LIBRARY University of California. RECEIVED BY EXCHANGE Class 1 UNIVERSITY OF CINCINNATI. Bulletin No. 13. Sekiks II. Publications of the University of Cincinnati. Edited by HOWARD AYERS. Vol.. II. LECTURES ON THE THEORY OF MAXIMA AND MINIMA OF FUNCTIONS OF SEVERAL VARIABLES. ( Weierstrass' Theory.) By HARRIS HANCOCK, Ph.D. (Berlin), Dr. Sc. (Paris), Professor of Mathematics. Tlie University Bulletins are Issued Monthly. Entered at the Post Office at Cmcinnati, Ohio, as second-class matter. (iNAn 1 PR£S& o •■ , '-'ft/*., p. University of Cincinnati. Bulletin No, J 3. Publications of the University of Gncinnati. Series II. Vol II * Edited by HOWARD AVERS. I i LECTURES ON THE THEORY OF 'AXIMA AND MINIMA OF FUNCTIONS OF SEVERAL VARIABLES. (Weierstrass' Theory.) By HARRIS HANCOCK, Ph.D. (Berlin), Dr. Sc. (Paris), . Professor of Mathematics. The University Bulletins are Issued Monthly. Entered at the Port Office »i Cincinnati, Ohio, as sccond-cUss matter. \0 f# PREFACE. In his lectures at Berlin the late Professor Weierstrass often indicated the necessity of establishing fundamental parts of the Calculus upon a more exact foundation. It has already been pointed out {Annals of Mathematics, Vols. IX., X., XI. and XII.) how the old rules and theories of the Cal- culus of Va?'iatio7ts soon led to perplexities which appeared almost insurmountable. Dirichlet's Principle is found to have been established upon a weak structure, and we very soon find innumerable fallacies and difficulties when we seek to discuss in this manner Minimal Surfaces and the allied theory. These difficulties may be overcome by subjecting the problems in question to a more rigorous treatment and by giving more emphasis to their analytic formulation. In every Differential Calculus which I have seen \cf. also Pierpont, Bull, of Amer. Math. Soc.,July, i8()8\ the Theory of Maxima and Minima is both inexact and inadequate, when several variables are treated. This subject, when made more rigorous, should evidently receive increased attention. Indeed, at the pre- sent state of mathematical science it seems that students should devote more attention to its study, for it has a high inter- est as a topic of pure analysis, and finds immediate applica- tion to almost every branch of mathematics. Further, the Theory of Maxima and Minima should receive more attention for its own sake — for example, in the solution of such problems as the deter- mination of the polygon which, with a given periphery and a given number of sides contains the greatest area, the deriva- tion of the shortest line from a point to a surface, etc. In the > Calculus of Variations its use is really essential, while in Mechanics it may be shown that all problems which arise, may (3) 175900 4 PREFACE be reduced to problems of Maxima and Minima; from it we may derive a proof of the existence of the roots of algebraic equa- tions as also a method for the reversion of series. I do not assume credit for the origin of any of the theories that are here set forth. In the presentation of the subject-matter I have followed Weierstrass' lectures delivered in the University at Berlin, my lectures being for the most part a reproduction of his lectures. In these lectures Weierstrass subjected to a more rigorous investigation the work which is in a great measure due to older writers, whom I have indicated in the context. Before entering upon the Theory of Maxima and Minima it seems advisable to give a short account of Weierstrass' Theory of Analytic Functions and to give more exact definitions of those functions to which the ordinary rules of differentiation are applic- able. This investigation is carried out only so far as the present treatise seems to require. Certain theorems are also introduced upon which the later discussion depends. The Theory of Maxima and Minima may be then presented in a clearer and more con- nected form. My thanks are due to Messrs. Harry H. Steinnietz and Harold P. Murray of the University Press for their care in the printing of this work. Harris Hancock. McMicKEN Hall, University of Cincinnati. Jan., 1903. CONTENTS. CHAPTER I. CERTAIN FUNDAMENTAL CONCEPTIONS IN THE THEORY OF ANALYTIC FUNCTIONS. ART. 1 Rational functions of one or more variables. Functions defined through arithmetical operations. One-value functions. Infinite series and in- finite products. Convergence. Art. 2. Uniform Convergence. 3 Region of Convergence. Differentiation 4 Many-valued functions. Functions of several variables. Functions which behave like an integral rational function 5 Arithmetical dependence. Art. 6. Many-valued functions. Art. 7. Possibility of expressing many-valued functions through one-valued functions. ............. 8 Analytic functions expressed through power-series. Analytic structures. 9 Analytic structures defined in another manner. Structures of the first kind, second kind, etc. .......... 10 Analytic continuation. Power-series closed in themselves. 11 Definition of analytic functions. Function-element. .... 12 Existence of the general analytic function 13 Extension of the above definitions to systems of functions of one or more variables. One-valued functions defined Definition of many-valued functions. Values common to n functions. 14 An important theorem for the Calculus of Variations. .... 15 The same theorem proved in a more symmetric manner 16 Application of this theorem and the definition of a structure of the (» — wi)th kind in the realm of n quantities. . \ 17 Property of power-series. Transformation of the expression of a structure. Continuation of a function. Coincidence of two structures. 18 A complete structure defined. Monogenic structures. .... 19 Boundary positions. Singular systems 20 More exact conception of one-valued and many-valued functions. 21 Points at infinity PAGE. 9 10 11 12 13 14 IS 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 CHAPTER II. THEORY OF MAXIMA AND MINIMA OF FUNCTIONS OF SEVERAL VARI- ABLES THAT ARE SUBJECTED TO NO SUBSIDIARY CONDITIONS. 1 Introduction. Art. 2. Nature of the functions under consideration. Defi- nition of regular functions, 3 Definition of maxifna and minima of functions of one and of several (5) 31 6 CONTENTS ART. PACK, variables 32 4 The problem of this chapter proposed. Taylor's theorem for functions of one variable 33,34 5 Taylor's theorem for functions of several variables 35,36 6 The usual form of the same theorem 37,38 7 A condition of maxima and minima of such functions. .... 39 THEORY OK THE HOMOGENEOUS QUADRATIC FORMS. 8 Indefinite and definite quadratic forms. 40 No maximum or minimum value of the function can enter, when the cor- responding- quadratic form is indefinite. When is a quadratic form a definite form which only vanishes when all the variables vanish ? . 41 9 Some properties of quadratic forms. The condition that the quadratic form <^ \X\, X2, Xxi)^^^ A\fiX\Xij, be expressed as a function of n — 1 variables. 42-44 11,12 Kvery homogeneous function of the second degree ^ (a*i, X2, A'n ) may be expressed as an aggregate of squares of linear functions of the variables 45-47 10-17 The question of art. 8 answered 44-50 APPLICATION OF THE THEORY OF QUADRATIC FORMS TO THE PROBLEM OP MAXIMA AND MINIMA STATED IN ARTS. 1-6. 18. Discussion of the restriction that the definite quadratic form must only vanish when all the variables vanish 51 The problem of this chapter completely solved 52 CHAPTER III. THEORY OF MAXIMA AND MINIMA OF FUNCTIONS OF SEVERAI, VARI- ABLES THAT ARE SUBJECTED TO SUBSIDIARY CONDITIONS. 1 The problem stated. Art. 2. The natural way to solve it. . . . 54 3-6 Derivation of a fundamental condition by the application of the theorem of arts. 14 and 15 of Chapter I. 55-57 7 Another method of finding the same condition 57 8 Discussion of the restrictions that have been made 58 9 A geometrical illustration of these restrictions 59 10 Establishment of certain criteria. 60 11 Simplifications that may be made. 60 12 More symmetric conditions required 61 THEORY OF HOMOGENEOUS QUADRATIC FORMS. 13 Addition of a subsidiary condition. 61 14, 15 Derivation of the fundamental determinant Ae and the discussion of the roots of the equation Ae^o, known as the "Equation of Secular Variations." 61-65 16 The roots of this equation are all real. ....... 65 17-19 Weierstrass' proof of the above theorem 66 20, 21 An important lemma. 67, 68 CONTENTS ART. 22 A g-eneral proof of a general theorem in determinants 23 The theorem proved when the variables are subjected to subsidiary con- ditions. ............. 24 Conditions that the quadratic form be continuously positive or continu- ously negative. PAGB. 69 70,71 72 25 26 27 28 APPLICATION OF THE CRITERIA JUST FOUND TO THE PROB- LEM OP THIS CHAPTER. The problem as stated in art. 1 solved and formulated in a symmetric manner. ............. 73 The results summarized and the general criteria stated 73, 74 Discussion of the geometrical problem: Determine the greatest and the smallest curvature at a regular point of a surface. Derivation of the characteristic diiferential equation of Minimal Surfaces. . . • 75-80 Solution of the geometrical problem: From a given point to a given sur- face, draw a straight line whose length is a ininimwin or a maximum. 80-84 CHAPTER IV. 7 8 9 10 11-16 11 SPECIAL CASES. 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. Difficulties that are often experienced. 85 Fallacies by which maxima and minima are established, when no such maxima or minima exist. 86 Definitions: Realm. A position. An n-ple multiplicity. Structures. A position defined which lies within the interior of a definite realm, on the boundary, or without this realm. • . . 87 Statement of two important theorems in the Theory of Functions. . . 87, 88 Upper and lower limits for the values of a function. Asymptolic approach. Geometrical and graphical illustrations. 88, 89 Cases where there are an infinite number of positions on which a function may have a maximum value 89, 90 Reduction of such cases to the theory of maxima and minima proper. . 90, 91 The derivatives of the first order must vanish. ...".. 91 The results that occur here are just the condition which made the former criteria impossible 91 The previous investigations illustrated by the problem: Among all poly- gons which have a given number of sides and a given peritneter, find the one which contains the greatest surface-area 92 Solution of the above problem. 92-97 Cremona's criterion as to whether a polygon has been described in the positive or negative direction. 93 CASES IN WHICH THE SUBSIDIARY CONDITIONS ARE NOT TO BE REGARDED AS EQUATIONS BUT AS LIMITATIONS. 17 Examples illustrating the nature of the problem when the variables of a given function cannot exceed definite limits. 97 8 CONTENTS ART. PAGB. 18 Reduction of two inequalities to one , . <» 97 19 Inequalities expressed in the form of equations. 98 20 Examples taken from mechanics. 98 gauss' principle. 21 Statement of this Principle. 99 22 Its analytical formulation 100 23 By means of this Principle all problems of mechanics may be reduced to problems of maxima and minima. 101 TWO APPLICATIONS OF THE THEOKY OF MAXIMA AND MINIMA TO REALMS THAT SEEM DISTANT FROM IT. /. Cauchy's prooj of the existence of the roots of algebraic equations. 24 Statement of the proof of this theorem. 101, 102 //. Proof of a theorem in the Theory of Functions. Reversion of Series. 25 Statement of the theorem 102 26 Recapitulation of what has been done in the previous investigation re- g-arding the reversion of series. 102 27 The theorem in its modified form is thus stated: (1) When the variables X and y are connected by the equations li=n [1] :>'x=2(^^''+ A^m)^^' M=l where dXfi and \ , are quantities which are subjected to certain conditions, then it is always possible to fix for the variables X^, X2, • • . .X^ and j^-i, y^, j>/„ definite limits g^, g^, g^ and hi, hi, . . . . h„ in such a manner that for every system of the y'sfor which \y\ I <^ h\ (A.=l. 2, «) there exists one system of the x'sfor which |;i;x|<^^x (\=1, 2, n) so that \\] is satisfied. (2) Thesolu- tion of [1] has a similar form as the equations [1] themselves, viz: — X\ = 2 (^Xm +Yxm ).>''' • (X=l, 2, fi) . ■ .103,104 28-36 Proof and discussion of this theorem. Determination of upper and lower limits for the quantities that occur. 105-114 32 Unique determination of the system of values of the .r's that satisfy the equations [1] above. 108 33 If the j/'s become infinitely small with the .ar's, the .r's become infinitely small with the jy's. The jr's are continuous functions of the jy's. . 109-111 34 The x's,, considered as functions of thejv's, have derivatives which are con- tinuous functions of the jv's. The existence of the first derivatives. . 111-113 35 The .r's, expressed in terms of the jt/'s, are of the same form as the given equations expressing the jv's in terms of the .^''s. .... 113 36 Conditions which must exist before the ordinary rules of differentiation are allowable in the most elementary cases. 113, 114 CHAPTE)R I. CERTAIN FUNDAMENTAL CONCEPTIONS IN THE THEORY OB' ANALYTIC FUNCTIONS. 1. In the development of the conception of the analytic func- tions if we start with the simplest functions which may be expressed through arithmetical operations, we come first to the rational /unctions 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 again 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. The necessity at once arises of developing the conditions of convergence of infinite series and products, since such an arith- metical expression represents a definite function only for values of the variables for which it converges. Mere convergence, how- ever, is no't 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 de- rivatives, and in order to have this property the above expres- sions of one variable must converge uniformly (gleichmassig) in the neighborhood of each definite value. 