liiir ;|H 111 llli \SP.^5^ CORNELL UNIVERSITY LIBRARY GIFT OF Est3te of Willipin E. Patten MATHEMATICS CORNELL UNIVERSITY LIBRARY 3 1924 051 165 938 DATE DUE GAYLORD PRINTED IN U S A. ■TtT Cornell University Library The original of this book is in the Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924051165938 THE NEW HAVEN MATHEMATICAL COLLOQUIUM LECTUEES DELIVERED BEFORE MEMBERS OF THE AMERICAN MATHEMATICAL SOCIETY IN CONNECTION WITH THE SUMMER MEETING, HELD SEPTEMBER 5tH TO 8th, 1906, UNDER THE AUSPICES OF TALE UNIVERSITY BY ELIAKIM HASTINGS MOORE ERNEST JULIUS WILCZYNSKI MAX MASON NEW HAVEN YALE UNIVERSITY PRESS 1910 Press of the new era printing company lancaster. pa PEEFACE. The American Mathematical Society held its Fifth Collo- quium in connection with its Thirteenth Summer Meeting, under the auspices of Yale University, during the week Sep- tember 3-8, 1906. At this Colloquium the following courses of lectures were given : Professor Eliakim H. Moore, of the University of Chicago : three lectures on " The Theory of Bilinear Functional Operators." Professor Ernest J. Wilczynski, of the University of Cali- fornia : three lectures on " Projective Differential Geometry." Professor Max Mason, of Yale University : three lectures on " Selected Topics in the Theory of Boundary Value Prob- lems of Differential Equations." The present volume contains these lectures as prepared by their respective authors for publication. The aim and scope of the CoUoquia of the American Mathe- matical Society are well set forth in the following extract taken from an ofBcial circular issued prior to the first Colloquium : * " The objects now attained by the Summer Meeting are two- fold : an opportunity is offered for presenting before discrimi- nating and interested auditors the results of research in special fields, and personal acquaintance and mutual helpfulness are promoted among the members in attendance. These two are the prime objects of such a gathering. It is believed however that a third no less desirable result lies within reach. From the concise, unrelated papers presented at any meeting only few derive substantial benefit. The mind of the hearer is too un- prepared, the impression is of too short duration to produce accurate nowledge of either the content or the method. . . . Positive and exact knowledge, scientific knowledge, is rarely increased in these short and stimulating sessions. *Cf. Bull. Amer. Math. Soo., ser. 2, toI. 3 (1896), p. 49. iii IV PREFACE. "On the other hand, the courses of lectures in our best uni- versities, even with topics changing at intervals of a few weeks, do give exact knowledge and furnish a substantial basis for reading and investigation. . . . " To extend the time of a lecture to two hours, and to mul- tiply this time by three or by six, would be practicable within the limits of one week. An expert lecturer could present, in six two-hour lectures, a moderately extensive chapter in some one branch of mathematics. With some new matter, much that is old could be mingled, including for example digests of recent or too much neglected publications. There would be time for some elementary details as well as for more profound discussions. In short, lectures could be made profitable to all who have a general knowledge of the higher mathematics." The colloquia preceding the fifth are the following : I. The Buffalo Colloquium, 1896. (a) Professor Maxisie BOchee, of Harvard University : Linear Differential Equations, and their Applications. This colloquium has not been published, but several papers appeared at about the time of the colloquium, in which the author dealt with topics treated in the lectures.* (6) Professor James Pierpoxt, of Yale University : Galois's Theory of Equations. This colloquium was published in the Annals of Mathematics, ser. 2, vols. 1 and 2 (3 900). II. The Cambridge Colloquium, 1898. (a) Professor William F. Osgood, of Harvard University : Selected Topics in the Theory of Functions. This colloquium was published in the Bulletin of the Amer. Math. Soc, ser. 2, vol. 5 (1898), p. 59. (6) Professor Arthur G. Webster, of Clark University : The Partial Differential Equations of Wave Propagation. * Two of these papers were : Regular Points of Linear Differential Equations of the Second Order, Harvard University, 1896 ; Notes on Some Points in the Theory of Linear Differential Equations, Annals of Math., vol. 12, 1898. PREFACE. V III. The Ithaca Coi.loquium, 1901. («) Professor Oskae Bolza, of the University of Chicago : The Simplest Tyj^e of Problems in the Calculus of Variations. Published in amplified form under the title : Lectures on the Calculus of Variations, Chicago, 1904. (b) Professor Ernest W. Bro\vn, of Haverford College : Modern Methwh of Treating Dynamical Problems, and in Paiiicular the Problem of Three Bodies. IV. The Boston Colloquium, 1903. («) Professor Henry S. White, of Northwestern University : three lectures on Linear Systems of Curves on Alge- braic Surfaces. (6) Professor Frederick S. Woods, of the Massachusetts In- stitute of Technology: three lectures on Forms of Non- Euclidean Space. (c) Professor Edward B. Van Vleck, of Wesleyan Univer- sity ; six lectures on Selected Topics in the Theory of Divergent Series and Continued Fractions. This colloquium was published in full, under the title : TTie Boston Colloquium. Lectures on Mathematics. The Macmillan Co., N. Y., 1905. The expense incurred in publishing the present volume was defrayed by a grant from Yale University. The editor wishes to express his thanks to The New Era Printing Co. for their efforts to secure the typographical excel- lence of the present volume, and for their unfailing patience in spite of many unavoidable delays in printing. TABLE OF CONTENTS Introduction to a Form of General Analysis BY Eliakim Hastings Moore Pages Preface 1 Fart I. Fundamental Closure and Dominance Froperties of Classes of Functions of a Genei-al Variable. §§ 1 Introduction 13 2 Classes and properties of elements. General notations 15 3 Properties of and propositions concerning ele- ments. General notations 18 4- 5 Functions on the class ^ to the class 21. . . . 24 6-13 Relative uniformity. Range and scale of uni- formity 27 14-17 Classes of functions. Closure properties ... 36 18—26 Dominance properties. Relative uniformity as to a class of functions 39 27 Extension of classes of functions with respect to certain closure properties 53 28-42 Excursus. — Extension of classes with respect to certain closure properties — in the theory of classes in general 55 43-44 The extensions: TIal; ^^ = CSUlU; Mijf = (2liji){x)y = fix) ^ 207 12. The transverse vibrations of a cord 209 13. The existence of normal functions for the equation y" + \A{x)y = 210 14. The expansion of a function in terms of normal functions 218 INTRODUCTION TO A FOEM OF GENERAL ANALYSIS. BY ELIAKIM HASTINGS MOORE. Preface. Especially during the last decade the study of Integral Equations has brought to light numerous analogies between the w-fold algebra of real w-dimensional space and the theory of continuous functions of an argument varying over a finite interval of the real number system and the theory of certain types of functions of infinitely many variables. These anal- ogies have their root in the classic analogy of a definite integral to an algebraic sum. We lay down a fundamental principle of generalization by abstraction : The existence of analogies between central features of vari- ous theories implies the existence of a general theory which underlies the particular theories and unifies them with respect to those central features. This formulation was the dominant note of my series of lectures : On the Theory of Bilinear Functional Operations, at the Collo- quium of the American Mathematical Society, held in Septem- ber, 1906, in New Haven under the auspices of Yale University. In the note : On the theory of systems of integral equations of the second kind* read December 29, 1905, before the Chicago Sec- tion of the American Mathematical Society, I gave a prelimi- nary indication of the point of view of the Colloquium Lectures. *Inabstract, Bulletin of the American Mathematical Society, ser. 2, vol. 12 (1906), pp. 280, 283-4. — A brief synopsis of the Colloquium Lectures will shortly be published in the Bulletin . 1 2 E. H. MOORE. The subsequent development of that point of view led to the determination of a form of General Analysis * to which the present memoir is an introduction. Intending in this preface to give a summary characterization and a precise definition of this form of General Analysis, for the purpose of orientation we first call to mind a few of the types of variables p, of ranges (classes, sets, ensembles, Mengen) ^ of the variable p, and of functions fj, of the variable p on the range ^ recognized in cur- rent Analysis. From the linear continuum with its infinite variety of func- tions and corresponding singularities G. Cantor developed his theory of classes of points {Punktmengenlehre) with the notions : limit-point, derived class, closed class, perfect class, etc., and his theory of classes in general (allgemeine Mengenlehre) with the notions : cardinal number, ordinal number, order-type, etc. These theories of Cantor are permeating Modern Mathematics.f Thus there is a theory of functions on point-sets, in particular, on perfect point-sets, and on more general order-types, while the arithmetic of cardinal numbers and the algebra and func- tion theory of ordinal numbers are under development. Less technical generalizations or analogues of functions of the continuous real variable occur throughout the various doctrines and applications of Analysis. A function of several variables is a function of a single multipartite variable ; a distribution of potential or a field of force is a function of position on a curve or surface or r^ion ; the value of the definite integral of the Calculus of Variations is a function of the variable function entering the definite integral ; a curvilinear integral is a func- * Cf. my papei On a Form of General Analysis, with application to Linear Differential and Integral Equations, read before the Section on A nalysis of the Eome Congress of 1908, Atti, etc., vol. 2 (1909), pp. 98-114. t Cf. A. ScHOENri.lBS, Die Entwickelung der Lehre von den Punktmannig- faUigkeiten, Bericht eratattet der DeuiscJien Mathematiktr-Vcreinigung, J ah res - berioht, Band VIII (1900), Erganznngsband II (1908). — A. Schoesflies, Mengenlehre, Encyklopadie der mathematischen Wissensohaf ten, Ij, art. IA5 (1899), pp. 184-207. — E. Baiee-A. Schoenflies, Theorie des ensembles, Enoyclop^die des sciences math6matiqaes, Ii, art. 17 (1909), pp. 489-531. A FOEM OF GENERAL ANALYSIS, 3 tion of the path of integration ; a functional operation is a func- tion of the argument function or functions ; etc., etc. A multipartite variable itself is a function of the variable index of the part. Thus a finite sequence: Xj^; •■■; x^, of real numbers is a function x of the index i, viz., x{i) = x^ (17= \; . ■ ■ ; n). Similarly, an infinite sequence : x^; • • ■ ; x^; • • ., of real numbers is a function x of the index n, viz., x(n) = x^in=l; 2; • • ■ ) . Accordingly, n-fold algebra and the theory of sequences and of series are embraced in the theory of functions. As apart from the determination and extension of notions and theories in analogy with simpler notions and theories, there is the extension by direct generalization. The Cantor move- ment is in this direction. Finite generalization, from the case «, = 1 to the case n = n, occurs throughout Analysis, as, for instance, in the theory of functions of several independent, variables. The theory of functions of a denumerable infinity of variables * is another step in this direction. We notice a more general theory dating from the year 1906.. Recognizing the fundamental r6le played by the notion Umit~ element (number, point, function, curve, etc.) in the various special doctrines, M. Fe^chet has given, with extensive appli- cations, an abstract generalization f of a considerable part of Cantor's theory of classes of points and of the theory of contin- uous functions on classes of points. Fr6chet considers a general class ^ of elements p with the notion limit defined for sequences of elements. The nature of the elements p is not specified ; the notion limit is not explicitly defined ; it is postulated as defined subject to specified conditions. For particular applica- *Cf. D. HiLBEET, Grundzilge einer allgemeine Theorie der linearen Integral- gleichangen, vierte Miiteilung, fiinfie Mitteilung, Gottinger Nachrichten, 1906, pp. 157-227, 439-480. Wesen und Ziele einer Analysis der unendlich- vielen unabhdngigen Variabeln, EendicoDti del Ciroolo Matematioo di Palermo, vol. 27 (1909), pp. 59-74. fM. Fbechet, 8ur quelques points du calcul fonciionnel, Paris thesis, 1906 j reprinted, Eendioonti del Ciroolo Matematico di Palermo, vol. 22 (1906), pp. 1-74. 4 B. H. MOORE. tions explicit definitions satisfying the conditions are given. More theorems are obtained by defining limit as usual in terms of the notion distance {icart, voisinage) postulated as defined, subject to specified conditions, for the pairs of elements. f We conclude this review of current Analysis with the remark that the functions considered are either functions /u. of variables p of specified character or functions jji, on ranges 5p with postu- lated features: e. g., limit; distance; element of condensation; connection, of specified character. In the form of General Analysis here in question we pro- pose to consider /wnc^ioris fJ' of a general variable p on a general range ^. This general is the true general, embracing every well-defined particular case of variable and range. For the present we confine attention to real- and single-valued func- tions fj,. A property of functions /a is said to be of general reference in oase it is defined for functions /it of a general variable p on a. general range ^. Thus^nifeness on the range ^ and constancy on the range ^ are of general reference. On the other hand, e. g., the property : continuity on the range ^, is of special refer- ence, its definition being with reference to some special feature, e. g., limit or distance, postulated as defined for the range ^. I proceed to indicate certain properties of general reference to be considered in this memoir. The function fi^ is dominated by the function fi^ in case i/^il = \l^2\> viz., for every value of ^ ki(p)| = |/^2(p)|. I I add two remarks. A contribution to Fr&het's theory is made by T. H. Hildkbeandt in his Chicago dissertation of 1910. The distance function ^{pi, p^) ia replaced by a relation Kp^p^m, where m is an integer, generalizing the relation: ''(Pij^j) Sl/2". This relation was suggested by the relation Kp^p^ of Part II of this memoir. Mr. Hildebrandt further investigates the interrela- tions of the various conditions. P. ElETZ, in his paper : Stetiglceitsbegriff und ahstracte Mengenlehre, read before the section on Analysis of the Rome Congress (Atti, etc., vol. 2(1909) pp. 18-24), indicates a more general theory involving, instead of Fr^chet's limit for sequences of elements, the notions : elemeni of condensation ( Ver- dichtungsslelle) ; connection {Verkettung), postulated as defined for subclasses of the class Tt ; for pairs of subclasses of the class 50J. A FOEM OP GENERAL ANALYSIS. 5 A sequence {/i^J of functions /x^ (n= 1 ; 2 ; •••) may have the property that there exists a function /m and a sequence {a J of real numbers a„ (n = 1 ; 2 ; • • ■) such that for every posi- tive integer n the function fi^ is dominated by the function a ^ IM , viz., for every n and p | /t*,, (p ) | = | '^„ A' (^ ) | • Further, the convergence on ^ of a sequence {/^„} of func- tions to a function /i, in notation : n the unifonn convergence on ^, in notation : and the uniform convergence on '^ relative to the function cr as scale of uniformity, in notation : n are properties of general reference. Here relatively uniform convergence* differs from uniform convergence only by the substitution, in the final inequality of the definition, for e of e|o-(p)|, e being the arbitrarily assigned positive number. Thus, uniformity of convergence is the special case of relative uniformity in which the scale function a is the constant func- tion 1. Since individual functions and pairs and sequences of func- tions are only special cases of classes of functions, we may * As an illustration of a proposition involving relative uniformity of con- vergence the following (cf. my Rome paper, loc. cit., p. 103) may serve. The range 5p being the class' of all continuous functions p (u) on the finite interval U : a^u^b, of the real number system, the necessary and suffi- cient condition that a sequence {pn} of continuous functions p„(«) converge uniformly on the interval U is that the sequence {/in} of functions /^n : P'>{p)= I Pn{u)p{u)du, of the variable continuous function p converge on the range ^ uniformly as to the scale function a : In this example, the argument p of the function /i heing itself a function of the variable u, the function ft is, more precisely, a functional operation. 6 E. H. MOOEE. define General Analysis as the theory of dosses 3Jf of functions jjl of a general variable p on a general range ^, and we consider properties of general reference of such classes 2ft of functions. For instance, the class Tl of all continuous functions /i on a finite interval ^ : a = p = b, of the real number system has the following properties of general reference : 1. Every function fi of the class 3Jl is finite on the range ^. 2. The class Tl contains every function 6 constant on the range 5p . 3. The class 3Ji contains every function of the form : = «i/^i + a^f^i, where fi^, fi^ are functions of 9Jt and a^, a.^ are real numbers. 4. The class 3Jt contains every function 6 of the form : n viz., every function expressible as the limit of a sequence {fi^} of functions fi^ (n = 1 ; 2; • • •) of the class 9Jl, the sequence converging uniformly on the range ^. Now the property 4 is for this class 3K (and for any class Tt having the properties 1, 2) equivalent to the property : 4'. The class Tl contains every function 6 of the form : ^ = Lm„ (5P; i^), n viz., every function 6 expressible as the limit of a sequence [fij of functions of the class 9JJ, the sequence converging on the range ^ uniformly as to a function /x likewise of the class Tt . Again, the class Tl of all absolutely convergent series has the properties 1, 3, 4' but not the properties 2, 4. It has also the property : 5. For every sequence [nj of functions of the class Wl there exists a function /^ of the class Tl and a sequence {a J of real numbers such that for every n the function /j,^ is dominated by the function a fi . A FORM OF GENERAL ANALYSIS. 7 The class SJi of all continuous functions on the interval ^: a=p = b, has also this property 5; indeed, the property 5 is implied by the properties 1, 2. A class 3Jl having the property : 3 ; 4'; 5, is said to be linear; to be dosed; to have the dominance property D, or for brevity, to have the property : L; C; D. In this memoir I study in some detail certain closure and domi- nance properties of classes 3)i of functions. Of these properties the most important are the properties L, 0, D. We speak of the genus (LCD) of all systems (^ ; 2)i) whose classes Tt have the composite property LCD. The property D plays an impor- tant role in situations involving double limits in connection with relative uniformity of convergence, and relative uniformity en- ters into the definition of the special closure property C. Part I is devoted to the consideration of classes 2R having a common range ^. — A class 3Jl gives rise to the class SO'ii , the class 2)i extended to be linear ; viz., the class consisting of all functions /a^ of the form : where jitj ; • • ■ ; fi^ are functions of the class 3Ji and a^; ■ ■ ; a^ are real numbers. Two classes 9)i, (S give rise to a class SJi^, the class 9)1 extended as to the dass ©, consisting of all functions 6 of the form : n viz., of all functions 6 expressible as the limit of a sequence {fi^} of functions /u.^ (n = 1 ; 2 ; ■ • ■) of the class 9)i , the sequence converging on the range ^ uniformly as to some function o- of the class © . The ^-extension Tl^ of the dass 9)t is the class (9)ti)m.3. A[a^ — a) = e, viz., for every positive number e it is true that there exists a positive integer m such that for every positive integer n> m 22 E. H. MOOEE. it is true tiiat the absolute value * of a„ — a is less than or equal to e. Thus, for the class 3£ of all sequence-individuals ({«„}, a), the property P : convergence of the sequence {a^} of real num- bers to the individual number a as a limit, is the implicational property derived from the class 3) = [y] = [e] and the prop- erty Q^ : belonging to 2), with a certain property M^ = i?^. This property R^ is the existential property derived from the class 3)^ s [y^] = [m] and the property Q'^^ : belonging to 2)^, with a certain property Ii[y^, = R'^^. This property i?^„ is the implicational property derived from the class 3)"„— [2/l„,] = [**] and the property Q"^^ : greater than m, with the property The negative of the convergence has the form : S e 3 m . 3 . 3" n> m 3 A(a^ — a) > e, viz., there exists a positive number e such that for every posi- tive integer m it is true that there exists a positive integer n'> m such that the absolute value of a — a exceeds e. Propositions concerning elements. — The prepositional prop- erties of elements x of the class X defined above are propositions concerning the elements. Such a proposition involves as con- stants the element x, the class 3), and the properties Q and P. In the examples cited the property Q of the pair (x, y) is independent of x, that is, for an element y the property Q either does or does not hold for the pair {pa, y) whatever be the element x ; in such a case the property ^ is a property of the element y, so that we may replace the notation Q^ by Q, or, more precisely, the notation y ^' or (x, y)'^ hj x . y^. Here X qua constant may be omitted. Special cases of prepositional properties. — It is logically instructive to consider the four special cases of implicational and existential properties derived from properties Q and P each independent either of x or of y. The definitions are given in the following table : * For a real number a the notation Aa denotes the absolute value \a\ of a. INTRODUCTION TO GENERAL ANALYSIS. 23 Implioatlonal P Existential P i2oii 3) -Bon X §on3) yKZi.y^ 2/«.3.x« Sy0), where P = J indicates that the property P holds (under the adjoined conditions on the properties Q, R) for "^^'^ element x, and Q ~i? > g indicates that the composite property Q ~R (of Q and ~P) holds for 3°^^ element y. The eight properties in reduced form are given in the follow- ing table : P=l (Q-i?=0) P = {q-R>o) p = p = 1 iQR>0) (gp = 0) p = p = = 1 (R=l) = -Q (R<1) P=Q (P>0) P = (P = 0) p= 1 P = R P = R P = («>0) («=0) P=-Q^R P=QR *0r: "for every element y (of the class D) having the property Q it is true that it has the property B." 24 E. H. MOOEE. Here, for instance, the entry -. P = -Q (i? < 1), § being on X and li on 3), means that P holds precisely for those ele- ments X for which Q does not hold, if R holds not for every element y, i. e., if ~R holds for some element y ; ~E > and i? < 1 being equivalent conditions on R. This table may be variously checked. First, as we saw above, the '"Snuaf' property from 3) ; Q ; ^ is equivalent to tbe "Sistentiaf' property from 2) ; 'b\~^, and the negative of the '"x'iSaf' property from 2); ^; i? is the j^^pSJli property from "^ ; Q; ~R. Again, the case : R independent of x and of y, viz., ^ = 1 or jB = 0, is a subcase common to the cases : R onf) ; R on dc, and similarly for § = 1 or § = 0. Functions on the class ^ to the class 31. §§ 4, 5. 4. A (single-valued) function F on the class ^ to the class ^' is a correspondence P of all elements p to (all or merely some) elements p' such that for every element p of ^ there is a definite corresponding element of ^', in notation : P{p) or Pp or P . This function P on ^ to ^' is G. Cantor's Belegung von ^ mit ^' ; Mathematische Annalen, vol. 46 (1895), p. 486. It is convenient to adopt the terminology of the theory of functions. Thus, p is the variahle of the function P ; the class 5|S is the range (of the variable) of the function P ; at the argu- ment value ov place p the function P has the functional value j;,etc. We indicate such a function P by the various notations : F, {F), (F\P), {J^,\P^), where the class ^' is to be understood from the context. The last notation may be read : " the system of (functional) values P^ where p varies over the class ^." The intention is to dis- criminate sharply between Junction and functional value. In view of the generality of ^ the preceding explanations cover the case of functions of two or more variables. Thus, a function : P^ {P^,^,) = {P,,,„\py) = (i^,,,„|/ V""), INTBODUCTION TO GENERAL ANALYSIS. 25 of the two variables p', p" which vary independently over the classes ^', ^" (which may be identical) is a function : of the variable p which ranges over the class ^, the element p being the bipartite complex p'p' or (p, p") of the elements p, p" and the class ^fj being the class of such bipartite elements p'p" , in notation : Such a function F on ^'^" to some class ^"' is likewise a func- tion F on 5p' to a certain class 3Jt" of functions M" on 5p" to ^"', viz., the class 9)1" of all functions : M"^F^.^{F^,^.,\p"'"), where p' varies over ^', in notation : W'^ [all(i^,.,.b"'"')|p'*'j. Conversely, a function F on ^' to a class SJi" of functions on %" to 5p"' is a function i^ on S^"^" to ^p'". 5. We are at present concerned with real-valued single-valued functions, that is, functions on ^ to the class 21 of all real num- bers. We denote these functions by small Greek letters : ^, fj-, etc. The theory of functions on ^ to St may naturally precede the theory of functions on ^ to more general classes 5p'. In partic- ular, in accordance with the remarks of the preceding paragraph, one readily extends many of the notions and propositions of the sequel to the theory of functions on 5p to classes 5^' of functions on ^" to 31, or to the theory of functions on ^ to classes ^' of functions on ^" to classes 5)3'" of functions on ^"" to 91, etc.; this holds true, for instance, of the fundamental notion of rela- tive uniformity of convergence (§ 6). The class ^ enters the theory by the mediation of a class 9)1 : of functions fi on ^ to 31, so that the theory relates to the sys- tem i: = (2i; ^; an). 26 E. H. MOORE. We notice in particular the following instances or cases of the'general system X '■ (I) ^^ is the class of a single element : e. g., p =1. 3)V is the class of all functions on ^^ to 21, i. e., in effect, Tt^ is the real number system 21. (IIJ ^^^ is the class of n elements : e. g., p = 1,2, • ■ •, n. Tl^^ is the class of all functions fj, on ^"" to 2t, viz., of all n-partite real numbers : (III) ^™ is the class of a denumerable infinitude of ele- ments : e. g., p = 1, 2, ■ ■ ■, n, • ■ ■. 9Jl'" is the class of all functions n on 5p"^ to 21, viz., of all infinite sequences of real numbers : /^=(^b) = (^U f^2> ■•■> f^n, •••)• (III„) 3Jt"^ is the subclass consisting of all functions /jl such that p p^=oo (IIIJ W™' is the subclass of all functions /u, such that JIM p:=l •converges.* (IV) ^^^ is the finite interval! (01) : O^p^l, of the real number system 21. 3)1^^ is the class of all continuous functions of p on ^^"^ to 2r. Thus, theory I is the theory of the real number system or one-dimensional analytic geometry ; theory 11^ is n-dimensional analytic geometry; theory 111,, is the theory of numerical sequences tending to the limit ; theory IIIj is the theory of absolutely convergent series ; theory IV is the theory of real- * Here, as in the sequel, A denotes the absolute value of; that is, for every real or complex number a, Aa = | o |. Further, e is a positive number. 501™" is a subclass of 3R^^^o, and e^ <[ e^ implies that SJimei is a subclass of 501^6! . t The theory relates equally to case IV for any interval {a^a^). INTRODUCTION TO GENERAL ANALYSIS. 27 valued single-valued continuous functions on the finite interval (01). The general theory of the sequel, duly particularized, belongs to each of these particular theories. A function /m on ^"" is a point or vector in n dimensions expressed analytically by its n coordinates associated with n independent directions and units. Thus a value of p specifies one direction and associated unit. Any class {Menge) SJt of objects /x each capable of characterization or specification by means of w measurements to scale may be represented as a class 2R of functions jx on 5]3"» or as a point-set in space aJt"» of n ■dimensions. We are at present interested in classes 2R = [/x] possessing various combinations of properties held in common by the classes : m'; 93?"" ; 9)1'"; a)i"'« ; ajj™«; m'^, and in fact by many other essentially distinct classes of which some will be indicated below.