one variable and then make its generalization for functions of several variables. If x v x 2 are two values of x situated sufficiently near each other within a given region, then the difference of the corre sponding values of the function is expressible in the form : where 6 denotes a quantity situated between and 1 ; or, if x 1 is written equal to x and x^x + h, [1] f(x + h)-f(x) = hf (x + h). From this theorem may be derived Taylor s theorem in the form,* [2] f(x + h) -f(x) = hf (x) + 1 A*/" (*)+ -- (x) + l h*fW (x -h 0h). (71 1)1 n. * See Jordan, Cours D Analyse, Vol. I, 249-250. 76 THEORY OP MAXIMA AND MINIMA In the two formulae last written, instead of x + h write x and write a in the place of x\ they then become [ 1] /(*) - /() = (*- and [2"] f( x )-/(a) Since f(x) is a regular, and consequently continuous, function, the same is true of all its derivatives. If f (a) is different from zero, then with small values of h = x a the value of / (a + Oh) is different from zero and has the same sign as / (a). According to the choice of h y which is arbitrary, the differ ence /(#) /() can be made to have one sign or the opposite sign, if / (a) is either a positive quantity or a negative quantity. Hence the function f(x) can have neither a maximum nor a mini mum value at the position x = a if f (a) ^ 0. We therefore have the theorem : Extremes of the function f(x) can only enter for those values of x for which f (x) vanishes (see 2). It may happen that for the roots of the equation / (x) = some of the following derivatives also vanish. If the nih deriva tive is the first one that does not vanish for the root x=a, then from equation [2 a ] we have the formula f(x) -/(a) =0 (a . + (a! - a)}, and with small values of h = x a, owing to the continuity of /(")(), the quantity /<">( +0h) will likewise be different from zero and will have the same sign as / (n) (a). If, therefore, n is an odd integer, we may always bring it about, according as h is taken positive or negative, that the difference f(x)f(a) with every value of f^ (a) has either one sign or the opposite sign ; consequently the function f(x) will have at the position x = a neither a maximum nor a minimum value. NO SUBSIDIARY CONDITIONS 77 If, however, n is an even integer, then h n is always positive, whatever the choice of h may have been ; consequently the difference f(x) f(a) is positive or negative according as / (n) (a) is positive or negative. In the first case the function f(x) has a minimum value at the position x = a ; in the latter case, a maximum. Taking this into consideration we have the following theorem for functions of one variable ( 3) : Extremes of the function f(x) can only enter for the roots of the equation f (x) = 0. If a is a root of this equation, then at the position x = a there is neither a maximum nor a minimum if the first of the derivatives that does not vanish for this value is of an odd degree ; if, however, the degree is even, then the function has a maximum value for the position x = a if the derivative for x = a is negative, a minimum if it is positive. 50. To derive the analog for functions of several variables, we start again with the Taylor-Lagrange theorem * for such functions. This theorem may be derived by first writing in f(x lt x 2 , > , x n ) A = A 4- u(x x - A ), (X = 1, 2, . ., n), where u is a quantity that varies between and 1 ; we then apply to the function <l>(u) = f(a 1 +u(x 1 -a l ), a a +tA(a^-a 2 ), ., a n +u(x n -a n )) Maclaurin s theorem for functions of one variable, viz.: [3] f() and, finally, in this expression write u = l, as follows : For brevity denote by f k (x v x 2 , - ., x n ) the first derivative of f(x v x z , -, x n ) with respect to x k and by f kl ,^(x v x 2 , . . ., x n ) the derivative of f(x l , # 2 , -, x n ) with respect to x ki and x kt , thatis f ,, ay^,^,...,^,.) A.nft>4. .*) gXk g Xki *See Lagrange, The orie des Fonctions, p. 152. 78 THEOKY OF MAXIMA AND MINIMA It follows, then, that k v k 2 ,^,k> n -i \ Hence, from [3] we have (f>(u)-f(a v a a , . . ., a n )= ^^{f fc (a lf a a> . . ., a n )(x t -a t )} * / /Wl ^1^ ^ a n + 6u(x n - a n }}(x ki - a ki ) . . . (x km - a km )} From this it follows, if we write u = 1, that f(x v a? a , . . ., a; w ) -/(!, 2 , -, )=]{/*(!, 2 , - - ., a M )(aj fc -a fc )} /; + 2! S ^A, *i( a i> V " " * /*p /." 2 + _ / Ajj, A* 2 , , k m _ i (%_!-%,_,) 2 {A 3 * 2 ,...,;u(a 1 + <% 1 - !>, 2 + l9(^ 2 - a 2 ), . . ., *t,*n -V*. NO SUBSIDIARY CONDITIONS 79 51. We are not accustomed to Taylor s theorem* in the form just given ; to derive this theorem as it is usually given, observe that upon performing the indicated summations each of the in dices k v k 2 , - - ., independently the one from the other, takes all values from 1 to ?i, so that the Xth term in the development is a homogeneous function of the Xth degree in x 1 a v x 2 a 2 , -,x n a n . The general term of this homogeneous function may be written in the form i D - .V fa - a^(x 2 - 2 )*. (x n - a n )^ where Xj + X 2 + + X,, = X, D is the definite differential quotient _ /(A 1 +A.+ .-. + X)/ fl n \ f (l 2 " and N is the number of permutations of X elements of which \ v X 2 , , \ n respectively are alike ; that is, "^l X 2 ! ...Xj Furthermore, writing x k a k = h k , we have, finally, [4] f(x v x 2 , . ., x n )-f(a v a 2 , . . ., a n ) *Stolz (Grundzuge der Differential und Integralrechnung, p. 247) ascribes this mode of expression to A. Mayer (see paper by him in the Leipz. Ber. (1889), p. 128). The form as presented here is found in Weierstrass s lectures delivered at least ten years before the Mayer paper. 80 THEORY OF MAXIMA AND MINIMA This is the usual form of Taylor s theorem for functions of several variables. In particular, when m = 1 the above development is [5] f(x v x 2 , . . ., x n ) -/(ap a a , - -, a n ) H The function f(x v ., a? TO ) is regular and continuous, as are consequently all its derivatives. If, therefore, the first deriva tives of f(x v x 2 , ., x n ) are all, or in part, ^0 for x l = a lt - - , %n = a n> tnen they will also be different from zero for x 1 = a 1 + 0h ly ., x n = a n + 0h n , where the absolute values of h lt & 2 , , h n have been taken sufficiently small ; these derivatives will also be of the same sign as they were for x 1 = a v x 2 = a 2 , . . .,x n = a n . If, now, we choose all the h s zero with the exception of one, which may be taken either positive or negative, it is seen that when the corre sponding derivative has either sign, we may always bring it about at pleasure that the difference f(x v x 2 , . . ., x n )-f(a v a 2) . . ., a n ) is either a positive or a negative quantity, and consequently at the position a v a 2 , ., a n no extreme value of the function is permissible. We therefore have the following theorem : Extremes of the function f(x v x 2 , ., x n ) can only enter for those systems of values of (x 1} x 2 , , x n ) which at the same time satisfy the n equations (p. 17) [6] f =0, f =0, .-, f = 0. 0x^ dx 2 dx n It may happen that for the common roots of the system of equations [6] still higher derivatives also vanish. In this case we can in general only say that if for a system of roots of the equations [6] all the derivatives of several of the next higher orders vanish, and if the first derivative which does not vanish for these values is of an odd order, the function, as may be shown by a method of reasoning similar to that above, has certainly no maximum or minimum value. NO SUBSIDIARY CONDITIONS 81 52. If, however, this derivative is of an even order, then in the present state of the theory of forms of the nth order in several variables there is no general criterion regarding the behavior of the function at the position in question. We therefore limit our selves to the case where the derivatives of the second order of the function f(x v x. 2 , , x n ) do not all vanish for the system of real roots a lt a 2 , , a n of the equations [6]. In this case we have a criterion in the formula [7] f(x lt z, 2 , . . ., x n ) -f(a v a. 2 , . . ., aj by which we may determine whether f(x v x 2 , , x n ) has an ex treme value on the position a lt 2 , , a n , since we may determine whether the integral homogeneous function of the second degree, 17 l\ in the n variables h v h 2 , , h n is for arbitrary values of those variables invariably positive or invariably negative. Denote this function by / AM (a 1 + 0h v ,#+ 6h n ). On account of their presupposed continuity the quantities /cy(x v x 2 ,..., x n )\ and /Pf(x lt x 2 , . . ., x n )\ dxjx^ / 1+ i I ,...,. + ffc, GX^ /a,,..., an with values of h lf h^,---, h n taken sufficiently small differ from each other as little as we wish and are of the same sign ; * hence with small values of the h s the functions have always the same sign, and we may therefore confine our selves to the investigation of the latter function. * If any of the quantities ( - , 1 * ) becomes zero, we may replace \ it by exM( a i> > a n)> which must of course be given the same sign as /A,H(I+ Ohi, a n + 0h n ), e\n denoting an infinitesimally small quantity. 82 THEORY OF MAXIMA AND MINIMA If it is found that through a suitable choice of h v h 2 , -, k n the expression can be made at pleasure either positive or negative, the same will be the case with the difference/^, x 2 , ., x n ) f(a lt a 2 , -, a. w ), and consequently f(x v x 2 , , x n ) has on the position (a v a 2 , , a n ) no extreme value. We therefore have as a second condition for the existence of a maximum or a minimum of the function f(x lf x 2 , , x n ) on the position (a v a 2 , > , a n ) that in case the second derivatives of the function f(x lt x 2 , , x n ) do not all vanish at this position, the homogeneous quadratic form \ must be always negative or always positive for arbitrary values of h v A 2 , ., h n . II. THEORY OF THE HOMOGENEOUS QUADRATIC FORMS 53. The three kinds of quadratic forms, viz., definite, semi- definite, and indefinite, were denned in 13. As we have already indicated in 13, it is seen that if the homo geneous function is an indefinite form, the function f(x v x%, > , x n ) has neither a maximum nor a minimum upon the position (a v a 2 , ., a w ); for if the right-hand member of [7] is positive, say, for a definite system of values of the h s, then in accordance with the definition of the indefinite quadratic forms we can find in the immediate neighborhood of the first system a second system of values of the 7^ s for which the right-hand side of the equa tion [7] is negative ; consequently, also, the difference f(x lt a? 2 , -, x n ) -/(!, a 2 , . . ., a n ) is negative, so that therefore no maximum or minimum is permis sible for the position (a lf a 2 , ., a. n ). NO SUBSIDIARY CONDITIONS 83 If, then, the second derivatives of the function f(x v # 2 , , x n ) do not all vanish at the position (a v 2 , , a n ), it follows, besides the equations [6], as a further condition for the existence of an extreme of the function f(x v x 2 , ., x 2 ) that the terms of the second dimension in [4] must be a definite quadratic form, if we exclude what we have called the semi-definite case. The question next arises : Under what conditions is in general a homogeneous quadratic form [8] +(0^...,.* a definite quadratic form ? 54. Before we endeavor to answer this question we must yet consider some known properties of the homogeneous functions of the second degree. Suppose that in the function (f)(x lf x 2 , - , x n ), in the place of (x v x 2 , - , x n ), homogeneous linear functions of these quantities [9] are substituted, which are subjected to the condition that inversely the x s may be linearly expressed in terms of the / s, and consequently the determinant [10] ll C 12 * C \n 21 C 22 " " C 2n The function </>(^, J 2 , . ., a; n ) is thereby transformed into [11] <f>(x v a? 2 , -, * w )= -f ( yi , y 2 , . . ., y n ). Owing to this substitution it may happen that does not contain one of the variables y, so that <f>(x v x%, is expressible as a function of less than n variables. ,y n ) 84 THEORY OF MAXIMA AND MINIMA To find the condition for this write If i/r is independent of one of the y s, say y n , so that conse quently = 0, then from the n equations we may eliminate the n 1 unknown quantities > > dojr ^ ^2 , We thus have among the <f> s an equation of the form [14] where the & s are constants. Owing to equations [12] this means that the determinant of the given quadratic form vanishes, that is, [15] We note here the following formulas: [16] and consequently [17] There exists, further, the well-known Euler s theorem for homo geneous functions : [I 8 ] It is also easy to show reciprocally that if, as above, the equa tion [15] is true, the function <f> consists of less than n variables. NO SUBSIDIARY CONDITIONS 85 For if we assume that equation [15], or, what amounts to the same tiling, an identical relation of the form [14] exists, and if we substitute in 4>(x ly # 2 , , x n ) the quantities # A -f- tk^ in the place of # A (\ = 1, 2, , n) and develop with respect to powers of t, we then have <t> (x l + tk v x 2 + tk 2 , , x n + tk n ) = $(x l9 a? 2 , , x n ) + 2 t {k^(x 1} x 2 ,..., x n )} It follows, when we take into consideration the equations [14] and [18], since the equation [14] is true for every system of values (x v x 2 , , x n ), that <f>(x l + tk v - ., x n + *&) = < (a^, ., ). Hence, if the equation [15] exists, or if the k s satisfy the equa tion [14] for every system of values (x v x 2 , , x n ), then ^to, x 2 , - ., x n ) remains invariantive if x^+tk K is written for # A , where t is an arbitraiy r quantity. Consequently, it being presupposed that k v ^ 0, if t is so determined that the argument . -+ ^ = 0, we have [19] ^(jjj, jL- 2 , ., ^^^Ui-^v, x<t--^x v , . ., /i/y _ -j ^. ft _i_ x v-l T ^ "> X v + l T ^ where $ is expressed as a function of less than n variables. We therefore have, the theorem The vanishing of the determinant^? A U A 22 A nn is the necessary and sufficient condition that a homogeneous quadratic function <t>(x v x 2 , , x n )=^A^x x x^ be expressible as a func tion of n 1 variables. A ** 55. We return to the question proposed at the end of 53, and to have a definite case before us assume that the problem is : Determine the condition under which the function <(#j, x%, , x n ) is invariably positive. The second case where <f>(x ly x 2 , , x n ) is to be invariably negative is had at once if (f) is written in the place of $. *^ii \ > x n~ ~T X v\> 86 THEORY OF MAXIMA AND MINIMA We shall first show, following a method due to Weierstrass,* that every homogeneous function of the second degree <(<#!, 2 , . . ., x n ) may be expressed as an aggregate of squares of linear functions of the variables. 56. In the proof of the above theorem it is assumed that (f>(x v x 2 , , x n ) cannot be expressed as a function of n1 variables ; it follows, therefore, that the inequality [20] ^A u A 22 ...A nn ^O is true and that therefore it is not possible to determine con- i = ")i stants k, so that the equation ^k i (f> i = exists identically. i = l If, then, y is a linear function of the x s having the form [21] y = C& + c 2 ^ 2 H ---- -f- c n x n , and if g is a certain constant, then the expression <$> gy* (=<$>, say), after the theorem proved above, can be expressed as a function of only n 1 variables if the constants k v k%, , k n may be so determined that or [22" A=l From the assumption made regarding [20] it follows, on the one hand, that the inequality [23] 2)^x^0 A must exist. This is the only restriction placed upon the c s. On the other hand, in virtue of the n linear equations [24] 2>W=& (X=l, 2,.-.,) ft * See also Lagrange, Misc. Taur., Vol. I (1759), p. 18, and Mtcanique, Vol. I, p. 3; Gauss, Disq. Arithm., p. 271 ; Theoria Comb. Observ.,p. 31, etc. NO SUBSIDIAKY CONDITIONS 87 the quantities x v # a , , x n may be expressed as linear functions of <f> v < 2 , - , <, and, consequently, by the substitution of these values of x v x 2 , - - -, x n in [21] y takes the form V= It [25] y where the e v are constants, which are composed of the constants A^ and C A . But from equation [22] it follows that Such a representation of the $ A , however, since we have to do with linear equations, can be effected only in one way. l 2)*A *. We therefore have y = --^ - =$) e *^ "SU A=1 M=I from which it follows that fe-^S 1 ** (X=*l,2,...,). ^=1 Through the substitution of these values in [22] it is seen that X = M A = it 5X*A-0y5Xx=o; A=l A=l consequently, owing to the relation [25], we have [26] = t This value of g may be expressed in a different form ; for from [25] and [17] it follows that V = H V *= 71 [26 a ] y =5)A( a; i a, , x n ) 88 THEORY OF MAXIMA AND MINIMA Comparing this result with [21], we have [27] c v = <(! 2 , , e n ) (v = 1, 2, ., TI), and consequently or, from [18], !,,, > Since the quantities c p c 2 , , c w are perfectly arbitrary except the one restriction expressed by the inequality [23], the quantities e lt 2 > > e n are i n consequence of the equation [27], completely arbitrary with the one limitation resulting from [28], viz., the function <p cannot vanish for the system of values (e v e%, , e n )- otherwise g would become infinite. 57. Reciprocally, if the quantities e v e 2 , , e n are arbitrarily chosen, but with the restriction just mentioned, and if g is determined through [28], it may be proved that the expression (f)(= $> gy*\ where y has the form [25], may be expressed as a function of only n I variables. For, form the derivatives of this expression with respect to the different variables, and multiply each of the resulting quantities by the constants e v e z , - -, e n . Adding these products and noting [26 a ] and [28], we have The expression on the right-hand side is zero from [25]. Hence n constants may be chosen in such a way that the sum of the products of these constants and the derivatives of the expression (f> gy 2 is identically zero, and also ^(e lt , e n ) = (cf. [18]). 58. Substitute x x +te x for a3 A (\ = l, 2, ., n) in <f) ; if one of these arguments is made equal to zero, we have, as in 54, e k e k e k e k or, if the new arguments are represented by x v x 2) , x n _ v <f>(x v x 2 , . . ., x n ) - gy 2 = $(x v x 2> - -, a/ n _j). NO SUBSIDIARY CONDITIONS 89 Employing the same method of procedure with </> (x v x 2 , ,x n _ x ) as was done with <f> (x v # 2 , , x n ), we come finally to the func tion of only one variable, which, being a homogeneous function of the second degree, is itself a square. Hence we have the given homogeneous function <$>(x v x 2 , , x n ) expressed as the sum of squares of linear homogeneous functions of the variables. If the coefficients of <f> are real, as also -the quantities e, the coefficients g are also real, and since the quantities e may with a single limi tation be arbitrarily chosen, it follows that a transformation of such a kind that the result shall be a real one may be performed in an infinite number of ways.* 59. If, now, the expression [29] </> (x lt ff a , , x n ) = gfl* + g$l H + g n yl is to be invariably positive for real values of the variables and equal to zero only when the variables themselves all vanish, then all the qualities g lt g 2 , -, g n must be positive; for if this were not the case, but g v say, were negative, then, since the y s are, independently of one another, linear homogeneous functions of the # s, we could so choose the x s that all the ?/ s except y l would vanish, and consequently, contrary to our assumption, <t>(x v x 2 , , x n ) would be negative. Furthermore, none of the gs can vanish ; for if (/j, say, were zero, we might so choose a system of values x v x 2 , . . ., x n , in which at least not all the quantities x v # 2 , ., x n were zero, that all the y s would vanish except y v and consequently <f> could then be zero without the vanishing of all the variables ^l> ^2 * * * ^n* Eeciprocally, the condition of g v g 2 , , g n being all positive is also sufficient for </> to be invariably positive for real values of the variables, and for <f> to be equal to zero only when all the variables vanish. 60. In order to have, in as definite form as possible, the ex pression of $ as a sum of squares, we shall give to the expression [26] for g still a third form. *See Burnside and Panton, Theory of Equations (1892), p. 430. In this connec tion it is of interest to note the Theorem of Inertia of Sylvester, Coll Math. Papers, Vol. I, pp. 380, 511. See also Hermite, CEuvres, Vol. I, p. 429. 90 THEORY OF MAXIMA AND MINIMA In connection with [12] it follows from [27] that v = ^ A^ M (v = 1, 2, . ., n). M=I Denote by A the determinant of these equations, which from [20] is not identically zero, that is, [30] ^ We have as the solution of the preceding equation It follows from this in connection with [26] that [31] 9 = ^ an expression in which the c s are subject only to the one con dition that is not identically zero. 61. It is shown next that we may separate from <f>(x^ x 2 , , x n ) the square of a single variable in such a way that the resulting function contains only n 1 variables. For example, in order that the expression (f> gx% be expressed as a function of n 1 variables, we may choose for g the value [31], after we have written in this expression c x =0(X = l, 2, , n 1), while to c tl is given the value unity. From this it is seen that A _A_ 3A ~A where A 1 is the determinant of the quadratic form $(x v x 2 , . . ., #_!> 0). Of course this determinant must be different from zero. XO SUBSIDIARY CONDITIONS 91 Hence we may write <t>(x v x v - - ., &) = -r-a+$(3i, 4> * ^ -i)> <*! where - ~ -- ~ ^ We may then proceed with </> just as has been done with <f> by separating the square of x^_ lf etc. After the separation of /-t squares from the original function <f>, we notice that the determinant of the resulting function in n /z variables is the same as the determinant of the function which results from the original function (/> when w r e cause the JJL last variables in it to vanish. If this determinant is denoted by A^ we have the following expression for </> : 2 , . ., x n ) - 1 62. If now <t> is to be invariably positive and equal to zero only when all the variables vanish, the coefficients on the right-hand side of the above expression must all be greater than zero. We therefore have the theorem In order that the quadratic form be a definite form and remain invariably positive, it is necessary and sufficient that the quantities A v A Z) ->,A n _ v which are defined through the equation A^ = ^A U A^ -A,^^^,,^, be all positive and different from zero. If, on the other hand, the quad ratic form is to remain invariably negative, then of the quantities A n _ v A n _ 2 , , A lt A, the first must be negative, and the following must be alternately positive and negative (see Stolz, Wiener Bericht, Vol. LVIII (1868), p. 1069). 