2. When we say a function of one variable converges uni- fortnly, we mean the following:* It is assumed that the function in question has a definite value for x — x^. We next consider all values of x for which x — x^^ does not exceed a definite quantity d. We shall then suppose that the series is to be convergent for a given value of x that lies within this interval. In order that this series converge uniformly, it must be possible, after we have *A different definition is given by Weierstrass (Collected works, vol. II., p. 202 and Zur Functionenlehre, \\.) (9) 10 Theory of Maxima and Minima assumed an arbitrarily positive quantity 8 and when a remainder R„ {pc) has been separated from the series, to find a positive in- teger tn so that I ^ n (^) i -^ ^ > where n > tn for all values of x in this interval. \^Cf. Dini, Theorie der Func- tionen,p. ijy. Translat^dinto German by Lilroth and Schepp^ 3. The conditions of uniform convergence being retained, it is essential that all the transcendental functions have a property in common : If we take a value x^ within the region of convergence (Convergenzbezirk) in which these functions considered as func- tions of one variable converge uniformly, then they may be repre- sented for all the values of x in the neighborhood of x^ as series which proceed according to positive integral powers of x — x^. For example, /(^)=/(^— ^o + ^o)=^o+«i {x—x^^a^ (x—XoY + , where ao, «i, a2, are definite functions of Xq. From this it follows that they may be differentiated and a number of other properties are immediate consequences. 4. Starting from the same standpoint as in the case of one- valued functions we may in a similar manner give to many-valued functions a definition which is far reaching in its generality. For this purpose the conception of the usual operations must be ex- tended to several variables. It is then easy to extend in the required manner the conception of uniform convergence. Let (21, (22' <^n be a definite system of values of the variables x^, X2, . . . . x„ within the region of uniform convergence, and con- sider only the values of x^, X2, . . . . x^ for which Xi — a^, X2—a2, .... ^n— <3^ii do not exceed certain limits d^, ^2> • • • -^n- l^he function may be then represented through an ordinary series which pro- ceeds according to integral powers of Xi—ai,Xi — a2,X2—a2,.-. Xn—a„, 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. 5. The definition of arithmetical functions being established as in the preceding articles, the conception 0/ arithmetical de- ER31TY of Functions of Several Variables. 11 pendence among several variables may be defined as follows: If we represent a function which has been formed as indicated above by i^(;ti, Xi,...x^, then F(xi, Xj, . . Xa)—0 expresses a certain dependence among the variables x^, X2, . ■ . x„; that is, among the infinite number of systems of values for which the function has a meaning, those which satisfy this equation, are to be taken. There exists, therefore, among x^, X2, . . . x„ a. depend- ence of a similar character, as in the case of algebraic equations. If we choose the quantities Xi, x^, . ■ ■ x^ such that the equations Fi=o, F-i,~o,. . . . F^=-.o, where ntn{ where F denotes a given function of Xi, X2, . . . . x^. With such a problem we have to prove before everything else that the required functions of time are analytic functions. If for the point /=4 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. 13. If we start with the definition of a function given in Art. 11, and if we have for a definite value x' a definite value of the function, then this value depends essentially upon the way and manner how we come to x', that is, upon the choice of the values Ui, a2, . . . Un, by means of which we have come from one element to the derived element in question. From this it may be con- jectured that one and the same function can have different values for the same values of the argument; these values can not be regarded as distinct from one another, but are to be considered as vahies of the function. If a, lies within the region of convergence of the first series, a^ in that of the second, .... a„ in that of the n^^ series, then we have a definite power-series P(^x — a^. If now we take instead of «i, ^2, . . . . a„_i, n — 1 other values a'l, a'2, .... a'„-i, then we can have another power-series P, {x — a^, so that the function for every value x in the region of convergence of this series has a different value than in the region P {^x — a„). Accordingly^ we may offer the following definition: A func- tion is called one-valued on the position a , if in the neighborhood of d we may derive a function-element from the original function- element and always have only one function-element along what- of Functions of Several Variables. 19 ever path we have come from a to a'. If we have several function-elements, then the function is said to be many -valued, • it may be infinitely many-valued. Suppose next we have a system of functions of one variable X. Let n function-elements of x be given. Each of the ele- ments determines an analytic function. It must be shown how the values of these n elements are arranged with respect to one another. If the functions are all or in part many-valued, the question arises how are we to arrange them with respect to their different values and with respect to one another. If, for example, we have two algebraic equations between the quantities x and y^ we have through the elimination of y an equation for the deter- mination of the possible values of x, and by analogy we may through the elimination of x form an equation for y. We can- not write down an arbitrary value of y for every x, but in general for each of the possible values of y we can take only one value of X. • This investigation is very simple in the case before us : We have values belonging in canmon to n functions that are defined in the given manner, if we always make use of the same interme- diary a's for the determination of the functions. In order to have a system of values belonging in common to the functions, for x^^x' , we take a' in such a manner that it lies within the region of convergence of all the function-elements, the latter becoming P{x — a'). In the same way we take a quantity a" which lies within the region of convergence of all the function-elements P{^x — a'). These become P{^x — «"). Continuing in this man- ner we have a system of values belonging in common to tHe function. In algebra and in all branches of mathematics, where the connection of functions is defined through equations, it must be shown that we have in reality by this procedure the values which belong in common to the functions. These definitions may be extended to functions of several variables and to systems of functions of several variables. We must )'et show how functions of one variable are to be defined for the boundaries. This inve.stigation will be reserved, however, for another occasion (Art. 19), where we shall present the conception of an analytic function in a different manner. 20 Theory of Maxima and Minima 14. We prove next a theorem extensively used in the Calculus of Variations. Suppose that between the variables x^, x-^,. . . x^ we have m equations given which may be represented in the form of power - series, and let these be: c,.i (;t,— ai) + .... +Ci,„ (;p„— a„) + V =o, Cu (^1— «i) + . • • • + C2.„ (^„— a„) + V =0, Cm.l(^l—«l )+.... +<:^m.n(^n— «u) + Y =0, where V , V , — V are also power-series of x^ — ^i, — x^ — a„, but of such a nature that each term in them is of a higher dimension than the first. The equations will be satisfied for x^ — ai, ;?;„=«„. We propose the problem of determining all systems of values {xi, X2, x^, which lie in the neighborhood ofi^a-^, a^, ■ ■ . .a,), and which satisfy the m- equations above; that is, among the systems of values for which \ x^ — a, | ^ | x^^ — a„ | are smaller than a fixed limit p, determine those which satisfy our m, equations. The quantity p is only subject to the condition of being suffi- ciently small. To solve this problem we consider the system of linear equations, to which the given equations reduce when we Through these linear equations m^ of the differences Xy — ax , X2, — iZj' • • • • ^m — ci^ may be expressed in terms of the n — m remaining, if the determinants of the m>^ order which may be formed out of the m rows of the c's are not all zero. If, say, Ci.i, ,c we have o. XI of Functions of Several Variables. 21 By means of these equations we ma}^ represent x^ — a^, x-i, — a-i,., .... ;r„, — a„, as power-series in the remaining n — m differences, the formal procedure being as follows* : We write V =o, .... V = o, and thus obtain for x-^ — a-^, .... x^ — «„ expressions which represent the first approximations. These are substituted in V , V and the resulting ex- pressions are reduced so as to contain only terms of the second dimension. Continuing this process, we may represent the re- quired expressions to any degree of exactness desired. We accomplish the same in the following manner : We write for m of the quantities x^ — Uxy-.-.x^ — a„ power-series with in- determinate coefficients; and it is seen that these coefficients may be uniquely determined. It is a fundamental theorem in the theory of functions that these power-series are convergent as soon as the differences x — a which enter into them do not exceed certain limits, and further, that these power-series satisfy the given equations. (See Chap. IV, Art. 25 et seq.) 15. 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. Besides the equations given above, we introduce others which are likewise; expressed in power-series: Let c:m+i.i(.^i— «i)+ +c„+i,„ (;»;„—«„) Y =A, J\. m+l The quantities c are arbitrarily chosen, in such a manner, however, that the determinant. '1.1 -2,1 , c > ^2, 1.2 1 2 > ^m , 1 ) '' m 2 > • ^m+l,li ''m + l,2> • ^n.l » ^n,2 > • ^l,m I ''I, m+l ) ^2,m ) ''2. m+l > ^m , m » '-m.m + l > ^m+l,m) ^m + l,m+l> ^ii,m > ^11, mil ) * See also Chap. Ill, Arts. 3-6 ; Chap. IV, Arts. 25 and 26, ''l.n ^2,11 ■-m + l, 11 o. 22 Theory of Maxima and Minima Proceeding in a similar manner as in Art. 14, we write the quantities V equal to zero, and we thus have a system of n linear equations through which we can express the n differences x^—a^, Xr,—a^ through t^, t^, /„_„ : Xv—qv = ey^^ /l + e^.2 ^2+ • • • • +ev.„-m 4-m + Y • (v=l,2 n). With the help of these equations we can express x^ — «!,.... ^n—cin as power-series in A.- ■ • • K-m- To do this we again write V = o, and have only terms of the first dimension. We write the first approximations that have been thus obtained in V and by retaining the terms of the second dimension derive the second approximations, etc. It may be proved that these power-series in t formally satisfy the given equation; that they possess a certain com- mon region of convergence if we give certain fixed limits to I ^1 I , I i^2 1 > • • • • I 4-m I > that they consequently in reality satisfy the equations; and finally, that all the systems of values {^x^, x-^., . . . .x„) which lie in the neighborhood of («!, a2, . . . . a„) and which satisfy the proposed equations are had in this way. In the theory of maxima and minima we shall give strenuous proofs of the statements just made, and in this connection a the- orem will be proved which is very important in the theor)^ of the Calculus of Variations. (See Chap. IV, Art. 25 el seg.) 16. We make the following application of the theorem given in Arts. 14 and 15. In accordance with this theorem, if between n quantities x^, X2, . . . .x„ there exist -m equations in the form of power-series, then the differences Xi — ai, .... x„, — a,„ may be ex- pressed through power-series of the n — m remaining variables. Weierstrass said: '''' Through the m equations a structure of the {n — ni)"' kind in the realm of the n quantities Xi, X2, . . . .x^ is defined." In virtue of the theorem of Art. 15 these structures may be expressed in manifold other ways. If we introduce here, as in Art. 15^ the quantities ^j, ^2 • • • • 4-m^ we have a symmetric representation of this structure, namely: of Functions of Several Variables. 23 xv—av=Pv{fi, t^, 4-m) (for v=^l, 2, n). It follows from the method of the derivation of these expressions that they satisfy the given equations. If we wish to find all systems of values which satisfy the given equations and in which | x^ — a^ \ x„ -«„ I are less than p, we may always assume p so small that the required sys- tems are represented through the above formula. A structure of the (« — mf^ kind is thus defined in the realm of the n quantities ,ri, Xi^.... x„. This theorem would be of little importance if the following was not true: the structure is not a closed one within itself, but a structure which may be continued over its boundaries; it is, as Weierstrass expressed it, only an element of a complete structure. The question arises, how are all the remaining ele- ments derived from this one. We shall direct our attention in the following paragraphs to the discussion of this question. 17. Suppose, both for simplicity and clearness, that n — w = l; then, if we agree to represent x-^, X2,....x„ as power-series of the last variable x„, the functions P(x,^ — £?