* These properties involve a con- venient generalization of the important notion of uniformity — the notion of relative uniformity. Relative uniformity. Range and scale of uniformity. §§ 6-13. 6. If the notation R p denotes the statement that a relation involving an element p ot the class ^ holds for the element p, we denote by ^. (P) the statement that the relation holds for every element p. On the basis of convenient definitions of (absolute) uniformity and of relative uniformity of validity of the relation in question, we denote by ^, m the statement that the relation holds uniformly on the range ^, and by K (^; ^) *Cf. ?§ 55 IV; 74 1, U;84. 28 E. H. MOORE. the statemeat that the relation holds uniformly on the range ^ with respect to the function cr as a scnle of uniformity. With respect to the relation in question the notions of uni- formity on range ^ and of relative uniformity on range ^ as to scale function a- are subject to definition, usually in such a manner that R^ (^) implies R (p) and that R^ (^) and jRp (^; 1) are equivalent. Here 1 denotes the (constant) func- tion (T having for every p the functional value tr^ = 1. For instance, with respect to a function ^ on ^ to 91, we consider the relation R^ that ^ belongs to 31, that is, P ' P Then, by definition of <^, We say that the relation ^ ^ holds uniformly on ^ in case the functional value (^ is constant (independent of p), that is, the two statements : : m-, n=K {p.p.), are equivalent. We denote by a(^) the constant a qua function of p, and by 2t(5p) the class of such constant functions on ^ to 21. Thus the statements : 4>; m-, v'""', are equivalent. Denoting by 2lo" the class of all numerical multiples acr of a function a- : 'JS.cr = [acr | a] , we might define relative uniformity for the relation (f> " as follows : 0/ (^; a) = cf,-''. From a numerical relation : involving numbers a^, a^, ■ ■ ■ we derive a relation : ^{^ip> 4>2p> ■•■)> involving an element p of ^ and functions ^j , (^^ , • • • on 5p to 21, by substituting for the numbers a^, a^, ■ ■■ the functional INTRODUCTION TO GENERAL ANALYSIS. 29 values ^, ■ ■ ■. Further we derive a functional relation : * m>v 4>„ ■ ■ ■), involving functions ;, ^, • • • by the understanding that that is, elliptically, numerical relations amongst functions are understood to hold identically in the variables of the functions, the context indicating the variables in which the respective functions are to be taken. Thus, (f3 = y{r means ^ =-\^ (^p), and (^ = 1 means ^p= 1 {p), i. e., h^ . 3 . A(a^ — a) = f, viz., for every positive number e (it is true that) there exists a positive integer n^ (dependent on e) of such a nature that for every positive integer n exceeding n^ (it is true that) the absolute value of a — a is at most e. For a fixed e there is a least integer effective as n^. 76. The sequence {/u,^} of functions fi^^ on ^ to 21 converges to the function 6 as limit : (1) L/^„ = ^, n * Derivation of functional relations from numerical relations is not com- mutative with 'negation. — The signs: =; 4=, denote primarily logical identity ; diversity. Thus, (p^f ia the negative of ^ = i/', i. e., fp — fp (p), and not the functional relation derived from the numerical relation : a 4= *• t L denotes always Li. 30 E. H. MOOEE. that is, in accordance with § 6, (1') Lm„, = ^, iP), n in case (2) p.e:^:3: n^^ b n> n^^.D . A(f.^^- 0^) ^ e, viz., for every p and e (it is true that) there exists a positive integer n ^ (dependent on p and e) such that for every n exceed- ing n ^ (it is true that) the absolute value of fi^ — 6 ^ is at most e. For fixed p and e there is a least integer effective as n ^ . For a fixed e the system of least integers n ^ where p varies over ^ may be finite (bounded from oo), that is, there may exist an integer n^ dependent on e such that w ^ = n^ (p). If this is the case for every e, the convergence for every p of ^ holds uniformly on 5p. Cf. § 7c. 7c. The convergence holds uniformly on ^ : (1) L/^„ = ^ (5P), n that is^ (1') L/^.,.=^. in n in case (2) 6:3:3" n 9 n>n^.3.^(/x„-^)^e, that is, (2') e:^:S n^^n>n^.^.A{^^^-e;)^e {p), Viz., for every e there exists a positive integer n^ such that for every n exceeding n^ the absolute of the function fj^ — 6 \s for every p at most e. Here, for a function o-, the absolute of a, in notation : A(t, is the function on ^ to 31 for which {Aa-) = Aia- ) for every p. Id. a- being a function on ^ to 31, the convergence holds uniformly on the range ^ relatively to or as to the scale o- : (1) Lm„ = ^ (^; cr), n in case (2) e : 3 : a" w^ 9 n > n . D . A{^l^ - e)^eAa, INTRODUCTION TO GENERAL ANALYSIS. 31 that is, (2') e : 3 : a- n. 9 » > «, . D . ^(/.„^ - O^) g eAa^ (p), viz., for every e there exists a positive integer n such that for every n exceeding n^ it is true that for every p the absolute value of fi^^ — 9^ is at most e multiplied by the absolute value of V Thus, the definition (§ 7c2) of convergence uniform on ^■ becomes the definition (§ 7d'2) of convergence on sp uniform as to a scale function o- by the substitution of eAcr for e in the- final inequality. Accordingly, uniform convergence is an in- stance of relatively uniform convergence, the scale function being the constant function 1 . In the formulas of absolutely or of relatively uniform con- vergence the elements p may notationally be suppressed, since they enter similarly ; cf § 6. Thus the formulas in the func- tions alone have essentially the simplicity of corresponding: formulas for convergent sequences of numbers. 7e. We notice the propositions : 1) For every a- it is true that the relation : n implies the relation : n and indeed in such wise that, if at an element p the scale cr vanishes : 0-^ = 0, the corresponding sequence {fM^^ } of func- tional values /Lt^^ are ultimately all equal to the corresponding functional value of the limit function d. 2) The relations : 71 n are equivalent. 7/". Examples.* — The cases III„; HIj; IV furnish illus- trations of absolutely and of relatively uniform convergence * As to proofs, cf. 1 16. 32 E. H. MOORE. of sequences of functioas. For case IV we have the well known theorem : A uniformly convergent sequence of continuous functions has as limit a continuous function. The corresponding proposition holds in case III^ but not in case IIIj. The scale function l(5p) belongs to Tt^^ but neither to m'"" nor to 9Jti"i. In terms of relative uniformity, however, for the three cases, we have the proposition : A sequence of functions of g)J™o;nii;iv converging uni- formly as to a function of Sli"^"' ^">' ^^ has as limit a function of g)iirio;"ii;iv. The corresponding proposition holds in case III^ and evidently in cases I ; II„ ; III, and in every case in which the class 9)1 is the class of all functions on ^ to 21. 8. Similarly, we speak of the convergence of the sequence {M„} on ^ ; uniformly on ^ ; uniformly on 5p as to cr : in notation, in case there exists a function 6 eifective as limit of the sequence : n 7i 7i for the respective modes of convergence. A condition necessary and sufficient for the convergence in the various modes is the validity in the respective modes of the so-called Cauchy condition. This condition for the relative uniformity (^ ; cr) is as follows : that is, for every e there exists an integer n^ dependent on e such that for every two integers n, n" each exceeding n it is true that the relation : A^/j,^, — fJi,^^„) = eAa; holds. The limit 6 is of course unique. To the same function 6 INTRODUCTION TO GENERAL ANALYSIS. 33 as limit converges in like mode every sequence of functions obtained from the original sequence by (1°) any rearrangement of terms, (2°) the omission of any finite or infinite number of terms, or (3°) the insertion of any finite number of functions as new terms. Thus, with respect to a function a, any class 9)1 = [yt*] of functions constitutes a class (i) of Fr^chet, on the under- standing that a sequence {/u.^} has in the sense (L) a limit if and only if there exists a /x (the limit) such that in our sense L^„ = A' (5P; a). n 9. Definition. — A function cr is said to be dominated by a function t in case Aa- ^ At, that is, in case Aa-^ = Ar^ for every p. 9a. Comparison of uniform and relatively uniform conver- gence. — We notice the propositions: 1) Uniformity (of convergence) as to SZnA'm the former dominates the latter. As a corollary : the convergence uniform as to o- of S„j4/u.^ implies like convergence of S„M„) and the function X)„-4m„ dominates the function S„/*„- In tbis case, the series Sn/^^ is said to be absolutely-uniformly convergent on ^ as to a: 13. Propositions. — The relations: imply (1) j^Af.:=Ae' (^;:.' = ^'^" (^ ; ^'^")> n where t' and r" are respectively common dominants of 0', a-' and of e", a". More generally, from the relations : 36 E. H. MOORE. where the functions ti\^, 6', -4<^' + ^-^".^ ; n (6) L/^>:; = ^'^"(^'^"; ^'o> n where t', t" on ^', ^" respectively are common dominants : r of ^', cr' and t" of ^", a". Here the functions: /a,'±/x;;; 6' ± 6" ; A(t' + Aa-" ; fi'^^; e'6" ; t't", are functions on 5|3'• jii • 15. Closure properties of classes of functions. — A class 2R is A = absolute, if 9)f contains AW ; multiplicative as to 9t, if 3Jt contains St-BJ ; additive, if 9)Z contains 9Jt + 2R ; L = linear, if m contains 31911 + 219)1 ; multiplicative, if 3)Z contains 9)J9)f ; C'g = closed as to ©, if 2)i contains 9J(s ; C = closed,'^ if 3}J contains 9)? 5,,. In accordance with § 2, the notations : 9Ji^; 9)1 "■^'""^•^ 9);^; etc., designate classes 9)Z respectively absolute; additive; linear; etc. * 1) For every class 9Jl and funotion a the class 9Ji belongs to the class 5)?a. 2) For every class 3Jl and class © the class 3Jl belongs to the class 3J(3. 3) For every class © and class 391' belonging to a class W the class SK's belongs to the class 3)J"s. 4) For every class 39J and class ©' belonging to a class ©" the class S)}^/ belongs to the class SKs". 5) For every class 3K and class S the classes SUj and 3)J;is are identical. f A class 9JJ having the property Cm of being closed as to itself has the property C, and conversely. The notation C and the designation c/osuce are preferable to the iS-Cand self-closure originally used. A class 3H on ^U" is closed if and only if the corresponding n-dimensional point-stt is closed, i. e., contains its derived point-set, according to the usual definitions. 38 E. H. MOORE. The linear classes are the classes additive and multiplicative as to constants. A linear multiplicative class is integral. Since for every ©, SWs contains TO, for a class TO the properties : closure ; closure as to ©, are respectively equivalent to invari- ance under extension as to TO ; ©, viz., TOgn = TO ; TO^ = TO. 16. The classes S!}^i:ii.; m; ino; m.; iv ^j^g absolute, linear and closed* — They are moreover multiplicative and so integral, but these facts do not enter the theory of the present paper. A class TO closed as to 2l(5p) is a class TO containing all func- tions expressible as limits of uniformly converging sequences of functions of TO, and conversely. The classes TO"'^ with the exception of TO'"", TO"^' contain 2l(^) and being closed they are closed as to 3[(^). The class TO'"" does not contain but is closed as to 2l(5|J). The class TO'"' neither contains nor is closed as to 2l(^). For consider the function and the sequence {/i^} of functions p, defined as follows : ^^p-'" ip); IM„^=p-"' (p = «), (p>n). Then obviously the functions ;tt„ do and the function does not belong to the class TO"S while the sequence {fij converges on ^ uniformly to the function 0. In the light of this remark as to TO"'" one may appreciate the importance for General Analysis of the generalization of uni- formity to relative uniformity. 17. The properties of classes TO defined in § 15 are closure properties, in the sense that a class TO having the property contains a certain class dependent on TO and the property in question. Thus, the (specific) closure C of a class TO is the property : n viz., a sequence {yu.^} of functions ofW converging on ^ uniformly as to a function fiofTl converges to a limit which belongs to TO. *Cf. the theorem of ? 23. INTRODUCTION TO GENERAL ANALYSIS. 39 That these closure properties enter fundamentally in the theory of functions of the classes 2R^"^^ is evident. An approach to the theory of convergent sequences of continuous functional transformations or to the theory of other forms of double limits reveals the presence of another fundamental prop- erty held in common by these classes, a dominance property D. Dominance ■properties. Relative uniformity as to a class of functions. §§ 18-26. 18. A function a- is dominated by a function t in case Aa = At (cf. § 9), and by a class X = [t] in case in the class X there is a function t^ (dependent upon a) by which o- is dominated. A class (S = [o-] is dominated by a function t or a class % in case its every function a is dominated by t or by %. We use temporarily the notations : ^ dominated by t . -. dominated t)y Z . o , o , fS^ dominated by t . ^ dominated by X 19. Dominance property D. — A class 9Jl has the dominance property D in case for every sequence {fi^} of functions fi^ = (/A^ \p)ofTl there exists a function ft = (yw.^ | p) of Tt such that every function fi^ of the sequence is dominated by the class 2l/it, that is, such that there exists a sequence {a J ot real numbers of such a nature that for every n and p Au = Aa « . In symbols : m'': = :m B {imJ .3.3" ({«„}, h) 3 Aim^^^ Aa^f^^, (np). 20. Relative uniformity as to a class of functions. — An analytic relation involving a parameter p is said to hold uni- formly on a range '^ofp with respect to a class of functions on ^ as a scale of uniformity in case it holds uniformly as to some function of the class. Thus, if © = [o-], 40 E. H. MOORE. and n are equivalent statements. 20a. Propositions of comparison. 1 ) Uniformity as to @ and uniformity as to 31© are equivalent. 2) If and only if* 21© is dominated by 212; does uniformity as to © imply uniformity as to X. 8) Uniformity and uniformity as to 2l(^) are equivalent. Here, according to § 6, 2I(^) denotes the class of constant functions of the argument p. 4) If and only if 2l(^) is dominated by 21© does uniformity imply uniformity as to ©. 5) If and only if 21© is dominated by 2l(^) does uniformity as to © imply uniformity. 6) The classes 3)t = 2)J'' ^'"' ^^ are dominated by and domi- nate 2l(5p), and accordingly uniformity as to 3Jt and uniformity are equivalent. The class 9Jt = 2Ji^" dominates but is not dominated by 2I(^), and accordingly uniformity implies but is not implied by uniformity as to 9)t. The classes 9Jt = 3^"^°' '^^' are dominated by but do not dominate 2l(^), and accordingly uniformity as to 9)^ implies but is not implied by uniformity. 7) The convergence uniform as to © of the sequence {fi^] and the corresponding validity of the Cauchy condition are equivalent. 21. The dominance property D enters the theory chiefly as a property of classes functioning as scales ofuniformiiy. Before- * The condition is evidently sufficient. In proof of the necessity, note that l.i=.0 (^), a.3. 1.1 = (5P;a), and further that L-=0 (5p;2) implies a' T 3 L- = o {^\t), so that, for n sufficiently large, Ac^ nAr-^ and accordingly, if uniformity as- to © implies uniformity as to %, every function a of © is dominated by %.%- INTRODUCTION TO GENERAL ANALYSIS. 41 indicating propositions in this direction I define other domi- nance properties of classes of functions. 22. Dominance properties D„, D'^, D^, D^. — A class 3)1= [/i] has the dominance property D^ : in case for every function fi of W there is a nowhere neg- ative function fi^ of 9)i which dominates ijl ; Dj : in case 3Jl contains every function such that there exists a nowhere negative function /a^ of W dominating 6 ; Z>j : in case every two functions of 3JJ have in WSl a common dominant function ; Dj : in case there exists a denumerable subclass SR = [v^ | n] of the class 2l3Jt such that 2R is dominated by 9?. 22a. Propositions * concerning the dominance properties D, D„D'„D„D,.— (1) m^.~. {^my (p = d, d^, d,). (2) 3)j„ ™ • ^ . a)t -""""'"'^^ ^y^^.D.m'' (P = A A. -^p A)- (2') mo"' ■''.^ •iornir.^t.i by «„ _ 3 _ gjj P {P=D,D^, D^). (4) m . ©0® . e dominated by K„ _ ^ _ gj^^^^ _ gjj__ (5) at^ . 3 . a}p°. * 1) For the respective dominaDce properties P= D; Z); ; Dj, a class 9B has the property P if and only if the class 3193J has the property P. 2) For the respective dotniDance properties P=D; D^; D^; Dj, if a class 3K is dominated by one of its subclasses SJfp hiving the property P, then 3)1 itself has the property P. 2') For the respective dominance properties P= D ; i>, ; Dj, if a class 33} is dominated by the class 313)Jq, where 3)}„ is a subclass of 3)J having the prop- erty P, then 3)1 has the property P. 3) If 33} is a subclass of 3}, then 3)t is dominated by 3}. 4) ]f ©0 belongs to and SI©;, dominates ©, then for every class 3)! the ex- tensions 33ii ; 3)Js of 31( as to ©o and © are identical. 5) Every absolute class has the dominance property Dq. 6) Every absolute linear class has the dominance properties jDj, D^. 7) If a class 3)i is closed and a class 31 has the dominance property T)^, then the subclass of 33? consisting of all its functions /i which are dominated by 313} is closed. 7') If a class ^ has the dominance property X»i, then the class consist- ing of all functions dominated by 3(3} is closed. 8) Every absolute linear class with the dominance property Dj is closed. 42 E. H. MOORE. (6) m^'-.'^ .m^o^K (7) 2)t '^ . 9f ■"' . 3 . [all fl do^'nated by ajl-j C In particular, 9JJ being the class of all functions on ^ to 21, (7') 9|-0i . D . [all (^dominated by «-j C (8) 3)i^-^^'''.3.2«^. Here (8) is a corollary of (6, 7'), since m = ^m = Am = [aii .^^"■"i-^'t^'i by-^n-j _ 23. The class W = % and similarly every class 3l(^) has the properties B, D^, D^, D^. Accordingly, by § 22a2, these properties belong to every class 3Di dominated by and containing 2((^), in particular to the classes m = W' ""' ^^. Further, the classes ?0i = gji"^ ; "^» ^ ™» have the properties A, L, D„ D'„ A- We formulate the Theobem. The properties A, L, C, D, D^, D^ are pos- sessed by the classes 3)i = 3J1^' ""■ "^' "'»■ "'•' '''• The property D^ is possessed by the classes 9Jt = "SH^ '• ^^" ' ^^', but not by the classes 5Di = gj^i"; i"o; "i«. The -property D'^ is possessed by the classes 3Ji= 3)F' ""■ "^' '"»' "^', 6w< noi by the class dJl^^- The composite propeHy -Dp-^j *^ possessed by the classes m = m^' "", but not by the classes m = 9)1™' ™°' '""• "'. 23a. The two dominance properties B'^, B^ determine four composite properties P : P^B'.D,; B--B,; -B',B,; 'B'- B„ and corresponding genera of classes 2R or systems (31 ; ^ ; W), the genus P consisting of all classes 9)t having the property P. Thus the properties B'^, B^ serve as branching properties * * These branching properties are properties of general reference. Another branching property of general reference is Cac-c) ; this property is possessed by the classes aR = 2Ri; ii»; "i: ™o: iv but not by the class 5m"i. (cf. §§ 7f, 16) Other branchings relate to forms of expressibility (e. g., linear expressibility) of the functions of the class SB in terms of a system of fundamental func- tions belonging to the class 9]t itself or to an auxiliary class ©. INTRODUCTION TO GENERAL ANALYSIS. 43 for the classes M = 9Jt'' "»■ '"' ™°' '"«■ '^ viz., Classes m-. SK^'"-; gjini ; ni„ , m. . gjjiv. Genus P: D',D,; D^-D^; ' D'^D^. 236. The class SJl of all functions on a class % is obviously closed. Thus the classes M = 'Sk^'' "»' "' are closed. The clo- sure of SR''^ depends upon two fundamental properties of the class 3)i^^ of all continuous functions /tt on the finite interval ipi^, viz., (1) gjjiv contains and is dominated by 2l(^) ; (2) a sequence {/x^} of functions fx^ of Jlt'^ converging uniformly on ^^"^ converges to a function of M^^. The closure of 3)1™° and W" depends upon their properties ALD'^ in view of § 22a8. That the classes 2Jt = Tl™' ™»' ™" have the property D but not the property D^ is proved in § 23c. 23c. Dominance jji'operties JT^, K^ of classes Tt on ^"'. — For classes M of functions on 5p = ^"' we define two domi- nance properties * JT^, K^^ having relations to the properties I), D,. A class 9Jt = [jii] has the dominance property K^ : in case for every function /(^ of 9)1 there is a function n^ of 9Jt such that for every e there is an element p^ (de- pendent on e) such that p > p^ implies Afi =eAfjL^ ; viz., fj,.-.'^ .-. 3' fig 9 e:0 -.S: p^ 9 p>p^ . 3 . Afi^ S eAfi^ Op ■ * The properties : K^ ; E„, qa& properties of classes 5^ of functions on the special class ^™, are of special reference. The property K^ is later devel- oped (of. §§ 65, 67, 72, 75, 77) into a property iT, of more general reference. The two properties : Ki ; K^, of § 77 are defined for classes 3)J of functions on any class % with respect to a development A of 5(5. The notion development (§75) is a generalization of the denumeration of 5|?™and likewise a generali- zation of the partition by sequential halving of 5piV. The property K^ is a generalization of the property of uniform continuity on 5piv. The proper- ties : Ki ; K^, play a central r&le in the theory of composition of classes of functions of independent variables (cf. Part II). 44 E. H. MOORE. K^ : in case for every sequence {/it^} of functions of 9Ji there is a function /i^ of 9Jt such that for every n and e there is an element -p^^ (dependent on n and e) such that P >Pne implies A^t.^^^^eAn^^^, viz., {IM^} .-. 3 .-. .7 /.„ 9 71 . e : D : 3[p„,9p>p,, ■ ^ • ^M„^ = e>lMo^. We have for ^ = ^™ the propositions : * (1) ajjio.s.gjjif.; (2) 9Jt-^»*^ : D : /i . 3 . ja""""""^'^ '-"" ; (3) m^>-^.:3.aJt^''»; (4) HJJ^-'-^" 9 3^ ^nowhei-ezero . D . S^J -^ j (.5) goZ = 9)1'" ' "'"' ""• . 3 . 2)t -s^i-f'^o-°"-°2. Proof of (1) : Tl^" implies Tt^'- — The property -ff'j is evi- dently a special case of the property ^„. Proof of (2). — The class 'SI, having the property Z)^, is domi- nated by a sequence {vj of functions of the class 2l3Jt. The class Tl having also the property ^, we see that for the sequence {vj there exists a function fi^ of 93J and for every 7i and e an integer p^^ such that n. e.p>p^^.'D .Av^^^eA/i^^. Novv the function /j,^ is dominated by a function, say v^^, of the sequence {v,,}, viz., P-^Ai^^^SAv^^. Accordingly e-P>P^e-'^ -'^f'0J.= ^^f^0p• *l) Every class 30? having the property K^ has the property -^i 2) Every class 3)t having the properties Kq, D^ consists of functions ft ultimately zero, viz., for every function fi there exists an integer n such that iU vanishes for every argument jb>m. We notice further that the class con- sisting of all such ultimately vanishing functions has the properties E^, D^. 3) Every class 9JJ having the properties K^, D has the property Kf,. 4) Every absolute linear class 3)J having the property Kf^ and containing a funption II novphere zero has the property D. 5) The classes 3Dt=-5Di"i; "lo; niehave the properties: Z, ; E^; D, but not the property D^- INTRODUCTION TO GENERAL ANALYSIS. 45 Hence, the function fi^ is ultimately zero. Accordingly, the functions v^ are ultimately zero, and the functions /i, each domi- nated by a function v^, are ultimately zero. Proof of (3) : M^^'^ implies M^". — Consider a sequence {/J.J of the class 'M^^^^- By the property D there is for this sequence a certain function /x and by K^ there is for this func- tion fi a certain function n^. This function fj,^ is effective as the function fi^ of the property ^^ for the sequence {m„}. Proof of (4). — Consider a sequence {fij of the class Tl. By the property JT^j there is for this sequence a certain function im^, and by the hypothesis there is in 3){ a nowhere vanishing func- tion fjL. The function A/i^ + Afi belongs to the class W, which is absolute and linear, and this function is effective as the function fi of the property D for the sequence {/"■„}. Proof of (5).— Since the classes 50t = 3)1™' '"»■"'• are abso- lute and linear and contain functions nowhere zero, in view of (1, 2, 3, 4), (5) follows from (6) g)^"^ 3 . 3)H''» ; (7) 9)r"«.3.aJl^^'»; (8) 9)Z™" . 3 . ?0H'^'^. Proof of (6) : 3)t"^ implies Tl^^°. — Consider a sequence {/x^J of SDt"^. For it there exists a function 6 on ^"' such that p:D -.n^p.D . JL^^ ^pJ/x„^. 3}t™ being the class of all functions on 'ip'", such a function 6 belongs to 9)1™ ; it is effective as the function fi^ of property A"^ for the sequence {fi }, p being any integer greater than n and 1/e. Proof of (7) : SR™" implies SJl ■'''". — Consider a sequence {fij of m™", that is, such that 46 E. H. MOORE. Hence there exists a double sequence {p^^} such that n . m . p=p . 3 . Au. = m~^, and so a sequence {p^ | w = 0, 1, 2, • • • } for which p^=l; nSm.Zi .p^^p^^; ™>0 • => •P™>P„-i- Hence it follows that nSm. p^p^.D . Afi^^ ^ mr\ Now the function 6 : connected with such a sequence {j:/^} has and belongs to 9Jt"^°. This function is effective as the function /Xp of the property ^ for the sequence {/^„}. For we have the relation : Take two integers n, m^ with m = m^ . From the preceding relation for the various integers m = m^ and the corresponding ranges p^^j > p =p^ of ^ we see that n^m^. p^p^^.-D . Afi„^ S m-^Ad^ . Accordingly, for given n and e, if we take m^ as the least integer exceeding the numbers : n ; e^\ and then define j^ne ^s p^ , we have the desired relation : Jr 1 ne "np p Proof of (8,) : 3)1™- implies 2Jf ^i. _ 5^™. is the class of all functions /a on ^"' for which converges. Since 3Jl^"' is absolute and linear and contains nowhere vanishing functions, the proposition for the class 93i^'^ is a corollary of the proposition for the subclass consisting of all everywhere JDOsitive functions of ^Ji"'" . Further, the propo- INTRODUCTION TO GENERAL ANALYSIS. 47 sition for the general case e = e is a corollary of the proposition for the case e = 1. We are then to prove the proposition for the class of convergent series of positive terms. Let /i be any everywhere positive function with convergent series : p Setting Pp = l^p + t^p+-i + ■ ■ ■ (P) we have JjPp = Oj f^p = Pp — Pp+i ■ p The function a : = v'pp - Vpp+i (p), is everywhere positive with convergent series : Z^ = ^''p., p while p p p Thus, the function is effective as the function /i, of the property K^ for the function /u. of the class of convergent series of posi- tive terms. -n '•' Ti in Proof of (Sj) : 9)t"^» implies 2)1 ". — Making the same reduc- tions as in the proof of (8J, we consider any sequence {^J of everywhere positive functions with convergent series. We set np P Let {6„} be a sequence of positive numbers with convergent series : n Consider the double sequence : |c I : c = 6 ^ {np). X^npS ' np « g V J / n We have. p ^ Jl p n 48 E. H. MOORE. and accordingly •p V Thus there is a function : n everywhere positive with convergent series : p This function 6 is effective as the function fi of the property D for the sequence {m,,}. For, setting «„ = r («). n we have the desired relation : since ii^^ = a^c^^ (np) and c„^ < 0^ (np). 23d. Remarks on the properties : K^; K^, and the propositions of § 23c. The properties : D ; D^, are of general reference while the properties : K^ ; K^, are of special reference, in that they refer to classes 9}J on ^"'. I introduce the properties K^, K^ in order to make the convenient arrangement (§ 23c) of the proofs of the propositions of § 23 concerning the properties : D ; D^, for ^ = ^™- Later, in the theory of functions of two variables, the property K^ recurs (§ 656) and is developed into a property K^ of more general reference (§ 77). So far as I know the properties D, D^ are new. The theory of orders of infinites and infinitesimals contains the propositions that the class 3}f"^ : of all numerical sequences, and the class 9)J"'° : of all numerical sequences with limit 0, have the properties iT, and K^ ; * cf. proposition (5). The proposition that the class 3)t"^' of all absolutely con- vergent series has the property A"^ is in effect proposition (Sj *A8 to the property ^0 of the class SUmo, cf. DU Bois-Reymond, Mathe- matische Annalen, vol. 8 (1875), p. 365. INTRODUCTION TO GENERAL ANALYSIS. 