92 THEORY OF MAXIMA AND MINIMA III. APPLICATION OF THE THEORY OF QUADRATIC FORMS TO THE PROBLEM OF MAXIMA AND MINIMA STATED IN 47-51 63. By establishing the criterion of the previous section the original investigation regarding the maxima and minima of the f unction f(x lt x 2 , - , x n ) is finished. The result established in 57 may in accordance with the definitions given in 52 be expressed as follows: In order that an extreme of the function f(x v x 2 , , x n ) may in reality enter on the position (a v a 2 , , a n ) which is deter mined through the equations [6], it is sufficient, if the second derivatives of the function do not all vanish at this position, that the aggregate of the terms of the second degree of the equation [4] be a definite * quadratic form ; if, however, the form vanishes for other values of the variables without changing sign (that is, is semi- definite), then a determination as to whether an extreme in reality exists is not effected in the manner indicated and requires further investigation, as is seen below. In virtue of the theorem stated in 53, an extreme will enter for a system of real values of the equation [6] if the homogeneous function of the second degree that is, if is a definite quadratic form ; in other words ( 62), there will be a minimum on the position (a v a 2 , - , a n ) if the quotients where ^ M = ^ / 1 i/ 22 + A-,.,-M> are al1 positive, a maxi mum if they are all negative. In both cases the quotients must be different from zero. *Lagrange, Thtorie des Fonctions, pp. 283, 286; see also Cauchy, Gale, differ., p. 234. NO SUBSIDIARY CONDITIONS 93 This last condition is only another form of what was said above, viz., that 2J/A*W^ mus ^ n t be a semi-definite form. For if, say, A * then the summation ^f^hji^ being denoted by (f>(Ji v h Z) ,&), I.* this equation would directly imply the existence of a relation of the form where the are constants which do not all simultaneously vanish. If, therefore, k nt say, is different from zero, we may write and with the help of this relation we have from the equation A=l X=l the following relation Now in this expression the arbitrary quantities li may be so chosen that ^ ht = -h n (X=], 2,... ,TI), * and consequently the function 0(7^, 7i 2 , ., A H ) would vanish without all the tis becoming simultaneously zero. This case we cannot treat in its generality. Neglecting this case, it is seen that the problem of this chapter is completely treated ; however, the conditions that a quadratic form shall be a definite one appear in a less symmetric form than we wish. It is due to the fact that we have given special preponderance to certain variables over the others. We shall consequently take up the same subject again in the next chapter. 94 THEOHY OF MAXIMA AND MINIMA 64. The question is often regarding the greatest and the least values (the upper and lower limits) which a function may take when its variables vary in a given finite or infinite region. If this value corresponds to a system of values within the given region, then for this system the function will also be a maximum or a minimum in the sense derived above. For example, let it be required to distribute a positive number a into n + l summands, so that the product of the ^th power of the first, the 2 th power of the second, etc., and finally the a n + i P wer f the last summand will be a maximum.* The quantities a v a 2 , . ., a n + l are to be positive numbers. Let x v x 2 , ., x n , a x l x 2 . . . x n be the summands in question and write U= x^xf* - - - x n a (a - Xl - x 2 O""* 1 . We must then determine when U or, what is the same thing, its natural logarithm, has its greatest value. If we put the partial derivatives of log U equal to zero, we* will -o, These equations may be written the last term being had through addition of the preceding proportions. If we caH x>, x^\ . . ., ^ 0) the values of the variables which satisfy these equations, we have a, * Peano, 137. NO SUBSIDIARY CONDITIONS 95 The. corresponding value of U is To recognize whether U is the greatest of the values of U, we may show that U is in fact a maximum for the system of values xf\ . . ., x$ and that this position lies on the interior of the realm of variability under consideration. For, let x v x 2 , -, x n be an other system of positive values of the variables, for which also a x 1 ... x n is positive, and substitute for the variables in log U the values af> + u (x l - af>), . . , x + w (* tt - zf), where < u < 1. Since the partial derivatives of the first and second order of log U are continuous for all these systems of values, we have through the Taylor development, observing that the first derivatives vanish on the position x^\ , x , 1 \rr fr r ( -\ovV -- \ l( l ~ l 2[ (^))2 (-P ----- 4 1} ) 2 where icj 15 , , x$ are values of the variables of the form ^+0x l -xf...,x ( +6x n -x ( , where 0<0<1. The expression within the brackets is positive and different from zero, since it is assumed that the system of values x ly # 2 , . . . , x n do not coincide with x> , x^ , , x . It follows that log U < log U Q or U < UQ, so that Z7 is, in fact, the greatest value which U can assume. We note that U takes a smallest value, viz., zero, if one of the summands into which a is distributed, vanishes. If we allow the summands to take negative values, it no longer follows that U Q is the greatest of the values U. CHAPTER VI THEORY OF MAXIMA AND MINIMA OF FUNCTIONS OF SEVERAL VARIABLES THAT ARE SUBJECTED TO SUBSIDIARY CONDITIONS. RELATIVE MAXIMA AND MINIMA 65. In the preceding investigations the variables x v x 2 , - ., x n were completely independent of one another. We now propose the problem : Among all systems of values (x v x 2 , ., x n ) find those which cause the function F(x l> %%,, x n ) to have maximum and minimum values and which at the same time satisfy the equations of conditions : [I] f^(x 1} x 2 , , O=0 (X = l, 2, . . ., m- m<n), where f^(x v x 2 , ., x n ) and F(X I} , x n ) are functions of ike same character as f(x lf # 2 , , x n ) in 47. 66. The natural way to solve the problem is to express by means of equations [1] m of the variables in terms of the remaining n m variables and write their values in F(X I} x%, , x n ). This function would then depend only upon the n m variables which are independent of one another, and so the present problem would be reduced to the one of the preceding chapter. In general, this method of procedure cannot be readily per formed, since it is not always possible by means of equations [1] to represent in reality m variables as functions of the n m remain ing variables. A more practicable method must therefore be sought. 67. If (a v a 2 , -, a n ) is any system of values of the quantities x i> x %> > x n which satisfy .the equations [1], then of the systems of values ( Xl = a 1 + fcj, &,= a a -f A,, , x n = a n + h n ), in the neighborhood of (a v , a n ), only those which satisfy the equations [1] may be considered; that is, we must have [2] / A (a 1 +^ 1 ,a 2 +^ 2 ,.",a n +^ w )=0 (X = 1, 2, . . ., m). RELATIVE MAXIMA AND MINIMA 97 Hence by Taylor s theorem the h s satisfy the equations [3] {/AM (1 a *> > a n) U + [h lt h 2 , ., h n ] I = where [Aj, h 2 , >, h n ]j* denotes the terms of the second and higher dimensions in the respective variables. 68. It being assumed that at least one of the determinants of the rath order which can be produced by neglecting n m columns from the system of m n quantities [4] /Ml J 22 * * *J /2n> _/ml>//n2> * Jmn* is different from zero, then (see 135 and 136) m of the quan tities h may be expressed through the remaining n m quantities (which may be denoted by k lf k z , ., k n _ m ) in the form of power series as follows : [5] h,= (k l , A- 2 , . . ., *_ J where the upper indices denote the dimensions of the terms with which they are associated. These series converge in the manner indicated in 136 ; they satisfy identically the equations [2] and furnish, if the quantities k v k 2 , ., k n _ m are taken sufficiently small, all values of the m quantities h which satisfy these equations. 69. The condition that one of the determinants in the preced ing article be different from zero is in general satisfied; there are, however, special cases where this is not the case. A geo metrical interpretation will explain these exceptions. Let F and an equation of condition f = contain only three variables x lt x 2 , and # 3 . The equation of condition f(x l} # 2 > x z) represents then a surface upon which the point (x l} # 2 , x s ) is to lie and for which F(x v x 2 , x 3 ) is to have a maximum or minimum value. 98 THEORY OF MAXIMA AND MINIMA The determinants of the first order in the development with respect to powers of 7^, h 2 , and h s cannot all be equal to zero; that is, all the terms of the first dimension cannot vanish, the single terms being these determinants ; and this means that the surface / = cannot have a singularity at the point in question. Take next two equations of condition / : = and / 2 = between three variables x v x 2 , and X B . Considered together they represent a curve, and the condition that the corresponding determinants of the second order cannot all be zero means here that the curve at the point in question cannot have a singularity. 70. If the values of the m quantities k^ are substituted in the difference ^/ x x \_p/ a a a \ this expression then depends only upon the n m variables &i> &2> *> kn-m> that are independent of one another and may consequently for sufficiently small values of these variables be developed in the form [6] F(x v x 2 , -, x n ) F(a lt 2 , , a n ) P =n-m -I It was seen ( 51) that, in order to have a maximum or minimum on the position (a v a 2 , - - ., a n ), it is necessary that the terms of the first dimension vanish, and consequently [7] <7 p =0 (p=l, 2, ...,n-m). 71. This condition may be easily expressed in another manner. We may obtain the quantities e if, in the development F(x v aj a , -, x n )-F(a v a 2 ,...,a n ) we substitute in the terms of the first dimension the values of the m quantities from [5] and arrange the result according to the RELATIVE MAXIMA AND MINIMA 99 quantities k v k 2 , ., k n _ m . In other words, the equations [7] ex- tL- n press the condition that ^FJi^ must vanish identically for all *-i systems of values of the h s that satisfy the m equations [3] after they have been reduced to their linear terms. These are the m equations "=o *=i> 2, .... Now multiplying these m equations * respectively by m arbitrary quantities e v e 2 , - - ., e m , and adding the results to the equation we have the following equation : [9] *2f{(^+ I/I, + 2/2 M + + * m f m j*>3= 0. M = l But the es may be so determined that those terms in this sum mation drop out which contain the m quantities h, which are ex pressed in [5] through the n m other h s ; by causing these terms to vanish, a system of m linear equations is obtained, whose determinant by hypothesis is different from zero. Since the terms which remain of equation [9] are multiplied by the completely arbitrary quantities k v k 2 , ., _,, it is not possible for this equation to exist unless each of the single coefficients is equal to zero. Consequently we have as the first necessary condition for the appearance of a maximum or minimum the existence of the following system of n equations, in the sense that if m of these equations exist independently of one another, the remaining n m of them must be identically satisfied through the substitution of the e s which are derived *This method is due to Lagrange, Theorie des Fonctions, p. 268; see also Gauss (Theoria Comb. Observ. Supp. 11). 100 THEOKY OF MAXIMA AND MINIMA from the first m equation, it being of course presupposed that the system of values (a lt a 2 , ., a n ) has already been so chosen that the equations [1] are satisfied. Taking everything into consideration we may say : In order that the function F (x v x 2 , - >, x n ) have a maximum or mini mum on any position (a v a%, , a n ), it is necessary that the n + m equations (/i = l, 2,-.. ,71), be satisfied by a system of real values of the n + m quantities 72. These deductions were made under the one assumption that at least one of the determinants of the rath order which can be formed out of the m n quantities [4] through the omission of n m columns does not vanish. This condition was necessary both for the determination of the quantities h, which satisfy the equations [2], and also for the determination of the m factors e v It may happen* that a maximum or minimum of the function F enters on the position (a v 2 , ., a n ) even when the above condition is not satisfied. For if it is possible in any way to determine all systems of values of the h s not exceeding certain limits that satisfy the equations [2], the equations [7] together with the equations [1] are sufficient in number to determine the n quantities a v a 2 , - , a n . When the above assumption is not satisfied, the equations [8] exist identically, and consequently the equations [3], which serve to determine the A s, begin with terms of the second dimension. We may often in this case proceed advantageously by introducing in the place of the original variables a system of n m new vari ables so chosen that when they are substituted in the given equations of condition they identically satisfy them. * See Stolz, p. 257. RELATIVE MAXIMA AND MINIMA 101 73. To make clear what has been said, the following example will be of service ; its general solution is given in the sequel ( 91). Find the shortest line which can be drawn from a given point to a given surface. Upon the surface there are certain points of such a nature that the lines joining these points with the given point have the desired property and, besides, stand normal to the surface at these points. If by chance it happens that one of these points is a double point (node) of the surface, so that at it we have f l = 0, / 2 = 0, f 3 = 0, then in reality for this point the terms of the first dimen sion in the equations [2] drop out and we have the case just mentioned. If the surface is the right cone we may write The equation of the surface is identically satisfied, and it is easily seen that we may express the quantities h lf h 2 , h 3 through two quantities l\ and & 2 independent of each other even in the case where the required point of the surface is the vertex of the cone, that is, the point x= Q = y = z, OT u = = v- and in fact in such a way that not only indefinitely small values of h lt h 2 , h s corre spond to indefinitely small values of k lf A 2 but also that all systems of values h v h%, h 3 are had which satisfy the equation The variables, however, must be given at one time real, at another time purely imaginary, values if the equations [11] are to repre sent the entire surface of the cone ; but in this manner the unavoidable trouble has taken such a direction that the proposed problem falls into two similar parts, which may be treated in full after the methods of Chapter V. In other cases we may proceed in a like manner. The special problem will each time of itself offer the most propitious method of procedure. 102 THEORY OF MAXIMA AND MINIMA 74. We must now establish the criteria from which one can determine whether a maximum or minimum oiF(x^ x 2 ,- - , x u ) really enters or not on a definite position (a 1? 2 , ., a. n ), which has been determined in 71 above. One might consider this superfluous, since in virtue of the cri teria given in the previous chapter a maximum or minimum will certainly enter if the aggregate of terms of the second dimension in [6] is a definite quadratic form of the nature indicated. It is, however, desirable to determine the existence of a maxi mum or minimum without having previously made the develop ment of the function in the form [6]; for in order to obtain the coefficients C pa . we must pay attention not only to the terms of the first dimension but also to the terms of the second dimension, when the values of [5] are substituted in the development of F(x ly x 2 , - ., x n ) F(a v 2 , 75. The above difficulty may be avoided if we multiply by the quantities e lt (fji = 1, 2, m) respectively each of the expres sions [2] which vanish identically, add them thus multiplied to the above difference, and then develop the whole expression with respect to the powers of h. Owing to equation [9] terms of the first dimension can no longer appear in this development, and we have, if we write ft = m [12] [13] F(x v * 2 , . . . , x n ) - F(a lf a ..., a n ) = G (x v x 2 , . . . , x n ) 1 v We have, accordingly, the homogeneous function of the second degree ^Ofjcje^ of the formula [6] if we substitute in ^Gr^kph, p,cr (JL,V the values [5] and consider only the terms of the first dimension RELATIVE MAXIMA AND MINIMA 103 in the process. If then the criteria of the preceding chapter are applied we can determine whether the function F possesses or not a maximum or minimum on the position (a v a 2 , . . ., a n ). 76. The definite conditions that have been thus derived are unsymmetric for a twofold reason : on the one hand because in the determination of the quantities h some of them have been given preference over the others, and on the other hand because those expressions by means of which it is to be decided whether the function of the second degree is continuously positive or con tinuously negative have been formed in an unsymmetric manner from the coefficients of the function. It is therefore interesting to derive a criterion which is free from these faults and which also indicates in many cases how the results will turn out. With this in view let us return to the problem already treated in the preceding chapter and propose the following more general theorem in quadratic forms. I. THEORY OF HOMOGENEOUS QUADRATIC FORMS 77. THEOREM. We have given a homogeneous function of the second degree [14] <t>(x v a g, ., aw)=2) w4 ** 2 * a > ( A ^ = A ^ A,M in n variables, which are subjected to the linear homogeneous equa tions of condition [15] A = 2> AM .i M =0 (X = l, 2,...,m; m<n)- t we are required to find the conditions under which <f> is invariably positive or invariably negative for all those systems of values of the variables which satisfy equations [15]. It is in every respect sufficient to solve this theorem with the limitation that the quantities x are subjected to the further condition [16] ** + .**+... +** = !; for if if + 4 + ...+* = p 2 , then ( ) +( - 104 THEORY OF MAXIMA AND MINIMA x, x Furthermore, if x v -, x n satisfy [15], then, also, >..., (x x \ 1 P P satisfy these equations, while, since </ -^ , , J = -^<t>( x v * > x n}> the signs of the two quadratic forms are the same. It is, therefore, in every respect admissible to add the equation [16]. We have, however, thereby gained an essential advantage: for owing to the condition [16] none of the variables can lie without the interval 1 4- 1 ; furthermore, since the function varies in a continuous manner, it must necessarily have an upper and a lower limit for these values of the variables x v x 2 , ., x n ; that is, among all systems of values which satisfy the equations [15] and [16] there must necessarily be one* which gives an upper limit and one which gives a lower limit of < (see 8). We limit ourselves to the determination of the latter, By trial we can easily determine whether $ reaches its lower limit on the boundaries, that is, when one of the x*s 1, while the others are all zero. If this lower limit is not reached on the boundaries, then (f> has a minimum value within the boundaries (cf. 64). 78. Through the addition of equation [16] the theorem of the preceding article is reduced to a problem in the theory of maxima and minima; for if the minimum value of </>(%!> x%, , x n ) is positive, <j> is certainly a definite positive form. Consequently, if we write [17] G = $- * then, in order to find the position at which there is a minimum value of the function, we have to form the system of equations |^=0 (X=l, 2,...,n). te A O J p=Wl O/J This gives |E^2i* A +2V<,p~0 (X = 1, 2, . . ., n), K rj f ex k or, [18] ^AA- ^A+ P 1)VV= (X = 1, 2, ., n). M=l p=l *Crelle s Journal, Vol. LXXII, p. 141 ; see also Serret, Calc. diff. et int., pp. 17 et seq. RELATIVE MAXIMA AND MINIMA From the n + m + 1 equations 105 [19] =0 (/> = 1, 2, ., m), the ?i + ??i H- 1 quantities a^, # 2 , , x n ,f v 2 , , e m , e may be determined. Since we know a priori that a minimum value of the function < in reality exists on one position, we are certain that this system of equations must determine at least one real system of values. Consequently the first n -+- m linear homogeneous equations of [19] are consistent with one another and may be solved with respect to the unknown quantities x v x 2 , - , x n , e v e 2 , - , e m \ their determinant must therefore vanish, and we must have [20] ^H , ^12> i AH J J ,, A xioi j "^^22 > j "^^2??