„) which have been so defined may be continued as in Art. 13. We need only assume a position a'n in the region of convergence P(Xa — a^) and trans- form this series into Pi{x„ — a'„). We thus have in general a con- tinuation as soon as the realm of convergence of the second series extends outside of the original realm. If we make use of the quantity t, the continuation may be expressed in a much more general and symmetric form. As this quantity / is in general arbitrary, it is possible to express Xi, X2, . . . .x„ through / in many ways. Under the assumption that n — m=l, we shall limit this investigation to a structure of the form, Xv—av=Pv(l) (v=l,2,....n). We are thus freed from an assumption tacitly made that the series Pv (/) begin with terms of the first dimension; for we can choose Pv (t) quite arbitrarily, the only condition being that it must vanish for l=o. Let us take instead of the variable t the auxiliary variable t. There clearly exists an equation between / and r, since, if in the 24 Theory of Maxima and Minima formula Pi^x^ — a-^, x^ — aj, . . . .x^ — «„), through which t is defined, we write then we have r expressed as P{t). Reciprocally we may repre- sent ^ as a power-series in t. Suppose that t=^a^ T-f-a2T^+ ...., where the a's must sat- isfy the condition that the series converge for certain values of T. We recognize from this that the same element which is rep- resented through xv — av=Pv {t) may also be represented through Xv — av=Pv{r) (j/=l, 2, . . . . w). From these equations it is clear that to every value of t there belongs a value of t, and we may assume r so small that the corresponding value of t lies within the region of convergence of the original series Pv{t). We see that both systems of formulae represent the same systems of values. On the other hand, if Oj , o, we may express t as P{t): If we substitute this expression in the second system of for- mulas Xv — av=Pv{T), there must again appear on the right-hand side Pv(t). We may now choose r and / so small that the power- series, which represent them, converge; hence to every value of T there corresponds a value t and to every value of t there corre- sponds a value t. Consequently the systems of values {xi, x^, ....x^ which belong to pairs of corresponding values /, t are identical. The structure may be expressed through the one or the other system of formulae. We shall say briefly that we have trans- formed the expressions of the structure — not the structure. We may accordingly define the continuation of such functions as follows: Consider a system of formulae having the form Xv = ^v(^t^ (^ = 1, 2, . . . ./^), of Functions of Several Variables, 25 where the *'s are functions of t which may be expressed in the form of power-series. In this manner a structtire is defined, if to t all values are given for which *i, . . . . *„ converge. A structure is defined through These two structures, which in general have nothing in com- mon, may have a common position; they will then agree for a definite value /„ of t and a definite value r^ of t, so that *v(4)=*v(to) {v=\,2....n). However, there is nothing of especial interest in this. Yet it may happen that in the neighborhood of these positions the two structures completely coincide; that is, there corresponds to every value of / in a certain neighborhood of 4 ^ value of r in a cer- tain neighborhood of Tq, so that we have continuously *v(^)=*v(t) (v=1,2, n). This is expressed analytically, if we write where s and o- are two new variables. In order that the two structures correspond in the neighbor- hood of the two positions under consideration, v and ♦f must so correspond to each other that ^v goes into ^v, if we write and vice versa, *v must become *v if we write cr equal to a power- series in s. Consequently, to every infinitely small value of 5 (or 0-), which is smaller than p, where p is a fixed limit taken sufficiently small, there must correspond an infinitely small value of cr or (s), so that we have the equation .(/o+5)=:^v(To+o ; ) ' In the same way we say a function f{xi, x^,. . .x„) of n variables * And not only formally. (See Art. IS.) t See Annals of Mathematics, Vol. IX, No. 6, p. 187. of Functions of Several Variables. 33 has at a definite position Xi^a^, X2=a2, . . . . x„=a„ a maxiimiin or a minimum, if the value of the function for Xi=^ai, X2=a2, . . . .Xa=aa is respectively greater or less than it is for all other systems of values situated in a neighborhood \x\—ax\<^x (X=l, 2, n) as near as we wish to the first position ; and the analytical con- dition that the function /'(;i:i, X2, . . . .Xa) shall have at the position Xi=^ai, X2^=^a2t • ■ • .^„=iz„ a maximum, is '.fixi, X2, . . . .x^^fi^a^, aj- • • • -^n) <{u)=cf>(o)+ -fpf (C7)+ -^"{o)+ .... (»^ — Ijl m^l and in this expression write u=l, as follows: For brevity, denote by f^ (x^, Xj,.... x^) the first derivative of f(xi, X2, x„) with respect to x^, hy fi,^^^^(xi, x^, x^ the derivative of f{xi, X2,... .x„) with respect to ^ttiand x^,^, i. e., /r /„ \ d^ f (^Xi, X2, . . . . Xr) , ox^^^dx^^ We have, then, k=n ^' (^)=2 V ''^^'+^^^i' lt-=l ^), a^ + u{x,,—a„))(x^—a^V,, 36 TTieory of Maxima and Minima ki,k2,...k„_i ki,k2,...kni Hence, from [3] we have k + ' *^1 J *^2» "• ** itt— 1 ,^^ (/kj,fcj, ...u„(«i + e «^(^i— «i), . ... ) w! -*^ I a^ + €U(x„—a„))(Xi,^—a^^) . .. (^k„— ^„)) k i,k3,...kn. From this it follows, if we write ^=1: k 'i.'^a + I 1 X://k,>k,...k^_^(gl,g3.--'gn) (^k— «ki)---- ) ki,k2...k^_l 1_^ (/ki,k2....k„(^i + e(^i— ^i), a^ + eixi—a^),.... \ kl,k2,--Km of Functions of Several Variable:. 37 6. We are not accustomed to Taylor's theorem in the form just given; in order^ to derive this theorem as it is usually given, we observe that upon performing the indicated summations, each of the indices /J^i, k-^,. . ■ ■■, independently the one from the other, takes all values from 1 to n, so that the X'" term in the develop- ment is a homogeneous function of the X^'' degree in x-^ — a^, x-^ — a^, .... x^ — «„. The general term of this homogeneous function may be vi^ritten in the form ~ D.N.{x^—a^ \xi—a^--{x^—a^ ", where Xj + Xj-f .... -f X„=X, 'd^ f{x^,XT,,...xy (a f yXx, Xii . . . ^„ ) \ -tI; — ;n; -> x I On _ -(A1 + A2+ + A.„) and N is the number of permutations of n elements of which Xj, X2, . . . . X„ respectively are alike, ^•^--^= X.!X2!....X„ !- Further, writing Xi, — «k = ^k> we have finally: [4] /(xi, X:i,...x„)—/{ai, a2,...a„) + + n. — u 2U d/(xi, X2, ;i;,.) \ U dx^ J k=l 2^\\ dx^x^ ) h^ a\ , a2, (In ai ,a2 , a n \l+\2- +2 {f'"'' Xl,X l.'^2. Xl+X2 + .X„ X,! X^L.-XJ. Xl, X2, x„ X,! \^\...\S 38 Theory of Maxima and Minima This is the usual form of Taylor's theorem for functions of several variables. In particular, vv^hen ^w=l, the above develop- ment is : , [5] f{x^, Xi_ ^„)— / («!, «2. «n) k=n k=l The function f {x-^,. . . .x„) is regular and continuous, as are consequently all its derivatives. If, therefore, the first deriva- tives oi /{xi, X2,. . . .x„) a.re all, or in part, - have always the same sign, and we may therefore confine our- selves to the investigation of the latter function. 40 Theory of Maxima and Minima If we find that through a suitable choice of Ai, h^,....h^ the expression ' can be made at pleasure either positive or negative, the same will be the case with the difference /{x^, x^,. . . .x„)—/'{ai, a^.,. . . .«n)» and consequently f {^x^, Xi,....x^ has on the position (ai, a^., . . . .a„) neither a maximum nor a minimum value. We therefore have as a second condition for the existence of a maximum or a minimum of the function XC^n -^21 • • • -^n) on the position («!, i?2>- • • -^n) that, in case the second derivatives of the function /"(j^i, ;i:2, . . . . ;i^„) do not all vanish at this position, the homogeneous quadratic form f^\\ dx^dx Iai,a2, an ' must be always negative or always positive for arbitrary values of ^1, /^2, /?„. THEORY OF THE HOMOGENEOUS QUADRATIC FORMS. 8. There are two kinds of integral homogeneous functions of the second degree, or, as they are usually called, quadratic forms, viz.: I., formce indefinita;'* which with real values of the vari- ables can become both positive and negative, and that, too, for values of the variables, whose absolute values do not exceed an arbitrarily small quantity; II., formce definitce, which with real values of the variables have always the same sign. We distinguish among the definite forms: [1] Those which only vanish, when all the variables become zero, and '^ Gauss, Disq. Aiithm., p. 271. of Functions of Several Variables. 41 [2] Those which may also vanish for other values of the variables. • If our homogeneous function is an indefinite form, it is clear that the function /(;ri, x^, x^ has neither a maximum nor a minimum upon the position (^i, a^,... .«„); for if the right-hand member of [7] is positive (say) for a definite system of values of the ^'s, then in accordance vi^ith 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 ^'s, for which the right-hand side of the equation [7] is negative; consequently, also, the difference /(^l- ^2 ^n)— /(«1. «2 «„) is negative, so that therefore no maximum or minimum is permis- sible for the position (^i, ofj, . . . . a„). If, then, the second derivatives of the function /"(;ri, x^,. ■ . x„) do not all vanish at the position (ai, (Zj, • ■ • ■««)> it follows besides the equations [6] as a further condition for the existence of a maximum or a minimum of the function /"(;tri, X2,... .x-^ that the terms of the second dimension in [4] must be a definite quadratic form, and, indeed, as will be show^n later, one which can only vanish when all the variables become zero. The question next arises: Under what conditions is in gen- eral a homogeneous quadratic form [8] {x^,X2, x„) = ^Ax^^Xxx^ a definite quadratic form of the kind indicated ? ' 9. 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 ^ {^x-^, x^,. . . .x„) in the place of {xi, X2, . . . . Xa) homogeneous linear functions of these quantities [9] :yx-^^cx^i,x^ {\=l,2,...,n) 42 Theory of Maxima and Minima are substituted, which are subjected to the condition that in- versely the x'^ may be linearjy e;i;pressed in terms of the 7's, and consequently the determinant [10] ^11 > ^12) • t^ln ^21 ' ^22 ) ^211 ^nl t ^n2 > ^1 ^ ^11 ^22- n ii< o. The function ^ (;iri, ;i;2, x„) is thereby transformed into [11] <^(^l. ^2. ^n)=«/'(J^'l.>'2.••••:^^n). Owing to this substitution it may happen that i/» (jFi, ;V2, does not contain one of the variables y, so that <^ (;Vi, x^,. is expressible as a function of less than n variables. In order to find the condition for this, let us write [12] ,^;, = |^|i_ = 2^x/.^/. (X=l,2....^). If i/» is independent of one of the y s, say y^ , so that conse- quently ^ y^ =0, then from the n equations i>.=n li==n M 2 ^.=2(1^ l^j-Sf^.x^) (A = l, 2,....«) we may eliminate the m — 1 unknown quantities ^ , *^ , '9>'n-l We thus have among the ^'s an equation of the form [14] 2'^'»*^' .=0, where the ^'s are constants. Owing to equations [12] this means that the determinant of the given quadratic form vanishes, i. e., [15] ^1 =t -^11 -^22 • • • •■^x, — O. of Functions of Several Variables. 43 Reciprocally, it is easy to show that if the equation [15] is true, it is possible to express the function <^ as a function of n — 1 variables. For we have [16] X X, /t 2l *^x(^'i. X^,...X^) ^x| = 2 ^X/x^V^A-' X X, /i and consequently [17] 2 { ^^ (^1' ^2. ^n) ^'\| = 2 { *^^ (^'i.^'a. • • . • ^'« )f \ |- There exists, further, the well-known Kuler's theorem for homo- geneous functions: [18] 2{'^^(^l' ^2. ^n)^x| = «^(^l, ^2, ^n)- If now we assume that equation [15], or, what amounts to the same thing, an identical relation of the form [14J exists, and if we substitute in (f> (xi^Xj,. .x„) the quantities ;t^x+/ X:x in the place of xx (X=:l, 2,....n) and develop with respect to powers of /, we then have {Xi + i ^l,Xi+t i;2^ ^n + f ^n)=(Xi,X2, X„) + 2 t 2{^X<^x(^1.^2.----^n)|+^^<^(>^l''^2----'^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^, x^,... -x^), that <^ {x^ + t k^, ;c„+ 1 k^)= {x^, x^). Hence, if the equation [15] exists, or if the ^'s satisfy the equation [14] for every system of values {x^, X2,....x„), then [x,— ^Xv,X^---^Xy, . . . .,Xv~\ ——-Xv, O, Xy+i — ~^X„,. . . .Xn -,- Xp], Kv kv kv I where 4> is expressed as a function of less than n variables. We therefore have the theorem: The vanishing of the determinant ^ --^n ^22- • • --^nn is the necessary and sufficient condition that a homogeneous quadratic function (ft (x^, x^, x^J^^A^^ x^ x^ be express- ible as a function of n — 1 variables. 10. We return to the question proposed at the end of Art. 8; and in order to have a definite case before us, we shall assume that the problem is: Determine the condition under which the function (j> (x^, x^,. . . .x„) is continuously positive. The second case where ^ {x^, x^,. . . .x^) is to be continuously negative is had at once, if we write — <^ in the place of (j). We shall first show, following a method due to Weierstrass,* that every homogeneous function of the second degree ^ ( x-^, x^, . . . .x^ may be expressed as an aggregate of squares of linear functions of the variables. 11. In the proof of the above theorem we assume that <^ (yXy, X2, . . . .x„) cannot be expressed as a function of n — 1 vari- ables; it foUows, therefore, that the inequality [20] 2±^"^22----A,„^o is true, and it is not possible to determine constants k, so that the equation ^ k,- ^^=0 exists identically. *See also Lagrange, Misc. Taur., I., p. 18, 17S9 ; Mecanique, T. I., 3; Gauss, Disq. Arithm., p. 271; Jheoria Comb. Observ. p. 31, etc. of Functions of Several Variables. 45 If, then, y represents a linear function of x having the form [21] y = C^X^\CT.X^^ +<^n^n, and if ^ is a certain constant, then the expression <^ — g y^, after the theorem proved above, can be expressed' as a function of only n — 1 variables, if the constants k^, k2,....k„ may be so deter- mined that \=i or, [22] ^K4>x-g}'^hc^=o- \=l X=l From the assumption made regarding [20] , it follows on the one hand that the inequality [23] 2^: Cx^O \ ^\< must exist. On the other hand, in virtue of the n linear equa- tions, [24] 2^'^A'^/*=*^x' (X=l, 2,...;^) the quantities x^, X2,....x„ may be expressed as linear functions of (f>i, (f)2, ....(()„ , and consequently by the substitution of these values oi Xi, X2, . . . . Xa in [21] y takes the form [25] >'=^e^., where the e, are constants, which are composed of the constants A^X and c\ . iBut fiTom equation [22] it follows that y 2 ^^ 'f^^ ^2 ^^ ^^ 46 Theory of Maxima and Minima Such a representation of the <^x' however, since we have to do with linear equations, can be effected only in one way. We therefore have J^'=^^=J ^"^^^(^^^ from which it follows that lt=n K=g^x^^^c^ (X=:l, 2,....«). Through the substitution of these values in [22] we have X=l X=l consequently, owing to the relation [25] , we have x=« X=l This value of g- may be expressed in a different form, for from [25] and [17] it follows that »=« »=« I/=l y=l Comparing this result with [21], we have [27] c, =(^,(e„ ej, e„) (v=l, 2,....n), and consequently, [28] or, from [18] of Functions of Several Variables. 