49 and the theorem of du Bois-Reymond : * There is no last convergent series of positive terms. This phrasing of the theorem and the proof given above seem to be due to TH0MAE,t the proof occurring however independently to Hadamard. % Of this proposition of du Bois-Reymond the proposition that the class 3)i'"' has the property K^ is a direct generaliza- tion and likewise a corollary, in view of (3, S^). Hadamard § proves the general proposition directly ; he considers however only those sequences {/u.^} which satisfy the condition that for every p /a increases with n. 24. Propositions. — (1) 9n^.3 .gji^'. (2) an^^.3.50i^»-°'. Further, by § 13, the relations : LK=0' (^; ©); Lm::=^" (^; ©), n n imply (3) a'<.D.LaK = ae' (^; ©); n (4) t^.d.Lt/^: = t^' {%; ^^); n (5) @ ^' . 3 . L (/^: ± ^;) = ^'± ^" (^ ; ©) ; n (6) t-©^'-=>-L/^>;:=^'^" (^; ®®)' where in (6) the condition f is that 6' and ^" are dominated by More generally, from the relations : * Eine neue Theorie der Convergenz und Divergenz von Reihen mii posilioen Gliedern, Crelle's Journal, vol. 76 (1873), pp. 61-91. Cf. p. 85. t Elemenlare Theorie der analytiscken Fandionen einer complexm Verdnder- lichen, ed. 1 (1880), p. 22, §25 ; ed. 2 (1898), p. 28, i 22. t Sur les caract&res de convergence des sSries d, termt posiiifs ei snr les fonc- tions indejiniment croissanies, Acta Mathematioa, vol. 18 (1894), pp. 319- 336, p. 421. Cf. p. 321, theorem (i). • i Loo. oit., p. 328, theorem (^). 50 E. H. MOORE. /i', 6', ©' being on ^'; /*", 6", ©" being on 5p", we have by §13 (7) t' ■ t" • ©'''• . ©"''' . =• . L M>: = 6'0" (5P'^" ; S'2"), where the conditions f, f ' ^I'e that 6', ^" are dominated by 21©', 31©" respectively. ©' ©" denotes the class of all functions (t' a" on ^'^" s [p'y]. In case ^' = ^" = ^, a corollary of (7), by the assumption p = p" = p, is (8) r . t" • ©"" . ©"''^ =» . L /^>: = 0'o" (^ ; ©'©") . 25. Theorems. — Of especial importance in the application of the property D to the theory of double limits is the following Theoeem I. If a class © -^ is the scale class of each of a sequence of instances of relatively uniform convergence, then the corresponding sequence {cr^} of scale functions may be replaced by a single scale function a- effective for each of the instances. Such a function a is the function a associated with the sequence {o-^} by the dominance property D of the class ©, the theorem being a consequence of § 9al, 2, 3. As such an application we notice the following Theorem II. If the double array or sequence (1) {/-.„} of functions fi^^ on ^ to 2t gives rise to an iterated limit (2) LL/-„„ 711 n convergent every inner limit and the outer limit uniformly as to a scale class © having the dominance property D, that is, for a cer- tain function T] and a sequence {6^j of functions 6^, (3) m.D.L/^.„=^™ (5P;©); Jje^ = r, (^;©), n m then there exist a function a and for every m an integer n and for every e an integer m^ such that (4) e . m S m, . n S «^ . D . A{ii^^ -v)= eAa,— INTRCJDUCTION TO GENERAL ANALYSIS. 51 a result expressible also in the form : for every m there exists an integer n^ such that where for every m /jl^ denotes the many-valued function and in particular, (7) Lm™„„ = '? (^; S). Proof. — By theorem I, we have, from (3) and dominated by sis /Q\ 9T?-^i (g dominated by aJR ^ 3 5m dominated by «!!)! qjj Di (\Q\ mO (g dominated by a2K 3 i,m dominated by am grj i> (11) {ATir^.{m,)^ = m^.^.m^^. (12) (^9JZ)^''^.(S^.3.ang A 'l-S • m^". . 3 . an^^^. m^". . D . m»^^^. Tl'-''. D , qi> LCD fJlALD . 3 . an^,^^^^. INTRODUCTION TO GENERAL ANALYSIS. 53 C13) ©-°1 . 3 . (g .dominated by as (14) aji.©^>.3.aK(,^) = m,. (15) a)t.©^.3.(aK^)e = a)^. (16) 3«.@^.=>.2)i, = (a}f3)3 = a«(e,). (17) ajj^^ . 3 . 3)i^ = aw^^^^ . (3jj^^^)^ = (aK,)^™^). (18) 2ji^ . 3 . aj^,, = Ti^^^^ = (an^)^ = (m^\^^y (19) (20) (21) (22) Here the proof of (10^) that Tl^ has the dominance property I) turns, with reference to (10,) and § 22a, on the considerations : Tl ^® ; arte'"'""'"'*''' "^ ™ ; a)i ^ . ~ . (313^)^ . Further, (12) follows from (11) with the use of theorem III of § 25, which is restated here as (15). We formulate (13, 19-22) as the following Theorem. If a class Tl has the dominance property D, then its extension aJtjB as to itself is closed, is dominated by 2ia)J, and has the dominance property D; further, if W is absolute, W^ is absolute, and if W is linear, Tt^ is linear. Extension of classes of functions mth respect to certain closure properties. § 27. 27. Extension of classes of functions. — Theorem III of §25 relates to instances in which the presence of the dominance property D in the scale class (.~.3?^^ Accordingly (3) P^^F,:~:(P,Z>P,).iP,^Pd. Implication is a transitive relation ; equivalence is a symmetric transitive relation. We notice that (4) (p, ~ p[) . (P, ~ p;) . (A => P.) • 3 • {P[ => Pd> that is, the relation 3 holding between two properties P^ , Pj holds between two properties P[ , P'^ respectively equivalent to the properties P,, Pj. Similarly, the class 9Jtp (§ 29), the derived property Pp (§ 29) and the transformation Tp (§ 33) are defined, with reference to a property P, in such a way as to be respectively unchanged. if the property P is replaced by a property P' equivalent to P. Naturally, instead of considering in this way (classes of equivalent) properties P of classes 9Jt, we might consider classes P of classes 9)i, a class P consisting of those classes 9Jl having the property P. Similarly, instead of considering properties (cf. § 30) of properties P of classes SR, we might consider classes of classes P of classes 3)t. On the other hand, instead of considering classes 9)1 of elements jit of the funda- mental class 50t, we might consider (classes of equivalent) prop- erties 9}i of elements /i, a property 9)t belonging precisely to those elements /I which belong to the class 2)i ; in this case, properties P of classes 9)t are properties P of properties 9)1 (of elements ja). Similarly, properties (cf. § 30) of properties P of classes 9)J are properties of properties P of properties 9)i (of elements /I). Thus the theory of the sequel is subject to modification of form. The form chosen is the form immediately available for the applications at present contemplated. However, the propo- INTRODUCTION TO GENERAL ANALYSIS. 57 sitions being written symbolically may be read in the form of interpretation desired. 29. The extension 'Sfl p. The derived property Pp. — For a property P and a class W we define a class Mp, the extension of 3R as to P* as follows : (1) aKp=n[all3i 9 3)t«.3r], viz., 3Rp is_the greatest common subclass of all classes '^ (sub- classes of M) containing 9)1 and having the property P. Evidently, Tip exists, if there exists any such class % since every such class contains 9)}. Thus, if 3)i has the property P, "iSR = Ttpi (2) m''.'D.m = mp. Further, a class 3JI having an extension 3JJp has a second exten- sion {Tlp)p identical with the first : (3) m 9 3:mp.z>.mp = {mp)p, or, otherwise expressed, (3') $R=a)ip.3.5R = 3tp, that is, the extension Tip of a class Tl is invariant under exten- sion as to P . For a property P, we define as the derived property^ Pp, the property of invariance under extension as to P, — in symbols : (4) m^"" . = .m 3 m = mp, i. e., (4') m""^ . = .m 3 m = n [aii 31 ? w.m^], or, what is equivalent, (4") m''^. = .m 3 3: [31^] 3 m = n[-}i]. *Thia extension ajj/. agrees, for properties P extensionally attainable (of. J§ 27, 30), with the extension Tip of § 27. In general, however, the exten- sion Tip need not exist, and when existent need not have the property P. t The property derived from a property Q has the notation Pq . Thus, Ppp denotes the property derived from the property Pp, viz., from the property derived from a property P. 58 E. H. MOORE. Then, from (3, 4), (5) m 9 3" 3)t^.*3.?0lp^P. Further, from (2, 4), the property P implies the derived property Pp : (6) P^Pp, viz., (6') m^.'=>.m^F. The class Pp : [all M ^-p] , of all classes 9)1 having the property Pp, includes the class P : [all 9JK] , of all classes Tt having the property P. Since the extension Tip, qua greatest common subclass fl [9t] of the class [9Z] of all classes dl containing 9Jl and having the property P, is unchanged if the class [W\ is enlarged, we have the relation : (7) 9)Z . 3 . 2}ip = n [all m 9 9)r" . 9Z = 9Jp], or, otherwise expressed, (7') 3« . 3 . aJJp= n [all 9^ 3 9}P . 9i^-P] = 9)ip^. Thus, a property P and its derived property Pp (definable by (4") without the mediation of the notion : extension) give for every doss Tl the same extension: Tlp= Ttpp- From (4, 6') we see that an extensionally attainable property P is equivalent to its derived property Pp : /Q\ r> extensionally attainable ^ P ^^^ P Further, it appears * that the derived property Pp is equivalent * From a class P of classes 3Jl we derive a class Pn of classes Tl by the greatest common subclass process D, viz., Pn coDsists of the greatest common subclasses 9J of the various classes Q-f (subclasses of P) of classes 3K- Evi- dently the class Pn contains the class P, and (Pn)n=Pn. A class P of classes Sll has the property D , in notation :Pn,inoasePn=P. Thus, for every P, Pn". Consider a class [P] of all classes P with common class Pn ; this common class Pn belongs to the class [P] and is the only class P with the property fl . INTRODUCTION TO GENERAL ANALYSIS. 59 to the second derived property Ppp : (9) P.^.Pp~Pp^. 30. Properties of properties P. — A property P may have one or more of the following interrelated properties : f (1) 50K; (2) 3R=n[9l^] .D.3}K; (3) m.^ .S Wp] (4) m 3 s wip . 3 . a)f/. We speak of properties P^, P ^*, etc.J A property P '* is extensionally attainable, in the sense that for every class W there exists a superclass Tip, the extension of W as to P, having the property P and contained in every superclass of Tl with the property P. (Cf. § 27.) A property PMs a property of the fundamental class %}. Every property P^ is equivalent to a closure property, in fact, to the closure Cj,^ (cf. § 34) as to the following transformation § In this notation, P being the class of all classes 3Jl^ having the property P, and Pp being the class of all classes Tl^J' having the property Pp, by {i") we have accordingly we have, in the terminology of classes, ■^Pp~ ( ^P )n = ( ^n )n = -ffi =" ^p< and hence, in the terminology of properties, as stated in the text, P. Z).PP~PPp- The theory of §§ 29-42 gains in perspicuity when considered in connection with the suggestions of this footnote and those of the closing paragraphs of ?28. fl) The fundamental class SU has the property P. 2) If 9K is the greatest common subclass fl [^K] of a class [9J] of classes 3^ (subclasses of 2R) having the property P, then fflt itself has the property P. 3) For every class 931 it is true that its extension SOI/, exists. 4) For every class 501 whose extension 3Kp exists it is true that the exten- sion 3Rp has the property P. X E. g., a property P^ is a property P with the property]2. JThis transformation Tq is "'t^'^ below in §§ SS.l^, 39.14, 41. 60 E. H. MOOEE. T^: Tj 9Jt = 9Ji or ^, according as 30t does or does not have the property P. 31. Funotiontheoretic examples. — Within a fundamental class ^ ^ of functions the properties P : P ^ A; L; multiplicative ; C^ ; C, have the properties 1, 2 and so the properties 3, 4. They are extensionally attainable. As to the corresponding extensions 2R^ of a class 2R we notice : (1) 3Jl^ is the least common superclass of 9)1 and A'^l, viz., the class of all functions of the form : II or Afi . (2) SR^ is the class of all functions of the form : 1=71 E a.n,., viz., linear homogeneous in a finite number of func- tions of 3Ji with coefficients from 31. (3) 9)Zmuitipiicative Js the class of all functions of the form : i=n 1=1 viz., the product of a finite number of functions of 9Jt, repetitions allowed. (4) 9)l.©^.3.2K^^ = 3Jte. (5) m'> .^ .mc=m^. As to (4, 5) cf. theorem III of § 25. 32. Propositions concerning properties P and their derived properties Pp . — P^'.'^.P^* P\'^.P^; (1) D extensionally attainable ,^ TJ12 ,^ r)14 ,^ p32 ,^ p34 ^^ TiViSA * This ( . -^ . ) is the eqnivalence of the two siatemenU : that a property P has the property 1 (§ 30) and that it has the property 3. The equivalence (~-) of two properties (of. ? 28) occurs, c. g., in (16, 24, 25). INTRODUCTION TO GENERAL ANALYSIS. 61 (2) an 3 a"9t 9 (aji^.aKj.s. a"a)ip. (3) aji 9 a- Mj, . 3 . 9}i ™^ . {mp)p = 3)^ . 9R 3 a" 3Jf p : 3 : (4) ajjp = a)z . ~ . an/' . ~ . an ^ a" a)r ? an; = a)z. (5) an = n [aK] . 3 . 2)jp = a)t. an^^.~.anp = an.~.an a (aranp.anp'^) (6) V p p ; ^ . ~ . an 9 a" a)r 9 an; = an. (7) an 9 a" anp . 3 . mp^j-. (8) an^ . 3 . anp = 9Ji . an^^; p ^ Pp. (9) an.3.a)Zp= n [aiiat 9 ajt^.at^^], that is, (9') an . 3 . a)ip = n [aii at 9 a)r . atp = at].* (10) an . 3 . a}ip = ajfp^. (11) an^^p . ~ . an^i' . ~ . anp= an; Pp^-^ Pp. (i2)t Pp3P:~:an^^.3.an^:~:a3i = n[aK].3.a}K, so that, by (8^) and § 30.2, (i3)t p^:~ :Pp~P:~:an^^.~.an^. *For 3)lp and n[S(l] exist simultaneously, and, when existent, SRp is one of the classes 3f while every class 91 contains aitp, and accordingly Tip = D [3t] . t As to a property P the three statements : (1°) The derived property Pp implies the property P ; { 2° ) Every class 3Di having the derived property Pp has the property P ; (3°) The greatest common subclass n[5t] of a class [31] of classes Si having the property P has the property P, are equivalent. (1°) and (2°) are equivalent by the definition of implication O) between properties (of. ?28). (2°) and (3°) are equivalent by the definition of de- rived properties (§ 29). J As to a property P the three statements : (1°) The property P has the property 2 of J 30 ; (2°) The property P is equivalent to its derived property Pp ; (3°) Every class 93} having the property P has the derived property Pp ; and conversely, are equivalent. 62 E. H. MOORE. (14) 9)l = n [3ff^p] .3.2«^J, that is, by (6), (14') 9JJ=n [5} 9 5Rp = 3t] .3.2Rp=3JJ.* (15) P.3.(P3Pp).P/.t (16) P^-^ .{Fj,-^ F).-^ .(PpOP). (i7)j p':D:Pj?''':m.-^.3:3nj>.m'"p.mp={mp)p=Tipp. (18) Pi:3 :gjli™^.3.9Jt,p'^2^. (19) P' : 3 : 3«i™^ . 9)l2p'"^ • = • aJiip""'. (20) F'-.-D : sutj^^ . ajf^"" . 3 . ajJip™^ (21)§ P,' . P2' : 3 : 9Kp,'= . 3 . 3)fp ™^i. (22) II P/ 3 P.i : 3 : 3}? . 3 . SJJp,™^! . (23) P,DF,.-=> .P^^DP^^. (24) P,~A.=>.Pp,~Pp^. (25) Pi^~P2':3:3)i.3.3JJp, = 3np,. * We have 3K=n[5)t 3 9j = n[all3^' 3 31'''. 3!' -P]], aKp= n [all yi" 3 3M9!". Sfi'-'-P]. 3K, the 31, the 31' for every 31 are supposed to exist. The classes W contain 3R and have the property P and are accordingly amongst the classes W. Thus 'Sip exists and is contained in every class 3}' and so in every class 3} and so in 3R = n [3J]. On the other hand, by (3i), since 3JJp exists, Tl is contained in SRp. Accordingly, under the hypothesis, 50lp = 3)1. t(15i) is (82). A comparison of (11, 13) yields (ISj). t If a property P has the property 1 of ^ 30, of belonging to the funda- mental class 3)1, then the derived property Fp is extensionally attainable and for every class 2Jt the extension 3Kp exists and contains 3K and is identical with the second extension {.3)lp)p and also with the extension SJtPp as to the derived property Pp. § If Pi and Pj are two properties having the property 1 of ? 30, then, if the (necessarily existing) extension TlPi of a class 3K as to the property Pj has the property P^, it contains the (necessarily existing) extension SKp^ as to the property P^- II If of two properties P,, Pj having the property 1 of § 30 the first implies the second, then for every class 3Ji the extension as to the first property con- tains the extension as to the second property. INTRODUCTION TO GENERAL ANALYSIS. 63 33. The transformation Tpfor a property P^. — The proper- ties P' : of the fundamental class fj, and the properties P' : for ever}' m. the extension 3Rp exists, are equivalent (§ 32.1^). For a property P' extension as to P is the transformation Tp transforming a class m into 2Kp, the class m extended as to^ P, viz., m.o .T^m^mp. 34. The closure Cy. — For a transformation T closure as ta T is the property Cj, : (i) ajt^r. = .r2«^ viz., a class Tt is closed as to T in case it contains its transform TW by T. Similarly, extensibility by T, invariance under- T are- the properties Ej., I.j,\ (2) m^T, = ,^T^^ (3) m'T. = .Tm = m, so that Jy is equivalent to the composite property CyP^, viz., (4) T::^:I^'^GtE^. 35. The extension 2)1 r- ^^e derived transformation Tj.. — The fundamental class fjl is closed as to every transformation T, so that (by § 32. IJ for every transformation T and class- W the extension Mcj, of SR as to the closure Gj. exists. For brevity we set (1) mc^^m^, (2) Ta^^T,, so that (cf. §§29, 33, 34) (3) a)i.3.any=n [all 9i 9 3)^". r^r], (4) an.3.n,aji = 9j,v. Thus, corresponding to a transformation T we have the de- rived transformation* Tj, and for every class 9)1 the extension 2)f y of m as to T. *The transformation derived from a transformation U has the notation Tu. Thus, Ttx denotes the transformation derived from the transformation. Tt, viz., from the transformation derived from a transformation T. 64 E. H. MOORE. 36. Propositions conGerning properties P^. — For a property P' : of the fundamental class '§1, we have, by §§ 29-35, (1) Cr^-Pi>; (2) ajj^i'-~.rpaji™.~.9)jp™.~.ajip=5W; (3) 9)i . 3 . (9Jtp)p = mp= Wp^ = mc^^ = ajty^ , and, accordingly, (4) T.Zi.Fa^-^Cr.^^C,^; (5) Pi : 3 : 3)1 . 3 . (ajlp)p = 3)Zp = 9)1^^= n [all 3^1 9 3)1'' . JJp''] . 37. Properties of trayisformations T. — A transformation T may have one or more of the following interrelated properties : (1) m.-^.m' (2) 3)t . 3 . TTm - (3) 3)i . => . TTm = Tm. (4) 3)1 . 3 . 9)tj.™- (5) 9Jt.3. T9){™^. (6) T=r^, that is, (6') 3){ . 3 . Ta)t = T.j,m = m, (7) 3)f,™= . 3 . T3)f,™=. (8) m,''' . Tm,"^' . 3 . Tm,'\ (Q) m.-^ .Tm=ii [all Tm, a 9)i„'^«], that is, with respect to a property Q of classes 9)t, a transforma- tion T has the associated property Q in case for every class m the transform Tm is the least common superclass of the transforms TS)?,, of the various classes m,, subclasses of m, having the property Q. The property Q of classes m is sup- posed to be such that every class m has one or raore subclasses m, with the property Q ; accordingly, § is a property of every element /I qu^ class m. As instances we '^ive the following properties : IX- INTRODUCTION TO GENERAL ANALYSIS. 65 Q : singular (consisting of a single element) ; finite (consisting of a finite number of elements) ; denumevable (finite or consisting of a denumerable infinitude of elements) ; in general, of cardinal number less than or of cardinal number not ex- ceeding a given cardinal number. We speak of transformations J", T"% T^, etc. Transforma- tions y are extensional. Transformations T^ leave "contains" invariant. Transformations T* are (or have the property Q) associated loiih the ■property Q of classes 3)t. Evidently the transitivity of contains implies that every transformation T* leaves contains invariant : (9) T« . 3 . T^. Every transformation T induces a transformation T' on single elements (qua classes) to classes 2)1. Conversely, any such transformation T' induces a transformation ysinguiar q£ which it is the induced transformation T'. — Further, every transformation T^'^^"'"'' is a transformation y flmte . deoumemwe 37a. Functiontheoretic examples of transfoi-mations T«'85_ — On respectively suitable fundamental classes 2)t of functions the seven transformations T: Tm = A-m ■,wst;m + m; %m + %m -, mm -, m^ ■, m^, , respectively, are transformations T'' : leaving contains invari- ant, and accordingly (cf § 39.7, 5) they are transformations T^^^. The classes m closed under these transformations T re- spectively are the classes m containing Tm. Hence (§ 15) the corresponding closure properties Cj, of classes m are the properties : Cj:= A; multiplicative as to % ; additive ; L ; multiplicative ; C\ ; C. Of these transformations the second, fourth, sixth, seventh are transformations T' : extensional, so that for them closure as to T and invariance under T are equivalent properties of classes 3)1. 66 E. H. MOORE. Further, the seven transformations are transformations T^, — the first two for Q : singular ; the next three for Q : finite, or consisting of one or two elements ; the last two for Q : denumerable. 38. Propositions concerning the transformations Tpi. — (1) * P 1 . 3 . Tpl2345678 _ g- yl234 ^ (^Cj.'^P.Tj.= Tp). (2) P,' . P„} .-. 3 .-. 3n . 3 . gjtp^ = 9Jlp, : ~ : 93tp,^ . ~ . 3)fp ™ : ~ : 3Kp, = aji . ~ . ^p, = 3Jt. 39. Propositions concerning transformations T. — /-l\ rp\2 2 /TI34 . -j- yS 3 y^Z . yl3 3 y24 (2) T:3 :9)f.3.2)t^n (3) T^ 3 . T'. (4) T^^ . ~ . T'* . ~ . T^* . ~ . T'^ . ~ . T^^^ (5) T= . ~ . T\ (6) T*^ . ~ . T" . ~ . T^ (7) T^ D . T'. (8) r™ . 3 . T^ (9) T . D . C^' . r^"'««'* § . Tr = Tc^ = T^^^ =T,^.\\ *For every property P having the property 1 (of § 30) the derived trans- formation Tp (ct. I 33) has the properties 1, 2, 3, 4, 5, 6, 7, 8 (of § 37) and there exists a transformation T having the properties 1, 2, 3, 4 (of § 37) whose closure Cy (of. § 34) is equivalent to the property P and whose derived transformation Ty (of. ? 35) is the derived transformation Tp. For the first conclusion, cf. ? 36 and I 32.17, 18, 19. For the second conclnsion, the transformation T„ defined at the end of J 30 is a suitable instance. taKr = n [all 31 3 2)1'^. rSR"] by definition. By hypothesis, T^S so that 5IK . 3 . 3R ^™ . TT3R '^^, that is, for every 2K Tm is one of the classes 31 whose greatest common subclass is TIt- Hence 2Jl . 3 . Tlj,''"^, that is, P*. § Cf. i? 39.9i, 35.2, 38.1. II We prove that m.o. lT'^=Tj.^m. For we have INTEODUCTIOK TO GENERAL ANALYSIS. 67 (10) T' .O .Cr-^Ir. (11) T'.:>.C,^^^*.C,^P,^^C,^* (12) T^ D 71123-15678 -j- (13) T^^ . D . T'^^ . 3 . 7"^^ D T*^ (14)t yl25 , ^ , yl27 _ ^ _ yl28 . ^ . ye _ ^ _ ^112346678 ry3R = TOy =0 [all 91 9 9JJ'». rjj"]; T'r^aK = 3Ryy = n [all Sii 3 2R "' . TySJ,'''] = n[all3li ^ to"'. 3?,/']. Now every 31 is closed as to T, so that 3fJT= 3J. Thus every ?! is an %. Hence ry2K=n [Sf] contains rry3!R= n [92]]. Further, since by (9,) Tt has the property 8 of § 37, (9K«i . rrSfi^i) implies TrSK"! . Thus Tr'm is contained in every 3Ji and hence in Tr 3Ki = n [3rf,]. *Cf. ?? 39.11,, 32.16, 36.4. t As to T' cf. ?? 37.6, 35.1, 39.9,, 32.1, 32.32- J The six triads : 125; 127; 128; 135; 137; 138, of properties are equivalent [l 39.14, Ij). Further, each triad is a triad of independent properties. This appears from the following instances of transformations. Let 3H be a linear class of functions containing a function /; =(= 0(5p). The three transformations : T^: T,m=^{m,2m.); T,: T,m=m (SR+p), T,p=m, have respectively and m"' .~. p^; 23K5W; TO + p. For the three transformations as to the eight properties (1-8} the table is as follows : 12345678 r,: - + + _ + _ + + T,: +--- + - + + T,: + + + + ----, where the tabular entry is -)- or — according as the transformation fias or has not the property. Instead of those three transformations one might consider these three : the first and the fourth of § 37a and that of J 30, the respective parameters : %; 2)1 ; Tt, P, being suitably determined. 68 E. H. MOOEE. (15) T,.T,.{C\~Gry^.T,^=T,^. (16) T^2;^(Cr,~c,J.^.T, = I;. (17) T/ . r/ : 3 : {I,^ ~ J^J . ~ . T. = T, .* From (11, 9, 12, 16j we have Theorem I. The transformations T^ being classified with respect to their closures C^, — tioo transformations T^, T^ with equivalent closures : Cj.^ ~ Cy^ , being of the same class, — in every class there is precisely one transformation y^sweys^ ^^^^^ ^^ ^j^^ class containing a transformation T^ tlie corresponding derived transformation Tj,. For example, one such class consists of all transformations f T such that every class 9)t is closed as to T; the corresponding unique transformation is the transformation identity, leaving every class invariant. Further, from (10, 11, 9, 12, 17) we have Theorem II. The transformations T^^ being classified with respect to their systems of invariant classes, i. e., with respect to their invariance properties Jj, or with respect to their closures Cj,, in every dass there is precisely one transformation y 12345578^ ^^^^^ ^^ the da^s containing a transformation T^* the corresponding derived transformation T^. The classes of transformations T'* are, in fact, contained in the classes of transformations T^. The systems of classes 9)i closed as to the T^ and the systems closed as to or invariant under the J"* are identical, being the various classes P'^si + of classes 9)i, viz., the classes P containing the fundamental class 5^ and having the greatest common subclass property n (cf. the last footnote of § 29). Theorem III. Under a transformation T^ : leaving "con- tains" invariant, the transform Tm' of the ^ri^TZo^ScS 3Ji' of a class [W] of classes m rc»S«.™ the ^/^tTJ^ZlYZfclZs of the transforms TTl of the classes 3R. *Cf. (12, 10). t To this class of transformations belong the Zkrmelo transformations, viz., transformations on classes. 3Jl to elements Ji, quS, classes, every class 2U transforming into one of its own elements ju. t Corresponding to the properties pi^s* ; extensionally attainable. INTRODUCTION TO GENERAL ANALYSIS. 69 The relations least common superclass and greatest common subclass are in general not invariant,* in fact not even under the transformations T'^'. We have, from §§32.1, 38.1, 36.1, 32.16, pSiZi. P1234 _ 3 _ ^^12345678 , Z> . C Tp ~ Pp~ P, and, from §§39.13, 39.12, 39.11, 35.2, 37.6, y 127 _ 3 _ J" 6 _ 3 Jl 12345678 3 Q 1234 3 rp rp q and accordingly Theorem IV. The correspondences: to every property P^'^* the derived transformation Tp; to every transformation J" "346678 ^/^g invariance property Ip or its equivalent the closiire property Cj,, constitute a one-one correspondence between the classes of equiv- alent properties P"" (the properties extensionally attainable) and the transformations T i23<5678 ^y g„g^ ^^ nature that for every class 50t, possession of a property P and invariance under the corre- sponding transfo7-mation T are equivalent. 39a. Propositions '\ concerning properties P and transforma- tions T. — * Consider the transformation T'" = Tadditive (of. ??37o, 33, 35.2, 39.9) and two non-zero functions /^i , fi^ on '^ between which there is identically in p no relation of the form n^fii ± Hj/'z ^ 0| tbe notations n denoting as usual positive integers. Taking the classes we have rSKi = [all «/ii], T%^= [all nfi^], TWa = [all n/^i, nfi^, n,/ii + n^/i^'], TWi= [all nfi^, 2n/i,, n^fi^ + 2nj//j]. SRj is the least common superclass of Sftj and 9J}j, while 9JJi is the greatest common subclass of SKj and SK,. TSUj contains but is not the least common superclass [all n^,, nfi,} of TTli and ^'SKj. TWi is contained in but is not the greatest common subclass | all vfij, Svii^, n^fii + ^"2^2] of T3Jla and jTSK,. 1 1) If of two properties : P^ ; P.^, having the property 1 of § 30 the first implies the second, then for every class SB the (necessarily existing) exten- sion as to the first property contains the (necessarily existing) extension as to the second property. 2) For two properties : Pj ; Pj, having the property 1 of § 30, the rela- 70 E. H. MOOEE. (1) Pi»3P2':3:9}t.3.3Kp™-Pi. (2) Pi 3 Pp^ : C : 3)1 . 3 . mp/'Pi' ■ (3) Pi' 3 P^'^ ~ : «0t . 3 . mp^n'^i'i' . (4) Pii~P„>:3:a)}.3.2)tpi = 3}ip,. (5) Pi ~ P, : C : a)! . D . ajip^i, = 3)ip^i, . (6.) P;^ ~ Pj" : ~ : a)i . 3 . 9)lpii2 = 3Jfp,i2 . From these propositions (in view of § 39.9, 5, 11 ; § 35.1 ; § 36.4) we obtain others involving Cj., Ttr, and then, using the conditions 4, 5, 6 of § 37 to replace Tir by TTl, we have the following propositions : (7) Pi 3 Cys : ^ : SK . 3 . Td)l ^'p. (8) Fo c^^:C:m.^ .rm'^pK (9) P' 3 Cye : ~ : a)J . 3 . T'M '^^'- (10) Cr,DP'::3:m.0.mp^^. (11) (7r 3 Pp : C : aR . D . 3}{p, ^^^. (12) (7r« 3 P" : ~ : 9Jt . 3 . 3)lpi. ^°^. (13) P'-~Cr8:~:9}t.3.a)tp:.= T«a)Z. (14) Cj.,4 3 Cy,, : 3 : m . 3 . ?;9)t ^i'". (1 5) Cj,, 3 Cr^^ : C : 3)f . D . T/gjf ^i^'^\ (16) Cr,o 3 Cr^c : ~ : 3)1 . 3 . T/a)t ^'""^ tion that the first implies the property derived from the second is implied by the relation that for every class 3Jl the extension as to the first contains the extension as to the second. 3) For tvyo properties : F^; P,, the first having the property 1 and the second the properties : 1 ; 2, of § 30, the two relations : the first implies the second ; for every class 3K the extension as to the first contains the extension as to the secoiid, are equivalent relations. 21) T being a transformation having the property 6 of •§ 37, for every class 3)1 the transform TSR is the least common superclass of the extensions Ttp of the class 3K as to the properties P which have the property 1 of J 30 and are implied by the closure as to T. INTRODUCTION TO GENERAL ANALYSIS. 71 (17) (7y^e ~ a^,c : ~ : an . 3 . T^m = T^'m : ~ : r.^ = T/. r? From (7, 10), in view of the properties of Tj.; Pp; Tj,, we have the propositions : (18)* P' : 3 : a)( . 3 . aJZp = U [all Tm b T'.