^ 11 > a m l a !2 * tt m2 a lP ia > a ln> ^vO a ml> a m1> * * tt ?H?j 0, -, = 0. The equation Ae = is clearly of the n mt,h degree in e. The minimum value of (/> is necessarily contained among the roots of this equation ; for if we multiply the equations [18] respectively by x lt x 2 , , x n and add the results, we have [21] <t>(x v x v ...,)=, it being presupposed that the system of values (x v x z , ., x n ), together with the quantities e v %,-, e m , satisfies the system of equations [19], which is only possible if e is a root of the equa tion A e = 0. Furthermore, among the systems of values x which satisfy the system of equations [19] that system is also to be 106 THEORY OF MAXIMA AND MINIMA found which calls for the minimum, and since the value of the function which belongs to such a system of values is always a root of equation [20], it follows also that the required minimal value of (f> must be contained among the roots of this equation. As already remarked, this minimal value must be positive if $ is to be continuously positive for the systems of values of the x a under consideration, and from this it follows that Ae must have only positive roots. For if one root of this equation was negative, then for this root we could determine a system of values x v %%,, x n , e v e 2 , , e m for which, as seen from [21], <f) is likewise negative. Hence, in order that <j> be continuously positive for all systems of values of the x s which satisfy the equations [15], it is neces sary and sufficient that the equation A# = have only positive roots* The question next arises, When does the equation Ae have only positive roots ? It may be answered in a completely rigorous manner by means of Sturm s theorem ;t but the inves tigation is somewhat difficult; and the symmetry, which we especially wish to preserve, would be lost when we applied Sturm s theorem. For develop the determinant according to powers of e as follows : [22] e n - m B 1 e n - m - 1 +B z e n - m - 2 (- (- l) n ~ m B n _ m = ; then if all the roots of this equation are real and positive, the coefficients B must be all positive, and, reciprocally, if the roots of this equation are real and the B s are all greater than 0, the roots of the equation Ae = are all positive. The form is then a definite quadratic form. The necessary and sufficient condition that the form be not a definite one is that e = be the smallest root of the equation above. * See Zajaczkowski, Annals of the Scientific Society of Cracow, Vol. XII (1867) ; see also Richelot, Astronom. Nachr., Vol. XLVIII, p. 273. t Burnside and Panton, Theory of Equations, chap, ix; Hermite, Crelle, Vol. LII, p. 43; Serret, Algebre Sup., Vol. I (1866), p. 581; Kroneeker, Berlin. Monatsbericht, February, 1873. RELATIVE MAXIMA AND MINIMA 107 79. We shall first show that all the roots of the equation Ae = are real for the case where no equations of conditions are present. (See J. Petzval, Haidinger s Naturw. AHh. II (1848), p. 115.) Equation [20] reduces then to the form [23] - s > = 0, where A^ u = ^ A^ ., A HH -e an equation which is called the equation of secular variations and plays an important role in many analytical investigations; for example, in the determination of the secular variations of the orbits of the planets, as well as in the determination of the principal axes of lines and surfaces of the second degree.* 80. Weierstrass s proof t, which is very simple, that all the roots of this equation are real, depends only upon the theorem that if the determinant of a system of n homogeneous equations vanishes, it is always possible to satisfy the equations through values of the unknown quantities that are not all equal to zero. Instead of the equation [16] we subject the variables to the somewhat more general equation where i/r denotes a homogeneous function of the second degree, which is always positive t and is only equal to when the variables themselves vanish. * In this connection the reader is referred to Laplace, Mem. de Paris, Vol. II (1772), pp. 293-363; Euler, Mem. de Berlin (1749-1750); Tfieoria molux corp. sol., chap, v (1765): Lagrauge, Mem. de Berlin (1773), p. 108; Poison et Hachette. Journ. de VEcole Polytechn., Cah. XI (1802), p. 170: Rummer, Crelle. Vol. XXVI, p. 268: Jacob!, Crelle, Vol. XXX, p. 46: Christoffel, Crelle, Vol. LXIII, p. 257 : Bauer, Crelle. Vol. LXXI, p. 40: Borchardt, Liouv. Journ., Vol. XII, p. 30; Sylvester, Phil. Mag.. Vol. II (1852), p. 138; Salmon, Modern Higher Algebra, Lesson VI; and see in par ticular Edward Smith, Solution of the Equation of Secular Variation by a Method due to Hermite. (Dissertation. University of Virginia. 1917.) Numerous other references are given in the paper last mentioned. t Weierstrass, Berlin. Monatsbericht, May 18, 1868. Cf. also Rrouecker, Berlin. Monatsbericht (1874), p. 1. t Note the lemma of 83. 84, and 85. 108 THEORY OF MAXIMA AND MINIMA 81. If we form the system of equations (see [12] of preceding chapter) [24] <k-^A = (X = l, 2,..., W ), then these equations may always be solved if their determinant vanishes. This determinant is exactly the same as that in [23] if we write We assume that e = k + li, where i = V^T, and that we have found ^A = ?A + ^ (X = l, 2, -,n) as a system of values that satisfy the equations [24]. We must consequently have - (A; + to) ^ (f j + V> f a + v, *,&+ V) = (X = l, 2,..., ra). Since the real and the imaginary parts of these equations must of themselves be zero, it follows, when we observe that < A and i/r A are linear functions of the variables, that <Mfl> f 2 fn)~ ^A(fl, f 2 I fn)+ ^A(^I, ^7 2J I 7n)= 0, *A(I?I, 1?2 > ^n)- ^A(^I, ^?2> ^)- ^A(?I, f a , , f n )= 0. 82. Next multiply these equations respectively by ?? A and f A , take the summation over them from 1 to 71, and subtracting one of the resulting equations from the other, then, since (see [17] of the preceding chapter) A we have or, f 1 , { -, ln)}= 0, [25] Jflrfa, ,..., ,)+ ^r(f f . . ., ,)} = 0. RELATIVE MAXIMA AND MINIMA 109 If it is possible to find systems of values of the quantities x v x 2 ,..., x n which satisfy the equation [24] under the assumption that e = k + li, then these values must satisfy at the same time [25] ; but since after our hypothesis the quantity within the brackets cannot vanish, it follows that I must be equal to zero; that is, every value of e for which the determinant vanishes, is real. Hence we have the theorem : In order that a quadratic form $(x v x^,-.-, x n ) be invariably positive, it is necessar.y and sufficient that the development of the determinant [23] ivhich admits of only real roots, when expanded in powers of e, viz. [26] ev-B^-i + Btf*-* ---- + (-l)"5 n = 0, consist o/?i + l terms and that these terms be alternately positive and negative. If the function is to be invariably negative, then the equation [26] must be complete and have continuation of sign. Thus for the case, where the variables are subjected to no conditions we have derived the criteria as to whether or not a homogeneous quadratic form is a definite one directly from the coefficients of the function and in a form that is perfectly symmetric. 83. Lemma. If a homogeneous function of the second degree ^(ajp x 2 , -, x n ) can become zero for any system of real values of the variables which are not all zero, then T/T may be both positive and negative, it being presupposed that the determinant of ^r is different from zero. Let the function ty vanish for the system of values (f v 2 , , f ) and instead of x lf x 2 , , x n write in i/r the arguments f l + CjA 1 , f 2 + cjc, ., f n + c n k, where the c s are indeterminate constants. Developing with respect to powers of k we have (f p &,,)+ Aty (c v c 2 , , c n ). (i) 110 THEORY OF MAXIMA AND MINIMA By hypothesis the f "s are not all zero, and the determinant of i/r being different from zero, it follows that ifr a (a=l, 2, -, n) can not all be zero. Since, furthermore, c a (a= 1, 2, , n) are arbitrary constants, we a= n may so choose them that^c,^^, f 2 , - - -, f w ) is not equal to zero. a =1 Now by taking k sufficiently small we may cause the sign of the expression (i) to depend only upon the first term on the right-hand side of that expression. Hence, if we choose k positive or negative, we have systems of values (x v x 2 , , x n ) which make ty positive or negative. 84. The determinant of the system of equations [24] is formed from the partial derivatives of <f)(x v x 2) . ., x n )-e^(x 1} x 2 , . . ., x n ), that is, from <t> a (x v x 2 , . . ., x n ) e^ a (x v x 2 ,..., x n )= (ii) ( = 1, 2, ...,7i), where < a and ty a denote - -^- and - respectively. If this 2 cx a 2> cx a determinant is equal to zero for a value of e, it follows that we can give to the variables x v x 2 , - - ., x n values that are not all zero and in such a way that the n equations (ii) exist. Let this value of e be e = k + li] then if I > 0, it may be shown that the function ^r can have both positive and negative values. Denote the system of values (x v x 2 , . . ., x n ) which satisfy the equation (ii) by fc , . ., x v / * x a =!; a +^r) a ( = 1, 2, ...,); then, as in 82, it may be proved that 1 tt (f v f j -i U + ^ (iv ^ i ^n)] = 0- (in) Since by hypothesis / is not zero, the equation (Hi) can only exist either when ^(fj, | 2 , . . ., f n ) and ^(^ ?7 2 , . . ., ?; w ) have opposite values (and then it is proved, what we wish to show, that i/r can have both positive and negative values), or when the two values of the function are both zero (and then from what was seen in the preceding section T/T can take both positive and negative values). RELATIVE MAXIMA AND MINIMA 111 85. In this connection it is interesting to prove the following theorem : If the determinant formed from the partial derivatives of the homogeneous quadratic form ^(p ^ 2 , ., x n ) is different from zero, and if among the infinite number of quadratic forms X 2 , , X n )+ fJ*^(x v X 2 , . ., X n ) there is one definite quadratic form, the determinant formed from the partial derivatives of <l>(x v x 2 , . ., x n )-e^(x v x 2 , . . ., x n ) vanishes for only real values of e. The theorem will also be true if the determinant of </> (and not as assumed of T/T) is different from zero. Let \^ -f ftji/r be a definite quadratic form, and write We shall further choose two constants X and /* in such a way that when we put <t>( x i> x v-> x n)> c/> is different from zero. We know from the previous article that the determinant formed from the equations can only vanish for real values of k. The equations *-^=0 (a = l,2,...,n) (iv) may be written in the form ^A t i)^= ( = 1, 2, ..-, w), or * =t ^ * C =1.2. ,*). () A,Q A^A-j If we eliminate x v x 2 , , # from these equations, we must have the same determinant for their solution as from the equa tions (iv}. 112 THEORY OF MAXIMA AND MINIMA Hence every k which causes this last determinant to vanish must also cause the first determinant to vanish. But the & s are all real. It follows that if we form from them the n expressions these quantities must also be real. Hence the determinant of the n equations <l> a -e^ a =Q (a = l, 2,.-. ,w) has always n real roots e. We may therefore say : If among all the quadratic forms which are contained in the form \(f)(X v X 2 , -, X n )+ fl^jr(x lt Xy , X n ), there is one which can have only positive or only negative values. then the determinant of </> ety will have only real roots, it being assumed that the determinant of (j> or of ty is not zero. The theorem in 80 is accordingly proved in its greatest generality. 86. The case where equations of -condition are present may be easily reduced to the case already considered. The determi nant [20] was the result of eliminating the quantities x v x 2 , , x n , i e v > e m fr m tne n + m equations [18] 2 X A - ^ +2<VV* = (X 1, 2, -.,), [15] O p = a p ^= (p = 1, 2, . ft=i Since the result of the elimination is independent of the way in which it has been effected, we may first consider m of the quantities x, say : x v x 2 , x m , expressed by means of the equa tions [15] in terms of the remaining n m of the a? s, which may be denoted by % v 2 , -, f n _ m . We thus have [27] ^="~iV^ (M = l,2,...,m). I RELATIVE MAXIMA AND MINIMA 113 Through the substitution of these values, let <f>(x v x 2 ,. . ., x n ) be transformed into <l>(i> %%> > Zn-m) an d the equation 2 ? L A 2 = 1 into VT (f 1? f 2 , . . , f _ m ) = 1. The function T/T is invariably positive and is only equal to zero when the variables themselves all vanish. The equations [18] may be written in the form: L UJL K p=l CJC \ dx Multiplying these equations respectively by - (X = 1, 2, . , n), and adding the results, then, since we have the following equations : The last term of this equation drops out if we substitute in it the expressions [27], since the p expressed in the f s vanish identi cally, and we have the equations [28] f|-^ =0 (, = l,2,...,^-m). V$v C Sv Now give v all values from 1 to n m, and we have a system of n m linear homogeneous equations, from which we may eliminate the yet remaining v 2 , -, f,,._ TO . The result of this elimination is an equation in e and must give the same roots in e as [20]. The % 114 THEORY OF MAXIMA AND MINIMA equations [28] are, however, created in exactly the same manner as the equations [24]. If, then, Ae is the determinant of these equations, it follows that the roots of the equation Ae = are all real. 87. As the solution of the theorem proposed in 77 the final result is : In order that the homogeneous function of the second degree be invariably positive for all systems of values of the quantities x v x 2 , -, x n , which satisfy the m linear homogeneous equations of condition n it is necessary and sufficient that the form of the equation [20], developed with respect to powers of e and which has only real roots, consist of n m + 1 terms and that the signs associated with these terms be alternately positive and negative. There must, however, be only a continuation of sign if $ is to be invariably negative. The above method was first discovered by Lagrange, who did not, however, sufficiently emphasize the reality of the roots of equation [20]. II. APPLICATION OF THE CRITERIA JUST FOUND TO THE PROBLEM OF THIS CHAPTER 88. We have determined the exact conditions necessary for a homogeneous quadratic form to be definite for the case where the variables are to satisfy equations of condition and in a manner entirely symmetric in the coefficients of the given function together with those of the given equations of condition. At the same time with the solution of this problem, the problem of maxima and minima which we have proposed in this chapter is solved. RELATIVE MAXIMA AND MINIMA 115 89. Having regard to the remarks made in 71 and 74 we have as a final result of our investigations the following theorem : THEOREM. If those positions are to be found on which a given regular function F(x v x 2 , , x n ) has a maximum or minimum value under the condition that the n variables x v x 2 , , x n satisfy the m equations M /A fa, * > *) = (X = 1, 2, - ., m), where / A are likewise regular functions, we write /p/P = G to, * 2 , p=i seek the system of real values which satisfy the n -f- m equations cG If(o>i, # 2 > -, a n ) is such a system of values of x l , x 2 , -, x n , develop the difference ith respect to powers of h, and have (since no terms of the first dimension can appear, oiuing to equations [c]) the following development : = 2^ G n"( a l> CI V a n)fyA+ M! We must next see whether the function is invariably positive or invariably negative for all systems of values of the tis which satisfy the m equations i \l n in 1 9. . . vn\ [ e > &12> ^ln> /11> /21> * " /ml l ^22 ~~ g > * "> ^2n> /12 /22> * * /m2 1, 2- .<?,-. /l-/2." * J mn . /i* >/!,, . 0. -, o /. /, . - -, o 116 THEORY OF MAXIMA AND MINIMA To do this we form the determinant 07] /ml TW /m determinant put equal to is a% equation of the m n degree in e, which has only real roots. Developing the determinant with respect to powers of e, we have to see whether the develop ment consists of n m + 1 terms with alternately positive and negative sign or with only continuation of sign. If the first is the case, the function cf> is invariably positive, and the function F has on the position (a^, a 2 , , a n ) a minimum value ; if, on the contrary, the latter is true, then </> is invariably negative, and F has on the position (a v a 2 , , a n ) a maximum value. This criterion fails, however, when < vanishes identically, because the quantities <7 MV vanish for the position (a v a 2 , . ., a n ); and it also fails when the smallest or greatest root of Ae = is zero, since in this case we may always so choose the h s that <j) vanishes without the h s being all identically zero (see 83). In the latter case the function </>(#) is an indefinite or a semi- definite form ( 78). In both of these cases the development [d] begins with terms of the third or higher dimensions, and for the same reason as that stated at the end of 63 we cannot assert that in general a maximum or minimum will enter on the position (a lt a 2 , ..., a n ). 90. We give next two geometrical examples illustrating the above principles. PROBLEM I. Determine the greatest and the smallest curvature at a regular point of a surface F(x, y, z) = 0. RELATIVE MAXIMA AND MINIMA 117 If at a regular point P of a plane curve we draw a tangent and from a neighboring point P on the curve we drop a perpen dicular P Q upon this tangent, then the value that 2 ^- = * pp 2 As approaches, if we let P come indefinitely near P, is called the cur vature of the curve at the point P. If the curve is a circle with radius r, the above ratio approaches as a limiting value and is, r therefore, the same for all points of the circle. Now construct the osculating circle which passes through the two neighboring points P and P of the given curve. The arc of the circle PP may be put equal to the arc PP of the curve, when P and P are taken very near each other, and consequently, if r is the radius of this circle, the curvature of the curve is determined through the formula [1] 2 P Q PP 2 FIG. 11 The quantity r is called the radius of curvature, and the cen ter M of the circle which lies on the normal drawn to the curve at the point P is known as the center of curvature at the point P. The curvature is counted positive or negative according as the line P Q, or, what amounts to the same thing, J/P has the same or opposite direction as that direction of the normal which has been chosen positive. If we have a given surface and if the normal at any regular point of this surface is drawn, then every plane drawn through this normal will cut the surface hi a curve winch has at the point P a definite tangent and a definite curvature in the sense given above. The curvature of this curve at the point P is called the curva ture of the surface at the point P=(x, y, z) in the direction of the tangent which is determined through the normal section in question. 118 THEORY OF MAXIMA AND MINIMA Following the definitions given above it is easy to fix the analytic conception of the curvature of a surface and then to formulate the problem in an analytic manner. If P = (x r , y , z ) is a neighboring point of P on the surface, the equation of the surface may be written in the form: [2] Q=F l (x - + i Wi X - x) (y - y) + 2 F^(y - y) (z - z) dF , dF dF where F \ = ^> F ^ == ~^~ ) F 3=> dx dy cz - F - F - 12 dxdy 9 23 dydz 31 dzdx The equation of the tangential plane at the point P is [3] J P 1 (f-) + ^(i-y) + Ji(r-*)=0, where , rj, are the running coordinates. Therefore, if we write for brevity and take as the positive direction of the normal of the surface at the point P that direction for which H is positive, then the direction-cosines of this normal are H Consequently the distance from P to the tangential plane is [5] The negative or positive sign is to be given to the expression on the right-hand side according as the length P Q has the same RELATIVE MAXIMA AND MINIMA 119 or opposite direction as that direction of the normal which has been chosen positive. In the first case, paying attention to [2], which has to be satisfied, since P 1 lies upon the surface, we have rei where S 2 = (x 1 - xf + (y f - yf + (* - z) 2 . In the case where the direction P Q is contrary to the positive direction of the normal, we must give the negative sign to the right-hand side of [6]. Now let P 1 approach nearer and nearer P ; then the quantities x -x y -y z -z - 9 - j - > s s s which represent the direction-cosines of the line PP t become the direction-cosines of the tangent at the point P of the normal sec tion that is determined through P . Representing these by a, ft, 7 P Q and the limiting value of 2 =^- by K, then PP [7] K = ~{F n a^F 22 ^ + F 33 j^2F l2 aft + 2F 2 ^j-{-2F B1 ya} > where the terms of higher degree in x x, etc. are neglected. In this formula K represents the curvature of the surface in the direc tion determined by a, /3, 7. This is to be taken positive or negativ.e according as the direction of the length MP, where M is the center of curvature, corresponds to the positive direction or not. If the coordinates of the center of curvature are represented by x Q) y , Z Q and the radius of curvature by p, then x-x =p H or, since /c = - P 120 THEORY OF MAXIMA AND MINIMA Since H does not appear in these expressions, we see that the position of the center of curvature is independent of the choice of the direction of the normal. Suppose that the normal plane which is determined through the direction a, ft, 7 is turned about the normal until it returns to its original position. Then, while a, ft, 7 vary in a definite manner, the function K of a, ft, 7 assumes different values at every instance, and since it is a regular function, it must have a maximum value for a definite system of values (a, ft, 7) and likewise also a minimum value for another definite system of values (a, ft, 7). The quantity has the same value for all normal sections H that are laid through the same normal.* We have, therefore, to seek the systems of values (a, ft, 7) for which the expression F u a* + F^P+ 7^72+ 2 F l2 aft + 2 F 23 fty + 2 F 31 ya assumes its greatest and its smallest value. We have also to observe that the variables a, ft, 7 must satisfy the equations of condition the first of which says that the direction which is determined through a, ft, 7 is to lie in the tangential plane of the surface at the point P, while the second equation is the well-known relation among the direction-cosines of a straight line in space. * See Salmon, A Treatise on the Analytic Geometry of Three Dimensions (Fourth Edition), p. 259. KELATIVE MAXIMA AND MINIMA Following the methods indicated in 89, we write [10] G = F ll a*+ 121 - 1)+ 2 e ^a + and we then have (89, [c]) to form the equations g =0 , 5=0, g-o, da cp dy from which we must eliminate a, , 7, and . These equations are [11] where F^=F^ (X, ft = 1, 2, 3). Through elimination we have = 0, [12] 13 2 = 0. This is an equation of the second degree in e t and consequently gives us two values e l and e 2 , which are maximum and minimum values, since both maximum and minimum values enter, as shown above. Multiplying the first three equations [11] by a, & 7 respectively and adding the results, we have [13] Hence, from [7] we have [14] 122 THEOKY OF MAXIMA AND MINIMA Consequently the two principal curvatures at the point P have the values [15] Pi H I.A; and the coordinates of the corresponding centers of curvature are found from the formulae [16] In order to determine e, let us write 02 and form from these the corresponding quantities through the cyclic interchange of the indices. Equation [12] may be written in the form* Z> u l? + D*F* + D SZ FI + 2 D 12 F^ + 2 D u F t F t + 2 D^F^ = 0. Developing this expression with respect to powers of e, we have [17] H*e*-Le + M=Q, where L = ff 2 (^ n + F 22 + F Ba ) - (F^F* + F^F* + F S3 Fj) + 2 ^i 2 ^ 2 + 2 V 2 F g + 2 F^F Z F, and M= (f u F M - Fj) F* + (F m F 11 - F,*) F* 77T 7yT \ T7f 77T i / TTf TJT T7T ET \ T7T 77T ~ ^31^22) ^3^1 T (^31^32 - ^12^33) ^1^2 From [17] we have at once the values of the sum and the product of the two principal curvatures, viz. (see equation [15]): 1 1 L [18] PiP* # * See Salmon, loc. cit., p. 257. RELATIVE MAXIMA AND MINIMA 123 We have thus expressed the sum of the reciprocal radii of curva ture and also the measure of curvature of the surface at the point P directly through the coordinates of this point. Although the formulae are somewhat complicated, they are used extensively and with great advantage. In the case of minimal surfaces* which are characterized through the equation we have L = 0. This is therefore the general differential equation for minimal surfaces. 