47 1 x=w 2^'^*^^ (^1' ^^2' — ^") \=\ (j) (ei, e^, e„) Since the quantities Cj, Cj, . . . . c„ are perfectly arbitrary ex- cept the one restriction expressed by the inequality [23], the quantities ei, e2,....e„ are in consequence of the equation [27] completely arbitrary with the one limitation resulting from [28], viz., the function '^ where ;>' has the form [25], may be expressed as a func- tion of only n — 1 variables. For form the derivatives of this expression with respect to the different variables, and multiply each of the resulting quanti- ties by the constants e^, e-i,.... e„. Adding these products and having regard to [28], we have ^e\'^\—gy ^ex<^\ (^1. ^2 en) = ^e\^\—y- The expression on the right-hand side is zero from [25]. Hence we may choose n constants in such a way that the sum of the products of these constants and the derivatives of the expression ^ — gy^ is identically zero. From this it follows by a similar method of reasoning as was given in connection with the equation [14] that the expression ^ — gy^ may be expressed as a function of only n — 1 variables. 13. If we represent ^ (x^ x^, . . .x,,)—gy^ by ^ {x^, x^, x„), we may derive this function of n — 1 variables from [19] by sub- stituting x\-yte\ for x\ (X==l, 2, . . . .«) in ^; if one of these arguments is made equal to zero, we have as in art. 9 48 Theory of Maxima and Mimvia (Xi, Xi Xn)—gy= (^1— — ^k, ^k ^^^^=^^k, O, Xi,+i — ^k+1 ^ „ ^n^ \ -t-k) • • • • -t-n 't'k l> ^k ^k / f or, if the new arguments are represented by x-^, x'2, .... x\-i, ^ (Xi, X2, x„)—g x^^ (f> (x\, x'2, x'„^i). Employing the same method of procedure with ^ {x\, x'2 ^'„-i) as was done with (f> {xi, X2,... .x„), we come finally to the function of only one variable, which being a homogeneous function of the second degree is itself a square. Hence we have the given homo- geneous function <^ (x^, X2, . . . .x„) expressed as the sum of squares of linear homogeneous functions of the variables. If the coeffic- ients of (x„ X2,.... x^=g^y^ + ^2 >'2H . . • . + gny^ is to be continuously positive for real values of the variables and equal to zero only when the variables themselves all vanish, then all the qualities g^ gv- ■ • -gn must be positive ; for if this was not the case, but^i (say) was negative, then since the jv's are inde- pendently of each other linear homogeneous functions of the ;*;'s we could so choose the x's that all the jy's except jFi vanished and consequently contrary to our assumption (f> {xi, X2,....x„) would be negative. Further none of the ^'s can vanish ; for if £^i, say, was zero, we might so choose a system of values x^, X2,....Xn in which at least not all the quantities x^, X2,....x^ were zero that all the ^-'s vanished except y-^, and consequently could then be zero without the vanishing of all the variables x^, Xj,.... x„. ^ See Burnside and Panton, Theory of Equations. 1892. Page 430. of Functions of Several Variables. 49 Reciprocally the condition of ^i, gz,... ■£„ being all positive is also sufficient that be continuously positive for real values of the variables, and that <^ be equal to zero only when all the vari- ables vanish. 15. In order to have in as definite a 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. In connection with [12] it follows from [27] that /*=« '=2^.M^^ ("=!' 2 n). c„ = M=l Denote by A the determinant of these equations, which from [20] is not identically zero, i. e., [30] A = ^± AnA^....A„„. We have as the solution of the preceding equation '' = A ^2 dA dA x=i \^i cx (/^=1, 2, n). It follows from this in connection with [26] that A [31] g dA \,ii dA c\ Cu xm an expression in which the c's are subject only to the one condition dA that ^1 ^~2 — ^^ ^/^ IS not identically zero. 16. It shall next be shown that we may separate from (f) {xi, X2,.... x„) 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 (ji — g x„^ be ex- pressed as a function of n — 1 variables, we have chosen for g the value [31], after we have written in this expression c\—o (X — 1, 2, . . . . n — 1), while to c„ is given the value unity. so Theory of Maxima and Minima From this we have dA A^ 9A„ where A-^ is the determinant of the quadratic form ^{^x^, X2,.... x„_i,o). Of course, this determinant must be different from zero. Hence we may write where '^(x\, x'2, .... x'„^i) =4> (^1—— ^„, X2—^X^, .... X„_i—^^=^X„, oj. We may then proceed with (f> just as has been done with (f) by separating the square of x'^^i, etc. If we notice that the determinant of the the function of n — /a variables which results from the seperation of /i squares is the same as the determinant of the function which results from the original function, when we cause the fi last variables to vanish (in this function), and if we denote this determinant by ^^, we have the following expression for <^ : 17. If now is to be continuously 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

A, /t de a definite form and rem,ain continuously positive, it is neces- sary and sufficient that the quantities A^, A2,. . . .^,,-1, which are defined through the equation A^ = ^ — ^u ^22- . . -^n-^, n-« of Fundiotis of Several Variables. 51 be all positive and different from zero. If on the other hand the qardratic form is to rem,ain continuously negative, then of the quantities A^-i, ^„-2, A^, A, the first must be negative, and the following must be alternately positive and negative. APPLICATION OF THE THEORY OP QUADRATIC FORMS TO THE PROBLEM OF MAXIMA AND MINIMA STATED IN ARTS. 1 — 6. 18. By establishing the criterion of the previous article the original investigation regarding the maximum and minimum of the function /(;t;i, x^, x^ is finished. The result established in art. 12 may in accordance with the definitions given in art. 7 be ex- pressed as follows : in order that a maxim^um and minimum, of the function f {^x-^, x^,... .x,^ may in reality enter on the position («!, fl5i, . . . .a„) which is determined through the equations [6], it i^ necessary, if the second derivatives of the function do not all vanish at this position, that the aggregate of the term,s of the second degree of the equation [4] be a DEFINITE quadratic form. We can more especially say, as already indicated in art. 8: If we have a definite form which only vanishes when all the variables vanish, the function has on the position in question in reality a maximum or m^inimum value; if, however, the fortn vanishes for other values of the variables, then a deter- mination as to whether a maximum, or minimum- in reality ex- ists, is not effected in the fnanner indicated and requires further investigation as is seen below. In accordance with the theorem stated in art. 8, a maximum or minimum will enter for a system of real values of the equation [6], if the homogeneous function of the second degree -S:// 9'/(^i, ^2.----^n) 'j h^h^ \ \/* X, M is a definite quadratic form ; that is (art. 17) there will be a mini- mum on the position {tZi, a^,. . . . a„) if the quotients hear y of Maxima and Minima F, F, F.-^ F,' F^ F.-,' 52 where 7^^ = ^ ±/ii y^2+ /^n-^, n-/x , ^^^ all positive, a maxi- mum, if they are all negative; and if in both cases the quotients are different from zero. This last condition is only another form of what was said above, viz., that ^/^m ^^ ^/^ cannot be a definite quadratic form, which vanishes for other values of the variables that are not all zero. This condition is in general necessary. For, if (say) then, the summation ^^fk,i. ^K ^^ being denoted by ^{h^, h^,,.. . . ^„), this equation would directly imply the existence of a rela- tion of the form : 2 ^Av\K h^., h^)^o, where the ^i,are constants which do'not all simultaneously vanish. If, therefore, k^ (say) is different from zero, we may write "=1 and with the help of this relation we have from the equation X=« \=n— 1 <^(Ai, h,,....K)^^^^(^k„h,,....h„) h^=^,K+ ^xhx the following relation X=u— 1 y Now in this expression we may so choose the arbitrary quantities h, that of Functions of Several Variables. S3 h ——h (\=l,2,....w), and consequently the function ^ (/^j, h^,.... A„) would vanish with- out all the A's becoming simultaneously zero. Hence with this system of values of the A's the difference f{xx,X2, . . .x„) — /{au az,--. .«„) would begin with terms of the third dimensions; and consequently the function /"(;»;i, X2,....x„) would have no maximum or minimum on the position {a^, Uj,... .a„) unless all the terms of the third dimension vanish, a further con- dition being also that the aggregate of the terms of the fourth dimension have a continuously negative or a continuously posi- tive sign. y We are not able to give the criterion for this; it is, however, the more improbable that such a case happens, the greater the number of variables that appear, since the necessary conditions (that the terms of the third dimension shall all vanish) continu- ously increase. The problem of this chapter is thus completely treated; how- ever 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. 54 Theory of Maxima and Minima CHAPTER III. THEORY OP MAXIMA AND MINIMA OF FUNCTIONS OF SEVERAL VARIABLES THAT ARE SUBJECTED TO SUBSIDIARY CONDITIONS. 1. In the preceding investigations the variables x-^, x^,. . . .x^ were completely independent of each other. We now propose the problem : Among all systems of val- ues (Xi^, X2,... .x^ find those which cause the function F(xi, X2, . . . .Xn) to have maximtt7n and minim^um values and which at the same tim-e satisfy the equations of conditions : [1] /x(^i. ^2. ^n) = o (X=l, 2, m; m ymn> is different from zero, then (see Chapt. I., arts. 14 and 15) m of the quantities h may be expressed through the remaining n — m quan- tities (which may be denoted by k^, ^2- • • • •'^n-m) in the form of power-series as /bllows : |_5J A^x = \K\i A^2> • • ■ • l^a-m) y^ + V^l) ^2) • • • • "^n-m ^ ^ ~^ ' (\=1, 2,...w) where the upper indices denote the dimensions of the terms with which they are associated. These series converge in the manner given in Chapt. I., art. 15; they satisfy identically the equations [2] and furnish, if the quantities k^, k-j,,.... ^„_„ are taken suf- ficiently small, all values of the m quantities h which satisfy these equations. 5, In accordance with Chapt. I., art. 14 one of the determi- nants of the ^wth order of the system [4] must be different from zero, in order that the considerations of the preceding article be true. This condition is in general satisfied ; there are, however, special cases where this is not the case. A geometrical interpre- tation will explain these exceptions. Let F and an equation of condition, /"= • • • • ^n-ra> it is not possible for this equation to exist unless each of the single co- efficients is equal to zero. 58 Theory of Maxima and Minima Consequently we have as the first necessary condition for the appearence of of a maximum or minimum the existence of the following system of n equation : and indeed in the sense that if m of these equation exist independ- ently of one another, the remaining n — m of them must be identic- ally satisfied through the substitution of the e's which are de- rived from the first m equation, it being of course presupposed that the system of values (^i, ^2, ....a„) has already been so chosjen that the equations [1] are satisfied. Taking everything into consideration we may say : In order that the function I^ (xi, X2, ■ ■ ■ . x^) have a maximum or m>ini- mum- on any position (a^, a^.,. . . .a„), it is necessary that the n-\-m equations 1^.. aA..lA + +..,|A=», [10] dx^ dx^ dxfi ' ' "' dx^ /\(xi, X2, x„) = (X=l, 2, m), be satisfied by a system of real values of the n-^m. quantities 8. These deductions are connected with the one assumption that at least one of the determinants of the m^th. 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^. It may happen that a maximum or minimum of the function /^enters on the position {a^, aj, . . . .a„) even when the above con- dition is not satisfied. For if it is possible in any way to deter- mine all systems of values of the A'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 quanti- ties «!, a-i,. ■ ■ .a^. When the above condition is not satisfied, the equations [8] exist identically and consequently the equations [3] , which serve of Functions of Several Variables. S9 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. 9. To make clear what has been said, the following example will be of service ; its general solution is given in the sequel (Chapt. III., art. 28). The problem proposed is : 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 = o, f=^o, fz=o, then in reality for this point the terms of the first dimension in the equations [2] drop out and we have the case just mentioned. If the surface is the right cone : fi^x, y, z^~ o=x'^+y — z^, then we may write : I x=2uv, [11] < y=u'—v\ \ ---u^ + zJ^. The equation of the surface is identically satisfied, and it is easily seen that we may express the quantities hi, h-^, h^, through two quantities k^ 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=o=y=z, or u—o=v. This representation may indeed be effected in such a way that not only infinitely small values of h^, hj, h^ correspond to infinitely small values of k^, k^, but also that all systems of values h^, hj, h^ are had which satisfy the equation f{x+hi,y + h^, 2+h:i)=o. We have, however, to give to the variables at one time real, at another time purely imaginary values, if the equation [11] is to represent the entire surface of the cone ; but in this manner the 60 Theory of Maxima and Mitmna 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 Chapt. II. In other cases we may pro- ceed in a similar manner. The special problem will each time of itself offer the most propitious method of procedure. 10. We must now establish the criteria from which one can determine whether a maximum or minimum of F{xi, X2,....x„) really enters or not on a definite position (ui, ^2, . . . .«„), which has been determined in accordance with the theorem cited in art. 7 of the present chapter. 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] ; since, in order to obtain the coefficients Cp . " M, » the values [5] and consider only the terms of the first dimension in the process. If we then apply the criteria of the preceding chapter, we can determine whether the function F possesses or not a maximum or minimum on the position (a„ a^, . . -cia)- 12. The definite conditions that have been thus derived are unsymmetric for a two-fold 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 prob- lem already treated in the preceding chapter and propose the fol- lowing more general theorem in quadratic forms. THEORY OP HOMOGENEOUS QUADRATIC FORMS. 13. Theorem. We have given a homogeneous function of the second degree [14]