(PD C^)] . (19) P»^ 3 : 3}i . 3 . 3Jtp = n [all Tm 9 T* . (C^ 3 P)]. (20)* T" : 3 : gji . D . ran = n [all 5Wp 9 P' . (P D C^)] . (21)* T'':3:9Jt.D.r2Jt = U [all fflfp 9 P'.((7y3P)]. 40. Composition of properties P. — As explained in § 3, any class [P] of properties P gives rise to a composite propeiiyf n [P] , the logical product of the properties of the class [P] , so that for a class Tl the two statements : are equivalent. The properties P of [P] are mutually consistent and the com- posite property fl [P] is with extension, in case there exists a class ajl with the composite property H [Pj . Obviously, equivalence of properties is invariant under com- position of properties. 40a. Propositions concerning composite properties. — (1) [P^].3.n[P]^ (i = l,2,3,4), that is, a class [P'] of properties P' with the property i (i = 1, 2, 3, 4) of § 30 gives rise to a composite property n [P] with the property t. Every composite property n[P'], qua a property of Tl, is with extension. We have, as a corollary, (2) [pi2].3.n[P] '2, and accordingly, by § 32.1, the Theorem. A class [P] of p-operties P extensionally attain- able gives rise to a composite property fl [P] itself extensionally *In(18; 20; 21) we may replace ( T^ ; P^ ; P^) by (2"; P"; P'^,. t In ease the class [P] consists of a finite number of properties : P=Pj; ■ • •; P„, we denote, as heretofore, the composite property n [P] also by the notation PiP^ ■ ■ • Pn- 72 E. H. MOOEE. attainable; the class [P] is a class of mutually consistent properties. The properties P^, P^ being extensionally attainable, the composite property Pj-Pj ^^ likewise extensionally attainable. For every class 9)1 the three notations : a«AP. (=afi(p.p.)); (3«pJp,; {mp,)r„ denote three classes, extensions of 2R definite and in general distinct. In the notation 3JipjPj,, as in W,^^^^, the interchange of symbols Pj , Pj does not alter the meaning. 41. Corn-position of transformations T. — Any class: of transformations T gives rise to a composite transformation : T Viz., the transformation * T^ for which for every class 9Jl ?;3)l = U[all Tm 9 T^], that is, the composite transformation T^ transforms the class 9Ji into the least common superclass of the transforms T3Jt of the class 2R by the transformations T of the class %. For instance, consider a property P' :of the fundamental class ^. The class % of all transformations T (or T^ : exten- sional) with closure Cy equivalent to P contains the transfor- mation t T^ : T^M = 3)t or ^ according as Tl has or has not the property P, and this transformation T^ is the composite trans- formation Tj. of the class %. 41a. Propositions concerning composite transformations. — (1) 2; = [T] . 3 . (7j,^ ~ n [all Or b T^], viz., for every class % of transformations T closure as to the composite transformation T^ is equivalent to the composite of the closures as to the transformations T of the class X, that is, *In case the class % consists of a single transformation T, the composite transformation Jj is the transformation 2' itself ; it is not the derived trans- formation Ty defined in § 35. tCf. USO, 38.1j, 39.14. 'sc^- '7 INTRODUCTION TO GENERAL ANALYSIS. IS- (1') 3; = [7^-] . aji ... 3 ._ jT^gjj ^:~:T\D .Tm^^. (2)* ' 3: = [r'].D.i;> (3) X=[r].Zi .T^' (Q) % = [T«] . D . T^Q. Thus, the properties 1 : extensional ; 7 : leaving contains in- variant ; Q : associated with the property Q of classes Tl, of § 37 are invariant under composition of transformations. Evi- dently composites of these properties are similarly invariant. As a corollary of (1) we have the Theorem. The transformations T being classified with respect to their closures, composition of transfomiations induces a corre- sponding composition of classes of transformations. A similar statement holds for the similar classification of transformations T\ i being a property invariant under composition of transformations,, for instance, 1,7, Q or any of their composites. 42. Multiplication of transformations T. Semigroups. — In the theory of ti'ansformations in general, two transformations Tj , Tj by multiplication give rise to the product transformation T^T^ such that for every class m {T^T^)m= T,(T^m). This- multiplication is associative and usually non-commutative. The powers of a transformation T are denoted by 7= r^''; TT= r(2) ; • . ■ ; T^, • • •, where T'"+" = TT^^l A class : ^=[T], of transformations T gives rise to or generates the product class.- T = [all T'], consisting of all transformations T' of the form : T'= TT , • • • T, , It 11—1 1 ' viz., the product of a finite number : n = l, of transformations T of the class ST. *A class S^^ [r'] of transformations T^ : extensional, gives rise to a. oomposite transformation Tz with the same property. 74 E. H. MOORE. The product class X' contains the class X. If X' = X, the class 2^ is a semigroup. Otherwise expressed, the class 2^ is a semigroup in case it is closed under multiplication, that is, in case the product of every two transformations T of the class X belongs to X. For every class X the corresponding class X' is a semigroup. The class X of all transformations T is a semigroup ; and the greatest common subclass of a class of semigroups is a semigroup. Thus, by § 32.1, the property of being a semi- group is extensionally attainable, in the sense of § 30, the class 9Ji being here the class X. A class X has as its extension as to this property the corresponding class X'. A transformation T, qua singular class X, generates a cyclic semigroup X'', viz., the class of all powers of T. In the present case of transformations T of classes 3)i, the subclasses of a fundamental class 9J^, there are certain relations between the processes of composition and of multiplication of transformations T. 42a. Propositions coiicerning multiplication and composition of transformations T. — (1) T,^ . T,i . D . (T, T,Y . Cr,r. ~ ( G,^ C^J ~ C,^,„_. (2) T^' .T,\0.(T, T,y . ( C^^ C^,) 3 C,^,^ . (3) Jj" . ±2" . 3 . (T2 Tj)^ (Q = singular, finite, denuineralle), that is, the specified properties Q of transformations (associated with the corresponding properties Q of classes Tl, c£. § 37) are invariant under multiplication of transformations. \^) '^ = L J'--^:-i .J. J I ^j^^7_ singular, finite, denumerable, /c\ cf. rj'i'] 3 T ' I or a composite of these properties) , that is, a class X of transformations T' with specified property i generates a semigroup, whose transformations T' and composite transformation T^, have the property i. (6) X = [Ti] . D . C^. ~ [all Cj, 9 T^-\ ~ Cy,,. INTRODUCTION TO GENERAL ANALYSIS. 75 As a corollary, comparable with the theorem of § 41a, we have Theorem I. The transformations T\ viz., the extensional transformations, (or T^'', viz., the extensional transformations leaving "contains" invariant) being classified as to their closures, multiplication of transformations induces a corresponding multi- plication of classes of transformations. This multiplication of classes is commutative as well as associative. Each class contains its composite.* Such a class of transformations T'^, viz., of extensional trans- formations leaving " contains " invariant, is a class of trans- formations T"* (§ 39.7) ivith closure extensionally attainable (§ 39.11) containing precisely one y 12345678^ viz., for every T thk derived transformation Tj. (§ 39, theorem II). This transforma- tion T12345678 ^g f^f^ compositc of the class. In proof of the last remark, denoting by T^ the composite of the class of transformations including a transformation 1^^'', we are to prove that T — 7' that is, for every class 9)1 : where mT, = rt [aii3j 3 aR'^.r^gj"], T„m=yj [all Tm a r' . (Cy-c^,)]. In the first place, by § 39.11, Ty^ is such a transformation T, so that T^M contains T^^M = 9)lr,. In the second place, TIt^ contains T^Wl, for every such dl contains every such T^l ; in fact, (501". r^ implies Tm'"'; and {T^m'\ Cy~ C^) implies T^''; so that, as stated, TM''. The following theorem concerns cases in which for a trans- formation T the corresponding transformation Tj, is the com- posite of the cyclic semigroup of T. Theorem II. The transformation T being extensional and associated with Q : singular, or finite, and accordingly f leaving * The class of these composites is not necessarily closed under multiplication. t Cf . ? 37.9. 76 E. H. MOORE. " contaiTis" invariant (§ 37.1, Q, 1), its powers T^"^ and the com- posite Tg of its cyclic semigroup : [^all T^"'] , viz., the transformation Tj for which for every class Wl (7) T„aJi= U [allT^-'m'], ^ have the same three properties and their closures are equivalent (§ 42a 1-6). Moreover, (8) T,T=1,= TT,= T,1,. Thus, T^ has the property 2 o/ § 37. Hence: * /0\ 7112345678 . Tl rp rp (,yj -'o ' J^r— -'-To — -^o- We proceed to the proof of (8) for the case Q : finite, the proof for the case Q : singular, being even simpler. T being extensional, T^'^'^W. is always contained in T^^+'^gjJ, so that, in view of (7), in an obvious sense, (10) T, = Lr<"', T^m = Lr(»'3R. n n Accordingly, for every class W, Tjm = T^Tt ; that is, TJ= T^ . Further, T^SJi is always contained in TT^. On the other hand (cf. § 37 Q), T being associated with Q : finite, the class : TT^m = U [all r5R 9 gfj J".^ • fl"«e-| ^ is contained in T^. In fact, such a finite subclass 9^ of T^ is a finite subclass 9J of some T<"'9)i, and accordingly every T^ is a subclass of some T'""*"''?)! and so of T„9J{. Hence, always T^m = TT,m ; that is, T^ = TT^ . We have, for every n and m, T^m = T^^^T^m, and so, by (7), T,m = T.T^m ; that is, T„ = TX- After this excursus (§§ 28-42) on the theory of classes in general, we return to the consideration of classes Tl of functions on a general class 5p to the class 3t of real numbers. The extensions: of classes 9JJ of functions of a general variable. § § 43—44. 43. The extension Tljn,. — The properties: A = absolute, *In view of § 39.14 and § 37.6, by theorem I. INTRODUCTION TO GENERAL ANALYSIS. 77 L = linear, and their composite : AL, are, as we saw in § 31 and § 40a, extensionally attainable within the class M of all functions on ^ to 31. The formulas for the corresponding ex- tensions Mji,, m.1, of a class m, were given in § 31. In terms of the corresponding extensional transformations T^ , Tj^ : (1) Tjm = m^, T^m = m^, which are associated with Q -. finite (§ S7 Q), and their prod- uct 1YJ\ : (2) {TLT^m=^T^{TM)^{m^h, in accordance with § 42aII, for the extension M^l we have the formula : (3) m^^ = U [all (T^Tjm], which is expressible also in the forms : (4) 3K^x = JuiT/fj-m = j^iT^T^ym. The formula (3) states that the class aJt^^, i. e., the class m ex- tended to be absolute and linear (cf. §§ 27, 29), consists of all the functions each of which belongs to some class (T T y"'a)i, * i. e., to the class arising from the class 3K by the transformation Tl Ta repeated n times. The transformation Tj^T^^ transforms any class 3Jt into the class : T^TJiJl, consisting of all functions of the form : D «,/^,-f E a Ail. . It is convenient to write (5) 3)i = [/.] , 3JJ^ = [mJ , 3JJ^ s [mJ , m^^ = [/.^J . Then /Lt^ is of the form : ^i or Aix., and /t^ is of the form : (6) Mx= lLa^il^, i=l while /Lt^£ is derived from a finite number of functions /a of 9Ji by a combination of operations T^ , T^ involving a finite num- ber of coefficients a of 31. *I. e., 9)Z^£ is the least common superclass of the classes (ri7'^)W5)2 (ct. §2). 78 E. H. MOORE. 44. The *- and if-extensions Tlj^, 9K#. — For a class Wl we defined in § 27 the ^-extension 3Ji^ and the ii-extension Tl^ : Accordingly ;a^, /ijf are of the respective forms : * * A function /^^ is the limit of a sequence {/J-im} of functions /^Ln of the class Tljj, which converges on ^ uniformly as to the class SR, and accordingly, as to some function fi of the class 3R. A function fizn is linear homogeneous in a finite number of functions of 3K, the coefficients being real numbers. Arranging the totality of functions of SB thus related to the functions of the sequence {|Uin} as a sequence {|"n}, we suppose that the function Min is linear homogeneous in the first m„ functions /is with coefficients Onk (fc = l, 2, • • •, mn), of which some or all may vanish. Hence, we may set mn (1) /■'* = Ij 2 amcMi i^;/^). In particular, for the case : (2) mn = n (») ; a„k = l (nk), we see that 3Jl* contains the class of all functions /i* of the form : (3) ^* = L . => . a)t, = {m,\^^, . m^ = (9Jt^.)(»^^). m" .^ .yi" . 9Jt/^^ . 9)j^ = m^c ■ 5»t** = a«* . 9K#^^^^ . 9Jt# = m^i,c . a«## = 9}t#. (6) a)t^.3.g«,/. (V) 9JL-' . 3 . 931/ . 9«, = 9Ki. These types are all of the form (6). The *-extension 3R^ of a class 3)i of functions contains the sum of every series of functions of 502, which converges for some modified type of convergence of the form (6) uniformly on % as to some function of 331, and in fact it consists of all such sums. (5) 80 E. H. MOOEE. <8) m^^.o.A{mj,y\ (9) an^.m/.3.2)f* = 3rt# = a)i^^<;. • 3K/. (11) m^^.Aim.r'.^.m^^. '^i, ' ' ■)^ 'y viz., in case the property P is invariant under multiplication of INTRODUCTION TO GENERAL ANALYSIS. 81 its various argument classes 'Sfi of functions by any (one and the same) function <^ on ^ which for TyeWpT^osm..- Uhipartite properties -P of classes 2)i are properties of classes Wt taken singly. Multipartite properties P of classes 501 are properties of classes 9)i taken in systems,* usually of a specified type ; they are thus unipartite properties of such systems of classes. Evidently the properties : L; C; D; D^; D^, are homogeneous unipartite properties of classes 9)t, while the 'properties : D,; A, are positively-homogeneous unipartite properties. Amongst the homogeneous properties P are moreover sys- tems of linear homogeneous relations with constant coefficients amongst functions (not amongst functional values), e. g., linear homogeneous transformations of ■n-partite functions on ^J} to 21, — the classes 3)1 being in this case singular : consisting each of a single function. For multipartite properties P, in the homogeneity just de- fined the various classes 9)tj, 3)ij, ••• having collectively the. property or relation P enter as of like weight or dimension ; the homogeneity is isobaric. By the introduction of discrim- inating weights more general types of homogeneity of multipar- tite properties may be defined. The complete independence of the properties : L; C; D; D^ or A. The complete existential theory of the proposes : L; C; D; i»„; A. §§46-48. 4f!. With respect to systems : we have been considering in particular the properties : *By tbe use of the numeral suffixes, 1, 2, ■ • ■ in the system (33}i, 2)1,, ■ ■ ■) employed in the definition no implioatiou of denumerability of the system of classes is intended. 82 E. H. MOOEE. L; C; D; D,; A, of classes 3)1 of functions on ^ to 2t. We know that 9)i^ im- plies 3)l'°''. It is interesting to note that no other general rela- tion holds amongst the five properties, and that, in a sense to be defined directly, the four properties : L; C; D; D^{pv A) are completely independent. 47. Consider in general m properties : (1) P.'>P.:-',P,n, of systems ^ of a certain type. A system ^ has with respect to the m properties (I) a definite one of the 2" m-partite characters : ^) ( +); ( ), the part i of the character of the system J^ being + or — according as the system 5^ has or does not have the property P^ . The complete {elementary) existential theory of the m properties (1) of systems J^ is the body of 2"* propositions stating for the various characters (2) that there exists or that there does not exist a system 5Z (of the type in question) having the character in question. The m properties (1) are completely independent (and mutually consistent) in case the 2'" propositions of the complete theory are propositions of existence. The m properties are independent in the usual sense in case there are m propositions of existence, viz., in case there exist m systems X failing to have each precisely one of the m proper- ties, so that no one of the m properties is implied by the re- maining m — 1 properties. Sets of independent properties (postulates) fundamental for various mathematical disciplines have recently been exhibited. There would be some interest in the existential theories of those sets and in the determination (if possible) of sets of completely independent fundamental properties. INTRODUCTION TO GENERAL ANALYSIS. 83 48. Theorem. For the five properties : (1) L; C; D; D,; A, of systems : (2) E = (3t;^;9K), viz., of classes Tl of functions on ^ to 21, the complete existential theory consists of 2^ = 8 propositions of non-existence, viz., that no system (3) Z--+ exists, and of 2^ — 2^ = 2-1 propositions of existence, so that, in particular, the properties : (4) L; C; D; D„, and likewise the properties : (5) L; C; D; A, are completely independent. The non-existences (3) are expressed by the proposition : (6) 5p . 3)t^ . => . m''", already referred to. We notice further that (7) ^«°"^ aii-^. 3 . 9)1''^; (8) ^:p*""^9Jt^^«.3.a)H; (9) ^^'°s"i^'-.«DK3.9«^; (10) ^^"'s""'^30J^.3.5DH. The effect of these propositions is to cut down the 24 characters existing for ^^s unrestricted to 15 for ^ finite; to 14 for ^ dual ; to 7 for 5p singular. A class 93i of functions on ^ *°"^ to 21 is a point-set in real flat space of say n dimensions. A linear class 3)Z is the locus of points whose n coordinates satisfy a system of linear liomo- geneous equations, whose coefficients all vanish in case the point-set is the space itself: 9)J = 2)t"". Examples of classes 9}J of the vai'ious characters will be given as follows : 84 E. H. MOOBE. (7; 7; 1) on ^"" (n=l; 2; 3) and q ^j, m denumerably infinite __ ^HI These examples are sufficient, since the character, as to the properties (1), of a class 9JJ on ^ is invariant under the enlargement of ^ to a class ^' by the adjunction of a class ^" having no elements in common with ^, the functions /i of 3)i being extended to functions fi of 3Jt' vanishing on ^". We observe that there is no loss of existing characters in passing from ^ unrestricted to 5p denumerable nor in passing from ^ finite to ^ of three elements, and that the 1 5 charac- ters on ^ of three elements include all the 12 existing non- linear characters. 48a. Examples for the seven characters for ^ singular. — Character. Functions of the class 2B. {+-K-h + -h) (a)i=3)t'=3listheonlyother linear 3«.) <- + + + +) 1 {-+ + +-) -1,2 (-++ ) -1 ( + + +) l/« («)* ( + +-) -l/n,2/n (n) ( -I- ) -l/*! («) 486. Examples for the seven additional characters for ^ dual. — Character. Functions of the class 3K. [-\- + + ) (a, -a) (a)* (- + - + + ) (1,0), (0,1) (- + - + -) (- 1, 0), (2, 0), (0, 1) (- + ) (-1,0), (0,1) ( + +) (1/n, 0), (0, 1) (n) + -) (- IM 0), (0, 1), (2/n, 0) (n) ) (-1M 0), (0,1) (n) *The symbols (n), (a) denote /or every n, for every a. INTRODUCTION TO GENEEAL ANALYSIS. 85 48c. Example for the one additional character for ^ of three elements. — Character. ' Functions of the class Tl. (+ + + + -) ("l J a2> «1 + «2) («1«2) 4Sd. Examples for the nine additional characters for ^ de- numerably infinite. — Of the 2* = 1 6 linear characters the relation 6 of § 48 cuts out one of every four. The remaining 12 linear characters exist, the three LOD for ^ finite, viz., (+ + + + +) for ^ singular, (+ + + H — ) for '^ of three elements, ( + +H ) for ^ dual, and the remaining nine for ^ denumerably infinite. Let m' ^ [m'I ; m" ^ [^"] ; m"' = [/^"'] be the respective classes of all functions : fi' on ^' = [/)' = 1, 2, 3, • • -J ultimately zero ;* fi" on ^" = l^p" = 1, 2, 3, • • -J ultimately constant ;'\ fx" on ^'?P" for every p" of class 9JJ' on ^' and for every p of class m" on ^". These classes have the characters : m': (++- + +)] m": ( + - + + +); m'": (+ + +), while we have above listed classes having the characters : m'= M: (++++-); m'^l/x']: (+ ++ — ), on ^^=[j=l, 2, 3] ; 5P' = [z = 1, 2] respectively. The nine classes SR to be exhibited are the three classes W, W, W" and the six classes arising from them by composition with 9)J^ and Character. Class 3K. (++- + +) 2«'on^' (+ + - + -) {Wm'h on f.'^' *That is, such that there exists Po' (dependent on /i') for which p' Spo' implies /J-'p' = 0. fXhat is, snch that there exists po" (dependent on /x'^) for which p"^Pf/'' implies /^"p" = /""po"- 86 E. H. MOOEE. Character. Class 331. (++ ) {mm\on^'^' (+- + + +) m"onS^" (+-++-) {mm")jr, on ^'^" (+-+ — ) {mm")x. on ^'^" (+ + +) W" on 5]3'sp" (+ — +-) {mm"')L on 5p35p'5p" (+ ) {mm"Y on 5p2?p'5p" Here the classes : 5^' ; ^'5P'; • • • ; ^'^'^", are obviously de- numerable. Further, the notations: (2R'3Jt')£, etc., denote classes of functions on ^^^', etc.; viz., e. g., 90'i^2R' is the prod- uct class consisting of all products /^V' and (9Jl^2)i')x is the class 'Sft^Tt' extended so as to he linear. Functional characterization of classes ^ of elements. § 49. 49. From the complete existential theory of properties of classes Tt of functions on 5p to 21 we are led to characterize certain classes ^ of elements functionally, by means of proper- ties of their classes 3Jt of functions. Thus, using the properties L, C, D, Dq, A positively, in the following theorem we char- acterize the classes (I) 5p singular, (II12) ^ singular or dual, (II) ^ finite. Theorem. In the following three sets (I, 11^^, II) of prop- erties of a class ^ of elements any two properties of the same set are equivalent. (I 1) singular. (12) m.'^.m". (12') gjic^.D.gjt^). (13) ajt^.3.9)i^^^«^. (13') a)i^^^.3.3Ji^». (11,2 1) singular or dual. (11,2 2) 2}t^^».3.9Jt<^^^. (11,2 2') an ^^^^«. 3.9)1^. INTRODUCTION TO GENERAL ANALYSIS. 87 (II 1) finite. (112) 2)t^. s.an^^. (112') a)t^^^.3.3)J^. (II 2") 2)t^^^ . 3 . 9JJ^. (II 3) (55i^ 9 p . 3 . a" M 9 M^ + 0) : 3 : a" /. 9 p . 3 . M, + 0. (II 4) (3)1 3 3" /x 9 ^ . 3 . /^^ + 0) . 3 . 3)^^. (114') (50?^ 3 a'/a 9 p.3./i^ + 0).3.2)f^. (115) Lm„ = ^.=>.L/^„ = ^ (^; 1) n n For a finite ^ the conditions (II 3, 4, 4', 5) obviously hold. That they do not hold for ^^" appears from the examples : ad II 3 : m is the class m' of § 48cZ ; ad II 4, 4': 9)1 is the class of all polynomials in p with real coefficients ; adII5:/x„, = | (np);e^ = (p). 49a. Propositions. — Denoting for a class ^ by 9)1 "" tha class of all functions on 5p to 31, for a denumerable class ^ we have the two propositions : (1) 9)t^".3.3)t^. (2) 9)i''":3:L/^„ = ^- = -L/^„ = /^ (^J^ of which the former has been considered in § 23c5. The question arises whether denumerable classes ^ are func- tionally characterized by the validity of one or both of these propositions. We proceed to prove the second proposition ifor a denumer- able class ^, an assumed convergence : (3) Jjt^„ = f^, n is uniform as to some scale Junction a. In terms of an auxiliary sequence {e^J: (4i, 4,) 0<6„^,<6„ (m); Le„. = 0, 88 E. H. MOOEE. of positive numbers, e. g., e^— 1/m (m), in view of (3) there exists a sequence [n^] of positive integers satisfying the con- ditions : (5i) «„>+! > «„ W ; and, accordingly there exists a function o- satisfying the con- ditions : (6,) ^,S1 {p); (62) p.n^n^.-H . A(fi^^ - /i^) ^ e^a^ . Any such function a is effective as such a scale function o- : (7) Lm„=^ (^; <-). n The proof of (7) runs through the relations : (8) pSm.n^n^.'D . A{fM„j, — fi^) ^ e„,a^ (S^, 6,) ; (9) p . n S n^ . D . 4(/i„^ - fi^) ^ e^a-^ (8) ; (10) p.n.^. A{^„^ - M^) ^ e^")] = [?>"], !. e., the class ^ = [p] whose elements p = (p', p") are bipartite, the first part p ranging over ^' and the second part p" ranging independently over ^". In practice and with occasional caution we replace the notation ( p, p") by p p". *These relations Ki, K^ are properties K^, K^ on 5{53j ^?3i where 3 de- notes the class [m] of positive integers, in terms of which are defined in § 72 the bipartite properties K-^ , K^ of functions and classes of functions on %, and their associated properties. INTEODUCTION TO GENEEAL ANALYSIS. 91 Similarly we obtain from a finite number of classes ^', ^", • ■ • , «P'"' the product class ^"^" ■ ■ ■ 5pf">= [p'p" ■ ■ ■ p^"'] . For present purposes this composition of classes of elements is associative. The classes ^', ^" may be the same. We write and similarly, for example, ^^'^r r^ [{Pv Pv p,, v", K)] ^ Uhp'.p.p" p'^^ ■ Thus p, p" are generic elements of the respective classes ^', ^" conceptually distinct but not necessarily actually dis- tinct, while Pj, p^ are independent (conceptually distinct) generic elements of the class ^. 51a. Remark. — It is to be noticed at once that the theory of functions and classes of functions on ^ to 31 as developed in Part I is applicable to functions and classes of functions on ^'^", etc., to 21, and this is to be thought of in the sense that ^ in its generality includes ^'^", etc. 52. Reduction. — ^ being a class of elements, a reduction R is the transformation of ^ into a subclass of itself, in notation ^^ = [^^J , the class ^ reduced by the reduction R. We may speak of the reducing condition R, the class ^jj being the class of all elements p of ^ satisfying the condition R : ^^\_p-]; ^^^[pj = [alli,^]. A reduction R transforms ^ into the reduced class ^^ and a function i^ on ^ to a class ^' (cf § 4) into the reduced function F^ on $p^ to %' : ■F^{F^\p'^); F^^{F^\p^^), and a class % of functions on ^ to ^' into the reduced class %jt. of functions on ^jj to ^' : %^[F^; %j,^{F^]. The reduction R is said to be applicable to the class ^ and accordingly to functions and classes of functions on ^. 92 E. H. MOORE. The reduction R is the identity reduction, in case the reduced class ^^ is the class ^ itself. 52a. Propositions. — li being a reduction applicable to 5|3 and 3)i and *. (12) ajt/ . an^^ . => . (yjf^)/ . {m^)^ = (3Ji«)#, Here (11) is a corollary and a generalization of § 44a8, in view of (1, 3, 4, 9), and (12) is a corollary of (11) in view of § 44al2. 526. A unipartite property P of classes of functions is in- variant under reduction of classes in case where R denotes a reducing substitution applicable to ^. Similarly, a bipartite property P is invariant in case A,) {m,<^r.R.^.{mn, on the classes ^', ^" respectively to 31, arises by multiplication of constituent functions the prodwt class : of functions : M>"= {K'^^P"\p'p"h on the product class ''^''i^>" to 9t. The similarity of notations and the disparity of notions for the two product classes, of elements and of functions, in practice with occasional caution cause no confusion. This process of multiplication of two classes of functions, as extended to any finite number of classes, is obviously associa- tive and commutative, it being understood that in the permu- tations of the constituent classes each class carries with it its own variable. 53a. Suppose the two classes ^', ^" are the same : Then a function /i on ^^ : /^= {l^p\p), may be considered as a function ^l on (^^)jj : 1^ = (f^pplPP) (f^pp = f^P (p))' where (5P^)a' is ^'^ reduced by the reducing substitution : ^•- Pl=P2=P- Conversely, B reduces a function fi on "^V^s to a reduced func- tion fiji on ('~^^'^)ij which may be considered as a function on '5)]. Accordingly, the product Tt'M" in the sense of § 14 of two classes of functions on 5p may be considered as the reduced product (3)f'3Ji")i2 of the product 9)t'3)f" of the two classes in the 94 E. H. MOOKE. sense of § 53. In the sequel, in the absence of specification to the contrary, the notations 9Jl'3Jl", etc., are to be understood in the sense of ^ 53. 54. Composition. — Of the classes of functions arising by composition of two classes W, Tl" of functions on 5p', ^" respectively we consider the following four : a) the product : m'm" on ^'^"; h) the general product :.(3Ji'3}t'% on {^'^")r ; c) the *-composite : {m'm")^ on ^'^" ; d) the general ^-composite : (9}t'9)l")ij^ on (5p'^'%. Here R denotes the general reduction applicable to the product ^'^", while the ^-composite (simple or general) is the it-ex- tension in the sense of § 44 of the product (simple or general). We note by § 52alO that {m"m")^R is contained in (9)1' 9)1'%^. In case the reduction R is the identity reduction, the general product is the simple product and the general »-composite is the simple *-composite. 54a. Propositions. — A unipartite property P of classes of functions is invariant under multiplication of classes in case a;) m'^'.m"'' .^.{m'm")^. A bipartite property P is invariant in case a;) {m', • (a)tw%(e's")B contains (an;, W;,)ier where the conditions f', t" are that 3R'^,, 3R'l>, are dominated by 31®', 31©" respectively. JJc • M . J . (JJCiJJiij^fiK'im")^ contains {{m',)U^l)^")ii= m'*^"*)M- (9) a«'''' . 9Ji"^' . 3 . (a«'9Ji'%* contains {m\m\)s. (1 0) ajt'^ . TO"'' . 3 . (to'3K")b^ . (TO'a}r')^,*=(an'3Ji")^=, . (11) W"" . W"" . => . {m'm'%^ contains (TO'=,TO"J^=,. (12) TO''' . TO"'' . 