91 . PROBLEM II. From a given point (a, b, c) to a given surface F(x, y, z) = draw a straight line whose length is a maximum or a minimum. Write G = (x-a)*+(y-b)* + (z-cf+2\F(x,y ) z). (i) Then the quantities x, y, z, \ are to be determined (see 89, [c]) from the following equations: x-a + \F l = 0," y-b + \F 2 =Q, z - c +\F 3 =Q, F(x,y,z)=0. } It follows, since F v F 2 , F 3 are proportional to the direction- cosines of the normal to the surface at the point (x, y, z), that the points determined through these equations are such that lines joining them to the point (a, b, c) stand normal to the surface. If P=(x, y, z) is such a point, then to determine whether for this point the quantity is in reality a maximum or a minimum, we substitute x H- u, y + v, z + w instead of x, y, z in the function G. The quantities ic, v, w are, of course, taken very small. *See papers by the author on this subject in the first numbers of the Mathematical Review. 124 THEOEY OF MAXIMA AND MINIMA We must develop the difference G(x + u, y + v, z + w)-G(x, y, z) in powers of u, v, and w. The terms of the first dimension drop out, and the aggregate of the terms of the second dimension is + 2 F 12 uv + 2 F 23 vw + 2 F 3l wu). (w) Since the point (x + u, y+v, z+w) must also lie upon the surface, the quantities u, v, w must satisfy the condition F^u, + F 2 v + ^ 3 w; = 0, (v) where the terms of the higher dimensions are omitted (see [8] of the present chapter). If we wish to determine whether the function -v/r is invariably positive or invariably negative for all systems of values (u, v, w) which satisfy equation (v), we may seek the minimum or maximum of this function i/r under the condition that the variables are limited, besides the equation (v), to the further restriction (cf. [16] of 77) that For this purpose we form the function T/T - e(y?+ v*+ w 2 - 1)+ 2 e ^u+Fjo +F 3 w), and writing its partial derivatives with respect to u, v, and w equal to zero, we derive the equations * + (F SS - RELATIVE MAXIMA AND MINIMA 125 Eliminating u, v, w,. from equations (v) and (viii) t we have here A, exactly the same system of equations as in [12] of the preceding problem, except that here - and e r stand in the place of e and e . A Denote the two roots of the quadratic equation in e, which is the result of the above elimination, by e l and e%, and the corre sponding radii of curvature of the normal sections by p 1 and p 2 ; then, since ^ - has the same meaning as e in the previous problem, ft where the positive direction of the normal to the surface is so chosen that H>Q. If for the position (x, y, z) a minimum of the distance is to enter, then both values of the e must be positive ; if a maximum, then e 1 and e 2 must be negative. It is easy to give a geometric interpretation of this result : Let PN be the positive direction of the normal and A = (a, b, c). Then from (ii) it follows that the length from A to P has the same or opposite direction as PN, according as X is negative or positive. Hence, from (ii), AP=-\H. If the centers of curvature corresponding to p 1 and p. 2 be denoted by ML and M 2 , then , , Hence e = and e 126 THEORY OF MAXIMA AND MINIMA If, then, M l and M 2 lie on the same side of P and if A lies be tween M l and M 2 , as in Figs. 12 and 13, then the es have different signs and there is neither a maxi mum nor a minimum. l * If M l and M 2 lie on the same side of P while A is without the inter- P Mi A M 8 val M 1 - M 2 , then a minimum or FIG. 13 maximum will enter according as A starting from one of the centers of 1 2 > curvature lies upon the same side as P or not (see Figs. 14 and 15). AM! M 2 P If the points J/j and Jf 2 lie on FIG. 15 different sides of P and if A is situ ated within the interval M l M 2 , l ~ as in Fig. 16, then there is always a minimum. If, however, A lies without the interval M l - M 2 , then there is neither a maximum nor a minimum. In whatever manner M l and M 2 may lie, if A coincides with one of these points, then one of the two values of e is equal to zero, and the general remark stated at the end of 89 is applicable. The above results are derived in a different manner by Goursat, Cours D Analyse, Vol. I, p. 118. The case may also happen here (see 72) that in the solution of the equations (ii) and (Hi) a singular point of the surface is found at the point P, at which f\= = F 2 = F 3 . We cannot pro ceed as above, since, there being no definite normal of the surface at such a point, the determination whether for this point a maxi mum or minimum really exist cannot be decided in the manner we have just given. The general remark of 73 indicates how we are to proceed. 92. Brand s problems. The two following problems taken from the theory of light were prepared by my colleague, Professor Louis Brand. PROBLEM I. Reflection at the surface F(x, y, z) = 0. A ray passes from a point J^ to a point P on a given surface and is reflected to a point P^ When is P^P -f- PP 2 & minimum ? RELATIVE MAXIMA AND MINIMA Write PP l = c?! and PP^= d 2 so that 127 We seek to find the condition that makes f^ + c? 2 an extreme when P is subjected to the condition of lying on the surface Using the Lagrangian method ( 89) we must find the extremes of the function (x, y> *) FIG. 1 < x, y, *=< Writing , for et c., the necessary conditions for an extreme, viz., <f> x = <f> y = <f> 3 = 0, give [2] d. Let the direction-cosines of the lines PI[ and PJ^ be l v m v n^ and / 2 , m 2 , 7i 2 respectively ; and let /, m, n denote the direction-cosines of the normal to the surface [1] at the point P. Furthermore, since F xi F IP F z are proportional to I, m, n, write \F X = kl, \F y = km, \F Z = kn. Equations [2] then become [3] m% = km, n n = kn. Designate the angle between PP^ and PP 2 by (1, 2); between PP l and the normal by (1, n); between PP% and the normal by (2, ri). It is seen then that cos(l, 2)= ^/g-f cos(l, n)= IJ + cos (2, n ) = Z 2 / -h 128 THEORY OF MAXIMA AND MINIMA Multiplying equations [3] by l^, m v n v respectively, and adding, it follows, since If + m* + n* = 1, that [4] 1 + cos (1, 2) = k cos (1, ri). Similarly, by multiplying equations [3] by / 2 , ra 2 , w 2 , respec tively, and adding, we get [5] cos(l, 2) + l = &cos(2, n). From [4] and [5] we have cos(l, ri)= cos (2, n), or [6] (1,) = (2,). Moreover, upon multiplying equations [3] by /, m, n, respectively, and adding, we get cos(l, n) -f cos (2, n)=k, or, from [6], k=2 cos(l, n). Substituting this value of ]c in [4], we have l+cos(l, 2) =2 cos 2 (1, n), so that cos(l, 2)= 2 cos 2 (l, n)-l= cos 2(1, n)= cos 2(2, n). It follows that (1, 2) = 2 (1, n) = 2 (2, 71), and that the lines P.ZJ, P^, and the normal must lie in the same plane, and it is further seen that the normal bisects the angle between PJJ and PP 2 . We have thus arrived at the condition which is an optical law : The incident and reflected rays must lie in a normal plane, and the angle of incidence must be equal to the angle of reflection. The above result is merely a necessary condition for an .extreme ; to find whether an extreme really exists, and if it does, whether it is a maximum or a minimum, let us choose the plane P^PP^ as the #2/-plane. If the curve cut from the surface by the plane I{PI^ has the equation [7] y =/(*), RELATIVE MAXIMA AND MINIMA 129 the problem now becomes to determine the nature of the point P which makes , = 0, where ax the y being replaced by /(#). | dx d^ 2 while the equation of the normal to the curve [7] at P(x, y) is and the distance of the point (x iy y 4 ) from this normal is h = ( x - Further, take the origin at the point P and the tangent to the surface at P lying in the plane Pfl^ as the #-axis. Then = shows that dx = and as h l and h 2 have opposite signs, since JJ and J^ lie upon opposite sides of the normal, sin(l, TI)= sin (2, n), or (l,7i) = (2, 7i), as stated before in [6]. Note that d l 130 THEORY OF MAXIMA AND MINIMA It follows that for the origin and the direction y = 0, Writing = (l,n) = (2,n), we note that ^1 = ^ = cos 0, d d ^-7^ so that = (-r + T- ]COS 2 0-2?/ cOS0. FIG. 18 From this it is seen that ^ according as y" = -( ( jcos 6. ft /y>& "^ * -^ O \ /7 /7 / \&\ a/ Since y = 0, we note that y" is the curvature of curve yf(x) at the I origin, that is, y" = > where p is the radius of curvature. Hence, when > 0, and the path is a minimum ; when da? [9] Y < 0, and the path is a maximum. To interpret this result geometrically it is seen that - + icos 2 Wi is the curvature of the ellipse whose foci are at J^ and P 2 and which passes through P (see Pascal, Repertorium der Hoheren Mathematik, Vol. II, 1, p. 245). The quantities d 1 and d 2 are its focal radii at P, and is the angle between either focal radius and the normal to the ellipse at P. Note that this ellipse is tangent to the curve [7], since the normal to the curve bisects the angle between the focal radii of the ellipse and hence is also the normal to the ellipse. RELATIVE MAXIMA AND MINIMA 131 When - = -( + )cos0, the ellipse and the curve [7] have P 2 \ d i **/ the same curvature at P, and the test for extremes is inconclusive. But here the conditions for a maximum or a minimum are obvious from geometrical considerations. For, remembering that d l 4- d 2 is constant for points on the ellipse, say d 1 -\-d 2 = k, then d 1 -\-d 2 < k for points within the ellipse and d 1 + d 2 > k for points without the ellipse. Hence the path of the ray will be a maximum or a mini mum according as the curve [7] lies within or without the ellipse in the neighborhood of the point P ; and it is seen that [#] and [8] are but special cases of this general condition. PROBLEM II. Refraction at the surface F (x, y, z) = 0. Using the previous notation, it is required to find the conditions that make the time of passage from P^to P^, that is, + >an extreme, where v l v 2 v^ and v 2 represent the velocity of light in the two media. The Lagrangian function is (89) , y,z) = \F(x, y t z). Proceeding as in the case of reflection, we find in place of equations [3] above A tf [1] ra, = km. = kn. From these equations we deduce that 008(1, 2) FIG. 19 [2] [3] [4] + l cos(l, 2) 7 = k cos (1, n), 7 -f = k cos (2, n), cos(l, n) cos (2, n) , ( A . 132 THEORY OF MAXIMA AND MINIMA Multiplying [2] by and [3] by > and subtracting, we have V l V 2 S ( 1 > n ) cos (2, substituting from [4] the value of k in this equation, it is seen that _!_ 1 _cos 2 (l, n) cos 2 (2, n) V^ V V? V 2 sin 2 (l, n) _sin 2 (2, n) It follows that sin(l, n) ^sin(2, n) /?1 /J1 V l V 2 From [2] and [4] it is seen that 1 cos(l, 2) cos 2 (l, n) cos(l, 7i)cos(2, n) 1 = 1 > Vi V V V sin 2 (l, n) cos(l, 7i)cos(2, n) cos(l, 2) or s = s Dividing this equation by [5] and then multiplying the result by sin (2, n), we find sin(l, 7i)sin(2, n)= cos(l, n) cos (2, n) cos(l, 2), or cos (1,2)= cos [(1, n) + (2, n)] t and therefore [6] (1,2) = (l,7i) + (2, 7i), so that the incident and the refracted ray lie in a normal plane. Equation [5] may be put in the form sin (2, n) = v^ = where c is the index of refraction of the second medium with respect to the first medium. The above is a generalization of a problem due to Fermat. RELATIVE MAXIMA AND MINIMA 133 The geometrical criteria for a maximum or a minimum involves a certain Cartesian Oval whose foci are at P 1 and P% and which passes through P. Its equation in bipolar coordinates is d l and d 2 being the variable radii vectores. For points on this oval + is a constant, say k ; for points within this oval + < A- v \ V 2 d d v 1 v 2 and for points without this oval + -2 > k. v \ V 2 Hence the time occupied by the ray in passing from P 1 to P is a maximum or a minimum according as the curve cut from the sur face by the normal plane through P l and P Z lies within or without this Cartesian Oval in the neighborhood of the point P. CHAPTER VII SPECIAL CASES I. THE PRACTICAL APPLICATION OF THE CRITERIA THAT HAVE BEEN HITHERTO GIVEN AND A METHOD FOUNDED UPON THE THEORY OF FUNCTIONS, WHICH OFTEN RENDERS UNNECESSARY THESE CRITERIA 93. The practical application of the established criteria is in many cases connected with very great, if not insurmountable difficulties, which, however, cannot be disregarded in the theory. For often the solutions of the equations 89, [c], cannot be effected without great labor, if at all, and therefore also the forma tion of the function </> is impossible. It also happens, even if the function <f> can be formed, that the discussion regarding the coeffi cients of Ae = is attended with much difficulty. Moreover, the formation of the function </> and the investigation relative to the coefficients of Ae are very often unnecessary, since through direct observation we may in many cases determine whether a maximum or a minimum really exists. If it then happens that the equations [c] admit of only one real solution (i.e. of a real system of values x i> x i> > x n)> we mav b e sure that this is in reality the maximum or the minimum of the function. In the same way, if we can con vince ourselves a priori that both a maximum and a minimum exist, and if it happens that the equations [c] offer only two real systems of values, it is evident that the one system must corre spond to the maximum value of the function, the other system to the minimum value. The determination as to which of the two systems of values gives the one or the other is in most cases easily determined. One cannot be too careful in the investigation whether on a position which has been determined from the equations [a] and 134 SPECIAL CASES 135 [c] of 89 there really is a maximum or a minimum, since there are cases in which one may convince himself of the existence of a maximum or a minimum, when in reality there is no maximum or minimum. For example, to establish Euclid s theorem respecting parallel lines, one tries to prove the theorem regarding the sum of the angles of a triangle without the help of the theorem of the parallel lines. Legendre was able, indeed, to show that this sum could not be greater than two right angles ; however, he did not show that they could not be less than two right angles. The method of reasoning employed at that time was as follows : If in a triangle the sum. of the three angles cannot be greater than 180, then there must be a triangle for which the maximum of the sum of these angles is really reached. Assuming this to be correct, it may be shown that in this triangle the sum of the angles is equal to 180, and from this it may be proved that the same is true of all triangles. We see at once that a fallacy has been made. For if we apply the same conclusions to the spherical triangles, in the case of which the sum of the angles cannot be smaller then 180, we would find that in every spherical triangle the sum of the angles is equal to 180, which is not true. The fallacy consists in the assumption of the existence of a maximum or a minimum ; it is not always necessary that an upper or a lower limit be reached, even if one can come just as near to it as is wished (see 8). On this account the assumption of the existence of a real maxi mum is not allowed without further proof. We therefore endeavor to give the existence-proof. For this purpose we must recall several theorems in the theory of functions.* 94. We call the collectivity of all systems of values which n variable quantities x lf x 2 , - -, x n can assume the realm (Gebiet) of these quantities, and each single system of values a position in this realm. If these quantities are variables without restric tion, so that each of them can go from oc to + oc, we call the * Note especially 137. 136 THEORY OF MAXIMA AND MINIMA realm considered as a whole (Gesamtgebiet) an n-ple multiplicity (n-fache Mannigfaltigkeit). If x v x z , , x n are independent of one another, then we say a definite position (a v a 2 , ., a n ) lies on the interior of the realm if these positions and also all their neighboring positions belong to this region ; it lies upon the boundary of the realm if in each neighborhood as small as we wish of this position there are present positions which belong to the realm, and also those that do not belong to it; it lies finally without the defined realm if in no neighborhood as small as we wish of this position there are positions which belong to the defined region. If the quantities x v # 2 , x n are subjected to m equations of condition, then we may express these in terms ot n m inde pendent variables u lt u 2 , , u n _ m , and the same definition may be applied to these variables. 95. The following theorems are proved in the theory of func tions: (1)* If a continuous variable quantity is defined in any manner, this quantity has an upper and a lower limit; that is, there is a definitely determined quantity g of such a kind that no value of the variable can be greater than g> although there is a value of the variable which can come as near to g as we wish. In the same way there is a quite determined quantity k of such a nature that no value of the variable is less than Jc, although there is a value of the variable that comes as near to Jc as we wish (see also 8). (2)t In the region of n variables x v x 2 , , x nt suppose we have an infinite number of positions defined in any manner; let these be denoted by (x[, x 2 , -, x n ). Furthermore, suppose that among the positions we have such positions that x n can come as near to a fixed limit a n as we wish. Then we have in the region of the quantities x v x 2) - -, x n always at least one definite position (a lt a%, , a n ) of such a nature that among the definite positions (x{, x 2 , - , x n ) there are always present positions that *Dini, Theorie der Functionen, p. 68. See also a paper by Stolz, "B. Bolzano s Bedeutung in der Geschichte der Infinitesimal Rechnung," Math. Ann., Vol. XVIII. t Biermann, Theorie der An. Funk., p. 81 ; Serret, Calc. diff. et int., p. 26. SPECIAL CASES 137 The case of a maximum lie as near this position as we wish ; so that, therefore, if 8 denotes a quantity arbitrarily small, x(-a,\<8 (X = l, 2,... ,TI). This position lies either within or upon the boundary of the denned region (x[, x 2 , , x n ). 96. This presupposed, let us consider a continuous function F(x lt x 2 , , x n ), and let the realm of the quantities x lt x z , , x n be a limited one, so that, therefore, we have systems of values which do not belong to it. If for every possible system of values (x l9 x 2 , - , x n ) we associate the corresponding value of the function, which may be denoted by x n+1 , then we have denned certain positions in the region of n + 1 quantities. For the quantity x n+1 there is according to the first theorem an upper limit a n+1 ; consequently, owing to the second theorem there must be within the interior or upon the limits of the defined region a position (a v 2 , , a nt a, l+ i) of such a nature that in the neighborhood of this position there certainly exist positions which belong to the region in question. Now if it can be shown that this position lies within the interior of the region, then there is in reality a maximum of the function on the position (a v 2 , -, a n ) ; on the con trary, if the position lies on the boundary, we cannot come to a conclusion regarding the existence of a maximum of the func tion x 1l+l . It may in many cases happen that one can show, if (x v x 2 , , x n ) is any position on the boundary of the realm and if x n+1 denotes the corresponding value of the function, that there are present within the realm positions for which the values of the function are greater than for every position on the boundary. Then the position which we are considering here cannot lie upon the boundary, and it is clear that the limiting value of the function FIG. 20 The case of a minimum FIG. 21 138 THEORY OF MAXIMA AND MINIMA Case of asymptotic approach x x z FIG. 22 position Xi can be assumed for a definite position within the interior, since the function varies in a continuous manner. The analogue is, of course, true for a minimum. If, however, it does not admit of proof that there are positions on the interior of the defined realm for which the value of the function is greater or smaller than it is for all positions on the boundary, then nothing can be concluded regarding the real exist ence of a maximum or a minimum ; the position (a v a 2 , - > , a n ) would then lie on the boundary of the region, and there might be an asymptotic approach to the limiting value a n + 1 without this value s being in reality reached. Such cases need especial attention. The figures give a plain picture of what Maximum on the limiting has been said for the case y =f(x), where x is limited to the interval (x 1 x - x 2 ). 