(xi, x^,. . . .x„) is positive when ^ (fi, ^2' • • • • fn) is positive. 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.... +1; further 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^, X2,....x„ ; that is, among all systems of values which satisfy the equations [15] and [16] their must necessarily be one for which there is a maximum and one which gives a minimum value of (j). We limit ourselves to the determination of the latter. 14. Through the addition of equation [16] we have reduced the theorem of the preceding article to a problem in the theory of maxima and minima ; for, if the minimum value of <^ (xi, x^, . . . .Xa) is positive, <}> is certainly a definite positive form. of Functions of Several Variables. 63 If we write [17] \=1 p=l 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 dG This gives o (X=l, 2,. . . -n). P=m 3^ -2exx+2ya^ p=i ^=0, (\=1, 2,..../?) or, [18] 11=11 P=m 2 '-^X/. ^/^— ^ ;i;a. + 2 ^P ^P^ ^ ^• /*=i p=i From the w+???+l equations (X=l, 2,....«) [19] li=n P=m 2^>-/^^M— ^ ^x+ 2^'' «px=^. (^=1' 2,....«) * M=i p=i ix=n 2^p^^/^=^' (p=i> 2,... .w) M=l X=« 2-5=1; the « + ^w-|-l quantities x^, Xj, . . . .x^, ^i, ^2. • • • -^mt ^ J^ay be deter- mined. 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 determined 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 re- spect to the unknown quantities x^^, Xj,.... x„, e„ ^2, . . . . e„^ ; their determinant must therefore vanish, and we mvist have 64 Theory of Maxima and Minima [20] Ae = A.x\ & , -<^12» • • • ^21 ' ^22 ^' -^2ii t ^12 > • <3t„2 a ml a mil ■ a„ o, . . . .0 The equation A e = o is clearly of the n — mth 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-^, X2,....x„ and add the results, we have [21] under consideration, and from this it follows that A e 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^, x^, . . . . Xn, ^1, ^2, . . . . ^n, for which as seen from [21] <^ is likewise negative. Hence in order that (f> be continuously positive for all sys- tems of values of the x's which satisfy the equations [15]^ it is necessary and sufficient that the equation A e = o have only positive roots* The question next arises when does the equation A e = ^22 ^ > ^211 A. A n2 .A„„ — e o. an equation, which is called the "'EQUATION OF SECULAR varia- tions " and plays an important role in many analytical investiga- tions, 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. f •Hermite, Crelle, bd. 52, p. 43 ; Serret, Alg^bre Sup. 1866, t. 1, p. 581 ; Kronecker, Berlin Monatsbericht, 1873. Feb. t In this connection the reader is referred to : Laplace, M^m. de Paris. 1772. II., pp. 293-363 ; Euler, M^m. de Berlin. 1749-50 ; Theoria motus corp. sol. 1765. Chapt. V ; Lagrang-e, M^m de Berlin. 1773, p. 108 : Poison et Hachette, Journ. de I'Ecole Polytechn. Cah. 11, p. 170. 1802; Kummer, Crelle, bd. 26, p. 268 ; Jacobi, Crelle, bd. 30, p. 46 ; ^ Bauer, Crelle, bd. 71, p. 40 ; Borchardt, Liouv. Journ. t. 12, p, 30 ; Sylvester, Phil. Mag. 1852. II., p. 138.; etc. 66 Theory of Mayitma and Minima 17. Weierstrass' proof*, 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 van- ishes, 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 the arguments $i + Ci ^, ^2+ C2 k, f„ + c„ k, where the c's are indeterminate constants. Developing with respect to powers of k we have a=n »/'(^i+ c^k, ^2+ c^k, f„ + <:„ k)^2k 2 ^^ V*" (fi. ^2. D 1=1 + k^y\i {c^,Cj, c^) (i). By hypothesis the f 's are not all zero, and the determinant of v|/ being different from zero, it follows that T/»a (a=l, 2, n) can not all be zero, since otherwise the equations i/»a = (xi, X2, x^) — e^( Xi, X2, x„), that is, from (?, we shall show that the function xjf can have both positive and negative values. of Functions of Several Variables. 69 Denote the system of values (;Vi, x-^,... .,-tr„) which satisf}' the equation (m) by Xa. = ^a + i r)a (a=^ 1, 2, .... w) ; then, as in art. 19, it may be proved that Since by hypothesis /is not zero, the equation (ui) can only exist either, when »/) (^i, ^3, . . . . f„) and i/» (tJj, tjj- • • • • "^n) have opposite val- ues, and then it is proved, what we wish to show, that i/( 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 pre- ceding article, «/» can take both positive and negative values. 22. In the same connection it is intere.sting to prove the fol- lowing theorem : If the determinant formed from the partial derivatives of the homogeneous quadratic form ^ {x^, x^,... .x„) is different from^ zero, and if am^ong the infinite number of quadratic forms : >i(Xi, X2, ^„) + /^ ^(^1, ^2. ^n). there is one definite quadratic form., the determinant formed from the partial derivatives of (x^, X2, X,,) — e xjj {xi, X2, x^) vanishes for only real values of e. The theorem will also be true, if the determinant of ^ (and not as assumed of i/») is different from zero. Let Xj <^ + /Ai i// be a definite quadratic form, and write We shall further choose two constants \ and /Aq in such a way that, when we put <^ is different from zero. We know from the previous article that the determinant formed from the equations ^a — k^^^o (a=l, 2, w) 70 Theory of Maxima and Minima can only vanish for real values of k. The equations <^a — k xjjci. = o (a=l, 2,....n) (iv) may be written, in the form (X,, — k \i) {x,,X2, X„) + flxl){Xi,X2, x„), there is one which can have only positive^ or only negative val- ues, then the determinant of(f> — e \jj will have only real roots., it being assumed that the determinant of(f>, or of\jj, is not zero. The theorem in art. 17 is accordingly proved in its greatest generality. 23. The cavse where equations of condition are present may be easily reduced to the case already considered. The determin- ant [20] was the result of eliminating the quantities x^-, x-^,. . . .x, Ci, ^2, ■ • • • ^m from the n-\- m equations : n. of Functions of Several Variables. 71 y-^n P=m [18] 2^^/^ x^,— e XX -V 2 ^p ^p^ ^° ^^=^' 2'- • • •^^' p=i M=l [15] ^p=2«.^^M=^ (p=l, 2,'....^). M=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 quanti- ties X, say : x-^, x^^. . . . x„, expressed by means of the equations [15] in terms of the remaining n — m of the x's, which may be de- noted by ^1, l^j, f„_„,. We thus have [27] X^ 2 O^^^ (/^ = l. 2, m). v=l Through the substitution of these values, let <^ (^i, x^, . . .x„) be transformed into ... , ^ dd. 2d XX — exx+ 2 p=i e^ -^-1^=0 (X-1, 2 n). d XX Multiply these equations respectively by and adding the results, then since ^ d 6p d x\ _d Op d x\ (X=l, 2 n), d x\ d^, d ^v' x=« •^ 9 <^ dx\ d (}) \^ d x\ d$, d ^/ 72 Theory of Maxima and Minima 'K=n -2 -^ 2 dx\ 1^ x=i x=i we have the following equations: 1 d 1,9^ ^2, ^^ 9^p ^wr2^x+2^^-a?7=^- 2 36 ~2 36 ^ X ' ^ " af, (v=l,2,....n) The last term of this equation drops out, if we substitute in it the expressions [27], since the 6p expressed in the ^'s vanish identic- ally, and we have the equations [28] ll—e^=o {v=l,2,....n-m). 3 6 o & Now give to the v all values from 1 to n — m, and we have a sys- tem of n — m linear homogeneous equations, from which we may eliminate the yet remaining 6, ^2» 6-m • '^he result of this elimination, which is an equation in e, must agree with [20]. The equations [28] are, however, created in exactly the same manner as the equations [24]. If then A g is the determinant of these equations, it follows that the roots of the equation A e= o are all real. 24. As the solution of the theorem proposed in art. 13, the final result is : /n order that the homogeneous function of the second- degree ixi, X2, x„)=^Ax^xx Xfj,, X, M de continuously positive for all systems of values of the quan- tities Xi, X2, . . . . X,,, which satisfy the m linear homogeneous equations of condition, ^P^2 ^PM^M^^ (/)=l,2,....w). of Functions of Several Variables. 73 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 — w + 1 terms and that the signs associated with these terms be alternately positive and negative. There m^ust, however, be only a continuation of sign, if <}> is to be continuously negative. The above method was first discovered by Lagrange, who did not, however^ sufficiently emphasize the reality of the roots of equation [20]. APPLICATION OF THE CRITERIA JUST FOUND TO THE PROBLEM OF THIS CHAPTER. 25. We have determined the exact conditions that a homo- geneous quadratic form be definite for the case where the vari- ables are to satisfy equations of conditions, 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. 26. Having regard to the remarks made in art. 7 and art. 10 we have as a final result of our investigations the following theorem. Theorem. 1/ those positions are to be found on which a given regular function T (x^, X2, . . . . x^) has a m-axim.um- or minim^um value under the condition that the n variables x-^, X2,.... x„ satisfy the m, equations [aj f^{x^,X2 x^ = o (X=l, 2, m), where f are likewise regular functions, we write P=ra \b-\ ^+ 2 ^p /p= G {x^, Xj, x„), p-i and seek the system of real values •^1) -''2> • • • • ■^nt ^l! 621 • ■ • • e„ , which satisfy the n^m equations 74 Lc] Theory of Mayiima and Mmtma -.0 (X = l, 2, n), fy^ = o (X=.l, 2 m). I/{ai, a^,... .a„) is such a system of values of x^, X2,... .x„, then we develop the difference G {a^-^h^, «2 + >^2' «n + '^,.) — G («!, a2, a„) with respect to powers of h, and have {since no terms of the first dimension can appear, owing to equations \c\) the follow- ing development: [^J G{a^^h^, a^Arh^, a„ + ^„) — G{a^,a2, a 11/ = ^2 ^M-(^l''^2' ^n) ^M ^0 + M, " We must next see whether the function M ^(KK....h:).^^G^,h^ K M, •' is continuously positive or continuously negative for all systems of values of the h's which satisfy the m equations li^7t [/] ^/p^iauav- -a.) h^^o (p=l,2, . . . .m). M-i To do this we form the determinant lg\ ^U ^' L7-12, U^jn, /ii, /21, . -frnx Gzx , Cr22 e, . . Cr2„, /i2) 7221 • • / m2 ^nl > ^n2i ^nn~^iyiii>y2n' • -/mn /n , /l2. /in 1 0, O ,. . O /ml . /m2 /mm C* , ,.. aw^ this determinant put =^o is an 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- tnent consists of n — m^\ terms with alternately positive and negative sign or with only continuation of sign. of Functions of Several Variables. 75 If the first is the case, the function ^ is continuously posi- tive and the function F has on the position (ci, a^,. . . . a„) a MINIMUM value, if on the contrary the latter is true, then is continuously negative and F has on the position {a^, aj^.-a^) a MAXIMUM value. This criterion fails, however, when <^ vanishes identically, because the quantities Gt^ v vanish for the position (oti, «2. • • • • ^n) 5 and it also fails when the smallest or greatest root of A e = o is zero, since in this case we may always so choose the h'% that <^ vanishes without the A's being all identically zero. In both cases the development [^/] begins with terms of the third or higher dimensions, and for the same reason as that stated at the end of the last chapter we may assert that in general no maximum or minimum will enter on the position (ati, a-i,... .a„). We give next two geometrical examples illustrating the above principles. 27. Problem I. Determine the greatest and the smallest curvature at a regular point of a surface. If at a regular point /^ of a plane curve we draw a tangent and from a neigh- boring point P' on the curve we drop a perpendicular P' Q upon this tangent, then P'Q _i^^ the value that 2 A5 approaches. . ^^^^ ^^Cr**^''* p M &Y 1 V / J PP' if we let P' come indefinitely near P, is called the curvature of the curve at the point P. If the curve is a circle with radius r, then the above ratio approaches — as a limiting value, and is 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 de- termined through the formula 2P'Q _ 1 76 Theory of Maxima and Minima The quantity r is called the radius of curvature and the center 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 accord- ing as the line P' Q, or what amounts to the same thing, yl/'/'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 in a curve, which 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 curvature of the surface at the point P={^x,y, z) in the direc- tion of the tangent which is determined through the normal section in question. 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' , y , z') is a neighboring point of P on the surface, the equation of the surface may be written in the form: [2] o=F, (x'-x) + P\ if-y) + P\ (z'—z) +y I ^n(^'—^y+-F^{y—yy+p22i^'—'^y+2F,,{x'—x){r—y) + 2F^(y'~y)iz'-z) + 2F,lz'-^){x'-x) | + , where P\ =. ,^— , /Tj = -— , F^ d F y^ _3 F j^ d F X ay a z /r_?!^ zr_3!^ /r_^ ^'^ -d^' ^~ dy^' ^^ dz^' „ d^F J. d^F zr ^'F dxdy ' dydz ' dzdx The equation of the tangential plane at the point P is [3] F, {$-x) + F, (v-y) + F, a~z)=o. of Functions of Several Variables. 77 If we therefore write for brevity and take as the positive direction of the normal of the surface at the point /"that direction for which //is positive, tlien the direc- tion-cosines of this normal are ^^ ^2 and ^3. H H H Consequently the distance from P' to the tangential plane is [5] PQ -^{x'-x)+ f (y-r)+ § (^'-^). The positive or negative sign is to be given to the expression on the right-hand side according as the length P' Q has the same 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' lies upon the surface, we have [6] 2P'Q PP' F,lx'-xf + FAy'-yf + FJ^ z'-zf + 2F,l x'-x) {y'-y) + where S^={x—xy-\r{y—yf^-{2'—z)\ In the case where 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' approach nearer and nearer P, then the quantities ^ — X y' — y S S z' — z which represent the direction-co.sines of the line PP become the direction-cosines of the tangent at the point /"of the normal section that is determined through P' . Representing these by a, /8, y and . . . P' O the limiting value of 2 by k, then is PP [7] /c=-|,| Fy,a^^F^^\F^f-\-'ZFr,o.^^2F^^y^2F,,yo. |, 78 Theory of Maxima and Minima In this formula k represents the curvature of the surface in the direction determined by a, )8, y. This is to be taken positive or negative 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 Xq, j/q) ^0 ^•iid the radius of curvature by p, then is x—Xo=pjj ; or, since k = — , P f X Xq Fr ^11^2 + ^22/8'+ +2F,,ya [8} { y-y^=p^^ ^.