3 . (TO'TO")^*=(TO'=,TO'^)^^=(TO'^TO"^)^=, . Here (7) is a corollary of § 24.7. it! being the identity reduction, (7) is precisely § 24.7, and from this special case the general case is securable by reduction in view of § 52a9, 4. Further, (7) with §§ 54al2, 44a2j implies (8); (8, 5) implies- (9); (I3) with § 52a8 implies (lOj); (lOj) with § 44a5^ implies (IO2); (9, IO2) implies (11), since *-extension leaves contains invariant; (6, 11) implies (12j); (12j) with § 44a5,, 6^ implies (12.). 55. *-composite of a number of classes. — Omitting now further consideration of reduction, in view of §§ 54al3; 44:a5^;. 54al2, we have Theorem I. TO', TO" being two classes of functions on ^',. ^" respectively, having the dominance pr-operty D, the *-compos- ite (TO'TO")^ is linear and closed, has the dominance property D,. and is expressible in the forms: {m'm'%; (TO'^TO")^; (TO'TO"J^; (TO'^TO"*),. The dominance property D is invariant under *-extensioa and under multiplication of classes. Hence, as corollaries of the preceding theorem, we have the following theorems : Theorem II. TO', TO", TO'" being three classes of functions on 5P', ^", ^'" respectively, with the dominance property D, the- ^-composite (TO'TO"TO"')^ is expressible in the various forms: 96 £. H. MOOEE. {m'm-'m"%; im'^m"m"')^; {m'm'^m"%; {mwm'\)^, {Tt'^m\m"%; {m'^m"m'\)^; (m'm"^m'\)^; derived from the original form by the ^-extension of one or more of the constituent classes, and also in the various forms derived from these by the ^-composition of pairs of constituent classes, e.g., from {W^W^Ti'")^ the forms: {{m'^m\)^m"%; {{m'^m"%m\)^; {m'^(m\m"%)^. Theoeem III. Tt', ■ ■ ; SJt'-"' being classes of functions on ^', • ■ •, ^^"^ respectively, with the dominance property D, the *-composite class: is linear and closed and has the dominance property D. It is expressible in the various forms derivable from the original form by a finite sequence of operations of ^-extension of individual constituents and of it-composition of two or more constituents of the original form or of a form already derived. Theobem IV. The genus LCD of all classes W linear, closed, and with dominance property D, is closed under ^-com- position of classes, that is, Characterization of the functions of the *-composite class {mWy^ . § 56. 56. Id the General Analysis of Functional Equations in- volving several independent variables the *-composite (9Jl'9}t")^, of two classes Tt', W enters fundamentally. For instance, the class 6 of all continuous functions of two real variables : 0=p', p"=l, is the ^-composite (S'®")^ of the classes ©', 6" of all continuous functions of the respective variables : O^p'^ 1,0 ^p" ^ 1. There arises the necessity of conditioning the constituent classes W' , Tl" of functions, perhaps with respect also to the INTRODUCTION TO GENERAL ANALYSIS. 97 constitueat classes ^', ^" of elements, in such a way as to secure functional characterizations of the functions : P= {Pp'P"\p'p") = {Pj>\p)' of the *-composite class : on ^ = ^'5p" to 31. In Part I we found that the classes 2)J^, • • •, 9K^^ have in common the closure and the dominance properties : L, C, D, D^, Z)j, A. They belong to the genus LCD of classes linear closed and with dominance property D ; this genus is closed under ^-composition of classes (§ 55 IV). In Part II (§ 82 II) we are to secure conditions on the class 3Dt' such that for every class Tt' of the genus LCD the *-com- posite (SR'^Ji")^ is the class j^ of all functions on ^ to 31 satis- fying certain two conditions : (1°) K[,mxm"); (2°) m"{p'). (Cf. §§ 82.8, 72.12.) The second condition is that the function for every p' belongs to the class 3Jt". The first condition is defined in terms of Tt', Tl" and certain postulated features : of the class ^', viz., certain suitably conditioned relations : K[ ; K'^ , on the classes ^'3, ^'^'3 (where 3 is the class [mj of positive integers), of which metrical instances are : K[ : 7i;,„ . = .Ap'>m; K', ■ ir;,^,,„. . s . A{p[ -;jQ S - . In terms of the respective relations: K[; K'^, we define (§ 72.7, 8) two properties : of classes 9)t' of functions on ^', and have the theorem that for every class gjj'-o-si'-^a' the class % contains the class {W'W')^ ; cf. § 73.2j2, 5j2. 98 E. H. MOOEE. On the other hand, we have the theorem (§ 81) that for every class SJi''^'^ the class (9)t'9H'% contains the class ^, where the property A' of classes M' on ^', and the relations : K[; K'^, and so the property J^[^W\M") of functions on ^ are defined (§§ 75, 77, 79) in terms of a certain postulated development : A', of the class 5p' of elements, of which instances for the cases ^' , 5p' are furnished by the denumeration of ^' and the partition by sequential halving of the linear interval ^' . Accordingly, we have (§82.7, 8) the Theoeem. ^' being a class of elements and A' a development of^', Here the superscripts A' on the notations K[, K\, K[^ are to denote that the corresponding properties are those defined in terms of the development A' of 5p'. As leading up to the general theory which has been sketched, we shall consider (§§ 60-67) the cases III and IV in detail, after explaining (§§ 57—59) a scheme of notation useful in the study of functions of two independent variables. Functions and classes of functions of two independent variables. General notations for p7-operties involving uniformity and relativity. §§57-59. 57. Properties of functions of two variables. Uniformity. — Denote functions to 21 by the small Greek letters as usual, for instance, 0' = {'p' I p')> f = (C" I p")^ = (.^p\p) = {p'p" I p'p") on the respective classes ^', ^", ^ = ^p'^p". P' denoting a property of functions on 5p', ^' denotes a function 0' having the property P'. INTRODUCTION TO GENERAL ANALYSIS. 99 A function p'p" \ p'p") on ^'^" reduces for every p" to a function ^p„ = {4>p,j,„ \ j)') on ^' and for every p to a function p,= {<^j,,j,„\p") on 5)3". (Cf. § 4.) In special cases, e. g., if 5p'=^", the notations ^^, , (^y, are obscure and require modification ; in general, however, they will be found perspicuous. In case the context specifies the particular p or p" we are at liberty to use instead of ^j,, or c^^,,, the notation itself. Thus the statement ; for every p" the function (f> qud function on ^' has the property P', we write in the three forms : (1) p.^-V) V {p")) V''"\ of which the last two may be read : the function 4> has the prop- erty P' for every p" . Now the function may have the prop- erty P' for every jo" of 5p" and in some sense uniformly * on ^"; this we indicate in the two forms : (2) ^P' C^s"); 0^'(*"). For every particular use of this notation the precise sense of the uniformity is to be defined. Usually uniformity enters with existential properties and holds in case the objects (or some of the objects) specified as existing for every p" of $p" exist inde- pendently of/)" on ^", i. e., uniformly on ^". For example, setting for brevity, B^ s is contained in; B^ ^ is dominated by, we give to the notations : (3) cf,'^""^'; (pJ3,fl"'iP">- 0-B„?KW the respective meanings : The function ^' belongs to the class 9JJ'; For every p" the function <^ , qua function on ^', belongs to the class 9)t'; The function cf>, quS, for every p" function on S^', is a func- tion of the class 331' uniformly on ^", i. e., for the various p" of ^" the same function of the class 9)i', and we similarly give to the notations : /t\ J.' •Si'"' .Bi'W^p") ,£iK'C¥'0 iv 9 ; 4> ; 9 , the respective meanings : * Cf . i 6. 100 E. H. MOORE. The function cf>' is dominated by the class 3)Z", i. e., by some function fj," of 9)i". For every 2>" the function

, qua for every p" function on ^', is domi- nated by the class Tt' uniformly on ^", i. e., by some function /a" of W the same for the various p". The definitions (3) generalize those of § 6. 57a. Relativity. — It is convenient to think of P'(p") and ^'(^") as properties of functions on ^ = ^'^". Similarly, with reference to a property P' and a class W it is fre- quently convenient to define a property -P'(9)t") of functions on ^ = 5p'sp". This property P'(a)i") is relative to the class W as scale ; in practice, it is relative uniformity. For example, let ^' be ^' : a'^ Sp'^a[, and let P' denote uniformly continuous on S^', viz., let ^' mean (1) e : => : a- d 9 ^(^; -p^ < c?, . 3 . ^(<^;, - c^;,) S e. Then <^-P'(i'") means and we give to P''-'V"">^ ^.p'cm") ^j^g respective meanings : e.-. D .-. a^d 9 (3) S ix" 3 e .-. -^ .-. S d 3 (4) A{p[-p',)^,^,) ^ eAf.';, . In accordance with § 6, omitting p" and understanding it as implied by the notations involving 0, we may write (3 ; 4) in the compacter forms : * *3') For every positive number e there exists a positive number rfe (depend- ent on e) such that for every two (real numbers) elements ^j', p/ of ^P'^^the absolute value of whose diSerenoe is less than de the absolute of the difference of the corresponding functions ippi', iPp/ is (for every p") at most «. INTRODUCTION TO GENERAL ANALYSIS. 101 (3') e : D : a^ cZ, 9 A{f\ -p'^),,)Se; 3[ fi" 3 e::i -.3: d^ 3 ^^ ^ A(p[-p',)^, - <^,,) ^ eA^". In the case (3) is said to be uniformly continuous on ^', uni- formly on ^", and in the case (4) is said to be uniformly continuous on ^', uniformly as to scale 9Jt" on ^". We shall prove below (§§ 62, 66) that, if 6' is the class of functions con- tinuous and so uniformly continuous on ^', and 2R" is a class, linear closed and with dominance property I), of functions on ^", the class (6'5[R'% is the class of all functions (^ oipp" such that <^ for every p belongs to 9)1" and for every p" belongs to 6', its uniform continuity on ^' being uniform as to 3Jt" on ^", that is, in symbols, (5) (6W)h = [all <^ ^'^™"' • 'So^''^^'''], P' denoting uniformly continuous on ^'. For a property P', the notions P'(^") and P'(2Ji") are usually determined, as in the preceding instance, in such a way as to satisfy the conditions : A) sjjj"-BiS" . 3 . ^p'(.w') _ 3 _ ^P'(6") . C) (kP'oir'n . 3 _ d)-P'«"\ 58. Properties of classes of functions of two variables. — The notational conventions of § 57 are extensible to classes %', g", g: of functions on ^', ^:p", ^ = ^p'^" and properties P of classes ^' on ^'. Thus enter the notations : c^'-f". gp'(y')j '^p'iV")- g;P'(3)!"). 59. Derivation of properties. — From a property P of func- tions on ^^s we derive (cf. § 2) a property P of classes of func- tions : the class M has the property P (of classes) if its every function /jl has the property P (of functions), i. e., 4') There exists a function fi" of the class W such that for every positive number e ■ ■ ■ the absolute ot the difference of the corresponding functions fpi't fpi' '8 ^^ most e multiplied by the absolute of fi" (for every p"). 102 E. H. MOOEE. (1) 2R^:= :M.3.M^- Thus, from the properties : (2) P : is contained in Q ; is dominated by Kp".D .P'. But no general definition of the notations ; P'(^"), P'{W), as properties of functions or of classes on 5p s ^'^" has been given. We agree that in defining the notations primarily as properties of functions we define them at once secondarily as the derived property of classes, so that the notations : (■7) CKP'(W'). cvp'(g)!") have the respective interpretations : (8) ; ^ S . 3 . ^'(^'". INTRODUCTION TO GENERAL ANALYSIS. 103 Properties B bipartite on functions and classes of functions. The derived properties B, B^ of classes of functions. §§ 60-63. 60. Bipartite properties B. — la the functional characteriza- tions to be secured there enter fundamentally properties B bipartite on (or relations B between) functions and classes of functions on ^. We use the various notations : (1) {, my or <\>Bm or <^^^. Such a bipartite property B gives rise in connection with every class 9)t to a property B'^\ of functions ^, and the third notation (1) expresses this fact. (We do not now need to at- tend to the property <^B of classes of functions.) Thus, in accordance with preceding conventions, we meet the notations : (2) ^'■®''^"', (f^B'M'ip")^ ^S'aK'PB'O^ ^B'W(W) . ('3^ c^'B-m' crB'W(p"} CV S'.TO'C¥") CV £'a)!'(!ffi") where B' is a property bipartite on functions and classes of functions on ^', while B'm'{^"), B"jm'(m") are properties of functions on ^ = ?p'^", subject to definition for the various properties B' to be considered. For example, the bipartite properties B : (4) B^^ : is contained in ; B^: is dominated hy, yield the properties BW, : (o) -B„2R : is contained in 501 ; B^^ : is dominated by 9)^, primarily of functions and secondarily of classes of functions on ^ in the respective senses already used. The corresponding properties* B^W!, B^' are the properties of § 57.3j, 4j respectively. Thus the properties : (6) B,mw)) B,mW), are already defined (§ 57.3, 4). We define (7) B^m\m") ; B^mw) as denoting (8) B^m'm"; B^m'm", *The properties B^, B-^ are of general reference, and for that reason we do not write B^', £/. 104 E. H. MOORE. and, as second definitions, for temporary nse in the examples of § 63, we define as denoting 61. The derived properties B, B^ of classes of functions. — A property B' binary on functions and classes of functions on ^' gives rise to two properties : (1) S, -B', of classes 9K' of functions on ^'. A class 9)t' has the property B' in case its every function U has with 9)1' the bipartite property B , that is, (2) 3Jt'^':= :3)i' 3 /^'.3.A^''''''', so that (3) 3JJ'*'.~.3)I'''T For example, we have the proposition : a)i' . 3 . m^'^', that is, every class 9Ji' is contained in and is dominated by itself. A class 9Jt' has the property B'^ in case for every class ^-jjj„ic£< ^^ every class ^" the *-coraposite class (9)Z'9Jf'% on ^ = "ip'^" may be characterized as the class of all functions on ^ having the properties : (4) Bmxm"); B^m'Xp), that is, (5) m"'*:^ -.m' 3 m"^^^ . 3 . {m'm'% = [aii ^^'msfo.^o^^'W]. Since Ti" is linear and closed, so that 9Jf"^ = 3)1", it is evident that every function of (WW)^ has for every p' the property B^W : is contained in Tl". For a particular bipartite property B' there is for every determination of the property 5'9)i'(9)i") of functions on ^^ a precise determination of the property B'^ of classes of functions INTEODUCTIOX TO GENERAL ANALYSIS. 1 OS- Oil 5p', and accordingly of the definition of the classes 3Ji' having the property B\. By way of illustration I consider in § 63 the properties B^^, B^^, B^'\, B^^\. It is evident that a property B'^ may be possessed by no (existing) class 9)i'. For instance, B' denoting is not contained in, and jB'2fi'(2)^") denoting is not contained in 3)Z'2JJ", there exists no class W with the property B'^, since every class (3Jt'3Jl")* contains m'm". However, as we shall see, the classes 9}Z possessing some property B^ are very numerous, including, in particular, the classes : m'; m''"-, m™; m^^^-, m"'--, m'"", and classes arising from them by sequential *-composition. Cf. §§651, 661, 84. 61a. Propositions. — As corollaries of the definitions of properties B, B^ , etc., we notice the propositions : * * 1) If 9)1' on ^' and SOJ" on %" are two classes linear, closed, having th& dominance property D and with the respective properties S'* and B'\, then every function ^ on ^ = %'%" with the properties : B'W{m") ; B^W'W), belongs to {m'W)^ and has the properties: B"m'\W) : B^Wi'p"), and. conversely, every function p having the properties £"5)J*(9JJ') ; B^W{'p"), belongs to (9Ji'9Ji")* and has the properties : B'm'{W) ; B^W(,p'). 2') In case the properties B', etc., are such that (a) for every function ?>' and class 931' on -]]' the properties 5'9K'(2t) and- £'3)!' of 0' are equivalent ; and (6) for every function ^ on 5p =^'^" and pair of classes 9}t' and ©' on %', W being dominated by 3[3', and class W^^^ on %" the property B'9K'{9K") of ?> implies the property B"S>'(W') of 0, it follows that (c) if 9)1' has the property B'», then 9)}'* is the class of all functions <^' with the property B'W , and 9jfl'* has the property B'9)J', and 9Jt' has the property B' , and accordingly the two properties: B^\\, -B'9)J', of functions <)' are equivalent ; and (rf) if 9JJ' has the dominance property I) and the property B\, then 9)1' has the property £', and 9Jt'£ is linear and has the dominance prop- erty Dand the properties: B' ; B\ ; £i3l9.'i', and '))\\ is linear and closed and has the dominance property D and the properties : B' ; B'* ; Bi3l9)J' ; and accordingly, (e) if 9)1' has the dominance property D, then the property B\ belongs to both or to neither of the classes: 9)?' ; 9JJ'^(, and {/) if 91i' is linear and closed and has the property B'^, then 9)}' has the property B', and the properties : BtjW ; B'W , of functions 0' are- equivalent. 106 E. H. MOOEE. i^iLCDB'i, ^iiLCDB"* . ^ . JN /'JjS'iffi'0K").5o9J!"(3JO _ 3 _ XB"W{Vt').Bo^'(,p").B4,WW%\_ fJjB"m"(W).Bo^'(p") _ 3 _ Jj£'5in'(!m").£o9K"(P').J!o(iW'!Di"),\ 2) In case the properties B', etc., are such that /o \ J.' cm' -« ,'B'm'(V) j.'B"Bt' (2a) 0.9)1:3:9. .~.(?) ; (26) (j) . 9Jt"^'"^' . ajt"^*^^ : 3 : <^^'^'<.^"'> . 3 . ^^'sw it follows that (2c) aJi'^''.D.g)t'^= [allf ^"''].9)t'/'™'-3«'^'.(^o3«'*~^W); (2d) g)t'^^'* . 3 . 3)1'^' . 9Ji;^^^'^'*- ^^''■^' . ^'^LCDB'B'^.B,^^' . (2e) 9}r^:3:9)r'''*.~.W/'*; (2/) ^x""^"'* . 3 . 3)r^' . {B^m' ~ B"m'). The first proposition is the basis of a body of theorems con- cerning functions of two variables falling in many cases under the general head of the interchange of order of limiting proc- esses. B" denotes a property bipartite on functions and classes of functions on ^". B' and B" may be instances in the cases ^' and ^" of properties of general reference, that is, like B^ : is dominated by, applicable in general ; that they are such instances is not assumed ; one or both may be properties of special reference, — to the class ^' or ^" of elements or to any other features of a particular case. As to the second proposition, we notice that the conditions (2a, b) on B', etc., implying (cf. 2/) for classes <^'^^^'* the equivalence of the properties : B^Tt'; B'W, of functions <^'on ^' do not imply for such classes Tt' the equivalence of the prop- erties : B^rnXW); B'mXm"),aad so do not imply that such classes Tt' have the property ^^^ . 62. An example. — In order to recognize that the functional characterization of {(S,'Tt")^ given in § 57a5 is an Instance of the functional characterization ©/"(SJt'SJt")^ given in § 61.5, we need merely to notice that the property P' of functions on 5p' (§ 57al) INTRODUCTION TO GENERAL ANALYSIS. 107 is for the case W = 6' the property K'^Ti' arising from a prop- erty * ^2 bipartite on functions and classes of functions on ^' = ^' , viz., from the property II'^ for which (f)'ir'^3}V or ^'Ki'^' means For this property IC'^ we give to ^-S^a^'f™") the meaning : These properties : B'm', B"m'{m"), for B' = K'^, satisfy the respective relations : aJc : -J :

;' is of g^'aW" and so of (Wm")^ and so of 3t0?") (/); hence m'=0 (/). Accordingly, aR'^«'"* implies Tl' = 0(5p'). The converse implication is evident. 2) a)t'^^'"*.~.a)t' = o(5p'). For m' ^^'"* with aJt""^^^ implies (90t'$Ol") = rall^^i'"™'('^")--So™"W] = [all ^-Bi^wo-^o^'V)! and accordingly (21? '^'y^'"'''*"', and so, unless W = 0{^'), <^"s,%v'\ Now m"= aJi"™ s the class of all functions on «|?"™ has the properties LCD but not the property Bj3I(5]3"). Hence ajt' ■^^ * implies 3R' = O(^). The converse implication is evident. 3) m'^«* . ~ . ajj' 9 (3- (V . M^') 3 m' = atx). viz., the class Tl' on $' has the property -B^^ if and only if it has the form : 21,, z*^, where 31^ is a class of real numbers and /u.^ is a function of W- In this case 3)1'^°* with 501"^^^ implies (a}r9Jt'%= [all ')] = [all 05oti«'5!«"j^ that is, {m'm")^ = m'm", so that m'^ = an'. We prove that in a)i' there is a function yu,^ of which every function fi' is a numerical multiple. Consider any two functions /uj, /it^ of 3)?'. Then, taking 3}?" = aJt""", since (ajran")^ = Tl'm", we "see that /ij(l, 0) + z^aC-*' 1) = KC'^'u '^2); t'^^*' ^S) there is a function yuij of which /Aj and yu.^ are numerical multiples : /a[=aj/i3, iJi'^=a^'^. INTRODUCTION TO GENERAL ANALYSIS. 109 Hence, either a^ = 0, so that /a^ = 0(^'), or every function ^l'^ is a numerical multiple of ii[ : fi'^ = fi'^a^/a^. Thus, Tt' = SI^Mo where /j-'^ is any function of 9)V not everywhere 0, or, if there is no such function, /j,'^ = 0(^') is the only function ; hence, in either event W has the form 2tj/x^. Conversely, every class 2)t' of the form Sl^/u.,^ has the property B^^ . 4) gjJ'"''" with m"''^'' implies {•m'm")^ = [all <^^.»'"^'".^om"(P')j^ and accordingly 4a) aK'^ = [allf^'"']; a«7'™'. Hence, since 2R'^ contains 3l5Di', 46) A£i5)!'!lJ!".ao5W'(j)') _ ^ _ J^Bm,m".Som"lp') . 4c) 9}r ''■• . w/'* . 3 . cmw)^ = (m'*3)i")* . an'* = a)?'** • Further, by § 54al2 and § 44al, we see that 4d) a}Z''':3:aJi''''*.~.a}t'/'*; 4e) gji'^^^^.. : =) : 3)i'^ = gn'= [all <^'^''"] : 0'^''"' . ~ . f ^°™', that is, in effect, ((?) a class 2Jf' having the dominance property D, the classes W, W^ either both have or both do not have the property JB^^ ; (e) a class W' linear closed and having the dominance property D and the property B^^ contains every function it dominates. We shall prove in § 65 that the classes have the property B^^, while by 4e the class 9)Z''^does not have the property B^^ . Classes ^ containing (W'W)^. § 64. 64. Theorem. The respective hypotheses . (1) a}r^'.aji"^\g^[aii m, . ^ . AfJ,'^, ^ eAfx.'^^,). This property ^j of classes 5!)i' of functions on ^' ' "' is the property K[ derived in the sense of § 61 from the bipartite property* K[ : (4) ^ [S ft,' 9 e : D : ar m^ 3 ;?' > m, . 3 . Aj)'^, = eA/u,^,), for which we at once define the property K[3}V{W') of : (4') 3[ {/m", /j,') 9 e : 3 : 3" m. 3 p'>m,.^. A^^, ^ e JAt>". We notice that (5) W^^' 3 3' fl' nowhere zero . ^ . ^"f''!'^' _ 3 _ ^' ^l^' . (6) ^' "' . m' ^>'"^*' : 3 : <^' ''''"" . D . L n, there behig no such elements p' in 5|3' ". c) Propositions. — <7) ^ ^ ^ ' g: = [all <^-Bo«"(;'')] . 3 . g;Bo(™'5«")£. <8) ^ ^ Here the ease 11^ is the case I, and for the case II„ proposi- tion (8) is a corollary of proposition (7). In proof of (7) for the case 11^ and of (8) for the case III, •consider a function (j> of the class %, whether of (7) or (8), and set /Qs /? _ ■^ s' J, /Iln: m = l, 2, •••, »\ y') %'p" = ^^ ^"p' 'Pkp" Un :»»--= 1 , 2 .•■•,»/ ' or, more briefly, (10) o^^tK^.- k=l From the hypotheses : we see that <11) 0jo(^'n")j, (m). Further, by the definition of the functions S^^ , we have (12) p'^m.Z>. e^^, - 4>^, = 0, (13) p'>m.^.e^^,- ct>^, = - 4>^,, so that (14) m.p'.D.A(e„,^,-4,^,)^Acl>^,. The desired conclusion of proposition (7) for the case 11^ fol- lows directly from (11, 12) for m = n. Proceeding with the proof of proposition (8) for the case III, from the hypothesis : (15) ^B.WR'W^ we have, for certain a; fi' ; yu", (15') Ac}> = Aa/x' /m" , INTRODUCTION TO GENERAL ANALYSIS. 113 and hence, in view of the hypothesis : (16) m'""'', for a certain fi'^, (17) e : 3 : a- p: 9 p' > p; . D . ^0^, S eA,i'^^,^". Hence, with reference to (14), we have (18) e.m. p'>p:.'2 . A{d^^, - cj>^,) ^ eA^'^^.f^". Also, from (12), we have (19) e . mS p: . p' ^p', . 3 . 0^^, - ,, = . Thus, from (18, 19), we have (20) e.m^p'^.p'.D. A(e^^, - cj>^,) ^ e^/x;,,/.", or, omitting explicit mention of the now unconditioned p', (20') e.m^pl.D. A{d^ - 0) ^ e^/.>", and hence we have (21) L^„ = <^ (^'^" ; m'm"). m Accordingly, from (11, 21) we have the desired conclusion : (22) ^B^«^'(^"\ Corollary 1. From (23) it follows that (28) W,= [allf^n-9Jt'/'''^^ (29) m'^'*. Corollary 2. i"'].aJf'^^*^ (32) ^" . 9JI" : 3 : Sf''^^"\ The preceding statements hold if in (23, 23', 23") we re- place m' hy'mt'. e) The classes m' ^ m'^' ""^ ™= "'°^ ™' satisfy (23, 23', 23", 30) and accordingly (28, 31, 32) and for them (24) implies (25, 26, 27', 27"). Thus we have, in particular. Theorem I: the classes m' = W' ""' ™' ™°' '"" have the property JB^^. Classes on ^P"^ with the property K^^. § 66. 66. Theorem I. The class Tl^^ of all continuous functions on a linear interval ^^^ of the real number system has the p^'operty K,^ defined in §§61, 62. This theorem for the general interval is readily deduced by transformation from the theorem for the interval (01), to the proof of which we now proceed, obtaining by the way other theorems similarly extensible. a) The development A' and its representative system. — 5p'_ ?P'iv (jguotgg the interval : (1) O^p'^l, * (31i) is expressible thus : the class 3)1' is such that the properties : £n5K') £i3R', of a function i/>' on ^' are equivalent. Evidently 0' "^'implies ^,B^'m_ ,j,jjg fjjjgg of (31^-) ig that, conversely, <^'-^>™' implies (f,'^""^', i. c, the class 3)1'' contains every function i^' which it dorjinates. Of this (Slj) : the class W is absolute, is an immediate consequence. INTRODUCTION TO GENERAL ANALYSIS. 115 of the real number system. For the system : (2) {(ml)) {l^m;m = l,2,3,--; 1 = 0,1,^, ■■■), of indices * ml we introduce the corresponding systems : (3) ((/"")); A'^(C^'-)), of rational numbers ?•'"" : (4)t r"-"^-; and of subintervals ^' "" : i c\ cn''"^ r /wi-i /7)^^— 1 ^:r / fll/ «c- / ,n 74.1 (5) 5p = [^ ], r ^p ^r ""+', where as an additional understanding : (4') /'"-'sO; /"""+' =1. The system : (6) A'^((5P'"")), is a development of the class ^', the stage vi of the development being the system : (7) A""=(5p""'), of m + 1 intervals '^'"" (0 ^ I S m). The interval ^"" is of length 2/???. (0 < ? < m) or 1/m (? = 0, m), and contains the number ?■'"' which is said to rep- resent it and its numbers ^Z"*. On stage m every number p' has two representative numbers ?•'"' for adjacent values of I; a number ?•"" (0 < ^ < m) however has three representative numbers, viz., besides itself both of those with adjacent values of the second part of the index. The superscript index ml used with notations p' is used to denote the following property : belonging to the interval ^"'' o/" the development A', of elements p'. Thus we have the nota- tions : /n\ '"'' f -ml (8) p ; p , the latter denoting an element^' not belonging to ^'"'. *Thronghont ? 66 an index ml is a pair m, I of integers conditioned as in- dicated in (2). t Tlie superscript ml is merely an index. It is not an exponent. 116 E. H. MOORE. b) The relation * IC'^ . — The relation f J^'2 ^^ ^ property f on ^'^^';3) where 3 — ["*]> viz., a property of pairs p[, p!^ of elements of ^' with integers m, having reference to the develop- ment A' of ^'. We write (9) J^pi'pi'm ) to denote that p[, p'^ are two elements belonging to the same interval (say 5p""°'°) of some stage m^ = m of the development A' ; in symbols : (9') S{m^^m.\^m^)^ (p'r'" ■ p'r'l- One understands how questions of infinitesimal nature on ^' may be treated in terms of the development A' of ^', the ex- pression : (10) Sd^B A{p[-p',)^d^, being replaced by (11) 3: m, 3 K'p^.p^,^^. Thus, with reference to the development A' in terms of con- ditions J^'p^>p,'m we give definitions to the properties : (12) K',; k;W; Jr',m'{Tl"); K',^, defined in § 62 ; the new definitions being equivalent to the former definitions. This property K'^ is a bipartite property of functions and classes of functions on ^' . c) Developmental systems S)'. — With reference to the devel- opment A' and its system ((r"" )) of representative numbers a developmental system % 2)' is a system : (13) 2)' = [S'l ^ ((8'"")), of functions on ^' such that * This relation K/ has reference to the development A'. For the general development A' of the general class ^' (cf. § 77) it is denoted by K,/'^ . fThe term relation is used tor this property K^' on ^'^'3, since we are to be concerned also with a bipartite property K^' of functions and classes of functions on %' . The property K^ is defined in terms of the relation K^' . X Cf. § 78 for the general definition of developmental systems S(3R) with respect to a development A and a class 3)1. INTRODUCTION TO GENERAL ANALYSIS. 117 (14) m.;.D.O^S""'Sl. 8""'=!° ^'^' )' ll (/ = ;•""',); l=m (15) m.D. X8'"" = l. i=0 Hence, for every stage m at an argument p' which is a rep- resentative number r'™ the corresponding developmental func- tion B"" takes the value 1 and the other developmental func- tions 8"" of stage m vanish, while at an argument p' not itself a representative number with two representatives r""', r'™'"''^ the corresponding developmental functions S"", 8""'+^ take values of the interval (01) whose sum is 1, and the other de- velopmental functions 8"" of stage m vanish. Such a developmental system ®' plays a role in the theory of functions on ^' like the r6le of the developmental system 5D' of § 65 in the theory of functions on 5p' ^ For the simplest developmental system f SD' of continuous functions the function 8""' has on 5p"" the following specifica- cation : nR\ .,mi_\nip -l+l {r ^p ^r ), ^^''^ '-' -\-mp'+l + l {r'-'^p'^r'-'^'). d) Developmental property A'. — A class 2R' is said to have the developmental property A' in case there exists a developmental system J)', relative to the development A' and the representa- tive system ((r' "" )), which belongs to the class 2)t'. (Cf. § 79.) e) Propositions. —We notice first the proposition : (17) ^' .m'^^ .