97. Analogous considerations of the above are fundamental in the very defi nition of an analytic function. For con sider a power-series of x assumed or given in any manner ; let x 1 le a definite value of x. Then there are three possibilities: (1) x may lie in the region of convergence of the given series or of a series that is derived ( 138) from the given series ; the value for x = x of this series is a value of the analytic function which is determined through the original series. In other words, if with Weierstrass we call the original series as well as any other series derived from this one with regard to the function which it repre sents a function-element (Functionenelement), then the first possi bility consists in that, if any function-element is given, the definite value x f lies in the region of convergence of a function-element which is derived from the given one. We admit here also the complex variable. (2) It may happen that x does not lie in the region of con vergence of any series that has been derived in this manner and O x z FIG. 23 SPECIAL CASES 139 that we cannot derive from the original function-element another function-element whose region of convergence can come as near to the point x as we wish. In this case the function does not exist for x = x . (3) Although we cannot find a power-series within which x lies, nevertheless, it sometimes happens that we may still derive elements ivhose regions of convergence contain positions which can come as near to the point x 1 as we wish. Whether we can then define the function for x = x by the consideration of boundary conditions must in each case be considered for itself. If we have case (1) before us, then the function is defined not only for every value x 1 but also for all values in the neigh borhood of x and has for these values the character of an integral function. The definition of an analytic function as thus given is prefer able to other definitions from the fact that the existence of general analytic functions is at once recognized ; in short, that we have under our control, in our possession, all possible analytic functions. Every possible power-series within a region of con vergence gives rise to the existence of a definite analytic function. Moreover, one must assume the duty of proving in the case of every example that it leads to just such functions. For this reason investigations are necessary of which formerly we find no trace. If we have a differential equation, we must begin with the proof that the functions which satisfy the differ ential equation arise from such function-elements as we have just explained ; that is, we must first show, if y is the unknown func tion and x is the variable of the differential equation, that this equation can be satisfied through y = P (x a). Reciprocally, if any variable quantity y is so connected with another variable quantity x that it satisfies the differential equation, we must show that it may be derived from one single function-element in the manner indicated. This last proof is of especial importance in the application of analysis to geometrical mechanics. When a problem is given in mechanics, we have to represent the coordinates of the moving point as functions of the time. 140 THEORY OF MAXIMA AND MINIMA Only real values are permitted in this problem. We cannot therefore a priori know whether the required functions are analytic or not. These functions are generally denned through differential equa tions. We shall give the simplest case as an example. Suppose we have a system of points that attract one another according to an analytic law, and let x l} x 2 , , x n be the coordinates of these points. If the motion is a free one, we have the differential equation in the form where F denotes a given function of x v # 2 , , x n . With such a problem we have to prove before everything else that the required functions of time are analytic functions. If for the point t = t Q the initial position and the initial velocity are given, then in the neighborhood of the initial position we can find power-series, and we have to show that through these power- series the required functions are completely determined. II. EXAMPLES OF IMPROPER EXTREMES WHERE THE DIF FERENCE F(a^ + h v a z +h z , . . ., a n + Ji n } - F(a v a v . . ., a TO ) IS NEITHER POSITIVE NOR NEGATIVE BUT ZERO ON THE POSITION (a v a a ,...,an) WHICH IS TO BE INVESTIGATED 98. We shall now consider a case which is not included in the previous investigations, but may be in a certain measure reduced to them: The definition of the proper extremes of a function consists in the fact that the difference F(a l +h l) 2 + & 2 , . -, a n + h n ) - F(a lt a z , - , a n ) (i) must be invariably negative or invariably positive. There are cases where an extreme does not appear on the position (a v a 2 , , a n ) in the sense that the above difference must be positive or nega tive, but in the sense that the difference must be zero. Suppose, for example, we have the problem: Determine a polygon of n sides with a given constant perimeter S whose area SPECIAL CASES 141 is a maximum, a problem which we shall later discuss more fully (see 101). If this maximum is attained for a definite polygon, then we may at pleasure change the system of coordinates by sliding the polygon in the plane without altering the area. For example, let n = 3, and (x v y^, (x 2 , y 2 ), and (x s , y 3 ) be the coordinates of the vertices of the triangle. Then the expression which is to be a maximum is where the variables are subjected to the condition fl VW v kj ~ I tX^n tX^i There will not only be one system of values which gives for F a maximum value, but an infinite number of such positions ; since, if we take the triangle in a definite position, we may move it in its plane at pleasure. This is therefore a case where the difference (i) is not positive or negative but zero. 99. Such cases, however, may be reduced to maxima and minima proper if we choose arbitrarily some of the variable quantities. In the special example of the preceding section we may assume a vertex of the triangle at pleasure; let it be the origin of coordinates, and we further assume that one of the sides coincides with the positive direction of the X-axis, so that we may write x l =y l =y 2 = 0. If we agree that the triangle is to lie above or below the A^-axis, the problem is completely determinate. In so far as the necessary conditions for the existence of an extreme are concerned we may proceed in precisely the same manner as we have hitherto done, since under the assumption that there are no equations of condition we have \-h v a 2 + h 2 , , a n +h n )- F(a lt a 2 , -, a n ) 5} W(* a a ,- -, ) + (&!, hi, -, k n )&. (u) a=l 142 THEORY OF MAXIMA AND MINIMA If a minimum is to be present, then this difference can never be negative, but may be zero. For this to be possible the first deriv- a = n atives must all vanish. Since, if the sum ^h a F a (a lt a 2 , >, a n ) had (say) a positive value for h 1 = c v h 2 = c 2 , . , h n = c n , then we could place h a equal to cji and then choose h so small that the sign of the right-hand side of (ii) would depend only upon the sign of the first term. If we then make h positive or negative the difference would also be positive or negative. If equations of condition are present, it may be shown, as above, that the derivatives of the first order must vanish, since, if all these derivatives did not vanish, we might express some of the tis through the remaining ones, and then proceed as we have just done. The required systems of values (x v x 2 , , x n ) will therefore be determined from the same equations as before. 100. If we have found a system of values of the # s which satisfy the equations of condition of the problem, then in the neighborhood of this position there will be an infinite number of other positions which satisfy the equations. These last are characterized by the condition that the difference (i) vanishes identically for them. This is just the condition that made impossible the former criteria, by means of which we could decide whether an extreme really entered on a position (a lf 2 , ., a n ) that was determined through the equations in a^, # 2 , ., x n . One must therefore seek in another manner to convince him self which case is the one in question. This is further discussed in the following problem : 101. PROBLEM. Among all polygons which have a given number of sides and a given perimeter, find the one which contains the greatest surface-area. (Zenodorus.) We see at once that the problem proposed here is of a some what different nature from the problems of 90 and 91, since the existence of the maximum value of the function is no longer the question, as was proposed in 49 and held as fixed through out the general discussions. For if the definition of the maximum SPECIAL CASES 148 is such that the function on the position (a lt a z , , a n ) must have a greater value on this position than on all neighboring positions, then in this sense our polygon could certainly not have a maximum area ; since, if we had such a polygon on any position, we might slide the polygon at pleasure without changing its shape and con sequently its area. Therefore only a maximum of the area can enter, in the sense that the periphery remaining the same an in crease in the area of the surface cannot enter for an indefinitely small sliding of the end-points. We consequently cannot apply our general theory without further restriction. 102. Let the coordinates of the n end-points taken in a definite order be . ,,..,.. T y 1 1> y\ > ^2 y 2 > > " The double area of a triangle which has the origin as one of its vertices and the coordinates of the other two vertices x v y^ and # 2 , ?/ 2 is, neglecting the sign, determined through the expression To determine the sign of this expression we suppose that the fundamental system of coordinates is brought through turning about its origin into such a position that the positive X-axis coin cides with the length 01. We call that side of the line 01 posi tive on which lies the positive direction of the F-axis : The double area of the triangle 012 is to be counted positive or negative according as it lies on the positive or negative side of the line 01. If the point has the coordinates # , y , the double area of the triangle is 2 A 012 - (^ - X Q ) (2/ 2 - y ) - ( yi - T/ O ) (,v 2 - # ), where the above criterion with reference to the sign is to be applied. For the polygon we shall take a definite consecutive arrange ment of the points (1, 2, , n) and, besides, we shall assume that no two of the sides cross each other. The last hypothesis is justifiable, since we may easily convince ourselves that if two sides cut each other we may at once construct a polygon whose sides do not cut one another and which, having the same perim eter as the first polygon, incloses a greater area. 144 THEORY OF MAXIMA AND MINIMA Within the polygon take a point = (x , y ) and draw from it in any direction a straight line to infinity. This straight line always cuts an odd number of sides of the polygon. Now if we follow the periphery of the polygon in the fixed direction (1, 2, , ?i) and mark the intersection of a side by the straight line with -f- 1 or 1, according as we pass from the nega tive to the positive side of that line or vice versa, then the sum of these marks is either + 1 or 1. In the first case we say that the polygon has been described in the positive direction, in the second case in the nega tive direction. It may be proved* that whatever point be taken as the point within the poly gon and in whatever direction the straight line be drawn, we always have the same characteristic number +1 or 1 if in each case the positive side of the straight line has been correctly determined. 103. The double area of the polygon is FIG. 24 2 F= - or X Q ) (2/2 - 2/ ) - fa - X Q ) (y l - y ) + fa - x ) (y 8 - y ) +K-o) (2/1-2/0) -(^1-^0 (a) where the positive or negative sign is to be taken according as the polygon has been described in the positive or negative direction. We may, however, eventually bring it about through reverting the order of the sequence of the end-points that the expression 2 F is always positive. 104. Suppose that this has been done. The function 2F ia to be made a maximum under the condition that the periphery has a definite value S. We may write where S A _ lf A = (^ A - X K _tf + (y A - y x _i) 2 . (7) * The proof is found in Cremona, Elementi di geometria projetiva. Rome, 1873. SPECIAL CASES 145 Form the function G = 2F+ e(* 1>2 + s 2 , 3 + + s ntl -S), (5) and placing its partial derivatives equal to 0, we have A+1,A A-1,A *A+1,A *A-1, (X = 1, 2, -, 71 ; however, for \ = n we must write X + 1 = 1). Take in addition the equation (/3) and we have 2 n + 1 equations for the determination of the 2 ?i -f 1 unknown quantities 105. To reach in the simplest manner the desired result from these equations, we adopt the following mode of procedure. If we write x ^ = (<*A- ^i)+*(jfA- 3h-i), (?) then z x , geometrically interpreted, represents the length from the point X 1 to the point X both in value and in direction. If, further, we write z( = (a? A - X K ..j) - i (y A - y A _j), (7;) then *A-4=*A 2 -i,A- (^) Multiply the first of equations (e) by t and subtract from the result the second ; then owing to (?) we have A-1. *,+! - -^-) = - N S A-1, A 7 Now, multiplying the last two equations together, we have from (0) and therefore s? ,. = ?. A 1) A A) A T 1 146 THEORY OF MAXIMA AND MINIMA 106. Since s v _ lfl , is an essentially positive quantity, it follows that s _ = s (K] and consequently the sides of the polygon are all equal to one S( another. Hence each side = - > and we have from U] n z x + -i ein 4- S * = = const. If we write 2 A = e^ 1 , n where < A denotes the angle which s A _ 1)A makes with the X-axis, then <*A + i-0A) = const, or < A + j c A = const. ; (X) that is, all the angles of the polygon are equal to one another, and consequently the polygon is a regular one. It is thus shown that the conditions which are had from the vanishing of the first derivatives can be satisfied only by a regular polygon ; that is, if there is a polygon which, with a given perim eter and a prescribed number of sides, has a greatest area, this polygon is necessarily regular. Our deductions, however, have in no manner revealed that a maximum really exists. 107. To establish the existence of a maximum we must apply the method given in 93-96. We note that an upper limit exists for the area of the polygon, from the fact that the number of sides and the perimeter are given ; for if we consider a square whose sides are greater than the given perimeter S, we can lay each polygon with the perimeter S in this square, and in such a way that the end-points of the polygon do not fall upon the sides of the square. Hence the area of the polygon cannot be greater than that of the square, and consequently there must be an upper limit for this area, which may be denoted by F . The question is, Can this limit in reality be reached for a definite system of values? The variables x lt y^ x 2 , y%\ ; x n , y n being limited to this square, there must be ( 96) among the positions (x v y^ x z , 2/2 ; ; x n> 2/%) which fill out the square a position SPECIAL CASES 147 (a lf ^ ; a 2 > & 2 ; ; a n , b n ) of such a nature that in every neighborhood of this position other positions exist for which the corresponding surface area F of the polygon formed from them comes as near as we wish to the upper limit. We may assume that this position is within the square, since if it lies by chance on the boundary, then from what has been said above, it is admissible to slide the corresponding polygon without altering its shape and area into the interior of the square. We assert that the value of the function F for the position (a v &j ; 2 , & 2 ; ; a n , b n ) must necessarily be equal to F . For if this was not the case, the inequality must also remain if we subject the points a v \\ 2 , 6 2 5 3 a n> b n to an indefinitely small variation ; and on account of the continuity of F it would not be possible in the arbitrary neighborhood Qi(a v \\ ; a n , b n ) to give positions for which the corresponding area comes arbitrarily near the upper limit F Q . This, however, contradicts the conclu sions previously made. Hence all n corners with a given periph ery not only approach a definite limit with respect to their inclosed area but this limit is in reality reached. Since, furthermore, the necessary conditions for the existence of a maximum have given the regular polygon of n sides as the only solution, and since we have seen a maximum really exists, we may with all rigor make the conclusion : That polygon which, with a given periphery and a given number of sides, contains the greatest area is the regular polygon. PROBLEM Among the regular polygons with a constant periphery, the one with the greatest number of angles has the greatest area. (Zenodorus.) 108. Hadamard s problem. If A l = (x v y v z^, A 2 = (x 2 , y z , 2 2 ), A 3 = (X B , 1/3, 2 3 ) are the rectangular coordinates of any three points from a fixed origin 0, the volume formed on the three lines OA V > 2/3 and if x? + y?+ z?= df (i = 1, 2, 3), 148 THEORY OF MAXIMA AND MINIMA where d v d 2 , d 3 are positive constants, it may be easily shown that A is a maximum when A = d l d 2 - d 3 ; or, of all parallelo- pipedons constructed on the three sides OA V OA 2 , OA 3 , the one having the greatest volume is the rectangular parallelopipedon. As the parallelopipedon may occupy an infinite number of positions without changing the origin, we have here a case of improper maximum which is of interest. The extension of this problem is due to Hadamard.* ^22> where x% + ^| H h ^ = a i (i= 1, 2, . . -, w), Ae d s 6em# positive constants, show that the maximum of the absolute value of A is \ __ f i J i This may be done as follows : Let the determinant be developed with respect to the elements of the iih line, so that A = AHXH + A i2 x i2 + +A in x in . (i) We then have to find the maximum or the minimum of the func tion A of the n variables aj n , x i2 , , x in which are connected by the relation ,2 i 2 i i ,2 _ ,72 / -\ The Lagrange method (89) leads at once to the conditions If x kl , x k %, - , x kn are the elements of another line of the determinant, we have Anx kl + A i2 x k2 -\ +A in x kn = ; (iv) or, from (ii), x tl x kl + x i2 x k2 -\ +{*= 0, (v) where i = k. * Hadamard (Bull, des Sciences Mathtmatiques, Second Series, Vol. XVII, 1893). Proof by Wirtinger (ibid., 1908). An interesting application of this problem is found in Bocher, Introduction to the Study of Integral Equations, pp. 31 et seq. SPECIAL CASES 149 From this we conclude that the determinant can only have an extreme value when it is orthogonal. When the conditions (v) exist, the square of the determinant is another determinant, in which all the elements are zero except those of the principal diagonal, which are df, d$, - , d%. It follows that A =#.#,...,<*>. Here again we have an improper extreme which it is interesting to consider further. III. CASES IN "WHICH THE SUBSIDIARY CONDITIONS ARE NOT TO BE REGARDED AS EQUATIONS BUT AS LIMITATIONS 109. Besides the problems already mentioned, those problems are particularly deserving of notice in which the conditions for the variables are not given in the form of equations but as restrictions or limitations. For example, let a point in space and a function which depends upon the coordinates of this point be given. Furthermore, let the point be so restricted that it always remains within the interior of an ellipsoid ; then the restriction made upon the point is expressed through the inequality ^ 7 ,2 -2 = + 4- < 1 ~ 2 + 52 + C 2 We have, accordingly, such limitations when a function of the variables is given which cannot exceed a certain upper and a certain lower limit. We make such a restriction when we assume that a function f l shall always lie between fixed limits a and ~b. 110. This limitation, which consists of two inequalities [a] *<A<&, may be easily reduced to one. For from [a] it follows necessarily that W and, reciprocally, if [j3] exists and if a < b, then /i must be situated between a and b and consequently [a] must be true. 150 THEORY OF MAXIMA AND MINIMA Every limitation of the kind given may be analytically repre sented as one single inequality of the form [/3]. 111. We must next find the algorithm for the cases under consideration. This may be done at once if we consider that such cases may be reduced to those in which occur equations of condition. For this purpose we need only establish the problem of finding the maximal or minimal values of a function whose variables are subjected to certain conditions as follows: It is required among all systems of values which satisfy the equations / A = (X 1, 2, ., m) to find those for which F is a maximum or a minimum. By proposing the problem in this manner, it is clear that all the variables x which appear in the equations of condition need not necessarily be contained in the function. Suppose further we have the limitation that [7] /t>0, then, through the introduction of a new variable x n+l , we may transform this limitation into an equation of condition. For, as we have to do with only real values of the variables, the equation [r] /* = denotes exactly the same thing as [7]. If, therefore, a function F(x v x 2 , - ., x n ) is to be a maximum or minimum under the limitations where the / s are functions of x v x 2 , > ., x w then we may solve this problem if instead of the r last restrictions we introduce the following limitations : Jm + 1 = x n + l> Jm + 2 == X n + 2> * > Jm + r == %n + r- The problem is thus reduced to the one of finding among the systems of variables x v x 2 , , x n + r those systems for which F is a maximum or a minimum. 112. Examples of this character occur very frequently in mechanics. As an example consider a pendulum which consists SPECIAL CASES 151 of a flexible thread that cannot be stretched. The condition under which the motion takes place is not that the material point remains at a constant distance from the origin, but that the distance can not be greater than the length of the thread. Such problems are more closely considered in the sequel. It will be seen that by means of Gauss s principle all problems of mechanics may be reduced to problems of maxima and minima. IV. GAUSS S PRINCIPLE 113. For the sake of what follows we shall give a short ac count of this principle : Consider the motion of a system of points whose masses are m v m 2 , , m n . Let the motions of the points be. limited or restricted in any manner, and suppose that the system moves under the influence of forces that act continuously. For a definite time let the positions of the points and the components of velocity both in direction and magnitude be determined. The manner in which the motion takes place from this period on is determined through Gauss s principle : Let A v A. 2 , ., A n be the positions of the points at the moment first considered; B v B^---, B n the positions which the points can take after the lapse of an indefinitely small time T, if the motions of these points are free; C v <7 2 , , C n the positions in which these points really are after the lapse of the same time T ; and, finally, let C[ t C& -, C n be the positions which the points may also possibly have assumed after the time T, when the conditions are fulfilled. If we form v=n g v=n 2 and VmJtlCL , it follows from Gauss s principle that from r = up to a definite value of T the condition [1] SVAC^SXlW v = l ,- = 1 v=n 2 is always satisfied ; that is ? m v B v C v must always be a minimum. 