^^^ ^_^ ;..... ^2/^31 yo. ' - F^ ^n<^'+^22/8H +2/^31 r« Since H does not appear in these expressions, v^^e 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, /S, y is turned about the normal until it returns to its original position. Then while a, /8, y vary in a definite man- ner, the function k of a, /S, y assumes different values at every instance, and since it is a regular function, it must have a maxi- mum value for a definite system of values (a, ^, y) and likewise also a minimum value for another definite system of values (a, /8, y). /TV • 1 The quantity — has the same value for all normal sections that are laid through the same normal.* We have, therefore, to seek the systems of values (a, ^S, y) for which the expression ^nct' + /^22/8' + ^337" + 2^i2«i8 + 2F^^y + 2i^3ir« assumes its greatest and its smallest value. * See Salmon, A Treatise on the Apalytic Geometry of Three Dimensions. Fourth Edition, p. 259. of Functions of Several Variables. 79 We have also to observe that the variables a, /8, y must satisfy the equations of conditions: ,2 I cP. s -Jl [9] I a^ + ^ + y2 _i. the first of which says that the direction which is determined through a, /3, y 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 directions-cosines of a straight line in space. Following the methods indicated in art. 26 we write [10] G=h\a}^t\^^ +2/^3. ya — g(aH/3' + r'— 1) + 26'(>^t« + ^2i8 + i^3V), and we then have (art. 26, [c]) to form the equations: dG 0, d G o, dG o. [11] da ^' d fi "' 8y from which we must eliminate a, /3^ y and e'. These equations are : -e) a+ F,2p + F^y + (F22—e)p+ Fj^y + F^ p + F^y where /^xm=/vx (X, /x=1, 2, 3). Through elimination we have •'' 11 ^ ) -' "* 12 > ■'' 13 •^21 . ^22 e, Ft2 ■^31 > -' 32 » -' 33 This is an equation of the second degree in e, and consequently gives us two valu'^s e^ and e^, which are maximum and minimum values, since both maximum and minimum values enter, as shown above. Multiply the three first equations [11] by a, yS, y respec- tively and adding the results, we have [12] j^F^e =0, ^F^e =0, + (^33— e)y +/^3e' =£?, = 0, F, F^ F. -'' o 80 Theory of Maxima and Minima [13] i^„ a' + /';2/8' + ^33 r' + 2/^i2 «y8 +2/^23/3 v ^2F^^ia=e. Hence, from [7] we have p h' Consequently the two principal curvatures at the point P have the values [14] [15] • 1 e. 2 Pz H and the coordinates of the corresponding centres of curvature are found from the formulae [16] '' 1\ h\ F. ^1 ^1 ^1 ) /'^i F. F, V ^2 ^2 ^2 In order to determine e, let us write Ai=(^'22— ^) {Fyi—e) — F^^ F>n= F^^F^^—Fi^iF^ — e), and form from these the corresponding quantities through the cyclic interchange of the indices. Equation [12] may be written in the form* F>nF,' + D^F} -V D^F} + 2D,,F,F^ + 2D^F^F, + 2D,,F,F,= o. Developing this expression with respect to powers of e we have [17] H^e' — Le + M=o, where Z = ^n^n + ^22 + /^33)-(^u/^lH/^22/'V + /^33^3^) + 2Fn F, F^ + 2/^^23 F^ F, + 2F,, F, F, and ^=(/^22^33-/'V)/^l'+(/^33^n-^V) F,' -{- {.-^ 11 ■'' 22 -''12 ) -^3 + (-''l2-''l3' '■''73 ^11) -^2 ■'^3 + (Fzi F^i—F^, F22) F^ F^ + (F^i F32—Fn F^^) F, Fj. * See Salmon, p. 257. of Functions of Several Variables. 81 From [17] we have at once the values of the sum and the product of the two principal curvatures, viz. — (see equation 15): [18] r '^ J 1 ^ ^L. V Pk92 ^' 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, thej^ are used extensively and with great advantage. In the case of -minimal surfaces^, which are characterized through the equation Pl + />2 = 0, we have L~o. This is therefore the general differential equation for mini- mal surfaces. Art. 28. Problem II. From a given point {a, b, c) to a given surface F{x,y,z)^o, draw a straight line whose length is a maximum or a minim-um. Write G^{x—ay+{y-by + {3-cy + 2)^F{x,y,z) {t) Then the quantities x, r, z, X are to be determined (see [c] art. 26) from the following equations: X — a + XFi=o, y—6+\F,= o, ^ (^..^^ z—c + XF3=. o, F {x, y, z) = o. It follows, since F-^, /s, F^ 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 {x-ay^{y-by+{z-cy See papers on this subject in the first numbers of the Mathematical Review. I 82 Theory of Maxima atid Minima is in reality a maximum or a minimum, we substitute x-\rU, y-\-v, z-\-w instead of x, y, z in the function G. The quantities u, v,w are of course taken very small. We must develop! the difference G{x+u, y + v, z + w) — G(x, y, 2) {in) 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/^12 ^z^-f 2/^23^^ + 2/^31 w u) ....... . .{iv) Since the point {x^ti, y^v, s+zu) must also lie upon the surface, the quantities Uj v, w must satisfy the condition F-^UArF^V^F^W^^O, {v) where the terms of the higher dimensions are omitted (See [8] of the present chapter). If we wish to determine whether the function »/» is continu- ously positive or continuously negative for all systems of values {u, V, vi) which satisfy equation (z^), we may seek the minimum or maximum of this function »/» under the condition that the vari- ables are limited besides the equation {y) to the further restric- tion [cf. (16)] that 'CP-^tP'^v? — 1=0 {vi) For this purpose we form the function «// — e^u^^iP'^vf- — \) -\-2^ {F-^u^ F^v ■{- F^ui) , . . . .{vii) and writing its partial derivatives with respect to u, v and w equal to zero, we derive the equations: / e-l\ e' \ F2iU+\F^— -^jv + F^i w + — F2=o, > {viu) F^iU + F^.v + IFs^— ^j w +-^ F2=o. Eliminating 2^, Z', Zf, —— from equations [z^] and {vin) we have A of Functions of Several Variables. 83 here exactly the same system of equations as in [12] of the pre- ceding problem, except tha there ——and e' stand in the place of e and e'. Denote the two roots of the quadratic equation in e, which is the result of the above elimination, by gj and e-^ and the corre- sponding radii of curvature of the normal sections by pi and pj, then, since e-\ has the same meaning as e in the previous problem: X ' H' Pi 1 e,-l Pi 1^ where the positive direction of the normal to the surface is so chosen that iY>- o . If for the position (x, y, 2) a minimum of the distance is to enter, then both values of the e must be positive , if a maximum, then ^1 and e^ must be negative. It is easy to give a geometric interpretation of the result just obtained : Let PN be the positive direction of the normal and A = (a, d, c). Then from («) 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 (w) A P=—X//. If the centres of curvature corresponding to pi and p^ be denoted by Ml and Mj , then is \// A P M,P = M^P e—1 \H e^ — 1 A P e,—l Hence ei= } and ^2 = ■iyi\ i If then J/i and M^ lie on the same side of P and if A lies be- tween Ml and M^ as in figs. 1 and 2, then the e's have differ- ent signs and there is neither a maximum nor a minimum. M-,A M^P M, A M^ P ^ Fig. 1. ^ P Ml A M, -> Fig. 2. M, *M^ P A Fig. 3. ^ A M^ M^ P ^^ Fig. 4. M, P A M, -^> 84 Theory of Maxima and Minima If J/i and M^ lie on the same side of P and A without the interval M^ Mt_, then a maximum will enter according as A starting from one of the centres of curvature lies upon the same side as P or not (see figs. 3 and 4). '^^-^ If the points M^ and Mj, lie on different sides of P and if A is situated within the interval M^. . . .Mj as in Fig. 5, then there is always a minimum. If^ however, A lies without the interval Ml .... yJ/j, then there is neither a maximum nor a minimum. In whatever manner M-^ and M2 may lie, if A coincides with one of these points, then one of the tv^^o values of e is equal to zero and then the general result stated at the end of art. 26 is applicable. The case may also happen here [see art. 8 of this chapter] that in the solution of the equations {ii) and {in) a singular point of the surface is found as the point P, at which Fi~o=Fi_^F^. For such a case we cannot proceed as above; since, there being no definite normal of the surface at such a pointy the deter- mination whether for this point a maximum or minimum really exists, cannot be decided. The general remark of art. 8 indicates how we are to proceed. of Functions of Several Variables. 85 CHAPTE)R IV. SPECIAL CASES. THE PRACTICAL APPLICATION OF THE CRITERIA THAT HAVE BEEN HITHERTO GIVEN AND A METHOD POUNDED UPON THE THEORY OF FUNCTIONS, WHICH OFTEN RENDERS UNNECESSARY THESE CRITERIA. 1. 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 [c], art. 26, of the previ- ous chapter cannot be effected without great difficulty, if at all, and therefore also the formation of the function <^ is impossible. It also happens, even if the function ^ can be formed, that the discussion regarding the coefficients of Ae=(? is attended with much difficulty. Moreover, the formation of the function <^ and the investigation relative to the coefficients of A e are very often unnecessary, since through direct observation we may in many cases determine whether a maximum or a minimum really exists. If it happens that the equations [c] admit of only one real solu- tion {^i.e. of a real system of values x^, x^,. . ■ .^„), then we may be sure that this is in reality the maximum or minimum of the func- tion. In the same way, if we can convince ourselves a priori that 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 correspond to the maximum of the function, the other system to the minimum value. The determination which of the two systems of values gives the one or the other, is in most cases easily determined. 2. As has already been indicated in the introduction (see art. 1, Chapt. I.), one cannot be too careful in the investigation whether on a position which has been determined from the equa- tions [a] and [c] of art. 26, Chapt. III. there really is a maximum ^ OF THE UN'iVERSlTY OF 86 Theory of Maxima and Minima or minimum, since there are cases in whicji one may convince himself of the existence of a maximum or 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 =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 ap- ply 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 = 180°, which is not true. The fallacy consists in the assumption of the existence of a maximum or minimum; it is not always necessary that an upper or lower limit be reached, even if one can come just as near to it as is wished. On this account also in our case the assumption of the exis- tence of a real maximum is not allowed without further proof. We therefore endeavor to give the existence-proof. For this pur- pose we must recall several theorems in the theory of functions*. 3. We call the collectivity of all systems of values, which n variable quantities x-^, X2,....x„ can assume^ the reahn (Gebiet) of these quantities, and each single system of values a position in this realm. If these quantities are variables without restriction, so that each of them can go from — oo to + oo , we call the realm considered as a whole (Gesamtgebiet) an n-ple multiplicity {n- fache Mannigfaltigkeit). For example, the points of a straight line in the plane form a simple (einfach) multiplicity. The points 'Cf. Chapt. I., especially arts, 16 et seq. of Functions of Several Variables. 87 in a plane, a double njultiplicity; in space, a triple multiplicity. All straio^ht lines in space form a \-pie multiplicity. For consider two planes, then in each of them a point is determined by two quantities, and a straight line by two points, i. e., by four quan- tities. Also all surfaces of spheres in space form a \~ple multi- plicity, since every sphere needs for its complete determination the three coordinates of the centre and the radius of the sphere. If the quantities are connected with one another by equations of condition, so that m of them (say) may be considered as inde- pendent, then we call the collectivity of all positions an m-ple multiplicity, or a structure* of the /wth kind in the region of these n variables. For example, a straight line in space is a structure of the first kind in the region of three variables. If x^, X2,....x„ are in- dependent of one another, then we say a definite position (aj, a2, ■ . . .a„) lies on the interior of the realm, if these positions and al- so all their neighboring positions belong to this region ; it lies up- on the boundary of the realm, if in each neighborhood as small as we wish of this position, there are present positions which be- long 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-^, X2,....x„ are subjected to nt equations of condition, then we may express these in terms of n — m independ- ent variables Ui, u-^, .... ^„_m. and the same definition may be ap- plied to these variables. 4. 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 k, although there is a value of the variable that comes as near to k as we wish. *See Chapt. I., art. 7 and art. 9. See also Chapt. I., art. 16. tDini, Theorie der Functionen, p. 68. See also a paper by Stolz, B. Bolzano's Bedeutung in der Geschichte der Infinitesimal Rechnung. Math. Ann., bd. XVIII. 88 Theory of Maxima and Minima (2)* In the region of n variables x^, X2, . . . . x^, suppose we have an infinite number of positions defined in any manner, — let these be denoted by (yx[, x^., . . . .x'„), — further suppose that among the positions we have such positions, that x'„ can come as near to a fixed limit a„ as we wish. Then we have in the region of the quantities x^, x^, . . . .x^ always at least one definite position (^i, ^2, . . . .a„) of such a nature that among the definite positions (x[, x'^, ....x'„) there are always present positions that lie as near this position as we wish, so that therefore, if 8 denotes a quantity arbitrarily small, I ^\— «x I f(x.) ^^^ ^^^ show, if {xi, X2, . . . . x^) is any *Biermann, Theo. der An. Funk., p. 81; Serret, Diff. et Int. Cal., p. 26. of Functions of Several Variables. 89 Th^ case of a minimum. Cose of Q£>\/'npfof'C approach UK) < fix) < /(X,) Fig. 3. Hittif"^^ on tfit JlmWiT^ poJltton X,. fix,) >flx) >fix^. position on the boundary of the realm and if x„+-^ denotes the corresponding val- ue 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, of course, the position, which we are considering here, cannot lie upon the boundary, and it is clear that the limit- ing value of the function can be assumed for a definite position within the inter- ior, since the function varies in a con- tinuous manner. The analog is, of course, true for a minimum. If, how- ever, 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 existence of a maximum or minimum; the position {a-^, a2, . . . . a^) would then lie on the boundary of the region, and there might be an asymptotic approach to the limiting value a„+i without this value being in reality reached (cf. Chapt. I., art. 11). This, however, need not necessarily be the case. The figures give a plain picture of what has been said for the case y^/(^x), where x is limited to the in- terval {Xi. .X. .X2). IT IS POvSSIBLE THAT THE DIPPERENCE F{ai+hi,a2 + k2,. . .a„ + h„) — F(ai, a^, a„) IS NEITHER POvSlTlVE NOR NECzATIVE BUT ZERO ON THE POSITION («!, a-i,, (a!