■:> .m'^''^'^''^^'\ t This system of continuous functions is the point of departure for Boekl's determination, by the mediation of the known theorem of Weiebstrass, of a system of polynomials "■''" such that for a continuous function 0' the sequence Wm] of polynomials vL : converges on 5p uniformly to ^'. Cf. Borbl, Sur V interpolation desfonctions corUinues par des polynomes, Verhandlungen des dritten interna- tionalen Mathematiker-Kongresses in Heidelberg, 1904(1905), pp. 224-232 ; and Lemons sur lesf auctions de variables rSeltes et les developpements en series de polynomes, 1905, p. 80. 118 E. H. MOORE. that is, 5)' being a developmental system,* every continuous func- tion ji' is expressible ' on ^' and set (18) c^,^'\>'r,^ M; (19) ^;-Zo..S'" (H r Jul ml SO that the sequence {^,'J belongs to S^. Then, from the definitional properties (14, 15) of ®', we have the relations : (20) y=r""'.3.^:,,=c„,; (21) p' of (r'"'V'-'+') . D . e:^, of (c„,c,„,^0, from which follows for every p' (22) ^(0;. - ei^,) ^^(<^;-<^;,) + A<^;,-^:,0 s^(<^;,-<^;,) + ^('^;' -;') S24(<^; -' follows the desired conclusion : (23) ' on the interval 5p' . Accordingly, since ^''^=[all''''''"'] .3.3=' ^'^'i^', readily obtainable by the use of the last inequality of (22). Similarly we obtain the proposition : (26)* ^''^^^'■^'^^"■^'^"■^^■■ 3= = [all ^■Ki'il!'Om").£„!™"(j,')-] _ 3 _ g;Bo(5>'5«")i(!!)!'Si")^ and its corollary : (27)* ^ The proposition (27) is to 6e compared with § 65.8. /) Propositions comparable with the propositions of § 64. — The respective hypotheses : (28) ^ ' ^^ . 2)r -^ . 5P" . 9K" ^ . g s [all <^ •K'/aK'ca'"' ■ s.^^'^i"-^ . (29) ^'^'^. ajt'-^.^^ar'"^*^^. 3^ = [all (f)S2'm^^").£im'm".Bom"(.p')-j. (30) ^ ' ^ = ["all ^■S^2'3"'(!0!'0.5iTO'!m".£o9)!"(j'0--Bo™'(j'")"] imply the simultaneous conclusions : (31) g^^.g = ?^*.3=^'"'^"; (32) a)i' ^^' . 3 . (aji'3R")*^»s . cji>. AVe recall the definition : Tl' ^^ = : 9}r ^^"'" : = : 2R' 9 m' ^"'^ • = • m' ^"'^ * 26) For every developmental system ® and class W on 5}?' '^ and class W on 5P" the class jj of all functions ^ on 5]3 = 5P'^" having the properties : K,'W{W) ; -B|,2l3Jl"(p'), is a subclass of the class obtained by extending as to 3)J'3)l" the class S'9K" extended to be linear. 27) For every class 9Jl' on ^' , containing a developmental system S', and class 3)1" on ^" the class 5 of all functions ^ on 5(5 = $'5{5" having the properties : K^'WiWl") ; Sa%m"{p'), is a subclass of the class (ajt'sBJ"}*. 120 E. H. MOOEE. and accordingly remark that (33) ^' '"^ . %' '^^''"' . ^" . an" . =) . (g'3Jt")^^'^W'). Then, in proof of (32^) in either case (28; 29; 30), we notice that Tl''''' implies {m'my^""^'^'^'"' ■ ^''^ and hence, since ajl' has the dominance property D and '^ is dominated by 3iaJl'3n", we see that '^ has the dominance property D. Further, since WW" belongs to g we see that (32 J : (mW)^ belongs to g, is a corollary of the fact affirmed in (31) that '^ is linear and closed. Further, (3I2) is implied by (31j), and (3I3) is obvious in either case (28 ; 29 ; 30). Further (31^) for cases (29 ; 30) is implied by (31i) for case (28), in view of § 64.4, 5. We prove directly (31j) for case (28). The linearity of g is readily proved ; we proceed to prove the closure of %. With the hypothesis (28) we consider a sequence {^} of the class ^ which converges on ^ = ^'5^" uniformly as to scale g to a function 6. By § 64.1, 9, (^^i^^s'-^". We are to prove ffK.'mxw')^ or its equivalent : The convergence is by hypothesis uniform to some scale : (35) L„=^ (^; «^), n and we have for every e a corresponding n^ . Further, cf) and the .A<\>^,^Aap.;y; * The following relations hold uniformly in p" on 5p". INTRODUCTION TO GENERAL ANALYSIS. 121 (37) e . ^;,,^,„,. D . A{4>^, - „^^, - (/)„^^,) ^ e J/^;,/*" ; (39) e . n ^ n, . D . J( ^ - ^^,) ^ e^./,^, . Now ^(^^x' - ^..') = A^, - K.) + ^(<^...>' - <^«..') + ^i(<^„,, - e^^). Hence, in case : n^n,; ^^,/p,/™„,, we have and, in case further K'^^,^^^^, ^{^..' - ^.') = <2^10^^, + 24/.;,/.") ^ e(2a + 2)4/.;,/.". Accordingly, taking m* as the greater of m, and ??(„£, we have the relation : (40) e:D: K'^,^,^, . 3 . A{d^, - 0^„) S eA{2a + 2)/.;,/.", so that, as desired, /41\ ^ AV}ig)!'(5!!")_ ^r) On comparing the results of (27, 29, 30, 31, 32) we have Theorem II. From (42) ^''"^ .m'^''''^' it follows that from (43) 35" . M"^'^^ . (^ = [all ,^^'r2'9'i'»i").Bom"(p')"| it follows that (44) (3nW% = g.g^^^; (45) (3)ra}r)^«5.5^'™'^"; mid, if further (42') a}?'"''', ii follows that Corollary 1. From (42) it follows that (47) aW'* = [all (/.'^^'™'] . m'/''^'' . aJt'* = 3«'# ; (48) W^''*. 122 E. H. MOOEE. Corollary 2. From <49) ^aY ^^,LCDK " it follows that (50) m' = [all f-^^'"^'] . gjj'^^c'MV/f^'*^ Under the hypotheses : (42 ; (42') ; 43) on ^', Tt', ^", m" we obtain (44^ ; 46 ; (46')) by observing that (3Jt'9)l")^ contains g:(43) _ g{27) ^jji(,ij contains g'^'^ which (contains g'^^^ which) contains {m'm'% . (44j) implies (45 J. (31) implies (44^ ; 452). The class : [all ^'■^^'™'], is obviously absolute and accordingly the classes : Ti'^ in (47) and SR' in (50) are absolute. Further, we notice that, the class (3Ji'2Ji")^ of (44) is absolute, if the class ^" is absolute. On analysis of the proposition : (42, 43) implies (46), we have the following propositions : (51) ?p''^.3Jt'^'^'.3.2l(5p')^i"-"«'; SO that, since £^^W{p') implies £j2l3)i"(2)'), ^''''.9Jr''"''.^".aJl"^':3: (53) A) We return finally to the class 3)t' of all continuous func- tions on ^' . In view of the uniform continuity of its func- tions fi 3Jl' has the property K'^, and since SK' contains the developmental system ©' given by (16) it has the property A'. Hence m' satisfies (42, 42', 49) and accordingly (47, 48, 50) and for it (43) implies (44, 45, 46, 46'). We have proved, in particular, Theorem I: Tlie class 9)1''^ has the property K'^^ defined in § 62 (cf. § 66.12). Comparison of the theorems o/ §§ 65, 66. § 67. 67. The notions and the propositions of §§ 65, 66 are of special reference. Thus, ^' is in § 65 of case I, 11^ or III, INTRODUCTION TO GENERAL ANALYSIS. 123 and in § 66 of case IV. As pointing the way to the generali- zations to be given subsequently we proceed to secure formula- tions holding uniformly for the cases I-IV of ^'. a) For cases I-III of ^' we introduce the development : <1) A'^{{^"")) {lSm-m,l = l,2,3,.,.) where the class ^"" consists of the single element p"" =1, which islikewise the representative element r' '^ . Stage m : A''"=(^'"'), ■of the development consists of the m elements p'Sm, said to be developed of stage m. As developmental system : (2) 2)' = ((8'"")), we introduce the system : 3)' = [S'] = {8^} of § 65 a), setting ^'""= g;. For this system §66.14 holds, and § 66.15 holds for p' = m, viz., for p' developed of stage m. Further, this developmental system 3)' is the only system 5D' satisfying in this way § 66.14, 15. We denote by W'^' a class M' contain- ing this system 2)'. In § 66.12 we defined anew the properties: of § 62, by means of the condition : having reference to the development A' of ^' . Similarly the properties : of § 65 6) are definable with reference to the development A' in terms of the condition : on an element p and an integer m, that at some stage m^ = m the element p is undeveloped, that is, for the present development, p > m. 6) The properties K[ and K'^ were originally metrically defined in terms of the respective conditions : p' >m. 124 E. H. MOORE. The properties K[ are properties of infinltary nature, in that they are relations holding for large values ofp', viz., asp' tends towards oo. In case IV p' is limited, and accordingly the bipartite property K[ and the derived properties : -K'[W{W), K[ , hold vacuously and generally, i. e., The properties K'^ are properties of infinitesimal nature, in that they are* relations holding for values of jOj , p'^ distinct and close together, viz., as p[, p'^ tend to equality. In cases I— III p[ and jSj if distinct differ at least by 1, and the properties K'^ hold vacuously and generally. The properties K[ and K'^ have been defined also with respect to the developments A', the metrical conditions being replaced by the conditions: K'^,^, K'j,^,^,^^. The foregoing remarks as to the properties K\ in case IV and the properties K', in cases I-III may be made from this standpoint. At every stage m the development A' of ^' is complete, in that every element p is an element of a class ^'^ of stage m; at stages m = n the development A' of 5p' " is complete ; however, at no stage m is the development A' of ^' complete. On the other hand, at every stage m the developments A' of ^' , ^' ", ^' are fine, in that every class ^"" consists of a single element, while at no stage m is the development A' of ^''"^ fine. c) Now comparing the theorems II of § § 65 and 66, we notice that in § 66 the properties : K'^ of W, K'^TIXW) of (f>, enter where in § 65 enter the properties : K[ of W, B^m'Tl" of ^. To remove this discrepancy we notice, for cases I-III of ^', that (3) Tl'^^''' . Tl"^^ •■ 3 : (i)-S'i'™'(™")--So'™"(p') . 3 . jjBi^^'w . (4) guj'-s"'' . gjj" : 3 . ^b.^ww _ ^ _ ^k.'uxw).^ (5) M'"^'^^'^' .'iSl"^^ .-. 3 .•.^■s«''™"<^'):3 .^(hBi'M'm" _ ^ _ fLKi'w'm")^ * Apart from relations for p/ ^p/ which hold generally. INTRODUCTION TO GENERAL ANALYSIS. ]25 Accordingly, in §64.4, 5 and §65.8, d) for ^'"" we may in the characterization of (^ replace B^'SLTl'W, B^WW by K[^'{m"), in case we impose the hypotheses : m'^'^''""', Tl"^'. As thus modified, the propositions * for cases I-III are closely comparable with the propositions for case IV, in the former K[ entering where in the latter K'^ enters. Then, in accordance with the remarks of b), we may frame propositions holding uniformly for cases I-IV by introducing throughout in place of K[ or JTj the notation K[^ , denoting the composite property K[K'^ ; JTij^ however denotes the property of classes W defined by means o{ K'^^^RXM") (cf § 61.5), a property which is quite distinct from the composite property ^j^^^^. Systems (31 ; ^ ; ITr, 'SR). Relations K^ . Composition of relations K^. § § 6 8-7 1 . 68. Systems (2t; ^; K^, K^; 9Ji). — In the first section f (§§51-55) of Part II we have defined the composition of classes of elements and of classes of functions. In the second section (§§ 56—67) we have obtained functional characterizations of the *-composite (2)i'9)i'% of the classes ^' = ^'^~^^ with arbitrary classes 9Jl" ; these characterizations are of the type B^ of § 61 and more closely of the type -fiTjj^ of § 67, related to certain developments A' of ^' ~ by the mediation of certain relations K[, K'^ defined in terms of the developments A'. Before taking up in the fourth section (§§ 75-84) the study of the general development A' of the general class ^', we consider in this third section (§§ 68-74) the general relations K^, K^ of the general class ^'. 69. The relations K^, K^, K^„. — For a class ^ of elements a * We notice the theorem : m/i; n„; m gjjz-Di-ffi'A' _ ^ . ajl'^'''*, with its corollary, 5jjj,l; il„; III; iiio; III, . 3 . sgj' -^i'*. tCf. ?2 50, 56. 126 B. H. MOORE. relation K^ is a relation * on 5P3 and a relation ^ is a relation on %%% Here ^ denotes the class [m] of positive integers. Accordingly the notations : ad H.^ : H-p^i ', ctd A2 : -^pip^m > denote pm ; p^ pjm, fulfilling certain respective conditions. Thus for ^^""^"^ we had the metrical instances : Kp^ . = .p>m; JTp^p^^ . = . A{p^ - p^ ^ - .f We frequently consider simultaneously a relation IC^ and a relation -ff^ for a class ^ ; in such cases we speak of the relation -ffjj. Thus we consider relations K^{i = 1, 2, 12), as we may say, on ^. At present we consider individual relations ^j or K^ or K^^ on ^, postponing the consideration of classes of rela- tions on 5p. 70. Properties of relations K^, K^, K^^. — In systems (31; 5p ; K.^, K^; 591) the relations K^, K^ ai'e used to define properties of classes 9Ji. They are of the type specified ; other restrictions on generality will be made as needed. We notice the extreme instances, in which the conditions K^^ , K^^^^ are (o) never, (a ) always fulfilled, viz., (o) -3" {p, m) 3 Kp^; - S {p^, p^, m) 9 Kp^p^„„ (n) p.m.'^ .Kj,^; p^.p^.m.'D .Kp^j,^^. * These relations K^, K^ are properties ^1, K^ on 5|53, ^$3i where ^ de- notes the class [m] of positive integers, in terms of which are defined in \ 1^ the bipartite propertiea K^ , -ST, of functions and classes of functions on ip, and their associated properties. t More generally, if d is a nowhere negative function on 5p5p to 31, a func- tion of the general nature of distance, c. g., Feechet's voisinage or ecart, an instance is K' = (J •= — J^piPim ■ — ■ "pip2 = ^• A relation K2 gives rise to numerous relations K^, e. g., for every element y^, of ^ there is a relation K^, viz., ^pm ' = • ^ppqui • INTRODUCTION TO GENERAL ANALYSIS. 127 Relations K^ , K^ determine uniquely and are uniquely deter- minable from the respective general systems : of subclasses of ^ such that The subclasses may be null-classes. A non-symmetric rela- tion K^ is similarly related to a second system : such that I'^'p.p.m ■ ~ • P2^° Relations K^, K^, K^^ may have, for instance, the following properties : (1) m„ < m . ^,„ . D . /^„.„ ; m^ • -^Pi'Pi'Pi'pii' 'm (2) (3) p . J^p^ip^im • ^ • ■^p'pi"p'p"i>n -^p'pi"mJ^ p'pi"m • We denote relations ^,', etc., so related as follows : (5) (1) {K;,K';,K,y; (2) {K'[,K'[,K,J; (3) {K\,K"„Kif; (4) {K',, K;, K.f ; (5) (ir;, ^,,)^ (^;', A';,)^ (^;, ^;', ^,,)»; or in other ways of obvious interpretation. For given relations K^, K'l (i = 1 ; 2 ; 12) there is precisely one relation K^ , the eomposile relation, in notation, ^'^ K'l or K\", satisfying the conditions : {i=\) {K[,K'[,K,r; (^=2) {K'„ K'^, K^f; (i=12) {K'[„ K'i„ K,,r\ The propositions of §§ 72-4 having reference to systems {K'i, K'l, K}) are available, as indicated in the theorems of § 74, for systems {K\, K"^, K'^"), and also later on for systems of relations derived from developments A, the composition of developments inducing for those relations a certain composi- tion. In this composition of relations the composite relation K^ is K[ ", while the composite relation K^ is in general distinct from K'^", the relation {K'^, K"^, K^^ holding but the relation (-STj, K'!^, K^ in general not holding. Thus this composition of relations satisfies the conditions : (*=1) {K\,K[,K,r; (i = 2) {K'„K';,K,r; (i = 12) (£:[^,K';„KJ^. We do not now consider more closely the interrelations of these two types of composition of relations derived from developments. INTRODUCTION TO GENEEAL ANALYSIS. 129 71a. Propositions concerning the composite relations K\" in connection with the properties 1-5 (§ 70) of relations. — ^;:.^;;.3.(A7,^;;r. (2) ir[:.x';:.z>.(K'„K;,K-r. (3) The properties 1-5 of relations K^ are invariant under com- position of relations, viz., (3.) k:^.k':\^.k-" (i=i,2,i2); (3,) K',\Kr.^.K-"> (,=3,4,5). Properties K^ , JT^j^ of classes W. Composition theory of classes with the properties DK., BK.^. §§ 72-74. 72. The properties K^, K^, Ky^, etc., relative to relations K^ , K^. — Using the notations ', " as usual, relative to relations : K^ , K^ on ^ ; K\ , K'^ on 5p', and classes of functions : 9)i on ^; 3Jl' on ^'; 9)i" on ^", we define properties: K^; K^; K^^, etc., generalizations of the properties of § § 65 and 66, as follows : (1) <^^'^'™. = .<^9(a-/.9 e:3:a-»ft 9^^..D.^0^^e^^^). ^^^ A4>^,^eA^;,^"). (5) ^i^i=™. = .^. (8) ajf-^'^ : = -.m^'^ := : /x . 3 . fi"^^"^. (9) gjl-^'^ : = : ajj-^"^ : = : /x . 3 . /x^'^^'' : ~ : ^Jt-^'^^X 9jj"^^^ . . {m'm")^ = [all ^ -5-i'^'(™") • ^o^"!?')] ^"LCD _ _j _ (^1'^"^^ _ Hall 0-Br,,'3K'(>m").5„w'(p')]. * For the relation K^^K° every class 9K and function ^ are in the bipartite relation K^. For the relation K^ = K° every class 2R and function

'"'' . 3 . 3=, 8, ^'™. (6.) Jf.^ . 9}i-° . ^^iSK'-Bis™ . D . 5^; Ki'm. Biiim _ cv iTjTiR . Siasu! (7.) A7 . 3R ^^'' . 3 . gjt^-D^-^' . 9}f^^^< . a)i^^/w_ ^; . 3}i . ^, = [all 0^''<^] : 3 : ^,.^<™-^ : g^ = 3tg^ : (8;)* ir,'.m.%,^ [aii <^^-»'] . d . b^^^ ~ s,g^,. (9,) ir,'.W:^:{WK^.m^''^. {J^,m ~ ^,3Jt j. (10.) K^ . g:-^'™w . ~ . gj-Ki™. ( 1 1 ,) /i7 . an ^'^* . 3 . 3Jt^ = [all <^^'-*»'] . (1 2.) ^.i . ajt^-^* . 3 . 3Jt ^''' . ajt LDKiiCi^.Bi^m _ 5jj^ icz»^7rtff;,.Si3ia!_ (13.) /r. 1 . 3Jt ^ : 3 : 2R ^w. . ~ . 2);/'^<. . ~ . m^"^. (14.) K' . m ''^"^* .^.m. "■'-' . {B^m ~ ir.ajt). *2)l being a class of functions on 5|J, and Z',(i = l; 2; 12) being a relation Ki on ^ having the property 1 of § 70, and % being the class of all functions ^ on ^ having the property A'^OT ; the class 5i is absolute and has the prop- erty .K'i J)i and is closed under multiplication by the class 21 of constants ; and further, if the class 3Jl has the dominance property D^, the class 5, is linear ; if the class 3K has the dominance property D, the class g'; is linear and closed ; and it the class 3)1 has the property K,, the class 9Jf belongs to the class 5, ■ and the class %i has the property Ki. — Further, in the case j = l, the prop- erties : 5o5i i J^\%v ^i^'i ^^ properties: belonging to the class ^^ ; dominated by the class ^i, vfhether of functions or of classes of functions on 5p, are equivalent. 132 E. H. MOOEE. 726. PropositioTis concerning relations and classes of functions mi ^', 5p", 5p = sp'?p". — The following propositions are for i=\; 2; 12. (1 .) k:.%' '"'''"' . 3rt" . => . {%'m") ^''^'<-^"\ ^^''* : = : (2^ . 2;'), viz., (2;) (^; , ^;', ^0^3. (M'm") ^> . (S'r) '^''"''"" • '''"''^"; (2;) (^; , /it;', ^2) ^ => . (3Ji'a«") ^> . (^'g") ^=™'^" ■ ^>"'^"; (2;,) (^;, , ^;; , kj ^*.d. {mw) ^^ . (g'r) ^"'"'"" • ^^''™"'"'; (2") K'"^ rSyt'SOt"! -^1 /CV'c>;"\7tV"!lR'aK".£iM ru"!iv»")-Bim'm".j3o'sii"(.p').}i„mij"n ^ imply the simultaneous conclusions : (4,) 55^^-^^™"^"'.?^ = 5.,; (5;) m'""' .^.{m'm'%^<'''.%''; (6,.) 3)1'''°^''' .3}t"''°.3.g^«, ancZ, in the respective cases (1.; 2^ ; 3,), (7.) %^; m"\ ■=>.%''■, m'^ .m"^ .^.%^. 74. Closure under extension and composition of classes. — With reference to relations K[ on 5|s', IC'- on ^", and ITf, in particular, ^^ ", on ^ s ^'5p" we have the following propo- sitions : ^^^ xj . (^; , k:, x,y' . 3 . {Wiwy" . K[l . 'm' '''''''* . K"l . m" ""''''"* . ^^^^ ^\, . {K[,, K'u, ^uV'' ■ => • {m'm'Y^^'-. (1') K[^ . gji'""^''* . K'[' . m" '''''"* .o .K["' . {mwy^'^"'*. (2') K','' . jjr^"^^'* . /^;'" . 3)i"''^^"^"* . 3 . A";"" . (9jra)ry^'^'="'*. (1 2') £:[" . m' "'''-* . K'['^ . Ti" '''■''""* . 3 . ic['; '\ {mwy- "* . Here, in view of preceding propositions, the proposition i' is a corollary of proposition i (i= 1 ; 2; 12). Proposition i states that its hypothesis implies the conclusion : ^,„ my" LCD m _ r^^JJ ^7fi5i)!'a)!"(5)!"'). £o3'i"'(P'P")1 3 , 134 E. H. MOOBE. where yjr denotes a function on ^'^"^"' and 9Jt"' denotes a class of functions on ^"'. Now the class ®^ contains the class (3 : m ^ rol] jf,lCi'!!t'm"(,m"').BiW ^'2') -^ifi of ^^^ hypothesis of proposition 2 is replaced INTRODUCTION TO GENERAL ANALYSIS. 135 by the less exacting condition : {K'^, K'^ , K^.^f. This reduc- tion in the hypothesis induces only a slight complication in the proof. The propositions i' and §72al2. (J = 1 ; 2; 12) state that certain genera of systems (31; ^ ; -ST ; 3)1) are closed under certain operations. There are thus three theorems of which the follow- ing is for ■1=12. Theorem I. The genus of all systems: (21; ^; K,\'; M^^^'^"'"'-*), is closed under the combinations * of the three operations A,By,B^; A : *-extension of classes 9}f ; £ : simultaneous composition of classes ^ and relations JT,^^ and jBj .- multiplication of classes 9Jt ; B^ ■• ^-composition of classes 9K. The subgenus of all systems obtained by the combinations of operations A and B^ is the genus of all systems : (2t ; ^ ; /r// ; ^^omAic^.m,,-^ . this genus is closed under the operations A, B^ and their combi- nations; to this genus belong the systems I— IV. Here the properties (^,2), (AJT^^) are properties implied by the other properties of the respective systems ; of. § 72all, 14. The theorem of § 84 concerning composition of developments A is based on the proposition (12) of § 74. Similarly from §§72a7j, 7263^' we have the Theorem II. The genv,s of all systems : (2t; ^; J^,;^^"'''), is closed under the combinations of the operations A, B^, B^ (defined above), and A^ , A^ : A^: extension of classes 3)t to Ttz) A^: extensio7i of classes 9){ to Ti^f The subgenus of all systems obtained by the combinations of the operations A, B^ is the genus of all systems : C-)r;^;/i,;a}t^-^^^'0; * The combinations are understood to include the individual operations. We note that B^ is a combination of A and ^i. 136 E. H. MOOEE. this genus is dosed under the operations A, A^, A^j B^ and their combinations ; to this genus belong the systems I-I V. These theorems of closure of genera are comparable with theorem IV of § 55 and the theorem of § 84. A 12 • Systems (91 ; ^ ; A ; Tl). Developments A. Relatione K^ Classes Tl''. §§75-80. 75. Developments A of classes ^. — A development A of a class ^ of elements is a sequence {A"*} of systems A"' of subclasses of ^, each system A" consisting of a finite number, say l^, of subclasses of ^. The system A" is stage m of the development A. The sequence [l^] of positive integers is the sequence of the development A. There is no question of order but there may be repetitions amongst the l^ classes of stage m, and it is to provide for this possibility of repetition that we say system A"* instead of class A™. With this understanding, we introduce an index V^ : l-=l; 2; ...,l^, of stage m, and omit the superscript m of the index Z" when, in conjunction with the index m of the stage, it occurs in the bipartite index ml. We denote the l^ subclasses of stage m respectively by the notations : T', [i'T a=i,2, ...,M. Then stage m of the development A is the system : A^s (^p-"'), and the development A is the system : A^iiT')), of subclasses ^"" of the class ^. Admitting the possibility of nullclasses ^'"\ we notice as extreme instances the developments A = A° ; A°, where in A° : every class ^"'' is the nullclass ; A" : every class ^"^ is the class ^. INTRODUCTION TO GENERAL ANALYSIS. 137 We recall also the developments : A = A\ A"", A"!, A'v, of ^\ ^"", «p™, ^iv respectively, which were defined in § 66 a) and § 67 a). A representative system : of a development A is a system of elements r"' of ^, the ele- ment r"" belonging to the class ^"" of the development A. The element r"" is said to represent the class ^""^ and its elements /J™'. It is understood however that only the existent classes ^""^ have representative elements r"''. 76. Various classes derived from a development A. — For a development A the class ^'" : is the class of all elements (in notation : p") developed in stage m, the least common superclass of the classes of stage m, and the class ^'^ : sp'^ = [p'^] = UA s U ((5p™')) = U (^p-), is the class of all elements (in notation : p^) developed in the development A, the least common superclass of the classes ^"" of A. The class ^"^ : is the class of all elements (in notation : p'"^) not developed in stage m, and the class ^^ : ^™^U(^-»o|m„^m), is the class of all elements each for some stage m^ = m not de- veloped, the least common superclass of the classes ^"™<' (in^ = m). As to an element p the indices I of existent classes of stage m are designated by the notations : g^;h", 138 E. H. MOORE. according as the class 5P"" contains or does not contain the ele- ment p. Then the class ^p™ : mpm = (J ^ -'^12*> of classes of functions. (Cf. § 72.) 78. Developmental systems S(2){) relative to a class M of func- tions and a development A. — A developmental system S)(3)J) : (1) ^m) = an), is a system of functions 8°'' on ^ for which there exists a repre- sentative system dl^ = (('''"')) ^^ ^he development A such that e:3 : 3[ m^ a (m^m^.p).'D . A{j: V' - 1 ) ^ <- . ^(E -1 V' - 1) = e ; a a Ix.-.-D .-.S fi^B e:Zi : S m^^ a {m^m^,. p) .Zi . (16) h Here the indices g, h are understood to be g'', h^ ; and the con- ditions la, 16 are understood to apply only in so far as such in- dices are available, that is, for the various stages m the condition la applies only for those elements p which belong to at least one class of stage m, and the condition 16 applies only for those ele- ments p which belong not to every existent class of stage m. Thus, for the development A° there is only the condition la, in- dependent of 3}t, on the developmental system S)(9.1f) ; so that, e. g., the system : S'"'= 1 ; B"" = (/> 1), is a system S^^DJi) for every W for this development. For the development A° every system ((8"'')) is for every class 3R a developmental system ®(3K). Thus, the system 5)(9}i) need not actually depend on the class 9)1. Indeed, a system T* : (1') 2^ = an), satisfying (la) and (1'6) m.79./i^.=>.S/'''' = 0, 140 E. H. MOORE. is a system SD(9JJ) as to every class Tl. Such are the develop- mental systems 2) of § 66 e) and § 67 a) for the cases I— IV, the condition (la) being satisfied in virtue of the relation : (I'a) m.p^'.D .'Z B/'» = X -iSp"^ = 1 ■ g g We notice that the systems ®(9!}l) are the systems ®(49)i) and they are the systems ®(3l9}i). Also, if 9Jtj belongs to 3}i and 9Jt is dominated by 2l2Ru, the systems S)(2)t) are the sys- tems S)(2R|,). Accordingly, 3)J^' implies that the systems S)(3}t) are the systems ©(SJtx), and M^ implies that the sys- tems ®(a)t) are the systems ®(aJl^). 79. The property A of classes 9)i relative to a development A. — By the mediation of the notion of developmental systems ©(SJt) a development A of the class ^ gives rise to an associated developmental property * A of classes 3)i of functions on 5^ : m^ . = .m s s 33(2Ji) 9 ®(a)i)^»^ viz., a class Tl having the property A is a class Tt such that there exists a developmental system S)(3Jt) (relative to Tt and A) which belongs to 3Jl. We notice the propositions : (1) 33t^.3.(2l3n)^; (2) a)J^i^.3.3)J;r^>^; (3) 9Jt^^ . 3 . an^^^. 80. Theorem I. If 3R is a class of functions on ^ having the dominance propei^ty D^ and the developmental pi-operty A associated with a development A of ^, then a function (f) on ^ having the property ^j^3Ji is dominated by the class 3l2)i and belongs to the class 501;^, viz., Generalizing theorem I in the sense of relativity, we have Theorem II. ^p' . a' . M' '°'^' . ^" .m"^':"^: *Cf. §§66d), 67 a). INTRODUCTION TO GENERAL ANALYSIS. 141 ft We proceed to the proof of the first theorem. a) The development A may be such that for every m there are stages m^ = m consisting entirely of null-classes. In this case, for every p and m the relation K^^^ holds, and accordingly the fact that ^ has the property K^'^m (cf §§ 77, 72) implies that (f) = 0, and hence the desired conclusions. 6) We suppose then that for a certain m^ every stage m = m^ has some existent classes, that is, for some elements p there are classes ^""'. (Cf. § 76.) The class 9Jl has the developmental property A associated with the development A. There is a developmental system 2)(3Jl) s ((S"')) which belongs to 3Ji and there is a representa- tive system 3^-^ = ((r"")) associated with 2)(3Jt) (cf. §§ 79, 78, 75). The function ^ has the property K^^ W, that is, the properties K^m, K^m (cf. §§ 77, 72). Hence for certain functions ti„ /ij and stages m^, m^, m^ we have (1) K^,^^.D.A^SA^,^; (2) ^^^^^ . 3 . ^(0,,, - ,,) S A,.,^^ ■ (3) m ^ wig .p-" . 3 . E ^a/'^ S i. Consider a definite stage m (m = m^, m^, m^) of the develop- ment A and an element p of ^. If the element p is in no class of stage m, by (1) we have (4) Acf>p^Afi,p. If, on the other hand, the element ^ is in a class ^P""" of stage m, by (2) we have (5) A4>p^A^,+ A/j,2p = Aao+ Afi2p, where a^ is a constant, independent of jo, such that Ap^AaiJip. Accordingly, is dominated by the class 3l3Jt. Now, introducing for the various stages m (m = 7n^) the func- tions 6^ : (9)*' ^„-E'<^.-n which evidently belong to Tlj,, we proceed to prove that (10) ='L0,. {^■,m, and accordingly that (j) belongs to W.^ . We are to determine a function /jl and for every e an integer m^ such that (11) m^m^.p.'D.A{d^^-4>;)^€Afi^. The function (f> is dominated by 3l9Jt and has the properties JT^Tl, J^fW- Hence, for certain functions /ji^,iJi,^, /x^; positive constant a; and, for every e, integers m^^, m^ we have : (12) p.Zi .A<^j,^aAii,Qp; (13) KX-^-Hp = eAt,,j,; (1 4) ^1;,«, .^.A{<\>^- .,S^„,^4>^^ (g). The pair of representative elements 7\ , r^ depends on p, m, (j>. There is always at least one such pair whose elements however are not necessarily distinct. Then, for this element p, separating the indices I of stage m appearing in (9) into the indices g, h, we have : h g + (K - 4>p) E B,'^'' + 4>, (E V" - 1)' so that by (19) ^(C-'/'.)