152 THEORY OF MAXIMA AND MINIMA 114. To make rigorous deductions from Gauss s principle, which was briefly sketched in the preceding section, we shall give a more analytic formulation of it : For this purpose we denote the coordi nates of A v by x v , y vt z v) the components of the velocity of A v by x l> yl) z l> an( i the components of the force acting upon A v by X vt Y v , Z v . The coordinates of B v are therefore n *>O * v + rxl +~Xy 9 + ryl + T -Y v) z v + rz v + ~Z v] and from Taylor s theorem the coordinates of C v are o 29 x v + rxl + ^xl + . . ., y T consequently we have [2] Instead of x", however (see preceding section), other values may possibly enter, say x"+ ,-, so that we have rqn X^, n _ W //^ _i_ t "F\2 L^J ^ m vA,% s JMMW T & "- -AT) v = l v = l It follows from Gauss s principle that the difference of the sums [2] and [3] must always be positive. Hence [4] >jj?m. \ 2 [fc,(av" - X.) + rj v (yl 1 - Y v ) + (*" - Z v )] + ?v 2 + ^ 2 -l-C 2 ?j + ...; that is, the quantities x" t yl , z ! J must be such that the sum [2] is a minimum. Hence, among all the x", y" , z" which are associated with the conditions of motion, we must seek those which make [2] a minimum. 115. We have reached our proposed object if we can show that the ordinary equations of mechanics may be derived from Gauss s principle. SPECIAL CASES 153 If there are no equations of condition present, then clearly [2] is only a minimum when r Y 11 " V z" Z <L V -A-v) y v -*-!/) 2 f ^v If, however, we have equations of condition, for example, if any of the variables are connected by a relation such as / (x, y, z) = 0, then these must hold true throughout the whole motion. They may therefore be differentiated. We have in this way equations in _^, _J^, anc i _?*. Differentiate again and we have equations dt dt dt in a;,", y v t and z" ; Hence, in conformity with the rules that have been hitherto found for the theory of maxima and minima, the quantities x l > Uv> z v are t be so determined that the derived equations of condition are satisfied, while at the same time [2] becomes a minimum. But in this case also, as is easily shown, we are led to the usual differential equations of mechanics. 116. The theory of maxima and minima may be applied to realms which are seemingly distant from it. An example in question is the proof of a very important theorem in the theory of functions. THE EEVERSION OF SERIES If the following n equations exist among the variables x lt - a 1 n x n = a 21 x 1 y n =a nl x l +a n where the coefficients on the right-hand side are given finite quan tities and the "JT s are power-series in the x s of such a nature that each single term is higher than the first dimension, and if the series on the right-hand side are convergent and the 154 THEOKY OF MAXIMA AND MINIMA determinant of the nth order of the linear functions of the x s which appear in [1], namely, [2] > a ln n2> is different from zero, then, reciprocally, the x s may also be expressed through convergent series of the n quantities y which identically satisfy the equations [1]. 117. As an algorithm for the representation of the series for the x s, we make use of the following methods (cf. 135, 136) : If we solve the equations [1] linearly by bringing the terms of the higher powers of the x s on the left-hand side, we have where A^ denotes the corresponding first-minor of It is seen that in general >L ~ n < x = M A [3] x. = M in [2]. r 1(2) A where [x v x%, , x n ]@) denotes terms of the second and higher dimensions in x v x%, , x n . We shall therefore have a first approximation to the result if we consider only the terms on the right-hand side of [3] which are of the first dimension. A second approximation is reached if we substitute in the right-hand side of [3] the first approxima tions already found and reduce everything to terms of the second dimension inclusive. Continuing with the second approximations that have been found, substitute them in [3] and, neglecting all terms above the third dimensions, we have the third approxi mation, etc. ; we may thus derive the x s to any degree of exactness required. SPECIAL CASES 155 Since A is found in all the denominators, the development converges the more rapidly the greater A is. 118. In what follows we shall assume that the quantities on the right-hand side of [1] are all real and that we may write where H^ is a homogeneous function of the ith degree in x v X 2> > x n> an( ^ consequently by Euler s theorem for homogeneous functions -_-._. GJj[\n Cjtl\n I VJrL\f) ^L rp A & I *& I I ,y, A ^ ^x ~ 9 1 ~?7" ^02 -oTT" ^ O ^ ^ r ^ t/ X/j Zi c/t/^2 * l/i * / . + o ^1 ~^T" + o ^2 -oT~ "" h o O Vtb-t tj (siA/n <-) + ^4 i ^4 i i , T *l "^ r T ^2 ~^ T" T T*n J. J ^7- 4. J fir 4- ^r i CXt 4 < ^ 6/X r where the quantities X (X = 1, 2, , n ; /u = 1, 2, , n) are continuous functions of the sc s, which become indefinitely small with the aj s. The system of equations [1] may then be brought to the form M =1 The theorem of 116 in this modified form may be expressed as follows : (1) It is always possible so to fix for the variable^ x lt & 2 , , x n and y v y 2 , - , y n , limits g v g 2 , -,g n and h v & 2 , , h n that for every system of the y s for which y\\<h^ there exists * one system of the x s for which X K \ < g K , and in such a way that the equations [1#] are satisfied. * See Biermann, Theo. der An., Funk., p. 234, and also Stolz, p. 172. 156 THEORY OF MAXIMA AND MINIMA (2) The solution of the equations [la] has a form similar to the equations [la] themselves, viz.: where the Y are continuous functions of the y s, which become indefinitely small with these quantities. To prove this theorem we make use of the theory of maxima and minima. 119. If we give to the y K the value zero, the equations [la] can only be satisfied if their determinant vanishes, that is, when except for the case where the a? s vanish. For sufficiently small values of the x s the determinant [4] is not very different from the determinant [2]. We may therefore determine limits g for the x s so that [4] cannot be zero unless [2] is also zero. A is, however, by hypothesis excluded. Accordingly the y s can only be zero in [la] when all the x s vanish, provided the x s are confined within fixed limits. These limits may be regarded as the boundaries of a definite realm. 120. Again, we write =2X ax * +XJ *> = (X=l, 2,.. and consider the function [6] 5 A=l In $ we shall write for the x s all the systems of values where at least one x lies on the boundary of the realm in question. The realm of the x s is thus the totality of the x s for which A S Zer (X = l, 2, - -,%; /*=!, 2,...,w); it follows then that [4] is not zero, since a A J is by hypothesis different from zero. SPECIAL CASES 157 When one of the x s reaches its limit, there is no system of values of the x s for which the function [6] vanishes, since the function can (as follows from definition [5] of the F^ and the con siderations of 119) only vanish if all the y s and consequently all the x s vanish. There is then a lower limit G which is different from zero for the values of [6] which correspond to a definite system of values (x v x%, , x n ) of the boundaries. 121. We come next to the determination of the limiting values of the y s. For this purpose we consider the expression m If we ascribe definite values to the y s, then there is for the values [7] in the realm of the x s a system for which [7] is a minimum. We wish to show that this system of values of the x s does not lie upon the boundary of the realm. We prove this by show ing that there is a point within the realm where the expression [7] has a smaller value than it has on the boundary. The expression [7] may be written Since $ is at all events greater than F k , and consequently A<>. it follows that vs < ^ ix < ^ fix i < ^ SI y. where the h s are the limits of the ^ s. From this it results that -n ju. = n and, consequently, for a greater reason [8] H = l 158 THEORY OF MAXIMA AND MINIMA The limits h^ must be so chosen that the right-hand side of [8] is positive. This choice can be made so that the expression on the right-hand side for a system of x s which belongs to the boundary does not become arbitrarily small but always remains greater than a certain lower limit (see the preceding section). The expression, however, on the interior of the realm of the x s may be arbitrarily small, viz., when x 1 = x% = = x u 0. For this system of values the left-hand side of [8] is equal to We have therefore found the following result: We can give limits g to the variables x, and to the y s the limits h, in such a way that the expression [7] for systems of values of the x s which belong to the boundary of the realm is always greater than it is for the zero position (x l = x 2 - = x n = 0). Hence the position for which the expression [7] is a minimum must necessarily lie within the realm of the x s ; and we may be certain that within the realm of the x s there is a position where [7] has its smallest value. 122. In order to find the minimal position of [7] which was shown to exist in the previous section we must differentiate the function [7] and put the first partial derivatives equal to 0. This gives > = ! X * These n equations can, in case the determinant [10] ( = l,2,.-.,n; /t=l, 2, ...,) is different from zero, exist only if the quantities within the brackets vanish. [10] is identical with the determinant [4]; and (see 119) it may be always brought about through suitable choice of the limits g of the x s that [4] is different from zero if only the de terminant [2], as by hypothesis is the case, is different from zero. SPECIAL CASES 159 Hence the equation [9] can only be satisfied if [la] or [5] y v =F v (x v x^ . . ., x n ). We have therefore found that, since there is certainly a system v=n of values of tJie x s for which the function ^(y v F v ) z is a mini- v = l mum, there must also be a system within the realm of the x s for which the equations [1#] are satisfied if to the y s definite values in their realm are arbitrarily given. 123. We must further see whether within the fixed realm there is one or several systems of values of the x s that satisfy the equations [1#] with prescribed values of the y s which lie within definite limits. To establish this, assume that (x[, x%, , x n ) is a second system of values that satisfy the equations [!]; we must then have the equations [11] F v (x[,xi, -, x n )-F v (x v x 2 , . . ., ^) = (v = l, 2, . . ., n). Developing by Taylor s theorem, we have, when we consider only terms of the first dimension, The X are functions which depend upon the x"s and x s and vanish with these quantities. We may determine the n unknown quantities x[ x lt x^ x 2 , - ., x n x n from the n linear equations [11 a]. For small values of x and x r the determinant will be little different from the determinant [10]. We may therefore make the limits g of the x s so small that [12] is different from zero for all the x*a and x"s which belong to the realm ; and when this has been done, the equations [11 a] are only satisfied for r _ r * //, 1 9 v x p Zp (p -L, A , n) , that is, there exists within the realm in question no second system of the x s which satisfies the equations [!]. 160 THEORY OF MAXIMA AND MINIMA We have therefore come to the following result : It is possible so to determine the limits g and h that with every arbitrary system of the y s in which each single variable does not exceed its definite limiting value, the given equations [1] are satisfied by one and only one system of the x s in ivhich these quantities likewise do not exceed their limits* The first part of the theorem given in 118 is thus proved. REMARK. We have assumed that we have to do only with real quantities. The discussion, however, is not restricted to such quantities, as it is easy to prove that the same developments may also be made for complex variables. 124. The values of the x s which were had from the equations may be derived in the manner given in 118. If we write |^P+X, P |= A ("= 1, 2, -, n ; p = 1, 2, . . ., n), the linear solution of the equations [la] is [36] V* ^f9, (p = l,2,...,), =! A where A vp denotes the corresponding first minors. Now A is a definite quantity which lies within certain finite limits ; the same is also true of A r vp is found in a similar manner. Hence A J the quantities ^ are finite quantities which lie between definite limits ; and, therefore, if the y s, become indefinitely small, the x s will also become indefinitely small; that is, those systems of values of the x s which satisfy the equations [1] under the named conditions are, as has also been shown in 119, so formed that they become indefinitely small with the y s. * See also Hadamard, " Sur les transformations ponctuelles," Bull, de la SocitU Math., Vol. XXXIV, 1906. SPECIAL CASES 161 We may now show that the x s are continuous functions of the ?/ s. Let (ftp 6 2 , , >) be a definite system of values of the y s, and let the system (a v 2 , , a n ) of the x s correspond to this system of the i/ s. If we then write noi \y\ = ^x + ^xl ~ ., [13] I (X=l, 2,... ,71), lA=A+fAj the system of equations [la] or [1] goes into (X = l, 2,... ,TI]. Developing this expression according to powers of the f s, we have where the C 1 ^ are functions of the # s and f s. If the f s are indefinitely small, we may limit the C 1 ^ to the first derivatives of F^. In this case we denote the coefficients of [Ic] by (7 AM , so that rT? <?A M = T^ for (^1 = a l ^2 = a 2> > ^w = a ) 0gp (X = l, 2, ., ?i; /* = !, 2, -, ?i), and the determinant of the equations [Ic] goes into for (x l = a v x 2 = a a , . . ., x n = a n ) (\ = 1, 2,..., 7^; ^=l,2,..-,n). If the # s lie within definite limits, this determinant remains always above a definite limit. We may therefore say that the determinant has a value different from zero. Consequently the condition that the equations [Ic] may be solved is satisfied, and it is seen that indefinitely small values of the f s must correspond to indefinitely small values of the TJ S. This means nothing more than that the functions x are con tinuous functions of the *s. 162 THEORY OF MAXIMA AND MINIMA 125. The above investigations are true under the assumption that the functions F^ are continuous, that their first derivatives exist and likewise are continuous within certain limits. We need know absolutely nothing about the second derivatives. Of the # s, of which it is already known that they exist as functions of the y s and vary in a continuous manner with them, it may now likewise be proved that they, considered as functions of the y s, have derivatives which are continuous functions of the 7/ s. The proof in question may be derived from the following con siderations : If from [Ic] we express the fs in terms of the ?; s, we have The ^ are continuous functions of the f s, and the f s are continuous functions of the ?; s. Hence ^ may be represented as continuous functions of the rfs. If the rj s become indefinitely small, then the f s become indefinitely small, and we have definite limits for ^. In general, if we have a function f(x v , x n ) of the n variables #i> %%> > x n> an d if we consider the difference /(i + fi> 2 + fa a + f) -f( a v a v" > a n)> it is seen that it may be written in the form where the ^T A depend upon the f s and become indefinitely small with these quantities, and the & A are, in virtue of the definition of the differential quotient, the partial differential quotients of / with respect to # A for the system of values (a v a 2 , , a n ). From the above it results not only that the x*s are continuous func tions of the y s but also that the derivatives of the first order of these functions exist. SPECIAL CASES 163 We have, indeed, the derivatives of the first order if in the expressions ^ we write the f s equal to zero. The quantities ^, however, become then, in accordance with [Ic], the quantities which we should have in [Ic] if we had at first written C^ instead of C AM . But the quantities C^ are continuous functions of a lt a 2 , - ., a n . We may therefore say that the differential quotients ^ are con- A tinuous functions of the variables x t and it is then proved that the x s are such functions of the y s as the y s are of the x s. 126. For the complete solution of the second part of the theorem in 116 we have yet to show that the expressions [36] may be reduced to the form [3 a]. , For this purpose we must bring the quantities ^f in [36] ( 124) to the form , ~ where 6 Aja is the value of -**& when all the x s are equal to zero. Y A/X is a function of the x s, but the x s are functions of the y s, so that Y A i s a function of the y s which vanishes when they vanish. We may therefore in reality write [36] in the form [3 a] A 1 127. There may arise cases hi which we know nothing further of the functions F^, as was assumed in 116, than that they are real continuous functions. We cannot then conclude, for example, that the x s may be developed in powers of the y s ; but we may reduce the equations to the form [3 a] and show that the equations [la] are solvable. The theorem which has been proved is of great importance when applied to special cases, even for elementary investigations. If, for example, the equation /(#, y) = is given, then it is taught in the differential calculus how we can find the derivative of y considered as a function of x. 164 THEORY OF MAXIMA AND MINIMA If we assume that the variables x and y are limited to a special realm where the two derivatives with respect to x and y do not vanish and therefore the curve f(x, y)= has no singular points, and if the equation is satisfied by the system (# , y ), we may write x = # 4- f , y = y Q 4- 17. We have then f(x Q + f , y + 17) = 0, and we may prove with the aid of the theorem in 118 that 77 is a continuous function of f and has a first derivative. Not before this has been done have we a right to differentiate and proceed according to the ordinary rules of the differential calculus. MISCELLANEOUS PROBLEMS 1. Show that the problem of determining the extremes of the function f(x, y) may be reduced to the determination of the upper and lower limits of this function under the condition that a; 2 + y 2 = r 2 . (Stolz, Wiener Ber., Vol. C, p. 1167.) 2. Find the shortest distance from the point P(x lt y^, z x ) to the plane Ax + By + Cz + D = 0. A ar, + By, + Cz. + D Answer. . \A4 2 + B 2 + C 2 3. Find the points on a given sphere which are the farthest from and nearest to a* given plane which does not intersect the sphere. (Pappus.) 4. Find the triangle of maximum area whose vertices F 1? F 2 , and V s describe respectively three given plane curves C lt C 2 , and C s . When the three curves reduce to the same ellipse, show that there are an infinity of triangles of maximum area (a case of improper maximum). 5. Find the ellipse of least area that may be drawn through the three vertices of a triangle. 6. Find the ellipsoid of least volume which may be drawn through the four vertices of a tetrahedron. 7. In a triangle of greatest or least area circumscribed about a curve, the points of contact are the mid-points of the sides. 8. Among the triangles whose vertices are situated respectively upon three given straight lines in space, which is the one whose perimeter gives a maximum or minimum? Also determine the triangle of maximum or minimum area. Answer. In the first case the bisectrices of the triangle are respectively normal to the straight lines described by the vertices ; in the second case the altitudes of the triangle are perpendicular to these lines. SPECIAL CASES 165 9. Upon a fixed surface find a point P such that the sum of the squares of its distances from n fixed points A lt A 2 ,---, A n is a maximum or a minimum. Answer. If the tangent plane at P is taken as the a^-plane and the normal to this plane at P as the z-axis, the center of mean distances M, say, of the points A lie upon the z-axis. It follows that the points P are the feet of the normals which may be drawn to the surface from M. 10. Show that the semi-axes of a central section of a quadric A^x"- + A 2 y 2 + A s z 2 + 2 B^yz + 2 B n zx + 2 B s xy + 1 = are the roots r 2 of the equation B 9 , B 2 , I, B 2 , B v m, = 0, where the section is made by the plane Ix + my + nz = 0. 11. Show that the axes of the quadric of the preceding example are the roots of the following cubic in r 2 : =0. 12. z \ v y is to be a maximum, where ?/ 3 nyx + x 3 = and v x y. (Hudde, 1658. See Descartes, Geom., Vol. I, pp. 507-516.) 13. The fundamental theorem of algebra. Let/(<) be an integral function of t with constant coefficients. Write t = x + iy, so that (1) /(O = />(*, y) + iQ (x, y) = P + iQ, with the identical relations (2) dx dQ , dP dQ and = dy dy dx Form the expression fj. ( x, y) /a = P 2 + Q 2 . Within the circle of radius r = Vz 2 + y 2 the function /A is everywhere continuous, so that ( 8) the function /x must reach its lower limit for values of x and y within or on the boundary of the circle. By taking r sufficiently large it is seen that the lower limit of /x must be reached within the circle, so that there must be a minimum value of /x. Show that this minimum value is zero, and consequently that there must be some value of t which nxakes f(f) = 0, provided that f(f) is not a constant. In particular the semi-definite case must be considered. CHAPTER VIII CERTAIN FUNDAMENTAL CONCEPTIONS IN THE THEORY OF ANALYTIC FUNCTIONS I. ANALYTIC DEPENDENCE; ALGEBRAIC DEPENDENCE 128. If in the development of the conception of the analytic functions we start with the simplest functions which may be expressed through arithmetical operations, we come first to the rational* functions of one or more variables. The conception of these rational functions may be easily extended by substituting in their places one-valued functions, and first of all those which may be expressed through arithmetical operations, viz., sums of an infinite number of terms of which each is a rational function, or products of an infinite number of such functions. Such a transcendental function is, for example, where u t (x) \i = 1, 2, - ] are rational functions of x. The necessity at once arises of developing the conditions of con vergence of infinite series and products, since such an arithmetical expression represents a definite function only for values of the variable for which it converges. Mere convergence, however, is not sufficient if we wish to retain for the functions just mentioned the properties which belong to the rational and the ordinary transcendental functions. All such functions have derivatives, and we shall restrict ourselves once for all to functions which have derivatives. Furthermore, the derived series of the above expressions of one variable must converge uniformly (cjleiclimassig) in the neighbor hood of each definite value, and every term of the derived series * See Hancock, Elliptic Functions, Vol. I, pp. 6-9. 