„) WHICH IS TO BE INVESTIGATED. 6. We shall now consider a case which is not included in the previous investigations but may be in a certain measure re- I 90 Theory of Mayiima and Minima duced to them: The definition of the maximum or minimum of a function consists in the fact that the difference /^(ai+>^i, a2+^2. a„+A„)— /^(ai, ^2, a„) (0 must be continuously negative or continuously positive. There are cases where a maximum or a minimum does not appear on the position («!, ^2, . . . .a„) in the sense that the above difference must be positive or negative, 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 6' whose area is a maximum, — a problem which we shall later discuss more fully (see art. 10). 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 /?=3, and (xi,j^i), (^2.J^'2) ^J^d (x2,y2) 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 S^V{x^—x,y + {y2—}'ry+ ^{x^—x^y^iyr-yif + v\x—x^y+{}'—yiy 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 (*') is not positive or negative but zero. 7. Such cases, however, may be reduced to maxima and minima proper, if we choose arbitrarily some of the variable quan- tities. In the special example of the preceding article 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 ^-axis, so that we may write of Functions of Several Variables. 91 x-^=yy=yT:=o. If we agree that the triangle is to lie above or below the ^-axis, the problem is completely determinate. 8. In so far as the necessary conditions for the existence of a maximum or minimum are concerned, we may proceed in pre- cisely the same manner as we have hitherto done; since under the assumption that there are no equations of condition, we have 0=1 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 de- rivatives must all vanish. Since, if the sum ^Aa/^o(«i, ^jt • • • -(^^ had (say) a positive value for h^^^c-^, hi=.CT_,. . . .h^^c^, then we could place ha. ^Cah, and then choose h so small that the sign of the right-hand side of {if) depends 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 showed 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 A's through the remaining, and then proceed as we have just done. The required systems of values {xx, x^,- ■ ■ ■ ^„) will there- fore be determined from the same equations as before. 9. If we have found a system of values of the ;i;'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 char- acterized by the condition that the difference (*') vanishes identic- ally for them. This is just the condition that made impossible the former criteria, by means of which we could decide whether a maximum or minimum really entered on a position («i, fZz, . . . .a^ that was determined through the equations in Xi, x^,.... x^. I 92 Theory of Maxima and Minima 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 example. 10. Problem. Among all polygons which have a given number of sides and a given perimeter , find the one which con- tains the greatest surface-area. We see at once that the problem proposed here is of a some- what different nature than the problems of arts. 27 and 28 of the previous chapter, since the existence of the maximum value of the function is no longer the question as was proposed in art. 4 of Chapt. II. and held as fixed throughout the general discussions. For, if the definition of the maximum is such that the function on the position ((Xj, a^,....a^ 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 consequently its area. Therefore, only a maximum of the area can enter in the sense that the periphery remaining the same, an increase in the area of the surface cannot enter for an infinitely small sliding of the end- points. We consequently cannot apply our general theory with- out further discussion. 11. Let the coordinates of the n end-points taken in a defin- ite order be The double area of a triangle, which has the origin as one of its vertices and the coordinates of the other two vertices x^, y-^ and Xi^,y-i is, neglecting the sign, determined through the expression x^y^ — x^y^. In order 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 falls together with the length 01. We call that side of the line 01 positive which corresponds to the positive direction of the K-axis: The double area of the triangle 012 is to be counted posi- tive or negative, according as it lies on the positive or negative side of the line 01. of Functions of Several Variables. 93 If the point lias the coordinates x^,, y^, the double area of the triangle is 2 Aoi2 = (;t;i— ;ro) (>'2— Jo)— (ji^i— To) {x^—x^) , where the above criterion with reference to the sign is to be ap- plied. » For the polygon we shall take a definite consecutive arrange- ment of the points (1, 2,....n) and besides shall assume that no two of the sides cross each other. The last hypothesis is justifi- able, 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 perimeter as the first polygon incloses a greater area. Within the polygon take a point ^ (■^0. j^'o) 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,. . . .n) and mark the intersection of a side by the straight line with -'- 1 or — 1, according as we pass from the negative 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 de- scribed in the positive direction, in the second case in the negative direction. It may be proved* that, whatever point be taken as the point within the polygon^ 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. 12. The double area of the polygon is 2F^{x—x^) {yz—y^—{x-r-x^) (jJ'i— To) + {x^—x^) {y^—y^) — (^3— ^o) iy2—yo)+ ■ ■ ■ •+(x„—Xo)(yi—yo)—{x^—Xo)(yn—yo); or, 2F=x, yr-x^y, + X2 yr-x^yz +....+ x^y^-x^y^ (a) * The proof is found in Cremona, Elementi di geometria projetiva. Rome. 1873. 94 Theory of Maxima and Minima where the positive or negative sign is to be taken according as the polygon has been described in the positive or negative direc- tion. We may, however, eventually bring it about through re- verting the order of the sequence of the end-points that the ex- pression 2F is always positive. 13. Suppose that this has been done. The function 2F is to be made a maximum under the condition that the periphery has a definite value 6". We may write: 51.2-1-5^,3+ . . . -|-5„_i,„-h5„.i=6',_ (/3) where 5\_i.x = V {x\—x\-^^ -j- (^\— jx-i)^ (y) Form the function G=^2F At &{s^,z + Hi^ +5„,i — S), (S) and placing its partial derivatives =0, we have dG , {x\ — x\+x , ^>- — x.\^\ y xh 1— r x-i + ^ + = o, . d X\ \ 5 \+i, \ 5 X_]. X (0 dy\ (X=l, 2, . . . .«; however for X^n, we must write: \ + l = l). Take in addition the equation (/8) and we have 2n + l equations for the determination of the 2n + l unknown quantities Xi, Xj, . . . . X„, jt'i, JK21 • • • -jl^ni ^• 14. In order to come in the simplest manner to the desired result from these equations, we adopt the following mode of pro- cedure. If we write £;\ = {x\ — ;t;x_i) + i (jFx^>'x^i), (0 then ^x , geometrically interpreted, represents the length from the the point X — 1 to the point A. both in value and in direction. If further we write ^^=(x\—x\-i) — i (^'x— jj'x-i) (v) then is ^x- ^x^-'x-i.X • ^^^ of Functions of Several Variables. 95 Multiply the first of equations (c) by i and subtract from the re- sult the second, then owing to (C) we have 2X^2\ . + ..•( •SX-i.X -Sx x^ or .(0 Now multiplying the last two equations together, we have from {&) •^x-i,x— -^x.x+i and therefore 15. Since 5 ^-^i. v is an essentially positive quantity, it follows that 5X-:.X= 5x,x+i, {k) and consequently the sides of the polygon are all equal to one an- other. Hence each side= — , and we have from (i) n 2\+i e^n + s -const. z\ = — e ^ n z\ em — 5 If we write where ^x denotes the angle which s x-i, x makes with the A'-axis, then e^*^^+-'^^)'=const., ' or (^x+i — ^x=const.; (X) that is, all the angles of the polygon are equal to one another, and consequently the polygon is a regular one. Weierstrass thus showed that the conditions which are had from the vanishing of the first derivatives can be satisfied only 96 Theory of Maxima avd Minima by a regular polygon; that is, if there i.s a polygon, which, with a given perimeter 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. 16. In order to prove the existence of a maximum we must apply the method given in arts. 2-5 of the present chapter. We see that a limit is given 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 peri- meter 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 conse- quently there must be an upper limit for this area, which may be denoted by /%,. It may yet be asked whether this limit can in reality be reached for a definite system of values. The variables Xi, yi, X2, y2,.... x„, jK„ being limited to this square, there must be among the positions (x^, y^, X2, y2^ ■ ■ • • ^ni ^n) which fill out the square a position {a^, d^, a2, ^2- • • • -^n^ ^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 itpper limit. We may as- sume 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 (fli, ^1, (22, 62,... .a„, d„) must necessarily be equal to /%,. Since, if this was not the case^ the inequality must also remain, if we sub- ject the points a^, di, 02, ^2> • • • • <^n» ^n to an infinitely small variation; and on account of the continuity of F, it would not be possible in the arbitrary neighborhood of (^i, d^,... .a„^5„) to give positions for which the corresponding area comes arbitrarily near the upper limit Fo- This, however, contradicts the conclusions previously made. Hence, all n corners with a given periphery not only ap- proach a definite limit with respect to their inclosed area but this limit is in reality reached. Since further the necessary conditions for the existence of a maximum have given the regular polygon of of Functions of Several Variables. 97 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. We have now given what we consider the general Weier- strassian theory of maxima and minima, and in the sequel we shall discuss a few special problems of maxima, and minima. CASES IN WHICH THE SUBSIDIARY CONDITIONS ARE NOT TO BE REGARDED AS EQUATIONS BUT AS LIMITATIONS, 17. 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 re- strictions or limitations. For example, let a point in space and a function which de- pends upon the coordinates of this point be given. Further let the point be so restricted that it always remains within the in- terior of an ellipsoid; then the restriction made upon the point is expressed through the inequality — a^ tr c^ We have accordingly such limitations when a function of the vari- ables is given which cannot exceed a certain upper and a certain lower limit. We make such a restriction when we assume that a function fx shall always lie between fixed limits a and b. 18. This limitation which consists of two inequalities \A a0,/„,,2>0, /m + r>0, where the/'s are functions of x^, x^, x„, then we may solve this problem, if instead of the r last restrictions, we introduce the the following limitations: The problem is thus reduced to the one of finding among the sys- tems of variables x^, X2 x^+,, those systems for which F is a maximum or minimum. , 20. Examples of this character occur very frequently in mechanics. As an example, consider a pendulum which consists of a flexible thread that cannot be stretched, then 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 dis- of Functions of Several Variables. 99 tance cannot 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' principle all probletns of mechan- ics may be reduced to problem, s of m.axim^a and minim,a. gauss' principle. 21. 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.^, m-2, . . . . ;w„ . Let the motions of the points be limited or restricted in any manner, and suppose that the sys- tem moves under the influence of forces that act continuously. For a definite time, let the positions of the points and the com- ponents of velocity both in direction and magnitude be deter- mined. The manner in which the motion takes place from this period on is determined through Gauss' Principle: Let Ai, A2,. . . .A„, be the positions of the points at the moment first considered; Bi, B2,....B„, the positions which the points can take after the lapse of an infinitely small time t, if the motions of these points are free; Ci, Cj, . . . . C„, the positions in which these points really are after the lapse of the same time t; and finally let C/, Cj', . . . . C„' be the positions which the points may also possibly have assumed after the time t, when the conditions are fulfilled. If we form ^ m^v By Cv and ^ ^^^ Bv Cj , '-I "=1 it follows from Gauss' Principle, that from t=c? up to a definite value of T, the condition [1] 2 ^>' BpCy < 2 ^'' B' ^^ "=1 "=1 is always satisfied, that is, ^ ^•' ^^ ^^ "lust always be a mini- mum. "=1 100 Theory of Mwsiima and Minima 22. In order to make rigorous conclusions from Gauss' Prin- ciple, which was briefly sketched in the preceding article, we shall give a more analytic formulation of it : For this purpose we de- note the coordinates of Ay by Xy , yy, Zy, the components of the velocity of A by x\ y'^, z'^, the components of the force acting upon Ay by Xy, Yy, Zy. The coordinates of By are therefore Xv + TX'y + —Xy,yy+ T / y + — K^ , Zy + TZy + —Zy \ and from Taylor's theorem the coordinates of d- are _2 _2 2 consequently we have * [2]2^v^^;^=2^''{«-^v)'+(X'-n)'+«-^v)^}^V.... Instead of x'^, however, (see preceding art.) other values may possibly enter, say x''^^^, , so that we have [3] ^myByC\^ y=\ It follows from Gauss' Principle that the difference of the sums [2] and [3] must always be positive. Hence [4] c.>2^''{2[^''«-^,) + ^,(X'-n)+C,«-Z/)] + "=1 that is, the quantities x'^ , y'J , z'J must be such that the sum [2] is a minimum. of Functions of Several Variables. 