=Z^'/'.-^/" + (20) " U,A{r-4>rd+Ar-4>,)'\^Ah;^^+A4>^A{j;^^-'-\). Hence, since LA{4>^-4>,;)+A{4>,^ - <^j ^ (C+ i)^(;)^aJ^A|x^^,„h-^+2{2l^+l)eAlJL.,^+aeA^l,^ ^e\aAix^^ + 2(2C + \)Aim^^ + aAi^J. *If there are no indices h (i. e., if the element p belongs to every class of stage m) this and the later summations as to h are understood to have the value 0. 144 E. H. MOORE. In (17, 21) the constant a and the functions n^, n^, fi.^, fj,^ are four fuactions of 3}i which are independent of e, p, m. 3R has the dominance pi'operty D^ ; thus there exists a function /Lt of 9Jl such that 2l/x dominates fi^, /x.^, jm^, fi^ Accordingly from (17, 21) which hold for every e we readily exhibit for this function /a and every e an integer m^ such that the required relation (11) holds. Hence, every function having the property Kf^W is dominated by the class 3l3}t and belongs to the class 3Jt^ , in case the class 3Jt has the dominance property D^ and the devel- opmental property A associated with a development A of the class ^, with which is associated also the relation Kf^ under- lying the property Kf^Tl of the function (p. Theorem I has been proved. The proof of the more gen- eral theorem II follows precisely the lines of the proof of theorem I. We note merely that the representative elements r[, r'^ corresponding to the representative elements r^, r^ of (18), depend not only on p but on p" , a fact however which does not interfere with the argumentation since they do not appear in the relation corresponding to (21). Classes '^ contained in (3Jt'3)i")^. Classes 9)1 with the property Kf,^. §§81-82. 81 . Classes '^^ contained in (3R'9)i")^. — As a corollary of the second theorem of § 80 we have the Theorem. The hyp)othesis : 5P' . A' . aJt''^'''' . ^" . W'^' . % = [all ^K'"''^'!.^") ■ -Bo«5ra"(i'')] , implies the conclusion : This theorem is essentially a generalization of the theorems of §§65.8, 66.27; cf. § 67 c). 82. Classes 3Ji with the property K^^^ . — From the theorem of § 81 and the theorems (2^2, Sj^) of § 73 we have Theorem I. From (1) ^'.A'.aW'^^"^' INTRODUCTION TO GENERAL ANALYSIS. 145 it follows thai (2) %" . m" ^"^ .% = [all 4> K',t'^'m") . B„5)i"(p')] implies (3) % = {m'm'\ . g ^^^ • ^''^'^"; (4) m''"' .m" "'.■:>. (an'ajt")*^»; (5) 9)1"^ . 3 . {m'm'%^; (6) ^-ff4'^V{!m").JSo™"(p') _ 3 _ ^B.ik'w^ and, if further- (1') W^", it follows that (3') g^oa«'(y') ; (6') 0'^^ ^ ] . {B,m^ ~ ^t2 m) . m^^^""^''^^^^-^^- ■ ^i™ • Theorem III. From (10) 5P . A . 3)i^'^-°^^^ it follows that (1 1) SK^'"-* . 9w = [all y') ■ ■ •) 2/'""'^ it shall be called a semi-oovariant. It is not difficult to show that all seminvariants are functions of the following n — 1 , and of their derivatives : the first two of which are, explicitly, (5) P2='P2-PI-P[> Pz = Ps- ^PiP2 + ^P\ - Pi- There are n— \ independent semi-covariants, besides y itself, VIZ. (6) y, = y'"' + ( 1 ) P,2/"'-'' + ( 2 ) p.y''-'' + ---+Puy (k = l,2, ■••,71-1). These semi-covariants are characterized by the property that if y be transformed by a transformation of form (3), and if the corresponding functions for the transformed equation be denoted by rj^, the relation between t;^ and j/^ is given by the equation 2/ft = H^)V^ 154 E. J. WILCZYNSKI. which corresponds to (3), so that yjy, y^/y, ■ ■ ; yn-ify are absolute semi-covariants. An arbitrary transformation of the independent variable <7) I = ^(.t), converts (1) into a linear differential equation of the same form. Lie and Stackel have shown that the combination of (3) and (7) constitutes the most general point transformation which does not change the form or order of the equation. Such combinations of seminvariants and semi-covariants which are left invariant by the transformation (7) are known as invariants and covariants respectively. For instance 6^= P^ — fPj is an invariant. Let t)„ ■ ■ •, t)„ be the elements of a fundamental system of solutions of equation (1). This equation, being integrated, Vv ' ">Vn ^i^l ^^ known as functions of x. Interpret X)^, ■■■,y„ as the homogeneous coordinates of a point P, in a space of n— 1 dimensions. As x changes, P^ will describe a curve C,, an integr-al-curve of the differential equation. The equation, however, has an infinity of such integral-curves. For, the n functions n Vk = 'E(^',.,Vi (fc = l, 2, ■•■,n), 1=1 where the determinant of the constants Cj., is different from zero, will also constitute a fundamental system of (1) ; and any fundamental system of (1) may thus be expressed in terms of any other. The properties of the curve C,, which are expressed by the coefficients of (1), are not, therefore, characteristic of any particular integral-curve ; they are common to them all. Since the transition from one integral-curve to another is made by means of the most general projective transformation of the space considered, these properties are, therefore, common to a curve and all of its projective transformations ; they are projec- tive properties. But the representation Vk=f,{^) (fc = l, 2, •••, n), for a curve 0^ of a space of n — 1 dimensions contains two PROJECTIVE DIFPEEENTIAIj GEOMETRY. 155 arbitrary (non-essential) elements. In the first place, since the coordinates are homogeneous, only the ratios T)i ■ V2 '•'"'■ Vn have a geometrical significance. These ratios are not changed by the transformation (3), i. e., y = X{x)r]. In the second place, the independent variable may be arbitrarily transformed -without changing the curve. In other words, the integral- curve of (1) is not changed by the combination of the trans- formations (3) and (7) ; these transformations only alter the form of its analytical representation. If it be required, there- fore, to characterize analytically a geometrical property of the curve C^^, iu a manner which shall be independent of any special representation, it becomes necessary to investigate the invariants of equation (1). Any projective pt^operty of the curve will be expressed by an invariant equation or system of equations, and any invariant equation or system of equations will be the ade- quate analytical expression of some projective property of the curve. For m = 3 and ji = 4 we obtain in this way a theory of plane and space curves, which contains a great deal of interest- ing detail which time will not permit us to expound. The geo- metrical interpretation of the semi-covariants is essential for this theory. Incidentally it may be mentioned that there is obtained an adequate geometrical interpretation for the reduc- tion of equation (1 ) to the Foesyth-Lagueeee canonical form, which is characterized by the conditions that the coefficients of the n — 1th and n — 2th derivatives are equal to zero, a reduc- tion which may always be accomplished in an infinity of ways by transformations of the form (3) and (7). Halphen's investigations on the differential invariants of plane and space curves are geometrically identical with the theory which has just been outlined. If we confine ourself to the case of plane curves, the analytical relation between the two theories may be described as follows : Consider the linear differential equation of the third order 2/W -I- 2>p{y" + ^p^y + p^y = 0, 156 E. J. WILCZYNSKI. and let y^, y^, y^ be the elements of a fundamental system. In- troduce non-homogeneous coordinates by putting 2/1' ^ 2/1 and choose x as independent variable, so that will be the equation of the plane curve. The invariants will become functions of y, y, y", etc., and these are the differential invariants of Hai^phbn. It is easy, therefore, to derive Halphen's differential invariants from the invariants of the differential equation. Halphen however obtains them in a different way. He starts with the equation y = f{x) of the curve, and determines functions of y, dy/dx, d'y/dx'^, etc., which are left invariant when x and y are subjected to the most general projective transformation ~ o„ + G^x + c^' ^ ~ c„ + G^x + G^ This unsymmetrical and unhomogeneous formulation of the problem is manifestly a disadvantage. It is easy enough to obtain the unhomogeneous form when the homogeneous is known, but the inverse process is far more difi&cult. If, at the time of Halphen's first researches, he had been aware of the connection between his differential invariants and the invariants of linear differential equations, he would probably have put them into the homogeneous form. As he did not notice this connection until later, when his point of view had become ana- lytical rather than geometrical, it was left to the lecturer to recast these researches into the more adequate and elegant form. It should be mentioned moreover that, for this purpose, the geometrical theory of the semi-covariants is essential and Halphen does not seem to pay any attention to them. More- over, for the theory of space curves, a general projective theory of ruled surfaces is a prerequisite, a theory not then in exist- ence. These remarks will suffice to explain the relations be- tween Halphen and myself in the construction of this theory. projective differential geometry. 157 Second Lecture. The theory of ruled surfaces is based upon the consideration of a system of differential equations of the form «" + P2^y' + Piz^ + (?2i2/ + fe2 = Oj which shall be spoken of as ihe system {A), for the sake of brevity. In connection with this system of equations, consider the infinite group G of transformations ri = a{x)y + ^{x)z, ^ = -i{x)y + h{x)z, ^=f{x), aS-^y + 0, where a, /3, then (2) / = M„ -h M22' J = "11^22 - ^^12^21 are seminvariants, i. e., have the same value for system (A) as for any system obtained from it by a transformation of the form 7; = a{x)y + §{x)z, K = 7(a')y + K^)^, 158 where a, E. J. WILCZYNSKI. (3) ' ■ ■ B are arbitrary functions of x. Put further ^12 = ^^2 + {Pn -P22>u -Pni^n - ^22)^ % = 2m;i - {Pn -P2>2i +i'2iK - ^22); '^22=2w;2-i'l2"21 +^'21^,2; and form a third set of four quantities w^^, w^^, w^^, w^^, from the quantities v.^^ and p^j^ in the same way as the quantities v^^ are formed from m^^ and p^^^. Then ^=''u'^22- L = W„W22 - «^12«'2l (4) are two further seminvariants, and all seminvariants are func- tions of I, J, K, L and of the derivatives of these quantities. The invariants are functions of the seminvariants. Those which are fundamental are : e^ = p- 4J, e,,, = 8^;^, - 9(^:)^ + Mei, (5) e,^=eiK-i'') + i{ej, e, = ^, where "ll - ^^22 ^*I2 «21 (6) A = «U - *'22 \2 "21 t«„ - W22 ^12 W, the expression of which in terms of I, J, K, L is not rational. All other invariants may be expressed as functions of these four and of certain others obtained from them by a certain process which involves differentiation. The index, in the above notation, indicates the weight of the invariant, except in the case of 6^^ which is of weight ten. If 6^ denotes the result of the general transformation of the group G upon an invariant 6^ of weight m, we have The fundamental covariants may be obtained as follows. PROJECTIVE DIFFERENTIAL GEOMETRY. 159' Put (8) p = 2y' + p^^y + p^^z, a = 2z' + p^^y + p^^ ■ then (9) C, = P=zp-y<7 is a covariant of weight one. Put further Then (11) c, = a, c, = 0,^-^;o are two further covariants. Put finally a 2) ""^ ^^"" ~ "''-"' "^ ^"'''^ "^ ^*^^'' ~ ^''^^ ''' *'''''' /S = 4«2iP - 2(w„ - M,j)o- + t^^iy - i(*;,i - «2j)2 ; then (13) q = az-^y is a fourth covariant. All of these covariants are quadratic ; their weight is indicated by their index. All other covariants may be expressed in terms of these four and of invariants. We proceed to consider the solutions of a system of form (A). Let the functions /),.,. and g-^^ be analytic in the vicinity of X = Xg. Then, there will exist two functions y and z, ana- lytic in the vicinity of a; = a;^, which satisfy the differential equations, and which, together with their first derivatives,, assume arbitrarily prescribed values for x = a5„. Such a system of two functions, involving four arbitrary constants, shall be said to constitute a system of general solutions of system (A)^ Now let (y^, z^ for (z= 1, 2, 3, 4) be any four systems of solutions of {A). Then, denoting by Cj, c^, Cj, c^ four arbitrary constants, 4 4 (14) 2/ = Ec.yi, 2 = ZcA will also form a simultaneous system of solutions. Moreover,, from (14) and 4 4 (15) y' = T,oa'i, z' = T,c/i, Vi 2/2 2/3 Vi < < < < Vi 2/2 2/3 2/4 \ h h h 160 E. J. A\'ILCZYNSKI. the constants o^- --g^ can be determined in such a way as to give arbitrary constant values to y, z, y, z for x = x^, provided that the determinant (16) D = does not vanish for x = x^. If, therefore, D is not identically zero, we can express a general system of solutions in terms of Vv ■"} Vi ^^^ ^v ■ " 'j ^4 ^7 ™eans of (14). We shall, there- fore, speak of four pairs of solutions (y., z?), for which the de- terminant D does not vanish, as a fundamental system of simul- taneous solutions. We may express the condition Z) =|= in another way. If J) ^ it is possible to find four functions A, fi, v, p of x, so that the four equations (17) Xy^ + ii.y[ + vz^ + pz^ = (fc = 1, 2, 3, )4 may be verified. If (y^, Zj^) form a fundamental system of solu- tions, it must therefore be impossible to find functions "S, fi,v, p so as to satisfy (17). It may be shown, without any difficulty, that conversely any four pairs of functions (y^ kJ determine uniquely a system of form [A), of which they are a fundamental system of solutions, provided that their determinant D does not vanish identically, i. e., provided that they do not satisfy a system of equations of the form (17). We are now prepared to see what all this has to do with the theory of ruled surfaces. Let us interpret (y,, • ■ •, y^) and (zp ■ ■ •, z^ as the homogeneous coordinates of two points P and P^ of space. As x changes, P^ and P^ describe two curves C and C ; the points of these curves, moreover, are put into a definite correspondence with one another, those being cor- responding points which belong to the same value of x. Con- struct the tangents of these two curves at corresponding points. PROJECTIVE DIFFERENTIAL GEOMETRY. 161 In general they will not intersect ; they will, if, and only if, the determinant D is equal to zero. In order, then, that the curves C and C^ may be the integral curves of a system of form {A), it is necessary and sufficient that the tangents of the two curves, constructed at corresponding points, shall not intersect. Join the points P^ and P^ by a straight line L ^. The locus of these lines is a ruled surface S, the integrating ruled surface of system {A). The transformation . -q = a{x)y + ^{x)z, ? = 7(33)3/ + h{x)z, aS — ^y ^ 0, which is contained in the group G, transforms the points P^ and P^ into two other points P^ and P^ of the line L^^. Since the functions a, • ■ -, S are arbitrary, it is possible in this way to convert the curves C and (7 into any other two curves upon the integrating ruled surface S. The correspondence of the points P^ and P^ remains such that the line joining correspond- ing points is a generator of the ruled surface S. A transformation of the form where f(x) is an arbitrary function, changes the parametric representation of the curves in the most general way, without altering either the curves themselves or their point to point correspondence. The transformations of the group G, there- fore, leave the ruled surface S invariant. The ruled surface S, however, is not unique. For, on account of the fact that (A) is a system of linear homogeneous equations, it is evident that any projective transformation of S will be an integrating ruled surface of the system. Let us speak of two systems of form (A) as equivalent if they can be transformed into each other by a transformation of the group G. Then we may make the following statement : If two systems of differential equations of form (^A) are equivalent, their integrating ruled surfaces are projective trans- formations of each other. Moreover, if the fundamental systems of solutions be properly selected, the ruled surfaces coincide. Conversely, if the ruled surfaces of two such systems coincide. 162 E. J. WILCZYNSKI. the systems are equivalent. The integrating ruled surface of a system of form (^A) is never a developable. It is easy to show that any non-developable ruled surface may be defined by a system of form (A). The general theory of such systems of differential equations is, therefore, equivalent to the general theory of ruled surfaces. Any equation or system of equations between p.j^, q.^, p'-j., etc., tuhich remains invariant for all transformations of the group G, expresses a projective property of the integrating ruled surface. For, such equations remain unchanged whatever may be the two curves C^ and Q upon 8 which are taken as fundamental curves, and whatever may be the independent variable. They express, therefore, properties of the surface itself, independent of any special method of representation. These properties are projective because the coefficients of (^A) are left invariant by any projective transformation. Conversely, any projective property of a ruled surface can be expressed by an invariant equation, or system of equations. It remains to introduce the principle of duality. At cor- responding points P and P^ of the two fundamental curves C and G^, let us construct the planes p and p^ which are tangent to the ruled surface 8. They will intersect along the straight line L^^ which joins P^ and P^. The four pairs of coordinates, determining these planes p^ and p^, will form a simultaneous fundamental system of solutions for a new system of differential equations, which shall be called the adjoint of [A). This system may be written as follows : ( loj F"+ p,^ U'+p,, F+(9^,-HiM,,) U+ [fe+i(M,,-M„)] F=0. The invariants of even weight are identical for (18) and for (-4), while those of odd weight differ in sign only. Systems (J.) and (18) are identical if (19) Wj2 = M21 = 1*11 - M22 = 0. From the relations between the solutions of the two systems it PROJECTIVE DIFFERENTIAL GEOMETRY. 163 follows that, in this case, the ruled surface is a quadric. The conditions (19) are, therefore, characteristic of such systems (^A) whose integrating ruled surfaces are quadrics. Such systems coincide with their adjoints. A further result of fundamental importance, which is a simple consequence of the relations between the solutions of a system of form (^) and of its adjoint, is the following. If, in a system of form (A), p^^ =^21 ~ *-*> ^^ integral curves are asymptotic lines on its integrating ruled surface. From this result may be deduced the theorem of Paul Serret : The double-ratio of the four points, in which a moving generator inter sects four fixed asymptotic curves of a ruled surface, is constant. The system [A) being given, its invariants may, of course, be computed. But, with a certain restriction, the converse is also true. Upon this fact is based the proof of the following theorem which, from our point of view, is the fundamental theorem of the theory of ruled surfaces. If 6^, ^4 1, 9^ and 6^^ are given as arbitrary functions of x, provided however that 6^ and 6^^ are not identically equal to zero, they determine a ruled surface uniquely except for projective trans- formations. We shall not stop to deduce further consequences from this theorem. It may be noted, however, that it corresponds to the fundamental theorem of the metrical theory of surfaces which states that the two fundamental quadratic forms suffice to determine a surface except for its position in space. Third Lecture. Important new ideas are suggested by the problem of provid- ing a suitable geometrical interpretation for the covariants and semi-covariants. Let us begin with the covariant (1) P = zp — ya, where (2) /J = 22/' + p„2/ + p^^z, a-=2z' + p^^y + p^^. 164 E. J. WILCZYNSKI. If we substitute y ^= yj^, z-= z^^ (^=1, 2, 3, 4) into these expressions, the two sets of four quantities p^ and o-^ may again be interpreted as the homogeneous coordinates of two points Pp and P^. It is at once apparent that P^ and P^ will be points of the planes tangent to the ruled surface S at P^ and P^ respectively. But we may describe the position of these points more completely. In the first place it may be shown that, if P and P^ are converted into two other points Pj^ and Pj of the line L^^ by the equations y = a.y + 0z, z = yy + Sz, then P^ and P^ are transformed cogrediently into P- and P-, where p = ap + ^a, (7=yp + Ba, i. e., into two points of the line L^^ which joins P^ to P^. We find, therefore, a line P ^P „ which has a definite point-to- point correspondence with the line P P^, the generator of 8. This point-to-point correspondence is a very simple one. Suppose that the system {A) has been reduced to such a form that ^j2 = P2\ = ^j ^"^ ^^'^ *^^ curves C and (7 are asymptotic curves on S. Equations (2) show that P^ and P^ will then be points upon the tangents of these curves. Let us speak of the tangents to the asymptotic curves of S as its asymptotic tangents. The totality of these lines along a given generator g oi a, ruled surface constitutes a set of generators of a hyperboloid H, the osculating hyperboloid of 8 along g. This hjrperboloid may also be defined as passing through g and two other generat«rs of S infinitesimally close to it. The generators of H which belong to the same set as ^^ or i ^ may be called generators of the first kind. The asymptotic tangents of S along g will then consti- tute the second set of generators of H. We may now say that the line L is a generator of the first kind upon the hyperboloid H which osculates S along L ^. The correspondence between the points of L ^ and L^^ is such that the lines which join correspond- ing points are the generators of the second kind on the osculating hyperboloid. PROJECTIVE DIFFERENTIAL GEOMETRY. 165 It is the totality of the generators of the first set upon H, rather than a particular one, which is of importance in this connection. In fact, if the independent variable be changed by putting I = ?(•-«), p and (7 are converted into (3) -p = |, {p + r,y), a = -^,(a+ -qz), where (3a) 'y = F • The factor 1/|' is of no importance, since we are concerned only with the ratios p^ : p^: p^: p^, etc. By choosing tj conven- iently as a function of x, we may clearly make L^- coincide with any generator of the first kind on H excepting only g itself. As X changes, the line L describes a ruled surface S'. The generators of this surface belong to the congruence T, which is formed by the generators of the first kind on the single infinity of hyperboloids which osculate 8. There is one generator of S' upon each of these hyperboloids. Clearly, by an appropriate choice of the independent variable, the surface S' may be made to coincide with any ruled surface of the congruence F which has one of its generators on each of the osculating hyperboloids. We, therefore, speak of *S" as being the derivative of S with re- spect to X. The derivative ruled surface may serve as an image of the independent variable. This image changes, in general, whenever the independent variable is transformed. It does not change, however, when x is made to undergo a linear transfor- mation ^ = a.r -|- b, since such a transformation does not change the value of rj as defined by (3a). A linear transformation of the independent variable is, therefore, without any geometrical significance. We leave the covariant P and the congruence T which it defines, in order to return to it a little later. The covariant (4) C = Mi2 z" - M^i y^ + (m,i - M22) yz 166 E. J. WILCZYNSKI. may be decomposed into two factors linear and homogeneous in y and a. It defines, therefore, two points on every generator of the ruled surface S. If the locus of these points be chosen as fundamental curves, the corresponding system [A) is charac- terized by the conditions u^^ = u^^ = 0. Geometrically these two curves are characterized as follows. Four lines in space have two real, imaginary or coincident straight line intersectors. Consider a fixed generator g, and three other generators ^j, g,^, g^, of the ruled surface 8, together with their two intersectors. Let the generators g^^, g^, g^ approach g &s & limit. The inter- sectors of g, g^, g.^, g^ will, in general, approach limits /' and /", which will determine two points upon g. We may say briefly that these are the points of g at which tangents to 8 may be constructed which have four consecutive points in common with the surface. Following a nomenclature due to Caylby, we shall call these points the fleonodes of g. The four-point-tangents shall be called the fleonode tangents ; the locus of the flecnodes on S is its flecnode curve and the locus of the flecnode tangents, a ruled surface of two sheets, is the flecnode surface of 8. It may be remarked at once that the flecnode tangent is not, in general, tangent to the flecnode curve, never, in fact, unless the latter degenerate into a straight line. Since the discriminant of the covariant C is equal to 6^, the significance of the condition ^^ = becomes apparent. We may recapitulate as follows : The flecnode curve is determined by factoring the covariant C. Its two intersections with the generators of the ruled surface are dis- tinct if 6^^ ; they coincide if 9^= 0. If the integral curves C and C^ of a sy stein of form (^A) are the two branches of the flecnode curve, this system of differential equations is characterized by the conditions "i2 = Si = 0- It is important for many purposes to establish the relations which exist between a ruled surface and the fundamental con- figurations of line geometry. This may be accomplished by setting up the linear homogeneous differential equation of the PROJECTIVE DIFFERENTIAL GEOMETRY. 167 sixth order which is satisfied by the Pliickerian coordinates of the generators of the surface. The result may be recapitulated in the following theorem. The necessary and sufficient condition for a nded surface be- longing to a single linear complex, which is not special, are e^ + 0, A = 0, ^,„ 4= 0, while all of the minors of the second order in A do not vanish. If ^j(j = 0, while the other conditions remain the same, the complex is sjyecial. The surface belongs to a linear congruence vdth distinct directrices if all of the minors of the second order in A vanish, while 6^ is different from zero. The directrices of the congruence coincide if 6^ also vanishes. In this latter case the surface is, or is not a quadnc according as the equations «11 — ^22 = "l2 = "21 = are, or are not satisfied. We return to the consideration of the congruence T. The first question, which suggests itself, concerns its developables and its focal surface. The differential equations of the surface S' , the derivative of S' with respect to x, show that this surface is developable if and only if the seminvariant / is equal to zero. The complete answer to the question in regard to the develop- ables of the congruence based upon this remark is as follows. The congruence contains two families of co^ developables, which coincide if and only if 6^ = 0, i. e., if and only if the tioo branches of theflecnode curve of S coincide. To determine any developable surface of the congi-uence, it is necessary and sufficient to find a solution of the equation 4{|,a:}^4-2J{|,.T}+/=0, lohere {^, x} denotes the Schwarzian derivative of ^ with respect to x, and to take this solution | = |(.t) as the independent variable of the defining system of differential equations. The derivative of S ivith respect to | will then be a developable surface, and all developables of the congruence may be obtained in this way. 3Ioreover, any /our developables of the same family intersect all of the asymptotic tangents of S in point-rows of the same cross-ratio. 