166 CERTAIN FUNDAMENTAL CONCEPTIONS 167 must be continuous in the same neighborhood. (Osgood, Lehrbuch der Functionentheorie, p. 83.) 129. When we say that a series whose terms are functions of one variable converges uniformly, we mean the following : * It is assumed that the series in question has a definite value for x = X Q . We consider all values of x for which x X Q does not exceed a definite quantity d. This determines a fixed region for x, within which we shall suppose that the series is convergent. This region is known as the region of convergence (Convergenzbezirk). We may, for brevity, put R k (x) for in the series above. In order that this series converge uniformly, it must be possible after we have assumed an arbitrarily small positive quantity 8, and when a remainder R k (x) has been sepa rated from the series, to find a positive integer m so that | R k (x) | ^ 8, where k > m for all values of x within the region of convergence.! 130. Proceeding in this way we may form more complicated expressions ; for example, we may let w (x) be a sum of an infinite number of terms where each term is a transcendental function like v(x) above, so that We may continue by forming similar expressions out of the transcendental functions w(x) y etc. It is clear that if we proceed in this manner, there is no end of such expressions, so that even if we limit ourselves to one-valued functions, we do not obtain a clear insight into the possible kinds and forms of such functions. It is essential that all such transcendental functions have a common property, and we note that if we take a value X Q within the region of convergence in which the series representing these * Weierstrass, Collected Works, Vol. II, p. 202, and Zur Functionenlehre, 1. + See Dini, Theorie der Funktionen (page 137 of the German translation by Liiroth and Schepp). 168 THEORY OF MAXIMA AND MINIMA functions converge uniformly, they may be represented for all the values of x in the neighborhood of X Q as series which proceed according to positive integral powers of x X Q ; for example, in the form F(x) = F(x - + x ) = + a^(x - x Q ) + a 2 (x - where , a v a 2 , ., are definite functions of X Q . From this it follows that they may be differentiated, and a number of other properties are immediate consequences. 131. We may next extend the conception of uniform conver gence to functions of several variables. With Weierstrass (loc. cit.) consider the infinite series v= oo F(x li x 2 , . . ., x n )=^u v (x 1 , x 2 , . . ., x n ) v = whose terms u v are functions of an arbitrary number of variable quantities x\, x 2 , -, x n . Such a function converges uniformly in any part (R) of its region of convergence when with a prescribed quantity 8 chosen arbitrarily small there exists a positive integer m such that the absolute value of for every value of k which is ^ m and for every system of values of x lt x 2) y x n which belongs to (R). Let !, a 2 , ., a n be a definite system of values of the vari ables x v x 2 , ., x n within the region of uniform convergence, and consider only the values of x v x 2 , , x n for which x 1 a 1) x^a^ - ., x n a n do not exceed certain limits d ly d 2 , , d n , as in 129. The function may then be represented through an ordinary series which proceeds according to integral powers of x 1 a lt x 2 a 2 , x 3 3 , ., x n a n , and consequently may be differentiated; in short, it behaves, as Weierstrass * was accustomed to express it, like an integral rational function in the neighborhood of a definite position within the interior of the region of uniform convergence. *In this connection see Weierstrass, Werke, Vol. II, pp. 135 et seq., and also Bier- mann, Theorie der Analy. Funktionen, pp. 429 et seq. CERTAIN FUNDAMENTAL CONCEPTIONS 169 132. We may next introduce the conception of analytic dependence. If we represent a function which has been formed as indicated above by F(x^ x^ t ., x n ), then F(x lt x 2 > > x n)= expresses a certain dependence among the variables x lf x^, , x n ; that is, among the infinite number of systems of values for which the function has a meaning, those only which satisfy this equation are to be considered. There exists, therefore, among x l ,.x 2) -, x n a dependence of a similar character, as in the case of algebraic equations. If we choose the quantities o^, x%, - ., x n such that the equations ^i=0, F 2 = 0, - , F m = exist where m < n, we have a dependence among the quantities x lf x 2 , , x n defined in such a way that at all events we can choose arbitra rily not more than n m of the variables, since the remaining m variables are determined. 133. The conception of the many-valued functions is at once suggested. Suppose, for example, a function of two variables x and y is given ; then we may consider all the systems of values (x, y) in which x has a prescribed value. For such a value of x there may exist several values of y. We are to regard y as a function of x, and this function is a many-valued function. By the introduction of one or more auxiliary variables it is often possible to express the many-valued functions* through one- valued functions, and indeed in algebraic form. The development of analytic functions from an arithmetical or algebraic standpoint seemed especially desirable to Weierstrass. He wrote (see Werke, Vol. II (Oct. 3, 1875), p. 235): "The more I consider the under lying principles of the theory of functions and I do this con tinually the stronger am I convinced that this theory must be built upon the foundation of algebraic truths." 134. To illustrate the remarks of the preceding article, con sider any analytic dependence existing between, say, two variables x and y and limit one of the variables x to a definite region. The other variable must be expressed through x and in a fofm that remains invariably true for all values of x in question. Now, if to the one variable there corresponds a transcendental function, * We might cite, for example, the Abelian transcendents. 170 THEORY OF MAXIMA AND MINIMA it does not seem possible to express one variable arithmetically in terms of the other. We may, however, introduce a third aux iliary variable and thereby express both of the original variables as one-valued functions of the third variable and in such a way that, if we give to this variable all possible values, we have all systems of values of (x, y). The simplest example is perhaps the one given by the equation z = &, where z and y are two independent variables. It is not possible to express the dependence between x and y in an arith metical form ; that is, one in which transcen dentals do not appear. But if we introduce a third variable t, and write x = e t t we have z e^, so that y ^- Thus x and y are expressed as one-valued functions of t t and such that for one value of x there is invariably one value of t and of y. Poincare* proved that if x and y are connected by an algebraic equation, then all systems of values (x t y) may be expressed in the form just indicated. He also showed that if any analytic dependence exists between x and y, it is always possible to represent x and y as one-valued functions of a third variable. An example in point is the expression of the integrals of linear differential equations through the Fuchsian functions. However, he did not show in this latter case that all the points of the region in question were thus expressed through t. On the contrary it seems that there exists an infinite number of isolated points which can be reached only when t tends toward certain limits. For example, in the differential equation of the hypergeometric series, we should have to exclude in such a representation the real values of x from + 1 to +00. (See the Paris Thesis of Goursat.) In this manner the study of many-valued functions may be reduced to the study of one-valued functions. However, it is not *See Bulletin de la Socitte mathtmatique de France, Vol. XI (1883). See also lectures II and III, delivered in the Cambridge Colloquium, by Professor Osgood (Bulletin of the Amer. Math, Society, 1898) ; and the Problemes mathfrnatiques of Professor Hilbert in the Comptes rendus of the Congress of Mathematicians, Paris, 1900. In his treatment of Algebraic Functions of Two Variables, Professor Picard has done valuable work in this connection. CERTAIN FUNDAMENTAL CONCEPTIONS 171 a simple task, for if we wish in reality to make this representation even in the case of a linear differential equation, we encounter many technical difficulties. Nevertheless, it is essential to prove that there exist such representations. Weierstrass asserted (May, 1884) that he believed the follow ing theorem existed in the theory of functions: It is always possible, where an analytic dependence exists, to express this dependence in a one-valued form which remains invariably true. 135. We may next introduce the folio whig theorem, which is extensively used, particularly in the calculus of variations : Suppose that between the variables x v x 2 , , x n we have .m equations given which may be represented in the form of power- series, and let these be - ) +X = where ^ 1 , X 2 > . . ., X, rt are also power-series of x l a v , x n a n , but of such a nature that each term in them is of a higher dimension than the first. TJie equations will be satisfied for n\ = a lt ., x n = a n . We propose the problem of determining all systems of values (&i> fyy ) x n) which lie in the neighborhood of (a lf a 2 , > , ) and which satisfy the m equations above ; that is, among the systems of values for which \x l a^ , \x n a n \ are smaller than a fixed limit p, determine those which satisfy the m equations. The quantity p is subject to the condition only of being suffi ciently small. To solve this problem we consider the system of linear equations to which the given equations reduce when we X l =o.x,=o...,X.=o. Through these linear equations m of the differences x l a v x^a^-", x m a m may be expressed in terms of the n m 172 THEORY OF MAXIMA AND MINIMA remaining, if the determinants of the mth order which may be formed out of the m rows of the c s are not all zero. If, say, 0, we have (117) x 2 - a 2 = By means of these equations we may represent x l a v x 2 2 , -, x m a m as power-series in the remaining n m differences, the formal procedure being as follows : We write Xi= 0, , X^ = > an( ^ tnus obtain for x 1 a v , x m a m expressions which represent the first approximations. These are substituted in Xj> * * XL an( ^ ^ n tne resu lting ex pressions only terms of the second dimension are considered. These terms added to the terms of the first approximations respec tively constitute the second approximations. Continuing this process we may represent the required expressions to any degree of exactness desired. We obtain the same results if we express m of the quantities x l a v x 2 & 2 , ., x n a n through power-series in terms of the remaining n m quantities with indeterminate coefficients. These coefficients may be determined without difficulty. As just shown, these power-series are convergent as soon as the differences x a which enter into them do not exceed cer tain limits, and, furthermore, these power-series satisfy the given equations. 136. The problem of the preceding article may be solved in the following more symmetric manner, in which none of the vari ables is given preference over the others (see Lagrange, Thorie des Fonctions, Vol. II, 58). CERTAIN FUNDAMENTAL CONCEPTIONS 173 Besides the equations given above we introduce others which are likewise expressed in power-series : 1,1, c l, 2, *> c l, 7/i> C 2,2, > c 2,/n, c l, wi+1, * *, ^1, n ii 7/1,2, *> 771,771, I, C /7l+l,2, * *, C 77l+l, J7l 7/1,7/1 + !, * " *, m, C J+1, 771+1, * * * C 77l+l, 71 < C,2. C nfW , 71, TO+I, * * *, n, n where ^, 3 , , _,, are ?i m new or auxiliary variables. The quantities c are arbitrarily chosen, in such a manner, however, that the determinant 0. Proceeding as in 117 we write the quantities ^ equal to zero, and we thus have a system of n linear equations through which we can express the n differences x 1 a lf ., x n a n through t 1) t 2) "- ) t n _ m) say, x v a v = e v ^ 4- e Vi2 tz H h e Vi1l _ m t n _ m +X V (^ = 1,2,..., ?i). With the help of these equations we can express x l a lt ., ^ as power-series in t lt , ^_ OT . To do this we again write ^1 = 0, anc ^ have only terms of the first dimension. We write the first approximations that have been thus obtained in ]J and by retaining the terms of the second dimension derive the second approximations, etc. It follows directly from the above that these power-series in t formally satisfy the given equation ; that they possess a certain common region of convergence if we give certain fixed limits to KI|> |^|> >Ki-m 5 that they consequently in reality satisfy the equations ; and, finally, that all the systems of values (x ly % 2 , -, x n ) which lie in the neighborhood of (a lf a 2 , . ., a n ) and which satisfy the proposed equations are obtained in this way. 174 THEORY OF MAXIMA AND MINIMA 137. Suppose that between two variables there exists an ana lytic relation which is expressed in the form where P denotes simply a power-series and where X Q , ?/ is a definite pair of values of the variables. In the neighborhood of (X Q , y Q ) there is an infinite number of systems of values which satisfy the equation. The collectivity of these pairs of values (x, y) is called an analytic structure, or con figuration (Gebild), in the realm (G-ebiet) of the quantities (x, y). We may next make an application of the theorem of the pre ceding article. It follows that, if between n quantities x l} x 2 , -, x n there exist m equations in the form of power-series, then the differences ^ a l} , x m a m may be expressed through power-series of the n m remaining variables. Weier- strass said : " Through the m equations a structure of the (n m) th kind in the realm of the n quantities x, x%, - - , x n is defined." As in the case of two variables, we may proceed in a similar manner with several variables, among which an analytic depend ence exists. Let this connection be of such a nature that m (<n) of the variables are in general determined through the remaining n m. If, then, (a lt a 2 , - - - , a n ) represents a definite system of values of the variables, there exist m equations of the form P(x l - a lt x 2 - a a , ., x n - a n ) = which are to be satisfied for x 1 = a 1} x 2 = a z , > - } x n = a n . In the neighborhood of the position (a^, a 2 , , a n ) there are, then, an infinite number of other systems of values (x^ x 2 , > , x n ) which satisfy the same m equations. These define an analytic structure* in the realm of the quantities x lf x 2 , - - , x n . A fundamental theorem in the theory of functions of the complex variable is that these structures may be continued over their boundaries. The power-series above constitutes an element of a complete structure (97). * Weierstrass, Werke, Vol. II, p. 236. It may be remarked that Minkowski in his Geometric der Zahlen advances similar ideas at considerable length. See in particu lar 19 of his work just mentioned. CERTAIN FUNDAMENTAL CONCEPTIONS 175 138. Analytical structures, as above defined, may be represented in a different manner. If the equation connecting x and y begins with terms of the first dimension, we may, on the one hand, either express y y through P(x-x Q ) or x-x through P(y-y ); or, on the other hand, if the coefficient of either x X Q or y ?/ is equal to zero, it is possible to express only y y^ or only x X Q as integral power-series of x X Q or y y Q . In order that this distinction may not be necessary, we introduce a function t which begins with terms of the first dimension in x XQ, y y Q ( see 136); we may then always express the two quantities x, y as power-series of t. Through the introduction of such a quantity t it is made possible to include within certain limits all the systems of values (x x Qt y y Q ) which satisfy this equation. These values must firstly satisfy the given equa tion, and secondly they must afford all the systems of values which satisfy it within these limits. These considerations may be extended at once to equations in several variables. If we have a certain number of equations in x l a lf -x% a 2 , ., x n a n , and if we limit these equations to terms of the first dimension, we have linear homogeneous equations of the first dimension, the number of which we assume to be m (< n). If we can express m of the quantities x l a lt x 2 a 2 , ., x n a n through the remaining n m, it is always possible so to derive n power-series of n m quantities t lt t 2) - , t n _ m that they, substituted for x l} -x^, ., x w firstly satisfy the given equa tions, and secondly, if we give to t lf t 2) , t n _ m all possible values, they offer all the systems of values (x ly x 2 , ., x n ) which satisfy those equations, when certain limits are fixed for the abso lute values of % a lt x 2 a% , x n a n ; or, also secondly, that with indefinitely small values of the t s they afford all the systems of values of the quantities x l} x 2) , x n which lie indefinitely near the position (a l} a 2 , , a n )(see again 136). Take n power-series ^(t), ^(O* > ^n(0 an d write x l = ^ 1 (t), x z = 3*2 W> "i x n = n(t) > tnen through these equations a struc ture of the first kind (Stufe) in the realm of the n quantities x 176 THEORY OF MAXIMA AND MINIMA is defined ; in a similar manner a structure of the second kind is defined through the equations In general, if we take % power-series in t v t 2 , , t n _ m and write these equal to x lf x 2> , a; w , the collectivity of the sys tems of values (x v x z , , a? w ) offered through these equations constitute a structure of the (n m) th kind in the realm of the quantities x v x 2 , , x n . We shall in the sequel limit the discussion of the general analytic dependence to the cases where this dependence is expressed through algebraic equations and to the structures which result from such equations, viz., the algebraic structures. II. ALGEBRAIC STRUCTURES IN TWO VARIABLES 139. Let F(x t y) be an integral algebraic function of x and y which does not contain repeated factors, so that F(x, y) has no common factor with either - or --- Further suppose that dx dy F(x, y) is not divisible by any integral function in which appears only one of the variables x or y. The system of values x, y which satisfy the equation F(x t y}= form the algebraic structure that is defined through this equation. If a; , y is a pair of values such that F(x Qt y ) = 0, we may develop the equation F(x Qt y Q ) in powers of x X Q and y y^ in the form (cf. Stolz, loc. cit., p. 177) [1] G(f, 1,) = dF(x Q , 2/ ) + d*F(x Q9 y,} where for brevity we put x X Q = % , y y^ rj, and where d n F(x Q , is the homogeneous function of the nth degree in f , 77, viz., r=n /Aj\ QW TJT [2] d*F(x Q , y,} = V ( v } r ~ V- CERTAIN FUNDAMENTAL CONCEPTIONS 177 cF cF If - and 7r do not both vanish, the position (or point) X Q , y is ex ex said to be regular or simple. But if they both vanish for x = X Q , y = y Q , and if for the same position all the partial derivatives of the 2d, 3d, -, (k l)st order of F(x, y) with respect to x and y vanish, while those of the &th order are not all zero, the position X Q , y is called a singular position, and, specifically, a &-ple singularity. In such a case the left-hand side of equation [1] begins with terms of the kih order with respect to and 77. In the following treatment not only the integer k plays an im portant role but also the smallest exponent of the terms that are free from 77, as also the smallest exponent of the terms that are free from , on the left-hand side of [1]. If we denote the first by p and the second by q, the equation [1] may be written in the form [3] F^+bys+ri) = e { + f /(?)} + 1?* {b + r,g (T?)} + frHfc T,) = 0. Here /(f) denotes an integral function of and g(rj) an integral function of T) ; a and I are constants different from zero, viz., 1 c p F I c q F ~~~~ ~~ 140. Developments of the algebraic function y in the neighbor hood of a regular position. It may be shown * that if on the position 7* 77* / x = X Q , y = y$ the expression does not vanish I so that, say, q =1 cF\ dy \ and b = - ) , there is one and only one convergent series in integral i/o / positive powers of which vanishes with and which substituted for rj in [1] identically satisfies [1], We may suppose that this series begins with f ^, so that [4] , =B _ff* + <i(+1 f+i + .... We have also to consider in the sequel fractional positive i powers of x x Q = j; , that is, powers, say, f **, where /*=!. A series * See Pierpont, Vol. I, p. 288; or Goursat, Coitrs D Analyse, Vol. I, chap. iii. 178 THEORY OF MAXIMA AND MINIMA of this kind is convergent if there is a positive quantity R such that the series for all values of \^ <R is convergent, and that is for all values of \%\<R*. If the series is convergent for one of the /-t values of the /xth root of , it is evidently convergent for all the other /-i 1 values of f . Accordingly, to each of the values of f whose absolute value is smaller than R* there correspond p different values of the series. If, for example, we denote a definite one of the values of f, for i example, the principal one, by f **, the others are expressed through the product yf **, where j is any of the ftth roots of unity. Hence a series may, corresponding to the different values of j y appear in the p I other forms = The theorem stated at the beginning of this article may be O J7I generalized : If on the position x = a? , y = y Q the expression 9 does not vanish, there is one and only one convergent power-series in positive integral or fractional powers of f which vanishes with f and which written for 77 in the equation [1] identically satisfies it, viz., the series [4]. For if besides the series [4] a series [5] with /-&>! satisfied [1], then the equation i [6] F(x Q + p, 2/o + >?) = 0, where t = fr, would be satisfied by two series which have no constant term and in which 77 is expressed in integral powers of t. This, by the previ ous theorem, is impossible, because in [6] the term in TJ really O TJT appears, and in fact multiplied by the coefficient - dF dF y 141. Suppose next that = 0, but that = 0, so that p = 1 fyo 3x o and q>l. Then from what we have just seen it follows that the CERTAIN FUNDAMENTAL CONCEPTIONS 179 equation [3] may be solved through only convergent series in which f is expressed in powers of 77 hi the form [7] = - -^ + dtf + < + - - = Q(n), say, where q>l and d f = ; in other words, there exists a positive quantity S such that if ! rj \ < S, we have the identity [8] Write f in the form and note that [Q-,(rj)]^ =( 1 ! rj s H- } q = 1 ^ 77* -|- terms of higher order. V b If then we put [9] * = -fL^lV /J by reverting this series we have [10] t .,.