101 Hence among all the xi^ , y'^ , z'l which are associated with the conditions of motion, we must seek those which make [2] a minimum. 23. We have reached our proposed object, if we can show that the ordinary equations of mechanics may be derived from Gauss' Principle. If there are no equations of condition present, then clearly [2] is only a minimum when If, however, we have equations of condition, for example, /"(;i;, y, s)=^o, then these must hold true throughout the whole motion. They may therefore be differentiated. We have in this way equa- tions in ^^. -^ and -^ • Differentiate again and we have at at at equations in x'^, y'^ and s'^ . Hence, in conformity with the rules that have been hitherto found for the theory of maxima and minima, the quantities ^v' X'' K are to 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. TWO APPLICATIONS OP THE THEORY OP MAXIMA AND MINIMA TO REALMS THAT SEEM DISTANT PROM IT. I. Cauchy's proof of the existence of the roots of algebraic equations. 24. An interesting application of the theory of maxima and minima (cf. art. 5) is Cauchy's existence-proof of the roots of an algebraic equation: Let/C^) be an integral rational function of a real or com- plex variable z. The function becomes, when z is put ~-=^ x -|- i y, f(z)=(l)(x,y) + iy\i (x,y), where (f> and \jf are real functions of the real variables x and y. Hence ^ may be arbitrarily great; but, if x^+y^~-the square of the distance of the positiion (x, y) from the origin — is infinitely large, then (f>^ + ^ is infinitely large. 102 Theory of Maxima and Minima Draw a circle about the orij^in ; then the function 4'^ + -^ is de- fined both for the interior and the boundary of this circle. We may choose the radius of the circle so great, that the function for every point of the circumference is greater than it is for any arbi- trary point within the interior. The function <^^ + i/»^ must also have a lower limit, and since it is a continuous function, there must be a position within the interior of the circle where (^^4-\|(^is a minimum. Let this be the case for the point {x^t^y^)- It is then easy to show that ^ + rji^ can have no other value for (x^,, ^o) than zero. /"(^) is also =o for {x^, jj'o); and therefore ^0=^0 + ^J^o is a root of the equation /"(^) = o. II. Proof of a theorem in the Theory of Functions. The reversion of series. 25. The theorem which was stated in arts. 14-15 of Chap- ter I., and of which an application has already been made in arts. 4-6 of Chapter III., is of great importance in the Calculus of Variation. If the n equations exist between the variables x^, Xj, . . . .x„, 7u yv--x„- yi= a.xx Xx + «i2 ^2 4- + «1„ ^n + Ai' \ yn-= a^nXi + a„ [1] t-„2 X2 + + a„„ x^ + A„, where the coefficients on th^ right-hand side are given finite quantities, and the j\^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 determinant of the nth order of the linear functions of the x's, which appear in [1], [2] A = ^11' ^12' '^lii ^21' '^22> '^211 ^ 11 1 1 a „7 , a nn of Functions of Several Variables. 10.^ 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]. 26. As an algorithm for the representation of the series for the ;c's, we made use of the following methods (see arts. 14 and 15, Chapt. I.): We solved the equations [1] linearly by bringing the terms of the higher powers of the x'^ on the left-hand side, and thus had X. A f(^.-X.)+-^"(^-X)^ ■•+^(^..-X.)- x„ A, A <-X!).t(--XO +■■ A^ A ■(^"-X") ; where Axp. denotes the corresponding first-minor of a\^ in [2]. It is seen that in ofeneral M^ii [3] x\^ A\^ A "^^ ^V ^, L"^l' '''2! .Tm ■^^' A ^=1 ^=1 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 approximations already found, and reduce ever3'thing 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 dimension, and we have the third approxi- mation, etc.; we may thus derive the ;i;'s to any degree of exact- ness required. Since A is found in all the denominators, the development converges the more rapidly the greater A is. 27. In what follows we shall assume that the quantities on the right-hand side of [1] are all real and that we may write L = Ax 2 + Ax, ■A\^+..:. . (^=1,2, n) 104 Theoiy of Maxima and Minima where A\i is a homogeneous function of the /th degree in ;c„ X2, . . . x^ and consequently : ^^ O Xi 0X2 + 3x,^+3x2 ^^^^ d Xi d X, 2 + Ax, ^+ 4X2 ^^'' + 2x„ a^\2 dx„ + 3x, 1 dAx, dx„ + Ax„ dAx, -. d Xi d X2 ' -1- ; (\=l,2,....n) or, A^^^^i Am + ^2Am-1- f^nAx" • The quantities Av (^=^» 2, . . . .n; )u, = l, 2, . . • .n) are continu- ous functions of the x's which become infinitely small with the ;c's. The system of equations [1] may be then brought to the form [1] (X=l,2,..../^) The theorem of art. 25 in this modified form, may be expressed as follows : (1) // is always possible so to fix for the variables x,, X2,... .x^ and y\, y2, ■ ■ ■ -y^, limits g,, g2,--- -gn (^'>^d h,, ^2> ■ • • • ^^n that for every systein of the y's for which \y\\ . M=l (X=l, 2.....;^) of Functions of Several Variables. 105 where the \\ii are cofftinuous function of the y' s which become infinitely small with these quantities. To prove this theorem we make use of the theory of maxima and minima. 28. If we give to the y\ the value zero, the equations [1«] can only be satisfied if their determinant vanishes, that is, when [4] a\)x \x^ I =^' (X, ^^=1, 2 n) except for the case where the ;t;'s vanish. For sufficiently small values of the Jt's, the determinant [4] is not very different from the determinant [2]. We may therefore determine limits g for the x'^, so that [4] cannot be zero unless [2] is also zero. A=.o is, however, by hypothesis excluded. Accord- ingly the y s can only be zero in \\a\ when all the ;p's vanish. The ;c's are thus confined within fixed limits which may be re- garded as the boundaries of a definite realm. 29. Again we write M=« (\=1,2.....^) /»=1 and consider the function [6] x^l -'' X ^'^'^ '''2' • • • • ^ni )• In 6* we will write for the ;r's all the systems of values, where at least one x lies on the boundary of the realm in question. We re- gard as the realm of the x'^ the totality of the x'% for which a\ M + y^ \ ^ is only zero, when (X,/*=l, 2,....«) a\i ^o\ it follows then that [4] is not zero, since \a\y\ is by hypothesis different from zero. R 106 Theory of Mai^hna and Minima In order to make this clear, it may be mentioned that as the boundaries of the realm, we must consider the totality of the ;ir's where at least one of the ;t:'s reaches its limit. For the limits^ that is, when one of the x''& reaches its limit, there is no system of values of the x'^ for which the function [6] vanishes^ since the function can (as follows from definition [5] of the F\ and the considerations of art. 28) only vanish, if all the jk's and consequently all the x'% vanish. There is then a lower limit G which is difFerent from zero for the values of [6], which correspond to a system of values {x-^, Xj, . . . .x„) oi the limits. 30. We come next to the determination of the limiting val- ues of the y s. For this purpose we consider [7] 2( X=l -/* X \Xi, Xj, .... x^ ) - y\ y 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 x=« x=» ^{FK-yxy=.S—2^Fx_yx 1-2 ^ X=i Since i S is at all events greater than /"'a. and consequently Fk ) S = <1, it follows that li=n f-=n /*=« /*-i Ai=i M=l ii=n H=^n M=l M=l of Functions of Several Variables. 107 where the A's are the limits of the ^s. From this it results that 2 ^F, -y, Y> s- 2 1 Ts-" 2 ^'^ + 2 yl ' Ml M-^l Ml and, consequently, for a greater reason [8] 2 (^'m->'m)^>^V-2i^1^2^'^- M-l M-1 The limits have to 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''^, which belongs to the boundary, does not become arbitrarily small, but always remains greater than a certain lower limit (see the preceding article). The expression, however, on the interior of the realm of the ;i:'s may be arbitrarily small, viz.: when Xi=X2 = . . . . =x„^o. For this system of values the left-hand side of [8] is equal to ^=11 M=l 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-^=Xi= .... =x^—o^. Hence the position for which the expression [7] ^5 a tnini- ■mum mttst 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 posi- tion where [7] has its smallest value. 31. In order to find the minimal position of [7] which was shown to exist in the previous article, we must, following the rules laid down in Chapt. III., differentiate the function [7] and put the first partial derivatives ='''=2 («"? + A-p)^'' (i'=:l,2,....n) 110 Theory of Maxima and Minima may be derived in the manner given in art. 26: If we write «.'/'+ A-p =^'' (v, p=l, 2,. ...«) the linear solution of the equations \la\ is x ^.^-^li^v.. (P=l,2,... .^) where A'vp denotes the corresponding subminor of A' . Now A' is a definite quantity which lies within certain finite limits ; the same is also true of — , . ^';. p is found in a similar manner. Hence A A' the quantities — ~~~ are finite quantities which lie between defin- ite limits ; and therefore, if the y s become infinitely small, the ;i;'s will also become infinitely small ; that is, those systems of values of the x'%, which satisfy the equations [1] under the named con- ditions, are — as has also been shown in art. 28— so formed that they become infinitely small with the jv's. We can now show that the ;t;'s are continuous functions of the y s. Let (^1, ^2^. . . .^„) be a definite system of values of the jf's and let the system (^i, a^, a„) of the ;i;'s correspond to this sys- tem of the ys. If we then write ■ [13] (jx=^x+,x the system of equations [la] or [1] goes into (X=l,2 n) of Functions of Several Variables. Ill or (X=l,2,....«) Developing this expression according to powers of the ^'s, we have /i=« [Ic] TJX = 2 ^'^-^ ^-^ ' (X=l, 2,. /*-=! .«) where the y4'x/x are functions of the i?'s and f's. If now the f's are infinite!}' small, we may limit the A'\fL to the first derivatives oiFx. In this case we denote the coefficients of [Ic] by Ax/x, so that A\^i = -^— ior{xi=-ai, X2=a2, ^n=«n), d Xpi (X, fc=l, 2,....n) and the determinant of the equations [1] goes into dFx dxn for (xi—a^, Xi=a2, . . . .x^^aa)- (X,,i=l,2,.. .n) If the x'^ 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 infinitely small values of the |^'s must correspond to infinitely small values of the •rj's. This means nothing more than that the functions of the x'^ are continuous functions of the ys, 34. Our 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 x'^, of which we already know that they exist as functions of the ^s and vary in a continuous manner with them, we may now likewise prove that they, considered as functions of the ys, have derivatives which are continuous functions of the y&. 112 Theory of Maxima and Minima We have then proved that the x'^, are such functions of the y^ as the ys are of the x^. The proof in question may be derived from the following con- siderations: — If from [Ic] we express the ^'s in terms of the t^'s, we have ()a=l,2,....^) The ^^ are continuous functions of the f 's, and the ^'s are A continuous functions of the tj's. Hence —^ may be represented as continnous functions the tj's. If the Tj's become infinitely small, then the ^'s become infin- itely small, and we have limits for ^-^• If we have in general a function of the n variables x-^, x-i, . . . .x„, and if we consider the difference it is seen that it may be written in the form 2(^x+^x)^^ where the H\ depend upon the ^'s and become infinitely small with these quantities, and the b\ are the partial differential quotients of f with respect to x\ for the system of values {a^, Uj, . . . .»„) . From this it results the ;i;'s are not only continuous functions of the jf's, but also that the derivatives of the first order of the functions exist. We have indeed the derivatives of the first order, if in the •essions — -^ we write the ^'s equal to zero. The quantities — ^ , however, become then in accordance with of Functions of Several Variables. 113 [Ic], the quantities which we would have in [Ic], if we had at first written A\^ instead of A'xi,.. But the quantities A\iL are continuous functions of aj, ^2, .. . .a^. We may therefore say that the differential quotients — ^ are continuous functions of the variables x. 35. For the complete solution of the second part of the theorem in art. 26, we have yet to show that the expressions [3^] may be reduced to the form \Za\\ For this purpose we must bring the quantities — ^ in [35] (art. 32) to the form where b\u. is the value of A' A\y. A' when all the ;i;'s are equal to zero. \\y. is a function of the ;«;'s, but the ;i;'s are functions of theys, so that \\^ is a function of thejv's which vanishes when they vanish. We may therefore in reality write [35] in the form [3a] x=« ^M-=2 ('^^'' + Yxm) y>^- (/i-l, 2,....«) [3«] X=l 36. There may arise cases in which we know nothing further of the functions F\ — as was assumed in art. 26 — than that they are real continuous functions. We cannot then conclude, for example, that the x'^ may be developed in powers of theys; but we may reduce the equations to the form [3c] and show that the equations [la] are solvable. The theorem which has been proved is of great importance 114 Theory oj Maxima and Minima when applied to special cases, even for elementary investigations. If, for example, the equation /"(;?;, ;k)=o is given, then it is taught in the differential calculus hovi^ we can find the derivative of y considered as a function of x. 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 /{x,y)=o has no singu- lar points; and, if the equation is satisfied by the system (^oijo). we may write x=X(, + ^, y^y^j^-q. We have then /{xQ + ^,yo+'n) =0, and we may prove with the aid of the theorem in art. 27 that Tj is a continuous function of ^ 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. ^ OF THE UNIVERSITY OF JFORti^ IN PRESS : Lectures on the CALCULUS OF VARIATIONS. By Professor Harris Hancock. y m UNIVERSITY OF CALIFORNIA LIBRARY BERKELEY Return to desk from which borrowed. This book is DUE on the last date stamped below. 2%'5-?! Ill f^ 18 1953^8 ^64-0 HREcrr 7'66-iPM a&m DEPT. 20-' ceb'^*--' 22Way'64DVI/ RECD UD RECD LD NOV 18 1957 Rfe.w ^ LD 1 1 JAN 2 6 ^OGG ^ S LD 21-100m-7,'52(A2528sl6)476 I ^ "on