+^i+.... and from this it is seen that a positive quantity K may be so determined that for all values of t such that 1 1 \ < K the above power-series in t converges. This power-series when written for rj in the equation [9] identically satisfies it. If, further, we raise the equation [9] to the qth power and multiply it by , we have t> b b b t q = r n q Q-i( n}= ^ -f d g if] q +*+...= Q(rf) above ; a a a and this equation will be an identical one if for ?? we write the power-series P(t). The same is true of equation [8] ; that is, we have the identity , -, for all values t for which the series P(t) is convergent. 180 THEORY OF MAXIMA AND MINIMA If we denote the radius of convergence of the series P(t) by R and put = ^, where t is any one of the ^-values of the qth a root of - f , we have the following theorem : V a ? , so that the series tii] >^-?~ + ijj(jiFjS* 1 + exist, j denoting any of the qth roots of unity, then this expression written for rj causes the function G(, TJ) to vanish identically. Furthermore, there is only one such convergent series in inte gral or fractional positive powers of f, without constant term, which when substituted for rj in equation [1] causes that equation to vanish identically. For if there were another such series in integral positive powers i of **, say, then in the manner given above we could express ^, and conse- i quently also f , through a power series in ^ which identically satisfied [1] ; but besides the series [7] there exists no such series, and conse quently there is no such series as [12] which is different from [11]. III. METHOD OF FINDING ALL SERIES FOR y WHICH BELONG TO A / C -PLY SINGULAR POSITION* 142. In equation [1] let dF(x Q) y ), d z F(x Q , y ), be zero, so that this equation becomes [13] G(f,,) = I where N is the dimension of F(x, y) with respect to x and y. * Besides Stolz, p. 182, see also Puiseux, Journ. de Math., 1st Series, Vol. XV, p. 365; Picard, Traitt etc., Vol. I, p. 392; Hermite s preface to Appell et Goursat, Fonctions Alg&briques etc. ; Konigsberger, Elliptische Functionen, Vol. I, p. 187 et seq. ; etc. CERTAIN FUNDAMENTAL CONCEPTIONS 181 There is, consequently, a &-ple singularity at x , y^ and we shall next show that we may derive all those convergent series without constant term which proceed in integral or fractional powers of f and which when substituted for 77 in [13] identically satisfy it, if we can derive corresponding series for any simple, double, up to (k l)-ple position of any algebraic structure. In other words, the problem of deriving these series for a k-ple singularity is made to depend upon the derivation of such series for a position that is less than &-ple. If for T) in the homogeneous function* of the nth dimension **(^jrj *.(& *) (it being supposed not identically zero) we write the series [12] and arrange in ascending powers of f , then if X = /i, this expres sion begins at least with f M , and exactly with this term if <>(!, C A ) does not vanish. If X ^ /*, this expression begins with f n only when this term in reality appears in <(, 77) ; otherwise with a term of higher or lower order than f w according as X > IJL or X < fM. If in [13] we next decompose the lowest differential into its real or complex linear factors, we have [14] <^.(^)=IIK^ r-l where l\ + & 2 + 4- &/ = k and where one of the two coefficients a r , /3 r may be zero. Assuming first that X = /u, if in the above expression we write we see at once that for at least one value of r we must have * The method given by Weierstrass, Werke, Vol. IV, pp. 19 et seq., is essentially the same as that found here : see also Stolz, loc. cit. 182 THEORY OF MAXIMA AND MINIMA For if this were not the case, then (f, C A + ) would begin with f * instead of vanishing identically. If, next, X>^) one of the quantities j3 v /3 2 , , ^ l must be zero; and if X</i, then one of the quantities a 1? a 2 , , a L must A vanish. For if they were all different from zero, then 6r(f , c A f **+ ) k\ begins with f ^ . If a series of the form [12], where X = JJL, satisfies the equation [13], we shall have, if in [12] we write ?; = (C A -f TJ^ , a relation I 2 between ^ and f, viz., ??!= c A+1 f A + c A + 2 f A H ---- . The expression G(%, (CA + ^I)!) contains the factor p, which may be neglected, so that -^ satisfies the equation If in the series [12] (when \> /JL) we write r? = 77^, we find that the equation -, is satisfied by the series If a series for ?; where X</i satisfies [13], we revert the process and make the substitution f = rj^ r 143. In giving the practical method of determining the series for r; which satisfies [13] we must make a distinction between two cases : The function <& k (|, rj) either may contain different linear factors to their respective powers or it is the &th power of one single such factor. First case. Among the quantities a lt a z , , cci there must be at least one which is not zero. For each a L which does not vanish /? we put = c and make in [13] the substitution *i [15] *; We may then write CERTAIN FUNDAMENTAL CONCEPTIONS 183 where G l is an integral function in and ^ which vanishes for = and ?/ = 0. If, for example, i = 1, we have From this it is evident that the position f = 0, 1^= in the structure (f, i^) is at most a & r ple singularity and conse quently less than a &-ple, so that the problem may be regarded as solved, since, by hypothesis, when A^< k we have supposed that we may derive all power-series which satisfy (f, 77^ = 0. For 77, through the formula 17 = (e^ + i^) f , we have series arranged in integral or fractional positive powers of f which substituted in G(%,r}) cause this expression to vanish identically. Besides these series there are no other such series for 77 which begin with the term c^f. If in [15] we let r take all the values where a r = 0, we have in this way all those series for 77, where X ^ /*, which satisfy the equation G(f, T?)= 0. Among the quantities a 1? a 2 ,..-,ai there may be one, for example a v which is zero. If we consider rj and f interchanged and then make in [13] the substitution f = T?^, we may derive all series which proceed according to integral or frac tional positive powers of 77 with constant term zero and which when written for f in the equation ({?, 77)= identically satisfy it, and whose initial term is ^77 x , where By reverting each of these series we may express 77 as series in terms of f which satisfy [13], where X<^. Further, we have all such series. For if [13] was solved by writing for 77 a series [12], then we also satisfy [13] by writing for f a series in integral positive powers of 77* whose initial term contains 77^, where is an improper fraction. X 184 THEORY OF MAXIMA AND MINIMA Second case. Let <&( , rj) = (ccrj /3f )* and suppose first that a = 0. We make in [13] the substitution and have, after division by f*, the new equation [17] ^%) = ^ If for this equation the position f = 0, ^ = is less than a &-ple singularity the problem is by hypothesis solved, or if it remains a &-ple singularity and if the polynomial of the terms of the &th order in f and TJ I may be decomposed into different linear factors, we may proceed as in the first case. It may happen, how ever, that the position f = 0, r} 1 = is a &-ple singularity whose terms again form the Jcih power of a linear expression in f and ??! which must necessarily be j-( ar )\ ~ if) fc - K \ If, further, we write in [17] ?? 2 instead of rj v where rj 2 is defined by the equation the expression will be divisible by f*, so that we may write (f where G,(fc % )= + f J5T,(f, ,,), ^2(?> ^2) being an integral function of f and ?; 2 . Noting (-i) and (ii) it is seen that if there is for 77 a series of the form [18] , = J| + 4p + c / + t + .. ?) then for f = the quantity ?? 2 introduced above must be zero, and T7 2 must belong to those series that vanish with and which are obtained from the equation 6r 2 (, rj 2 ) = 0. CEKTAIX FUNDAMENTAL CONCEPTIONS 185 This equation may be solved as above for rj 2 if the position | = 0, 7; 2 = for the structure (r 2 (f, 7/ 2 )= is less than a &-ple singularity or if it is a &-ple singularity in which the terms of the kth order do not constitute the &th power of a linear func tion of f and 7? 2 . We further have all series, proceeding according to powers of f without constant term, which when substituted in [13] satisfy it, if we solve the equation with respect to ?7 2 in all possible ways through power-series in f without constant term and substitute these series for 7? 2 in the expression (cf. (i) and (ii)) But if the position f = 0, ?? 2 = is a &-ple singularity in the structure 2 (f, ?? 2 )=0, and if the terms of the kih order form the kth power of a linear expression in f, ?; 2 , which must have the form (arj 2 /3. 2 f ) A , we must write rj 3 instead of rj 2 , where ?? 3 tv I is denned by //3 2 \ ..... 7 ?2 = (-^-h7 73 jf, (tit) and proceed in a similar manner as above. Continuing in this manner it is evident that if a = we may derive all power-series in f without constant term which written for 77 in the equation [13] identically satisfy it, if through a series of transformations we may from the given equation G (f , 77) = derive an equation Gh(> 7 7/<)=^ whose left-hand side does not begin with the kih power of a linear expression in f and rj h . We must finally come to such an equation if F(x, y) and - cy have no divisor in common. For, since the factor f * appears with each of the substitutions [19], it is easily shown that the integer 186 THEOKY OF MAXIMA AND MINIMA li in [19] cannot pass a fixed limit. For if F is of the n\h degree in y, we may always find two integral functions U and V in x and y where U is at most of the (n l)st degree in y and V at most of the (n 2)d degree in y such that there exists the identical relation C Tfl [20] rJ-(a,y) + 0J^=.D(*), where D(x) is an integral function in x. Furthermore, since it is seen that We also note from the formula J^(aj, y + v} = F( if we make the substitution x = x + f , ?/ = y 4- 77, since that Expanding the left-hand side of. this expression, it is seen that It follows that after the substitution of X = X Q + , y = y + 7j, where from [19] [21] r, = + CERTAIN FUNDAMENTAL CONCEPTIONS 187 the left-hand side of [20] is seen to be divisible by f^ -D. But on the right-hand side D(x Q -f f) is of the same degree d, say, in f as D(x) is in x. It follows that h (k 1) ^ d or h ^ - -- K JL If, secondly, a = 0, or <1> A . (f , 77) = ( /3f )*, it is seen that through a corresponding change of the method given above, all series which proceed according to powers of 77 without constant term may be found which when written for f in the equation [13] identically satisfy this equation. Through reversion of these series we derive series in powers of f without constant term which satisfy the equation [13] with respect to 77, and in fact all such series. 144. The following theorem is proved by Stolz (Math. Ann., Vol. VIII, p. 438) : If X Q , y^ is a position of the structure F(x, y)=Q and if this equation is brought through the substitution x = X Q -f f , y = y -f- 77 to the form [3] above, viz., [3] F(.c Q + f, #) + *?) = ( p ( a then the collectivity of the convergent series hi integral positive i A powers of or f M , viz., c x f ** -h - , which vanish with , and when written for 77 in the equation [13] satisfy it, are charac terized through ^ _-^ 2> = ?> 2, x= *- In these expressions /u is the smallest of the roots of f which are contained to an integral power in each term of a series in i question, and X is the least exponent of f* 4 in this series. This is illustrated in the example of the next section. 145. The above theorem offers a check for the determination of all the series which belong to a singular position of a function, as is illustrated in the following example. Example. For the algebraic structure denned through the equation 4 x z y 3 - 9 x*y~ + 2 X G I/ - 21 xf + 8 y 1 - 10 a: 10 = (i) the point x = 0, y = is a 5-ple singularity. The terms of the fifth order in (/) are 4 x z y z and consequently may be decomposed into the factors x and y. 188 THEORY OF MAXIMA AND MINIMA Corresponding to the factor y, write in (i) y xy r The result of the substitution is, after division by x 5 , 4 y} - 9 xy* + 2 x\ - 10 a* - 21 x*y* + 8 x*y} = 0. () The point x = 0, ^ = is a triple singularity for this structure, the terms of the third order being 4 y - 9 xyf + 2 afy = ^(4 ^ - x) (^ - 2 x). (m) Corresponding to the first factor, write in (ii) y l = xy z and divide the resulting equation by x s . We then have 2 y 2 - 10 x 2 - 9 #2 + 4 y - 21 afy + 8 xyj = 0, where x = 0, y 2 = is a simple point. From this equation we have We thus have as a solution of (i) y = xy l = x 2 y% = 5 x 4 + terms of a higher order. (fy) Corresponding to the second factor in (Hi) write in (ii) y l = x(% + y 2 ) and divide the result by x 3 . We then have - |.v a - 10 x 2 - 6 yl + 4 3/| + . . = 0, and from this we have ?/ 2 = 4^- x 2 + - . It follows that (i) is satisfied by the series Corresponding to the third factor of (m), write in (zi) ?/ x = z(2 + y 2 ), and dividing the result by x s we have From this it follows that y z = f x 2 - + - ; and the corresponding value f y iS y = 2^ 2 +f^+.... ( y /) Returning to (i) write a; = yx l so that ( ) becomes 4 ar + 8 y - 21 ^ - 9 yar* + 2 y8a? - 10 a:"^ = 0. (mi) For this structure y = 0, x l = is a double point, the terms of the second order being , / . , /-. 4 x* 4- 8 y 2 = 4 (x^ + *^V2)(a?j iyv2). Corresponding to the factors of this expression we make in (vii) the substitutions (viii) CERTAIN FUNDAMENTAL CONCEPTIONS 189 If then we divide through by f, we have the equations (8* 2 - 21^) *V2 - 2lyxi + 4 xl + = 0. From each of these equations we derive series which begin with the same term, viz., x = Qy + , so that we derive the two series i V2 + z) - By inverting these series we have the series which proceed in ascending powers of x$, viz., - : y=^ T -H = +.... (it) and (a;) We have thus derived five power-series which proceed in integral or frac tional powers of x without constant term which satisfy (i), viz., (tV), (r), (i-O, (), and (x). It is further seen that 2/ot = l + l + l + 2-f2 = 7, which is the smallest exponent of the terms that are free from x, while 2A = 4 + 2 + 2 +1 + 1 = 10, which is the smallest exponent of the terms that are free from y in (i) (Stolz, p. 195). INDEX (The figures refer to the pages) Abelian transcendents, 169 Algebra, fundamental theorem of, 165 Algebraic curve expressed through power series, 36, 53 Algebraic function, its development in series, 177; at a singular point. 180 et seq. Algebraic structures, 176 et seq., 181, 187 Ambiguous case, the, iv, 27. See Semi- definite form Analytic dependence, 166 et seq., 169 Analytic function, 73 ; defined, 138 Analytic structure, 74, 174 Appell, 180 Area, maximum area, 143 Asymptotic approach, 138 Auxiliary variable, 101, 173 Baltzer, 2 Bauer, 107 Bertrand, v, 33 Biermann, 136, 155, 168 Bocher, 148 Bohlmann and Schepp, iv Bois-Reymond, Paul du, 2, 7, 74 Bolzano, 12, 136 Borchardt, 107 Boundary, 136 et seq., 156 et seq. Brand, 126 et seq. Burnside, 89, 106 Calculus of variations, iv, 171 Cantor, Geschichte etc., 15 Cartesian Oval, 133 Cauchy, 3, 7, 92 Cavalieri, 27 Center of curvature, 117, 125 Christoffel, 107 Complete differential quotient, 6 Contact of indefinitely high order. 36 Continuation of an analytic function. 174 Continuous function, 161, 162 Convergence, 166, 167, 168, 172, 178 Cremona, 144 Curvature, 117 Cusps, appearance of, 30 Cylinder, trace of. 31 Dantscher, Victor von. v, 36, 69 ; method of, 39, 62 et seq., 72 Definite form, 19; necessary condition. 21, 49, 50, 51, 64, 68, 82, 83, 91, 92, 109, 111, 114; conditions for, 91. 103 Derivative, existence of a, 161, 162. 163, 166 Descartes, iv, 165 Determinant, the sisn of the, 25 et seq.. 28, 29, 30, 32, 38, 51, 52, 59, 60, 83. 85 et seq., 90. 91. 92, 93, 97, 100. 107, 111; orthogonal, 149 Differentiation, one-sided, iv, 7. 11 et seq. Dini, 12, 136, 167 Distinctness as characteristic of an extreme, 37, 38, 47, 50 Double curve, 54 Double point, 101 ; with distinct tan gents, 29 ; isolated, 29, 30, 31 Element of a complete structure, 174 Equation of secular variations, 107 Euclid, iii, 15, 135 Euler, 16, 18, 107; theorem of, for homogeneous functions, 84, 155 Exceptional cases involving a squared factor, 54, 58, 68, 97 Existence of an extreme, proof of, 135. 146 Extraordinary cases of extremes, iv. 19 Extraordinary maxima or minima, 1, 6 et seq., 17, 19, 43 et seq., 74 Extreme, or extreme value, v, 2, 53. 54; criteria for, 4, 6, 26, 92. See Maxima and minima Extreme curves, 53. 54 Failure of general criterion, 55 Fallacious conclusions. See Incorrect ness of earlier theories Fermat, iii, iv. 15, 132; method of determining maximum and mini mum, iii Form. See Definite form Fourier, iii Fourier series, 74 191 192 THEORY OF MAXIMA AND MINIMA Fractional powers, 178, 182 Fuchsian functions, 171 Function, rational, 166; one-valued, 166; many-valued, 169 Function-element, 138, 174 Fundamental theorem of algebra, 49 Gauss, 19, 86, 99 ; principle of , 151 et seq. Genocchi-Peano, 1 Geometrical interpretations, 6, 24, 31, 46, 69, 71, 97, 125 Geometrical mechanics, 139 Geometry of numbers, 174 Gergonne, 19 Goursat, 6, 28, 27, 29, 31, 126, 170, 177, 180 Greatest value, 1, 48, 94. See Upper and lower limits Hachette, 107 Hadamard, 147, 148, 160 Hancock, 123, 166 Hankel, 74 Harkness, 12 Hermite, 89, 106, 180 Hilbert, 170 Homogeneous functions, 49, 155 Homogeneous quadratic forms, 82, 85, 103 et seq. ; expressed as a sum of squares, 86, 89, 91 ; with subsidiary conditions, 114 Hudde, 165 Huygens, 16 Hypergeometric series, 170 Improper maxima and minima. See Maxima and minima Incorrectness of earlier theories, 33 et seq., 52 Indefinite form, 19, 49, 50, 51, 64, 68, 82, 106, 116 Indeterminate coefficients, 172 Inflection, point of, 6 Integral rational function, 168 Isolated point, 29, 31 Jacobi, 107 Jordan, 75 Konigsberger, 180 Kronecker, 106 Rummer, 106, 107 Lagrange, iii, v, 4, 18, 22, 26, 33, 43, 77, 86, 92, 99, 107, 114, 127, 131, 148, 172 Laplace, iii, 107 Least squares, 26 Least value, 1, 50, 94. See Upper and lower limits Left-hand differential quotient, 7, 11 Legendre, 135 Leibnitz, 3, 15 Limitation expressed through an equa tion, 150 Lipschitz, 2, 74 Lower limit, 63, 94, 104, 136. See Upper limit Liiroth. See Dim Maclaurin, iii, 3, 4, 15, 22, 77 Maxima and minima (see also Extreme value), one of the most admirable applications of fluxions, iv ; condi tions for, iv, 4, 40, 99 ; inaccuracies in, v ; maximum defined, 1 ; mini mum, 1; ordinary (see under Ordi nary etc.); extraordinary (see under Extraordinary etc.); proper, 2, 5, 11, 17, 23, 26, 44, 45, 60, 61, 63, 74, 75; improper, 2, 5, 17, 23, 26, 31, 50, 59, 60, 63, 75, 140 et seq., 164; abso lute, 2 ; relative, 2, 21, 96 et seq. ; criteria for, 4, 7-12, 40-42, 43 et seq., 48, 51, 55, 64, 67, 68, 77, 80, 81, 82, 92, 100, 102, 115, 116; geometrical interpretation of , 6,46, 71 ; erroneous criteria, 33 ; condition for proper extremes, 40, 42; condition for im proper extremes, 41, 42, 140 et seq. ; criteria for relative maxima and minima, 115 Mayer, iv, v, 2, 79 Mechanics, problems in, 139, 150 ; derivation of the ordinary equa tions of, 152 Minimal surfaces, 123 Minkowski, 174 Morley. See Harkness Neighborhood of, in the, 65, 173 Newton, discoverer of the calculus, iii One-sided differential quotient, 7, 11 et seq. Orbits of planets, 107 Order of a curve, 54 Ordinary maxima and minima, 1 etseq., 17 et seq. See Maxima and minima Osculating circle, 117 Osgood, 167, 170 Panton. See Burnside Pappus, 15, 164 Pascal, Exercici etc., 11 ; Bepertorium etc., 130 IXDEX 193 Peano, iv, v, 2, 6, 12, 18, 21, 31, 33, 34, 52, 61, 68, 94 Pendulum, 150 Petzval, 107 Picard, 170, 180 Pierpont, 3, 7, 15, 34, 37, 177 Poincare, 170. 180 Poison, 107 Polygon. See Regular polygon Position, 135 et seq. Power-series, 171 et seq., 175. 179 Proper maxima or minima. See Max ima and minima Puiseux, 180 Quadratic form, 19 ; expressed as a sum of squares, 86 et seq., 89 ; ap plication of, 92 et seq. See Homo geneous quadratic forms Radius of curvature, 117 Realm, 135, 174 Reflection of a ray of light, 126 et seq. Refraction of a ray of light, 131 et seq. Regiomontanus, 16 Region of convergence, 167, 168 Regular function, 73 Regular point, 177 Regular polygon, 140, 142 et seq., 147 Relative maxima and minima. See Maxima and minima Reversion of series, 153 et seq. Richelot, 106 Right-hand differential quotient, 7, 11 Roots of unity, 180 Salmon, 107, 120. 122 Scheeffer, v, 19, 27, 35. 36, 39, 46. 48, 50, 62, 70 Scheeffer s method, 37 Scheeffer s theorem, 43, 46, 55, 59, 60, 61, 62, 70, 72 Scheeffer s theory. 43 et seq. Schepp. See Bohlmann ; see also Dim Secular variations, equation of, 107 Semi-axes of a central section, 165 Semi-definite case, iv Semi-definite form, 19, 49. 50, 51, 52, 64, 65, 68, 70 et seq., 82, 83, 92, 93, 106, 116 Serret, v, 33, 104, 106, 136 Severus, 16 Shortest distance to a given surface, 101, 123 Simple point, 177 Simpson, 16 Singular point, 164, 177, 180 et seq., 183, 184 Sluse, Ren< F. W. de, 16 Smallest value, 1 Smith, Edward, 107 Spherical triangle, 135 Squared factor. See Exceptional cases Stolz, v, 2, 6, 11, 14. 43, 45, 46. 50, 60. 70, 79, 91. 100, 136, 155, 164, 180, 181, 187, 189 Stolzian theorems, 39 et seq., 55, 58, 70, 72 Stolz s added theorem, 45, 60 Structure of the first kind etc., 175 Sturm s theorem, 51, 106 Surfaces of second degree, 107 Sylvester, 89, 107 System of m equations, solution of, 171 et seq. Tangent, parallel to z-axis, 6 ; com mon to two curves, 34, 35, 36 Tangential plane, 28, 31 Tartaglia, 16 Taylor s development in series, v, 4, 5, 9, 10, 19, 24, 33, 75, 79, 80, 97, 152, 159 Taylor-Lagrange theorem, 43, 47, 77 Todhunter, 33 Transcendental curves, 36 Transcendental functions, 167, 169 Uniform. See Convergence Upper and lower limits, 2, 12 et seq., 55, 57, 94, 104, 136, 137 Variations, calculus of, iv, 171 Von Dantscher. See Dantscher Voss, 2 \Veierstrass, iv. 73, 79, 86, 107, 138, 167, 168, 169. 174, 181 Wilson, E. B., 12 Wirtinger, 148 Zajaczkowski, 106 Zenodorus, 142, 147 RETURN Astronomy Mathemotics/Statistks/Computer Science LibrJL} 3 5 O TO ^ 1 00 Evans Hall 642-3381 LOAN PERIOD 1 7 DAYS 2 3 4 5 6 ALL BOOKS MAY BE RECALLED AFTER 7 DAYS DUE AS STAMPED BELOW NOV I MAR a 4 U UV254 20 Rec d UCB A/A NOV 1 2003 UNIVERSITY OF CALIFORNIA, BERKELEY FORM NO. DD3 BERKELEY, CA 94720 $ OA3OG U.C. BERKELEY LIBRARIES ... JTAT. "