168 E. J. WILCZYNSKI. The locus of the cuspidal edges of these two families of de- velopables is the focal surface of the congruence. It may be easily shown that the focal surface of the congruence coincides with the flecnode surface of S. For this reason we shall speak of the congruence T as the flecnode congruence of 8. Each sheet of the flecnode surface of S, has 8 itself as one of the sheets of its flecnode surface. The second sheet of its flec- node surface, however, is never a surface of the congruence F except in degenerate cases. It is possible, nevertheless, to find two families of oo^ non-developable surfaces of the congruence r each of which has one of the branches of its flecnode curve on one of the sheets of the flecnode surface of 8. Both branches of the flecnode curve of a surface of T can be situated upon the flecnode surface of 8 only if 8 belongs to a linear complex. Fourth Lecture. If we trace any curve on a ruled surface 8, there will corre- spond to it a perfectly definite curve on the derivative 8' ; the lines joining corresponding points being the asymptotic tangents of 8. The investigation of such correspondences gives rise to a number of important results. It may be shown that there exists a single infinity of ruled surfaces in the flecnode congru- ence two of whose asymptotic lines correspond in this way to the flecnode curve on 8. In other words this may be expressed as follows : If 8 is a ruled surface with two distinct branches to its flec- node curve and not belonging to a linear congruence, there exists just a single infinity of ruled surfaces in the congruence T, whose intersections with the two sheets of the flecnode surface of S are asymptotic lines upon them. They are the derivatives of 8, when the independent variable is so chosen as to make the seminvariant I vanish. Moreover, the point-rows, in which any four of these surfaces intersect the asymptotic tangents of 8, all have the same anharmonic ratio. A system of form (A) may always be reduced to the so-called canonical form, for which Pa = ^> ?ii + ?22 = 0. PROJECTIVE DIFFERENTIAL GEOMETRY. 169' r The significance of this reduction is now apparent. The funda- mental curves G^ and C^ are any two asymptotic curves on S, and the independent variable x is chosen in such a way that the flecnode surface of S intersects the derivative S' of S with respect to x along a pair of asymptotic curves. If ^^ = the family of surfaces just characterized coincides with the (single) family of developables of the congruence. If the ruled surface 8 has two straight line directrices, the reduction to the canonical form has a different significance. There exist, in that case, oo^ ruled surfaces of the congruence V whose asymptotic lines correspond to those of *S'. These may be arranged in cxj' families of oo' surfaces. One of these families, not very essentially distinguished however from any of the others, is obtained by the reduction to the canonical form. The process of forming the derivative of a ruled surface with respect to an independent variable may be repeated. We thus obtain S" the second derivative of 8 with respect to x. This may coincide with 8 itself. We then have the following result : Evei'y ruled surface which has a second derivative coinciding with S itself may be defined by a system of form (^A) for which p^^. = 0,. and for which the quantities q.j^ are constants. If the independent variable is so chosen that the second derivative 8" coincides with 8, the first derivative S' is a projective transformation of the original surface, and its asymptotic lines correspond to those of 8.. Such surfaces are either of the form where X and fi are constants, or of the form 2q{x^x^ - x^x^) = x^x, log -? , where 9 is a constant. All of the asymptotic lines of such a sur- face are anharmonic curves with the same invariants. We have seen that the derivative ;S", of 8 with respect to x,. is cut by the flecnode surface of 8 along a pair of asymptotic lines if 1= 0. But this relation turns out to be reciprocal, i. e., the surface 8' cuts out an asymptotic curve upon each sheet of 170 E. J. WILCZYNSKI. the flecnode surface of 8. The asymptotic lines upon the two sheets of the flecnode surface, therefore, correspond to each other. Since the flecnode surface is also the focal surface of the congruence T, we see that the flecnode congruence is a so- called W-congruence. It will be clear without lengthy explanation what we mean by the osculating linear complex of a ruled surface. Every linear complex gives rise to a point-plane correspondence. Thus, there corresponds, in the osculating linear complex, to every point of the generator '5' of a ruled surface, a plane con- taining g. There corresponds to every point of g another plane containing g, namely the plane which is tangent to 8 at that point. There will be two points on g (in general) at which these two planes coincide. We shall speak of these as the complex points of g, and of their locus on ^S" as the complex curve. The complex curve, like the flecnode curve, intersects every generator in two points. These flour points form a harmonic group on every generator. Of course the complex curve may be determined by factoring a quadratic co variant, viz.: [(«ll-'^22)'^12-Kl-^'22)«lJ»'+[(«ll-«22Kl-(«'u-«'22)%j2/' + 2(Mi2"21-«2i'"i2>2- The complex curve becomes an asymptotic curve if the surface S belongs to a linear complex. The covariant C^ of Lecture II determines a pair of points which divides both flecnodes and complex points harmonically. Of course not all three point- pairs can be real simultaneously. Let H be the hyperboloid which osculates the ruled surface 8 along one of its generators g. Let g, a generator of the first set on S, be the corresponding generator of the derived ruled sur- face 8', and let H' be the hyperboloid which osculates 8' along ^'. Since g is situated entirely upon H' as well as upon H, the rest of the intersection of these two hyperboloids is, in general, a space cubic. This cubic is called the derivative cubic. We obtain, in this way, associated with every ruled surface, surfaces con- taining a single infinity of space cubics. The derivative cubic PROJECTIVE DIFFERENTIAL GEOMETRY. 171 degenerates only if the two branches of the flecnode curve of 8 coincide, if 8 has a straight-line directrix, or if the derivative surface 8' is one of the developables of the congruence F. The relations of the derivative cubic to the ruled surface are numerous and of considerable interest. We shall mention only the following theorem. Let the derivative surface 8' be one of the oo' surfaces of the congruence T which intersects the flecnode surface along an asymptotic curve. The derivative cubic will then intersect the generator of 8 in its complex points. If the two branches of the flecnode curve of 8 coincide, there is a derivative conic to consider instead of a derivative cubic. The flecnode curve being of fundamental importance for the projective theory of ruled surfaces, it becomes essential to find out to what extent it may be arbitrarily assigned. We find the following results : An arbitrary space curve being given, it can be considered as one branch of the flecnode curve of an in- finity of ruled surfaces into whose general expression there enters an arbitrary function. Two curves taken at random cannot be connected, point to point, in such a way as to constitute the com- plete flecnode curve upon the ruled surface thus generated. From any ruled surface a single infinity of others can be derived, each of which has one branch of its flecnode curve in common with it. This gives rise to an interesting configura- tion which we proceed to describe. Let C be a branch of the flecnode curve of a ruled surface 8, and let the developable surface formed by its tangents be called its 2>rima7-y developable. There exists another important developable surface containing C, which we shall call its second- ary developable, as indicated in the following theorem. If at every point of the flecnode curve of 8 there be drawn the generator of the surface, the flecnode tangent, the tangent of the flecnode curve, and finally the line which is the harmonic conjugate of the latter with respect to the other two, the locus of these last lines is a developable surface, the secondary developable of the flecnode curve. 172 E. J. WILCZYNSKI. We can find a single infinity of ruled surfaces, each having one branch of its flecnode curve in common with that of S. This family of oo' surfaces may be described as an involution, of which any surface of the family and its flecnode surface form a pair. The primary and secondary developables of the branch of the flecnode surface considered, are the double surfaces of this involution. In fact, the generators of these surfaces, at every point of their common flecnode curve, form an involution in the usual sense. In the course of these few lectures I have been able to touch merely the most important points of this new theory. Many problems remain to be solved. The field is a fruitful one whose cultivation promises further vahiable results. I am, at present, occupied with the construction of a general theor}'^ of surfaces from the point of view of projective diiferential geometry. I hope that I may be in a position to discuss that more general subject with you upon a future occasion. For the present I must close, thanking the Society and all of you for the precious oppor- tunity of explaining this theory to an audience of competent mathematicians. SELECTED TOPICS IN THE THEORY OF BOUNDARY VALUE PROBLEMS OF DIFFERENTIAL EQUATIONS. BY MAX MASON. CONTENTS. ? 1. The solution of certain functional equations 174 I 2. Fundamental existence theorem for the ordinary differential equa- tion 176 ? 3. The Cauchy-Kowalewski existence theorem for the partial differ- ential equation 178 ? 4. Classification of the partial differential equations 179 ? 5. The boundary value problems of the equations of hyperbolic type. 182 §6. The potential equation , 188 ? 7. The equation Ak =/ ; Green's functions 194 ? 8. Doubly periodic solutions of the equation A«=/ 196 ? 9. The equation Au = cu +/ 202 ? 10. The analytical character of solutions 206 ? 11. The boundary value problem for the equation y" + (l>{x)y =f{x). 207 ? 12. The transverse vibrations of a cord 209 ? 13. The existence of normal functions for the equation /■'+A4(a;)y=0. 210 2 14. The expansion of a function in terms of normal functions 218 The title of these lectures may, I hope, be sufficient apology for their fragmentary character. The theory of boundary value problems is a field too extended to admit of systematic discus- sion in four lectures.* But the interesting nature of the prob- lems and their importance in the applications form perhaps a sufficient reason for a presentation of even a fragmentary treat- ment. I have chosen the topics and the methods of treatment entirely according to my individual interest, feeling that a per- sonal point of view is expected, and perhaps desirable, in the Colloquium Lectures of the Society.f I have attempted how- * Excellent systettiatic accounts of the progress in this field up to the year 1900 are given by BOchee and Sommerfeld in the Encyklopddie der Mathe- matischen Wissenschaften. tl regret that the application of integral equations of the Frkdholm type, with the results of Hilbert and others, finds no place in these lec- tures. This phase of the subject constitutes an extended field, and demands more adequate treatment than could be given here. 173 174 MAX MASON. ever to link the separate topics together as far as possible, and to produce some mild degree of homogeneity, even though this course necessitates the discussion of some very familiar topics. The solution of functional equations of a certain type is first considered, and the result is afterwards used freely to unify the treatment of diiferent problems. The method is essentially that of successive approximations, but may have some advan- tage over the usual form in directness of application. It is applied in § 2 to prove the fundamental existence theorem for the ordinary linear differential equation of second order ; in § 5 to solve a new boundary value problem for the linear par- tial differential equation of hyperbolic type — a problem which includes as special cases a number of those previously studied ; in § 6 to present Neumann's method of solution of Dieich- let's problem, and in § 9 to discuss the boundary value problem for the equation Aw = cm +/ for a small region. In § 8 the existence of doubly periodic solutions of the equa- tion Am =/ is treated. The method illustrates the construction and application of a Green's function for cases other than those well known in the potential theory. The concluding sections contain a discussion of the ordinary linear differential equation of second order, which includes the first boundary value problem, the existence of normal functions for the equation y" + XAy = 0, and the expansion of an arbitrary function in term of normal functions. The properties of the normal functions as solutions of minimum problems are made prominent by the method which is employed. § 1. The solution op certain functional, equations.* Consider the equation (1) f=g + sf, where ^^ is a known continuous function for values of its * The method presented in this section has been employed in several par- ticular cases, probably for the first time by Liodville. See BOchee, An In- troduction to the Study of Integral Equations (Cambridge Mathematical Tracts), 1909. THEORY OF BOUNDARY VALUE PROBLEMS. 175 arguments in a region R, and S a linear operator, so that 8{u + v) = 8u-]r 8v. It will be shown that a continuous solu- tion/may be found in a simple manner if the operator /S is of a certain type — is namely convergent according to the follow- ing definition. Let <^ be any function continuous in the region R considered : The operator S will be called convergent if the following conditions are satisfied: 1°. S(f) is a continuous function in R. 2°. The infinite series /j and Tj^ are obtained, which are linearly independent and are connected by the relation The general solution of (1,,) may then be written in the form y = c,»?, + o^r)2> when Cj and c^ are arbitrary constants. The general solution of the non-homogeneous equation (1) is expressed in terms of 178 MAX MASON. these solutions of the homogeneous equation by the well-known form obtained by the method of variation of parameters. If the coefficients in equation (1) are analytic at a; = a, i. e., are expressible as power series in {x — a), then the solution of the differential equation (1) under the initial conditions (2) may be obtained by the power series method as an analytic function. On account of the uniqueness of the solution it is evident that every solution of equation (1) is analytic if the coefficients are analytic. § 3. The Cauchy-Kowalewski existence theorem foe the partial differential equation. The general linear partial differential equation of second order is fl) Au + 2Bu + Cw + Du +Eu + Fu+ G = 0. \ y XX ' xy ' yy ^ y A problem analogous to that of the preceding section is to de- termine a solution u of this equation such that along a given curve in the {x, y) plane u and du/dn, where n is the normal to the given curve, take on assigned values — in other words, to determine an integral surface u = u{.r, y) of the equation (1), which passes through a given space curve and is tangent along this curve to a given strip. In case the coefficients of the dif- ferential equation and all functions which enter into the initial conditions are analytic the power series method as used by Cauchy, and extended by Kowalewski, gives the following solution of the problem.* Analytic case : There exists a unique analytic solution of the problem, except in case the given curve in the (x, y) plane (^pro- jection of the given space curve) satisfies the differential equation Ady^ - 2Bdxdy + Cdx? = 0, * See, e. g., Hedrick, On the characteristics of differential equations, Annals of Mathematics, 2d series', 4 (1903), p. 145. THEORY OF BOUNDARY VALUE PROBLEMS. 179 i. e., except for curves ^{x, y) = const., where ^ satisfies the equation (2) jr/ + 25?x+cr;=o. Can a solution of the differential equation (1) be found which satisfies the initial conditions if the functions involved in these conditions are not analytic? For some equations this may be done, as will be seen in § 5. On the other hand, for the equation U -f M =0, the values of u and dujdn can not be assigned arbitrarily along a given curve.* There is therefore a real difference between the cases of analytic and non-analytic initial conditions, and the limitation of the Cauchy-Kowalewski existence theorem to the analytic case is not caused by the method of proof alone. In the further developments no restriction as to the analytical character of the functions which occur will be assumed unless the fact is expressly stated. These functions will however be assumed to be continuous in the region under consideration. § 4. Classification of the partial differential equations.! Let the independent variables in the general equation (1) Au^^ + iBu^^ + Cu^^ + Du^ + Eu^ + Fu + G = be replaced by the new variables ^, »?, the equations of trans- formation being ? = l(^; y), V = v{^, y)- Then with similar equations for u^, u^^, u^^. After substitution of these values equation (1) takes the form * This was pointed out by Hadamard, Bulletin of Princeton Uni- versity, April, 1902. So far as I know no general treatment of this ques- tion has been made. fLAPLACB, (Euvres, 9, p. 21. DuBois-Eeymond, Crelle, 104 (1889), p. 241. 180 MAX MASON. (2) au^^ + 2/3mj^ + 7w^, + 8?^^ ^ eu^ + a7 — /3^ = — /3^ and therefore /8 does not vanish. The equation (2) may then be divided by 2/S, the result being ^h + 2^(H + ^% + (^w + -f) = 0. On changing the notation this equation takes the form u = au 4- bu 4- cu + f, the normal form for equations of hyperbolic type. 1{ AC — B^ = in the region considered, the equation (1) is said to belong to the parabolic type. In this case the first * According to the elementary theory of partial differential equations of the first order these solutions are obtained by the integration of ordinary differential equations. Their functional independence is readily seen as a consequence of the fact that AC — B'=^0. THEOEY OF BOUNDARY VALUE PROBLEMS. 181 member of equation (3) is the square of a linear expression. One function r]{x, y) may be found which satisfies (3), but no other solution functionally independent of 7?. In the equations of transformation choose -q as this solution. The coefficient 7 is then zero. Now ay - B^ = 0, and therefore /3 = also. On changing the notation as before, the equation takes the form «... = «w^ + bu^ + cu +/, the normal form for equations of parabolic type. If J. C — 5^ > 0, the equation belongs to the elliptic type. In this case two complex solutions (/>, -«|r, of (3) are to be found of the form fp = ^ + iv, -^ = ^ — in- Choose the functions |, 77 to define the transformation. Now ^ satisfies equation (3), that is, + c{i; + 2i^,^v, - V) = 0. Therefore /3 == 0, and a = 7. But 0:7 — /3^ = a7 > 0, so that a =1= 0. After dividing by a and changing the notation, the equation (2) takes the form ""x. + «,, = aWx + bu^ + cu +f the normal form for equations of the elliptic tyjje. It is evident that this classification applies as well to equa- tions of the form Ait^.^ + 25m^^ + Cm^^ = F(x, y, u, u^, mJ, which are not linear with respect to u, u^, m . In an important memoir on the subject of the analytical character of solutions of partial differential equations, Serge Berkstein * has made a similar classification of the general non-linear differential equation of second order, F{x, y, u, u^, u^, M„, M^^, uj = 0, based upon the value of the determinant . dF dF / dFV VduT }' du du , WW XX yy \ xy . *Mathematisohe Annalen, 59 (1904), p. 20. 182 MAX MASON. § 5. The boundary value peoblems of the equations op hyperbolic type. The different boundary value problems which arise in con- nection with the partial differential equation of hyperbolic type, (1) w^^ = au^ + hu^ + cu+f, have been treated by distinct methods, based for the most part on successive approximations and on Riemann's method.* A new boundary value problem will be treated in this section, which includes as special cases many of those previously studied. Consider a rectangle It containing a point {x^, y^ and bounded by the lines x = x^, y = yv a; = 352) y = yr Let C7. and C be two curves intersecting at (x^, y^, represent- able in the form (7,: y = 4>{x), C^: x = y{r{y), where it is assumed that for Xi = x = x.^, y^ = y = y^, the func- tions ^(cc) and ■^{y) are continuous and single- valued, that the derivative <^'(a;) is continuous, and that the inequalities (2) _ y^^^(^:c)^y^, x,^f(y)^x, are satisfied. i^2, Vi) V' c. ^ {^i, 2/1 ) ^ N *See for example Sommbrfkld, loo. oit., and R. d'Adkmae, Les equations aux derivees partielles d, earacterisiiques reelles, Paris, 1907. THEORY OP BOUNDARY VALUE PROBLEMS. 183 It is required to determine a solution* of the equation (1) u^^ = au^ + bu^ 4- CM +/ in R, which satisfies the conditions The coefficients of equation (1) are supposed continuous in R, ^{^)> ^{y) ^'■6 given functions continuous in the intervals (x^, x^ a° v¥^<^^ + r nv)^^ + u{x), r'V + v{x, v)dv - ij^X'^)Y[cj,{x)-] + U'(x). If V is any continuous function in R, the function u, defined by the second of the above equations, is continuous in R together with the derivatives u^, u^ , u^^ , and satisfies the conditions (3). The necessary and sufficient condition that u satisfies the differ- ential equation (1) is the following functional equation for v, found by substituting the above expressions for u, u^, u^, u^^ in (1) = mi v{^> y) =ff{^, y)+(.x) (4) - a(x, y)'{x) f t-[|, {x)]d^+b(x,y) f v{^, y)d^ ny nx + c(a;, y) | I v{l^, v)d^dv, J^(.x) J^nn) * I. e. , a function u which satisfies (1) at each point of B. This assumes the existence of Ux, at each point, and hence implies the continuity of u, Ux, Up in S. 184 MAX MASON. where g{x, y) = a{x, y){ U\x) - 4>'{x) Y[(x)] } + b{x, y) Y{y) ^^ ^ + c(c«, y) \ r Y{v)dr, + U{x)\ +/(x, y). This equation has the form V = g + Sv, where 5^ is a known continuous function and 8 a linear operator. There exists a unique solution if the operator S is convergent, as defined in § 1, i. e., if the series (5) z + Sz + 8h + .-; formed for any function z which is continuous in R, converges uniformly, and if the operation 8 may be preformed on this series term by term. From the definition of the operatoi' *§ by equation (4) it is seen at once that limit 8R = 0, if R^ approaches zero uniformly in i? as w increases indefinitely. It follows that the operation 8 may be preformed term by term on the series, if the convergence is uniform. The operator 8 is therefore convergent, and a unique solution of equation (4) exists, if the series (5) converges uniformly. That this is the fact, will be shown by forming a " majorant operator " to 8, i. e., an operator which gives rise to a series of the form (5), each term of which is positive and greater than the absolute value of the corresponding term of (5). This new series will be shown to be uniformly convergent. Let 7 be the greatest of the maxima of I a |, 2 I a(f)' |, 2 I 6 |, 1^| c | in R, and write THEORY OF BOUNDARY VALUE PROBLEMS. 185 Then it follows oa account of the inequalities (2), that for all cc, y in R, \y - 'f>{x)\ = y2-y, = y - ^. Define three operators S^, S^, S^ by the equations S^w=:y i w(f, y)d^, S^w^ry I w{x, r))dv, where m>(x, y) is a function defined for all values of x, y in the intervals (a, x^), (/3, y^). Let 2(0;, y) be any function continu- ous in a, and suppose that w{x, j/) s in its region of defini- tion, and that w{a+\x'\, /3 + \y'\)^\z{x,y)[\=\z[f{y) + x', 4>ix) + y"\\, where x' = x — if-(y), y = y — ^{x), for all values of x, y, in R. Then the functions {8^ + S^ + 8^)w and Sz satisfy this same inequality,* i. e., fi + \v'l 3+1/1 It follows, therefore, that Since w^O the function (/S, 4-/82 + S^y'w is an increasing func- * The proof of this fact is obtained at once by comparing the values of the integrals -which occur in Sz with the integrals of SiW, S^w, S^w ; for ex- ample, the integrals «(f, y)dy =- z{^,y)d^= i zmy) + x', y^dx' and /" «>(?, i8+l3/'l)(j!) 186 MAX MASON. tion of both x and y, and therefore {S^ + S, + 8^)^11,^18^1 for all points (x, y) in R. Choose for w the maximum S of I 2 I in i2. Then the following statement is justified : The series (5) converges absolutely and uniformly in R, if the series of positive terms (6) T.{.S, + 8, + 8,fh converges uniformly in R. The operators 8^, 8^, 8^ are commutative and associative, so that in the series (6) each term of the form S'j*i*S'/^>S3*'S occurs with the factor {k^+k^+k^)\ k^lk,lk,\ ■ Now S^w = S'jSjW = 8^SjW. Then T.{S, + S, + 8,YB= ± A^_,8-S,-S, *=0 m, n=0 where {k, + k, + k,)\ ™'"~^ \l k^\ kj ' the summation being extended over all positive integers (in- cluding zero) which satisfy the equations 4^-1-^3 = m, k^ + k^= n. A^„, is increased if these two conditions be replaced by the single equation ^1 + ^2 + ^K = '>n + n. Now -4^ „ is increased if each value of k^ is replaced by a greater value, and is therefore increased if the above equation be replaced by k^ + k^ + k^ = m + n. The summation under the last condition gives (1 + 1 + 1)""+" = 3""+". Therefore THEORY OF BOUNDARY VALUE PROBLEMS. 187 Now « »* "^ - 'y"'('" - ")"' 'vXy - ^T . 1 ^ m ! ■ n ! ' and therefore the terms of the series m, 71=0 are less than the terms of the series ^ [37(^-«)]-[37(y-^)]'' s_Sc-.._„..,_.. The series (6) and (5) therefore converge uniformly and absolutely in R, and the operator S of equation (4) is convergent. The following existence theorem is therefore proved : There exists a unique solution of the differential equation u = aw + bu,, + CM + / in the rectangle R, which satisfies the conditioTis This solution is given by the equations u= r r v{^, v)d^dv + r Y{r,)dv + U{x), v = g + Sg + 8'g+---, where g is given by equation (4') and 8 is the operator which occurs in equation (4). By specializing the boundary conditions in this problem solutions are obtained for various boundary problems which have recieved special attention in the past. The following are examples : 1°. The value of u is given on the intersecting curves y =<^(x), Of* ^~~ CC ' (w),^^(,)= U{x), (»).=.„ = %). The second condition may be replaced by the equation (mJ.=.„ = V'{y), 188 MAX MASON. since the value of u is known at one point of the line x = x^^ namely at the intersection of 2/ = y = v{t), where t is the length of arc and varies from to /. It will be assumed for simplicity that the functions ^ and tj have continu- ous first and second derivatives.f The function where .(.,,).ijrV(o?%iM2-m„, y -V ^{x, y, ^, V) = arc tan is a solution of Aw = at all points not on the boundary, what- ever be /. This may be verified immediately, since at all points not on C the integrand is continuous together with all its derivatives, and the differentiation may be preformed under the sign of integration. The function tt is a "potential of a double distribution." As the point (x, y) in the interior of R approaches a point on the boundary, corresponding to the parameter value i = s, the value of u approaches a definite value u^^ independent of the direction of approach. These values M;, form a continuous function of s, and satisfy the equation % de(s, t) where Uao^I-'-^m The necessary and sufficient condition that the function u as- *Leipziger Beriohte, 1870, p. 50. fit may be readily seen from this assumption that the function dd{s, t)/dt ■which occurs below Is continuous. tSee a. g , C. Neumann, Uniersuchungen iiber das logarithmische und New- ion' sche Potential, Leipzig, 1877, p. 139. 190 MAX MASON. sume the given continuous boundary values g(s) is therefore the following functional equation for / : (2) f^s) = g(s)-l£f{tf-^dt On account of the fact that this equation may be written in the form (3) /W-f + ^X'[/(,)-/(0]?%5* or where /S' is a linear operator, and is of the type considered in § 1. The operator 5' is in this case convergent, according to the definition of § 1, if the series where ^ is any function of s continuous in the interval (0, I), is uniformly convergent. From the form of 8 it is seen that each term in this series depends only on the variation of the preceding term. If m and M are the minimum and maximum respectively of the variation of (f> in the interval (0, I), i. e., the difference between its maximum and its minimum : then, from (4) and (5), (6) Bl£m'^. Furthermore, as has been seen, \S''\, so that |^"|<,o'-i-S0. Since /o < 1 it follows that the series in question converges uni- formly, and that ;S is a convergent operator. The following theorems may therefore be stated : There exists a unique solution f=i{9 + S9 + S'g+...} of the equation The function satisfies the potential equation in the region JR, and assumes the values g(s) on the boundary of R. A solution of Dirichlet's problem is thus obtained for a con- vex region, when the assigned boundary values g(s) are con- tinuous. It was pointed out by Schwaez that if the function g[s) is in general continuous, but suffers a finite number of finite dis- continuities, the problem may easily be reduced to the previous case. Suppose, for example, the function g has the discon- tinuity d at the point (a, /3) of the boundary, but is elsewhere continuous. Now the function d y-/3 — arctan TT X — a is a solution of the potential equation within M and assumes values on the boundary which are continuous except at (a, 0) where they have the discontinuity d. Therefore, if v(x, y) is THEOEY OP BOUNDARY VALUE PROBLEMS. 193 the solution of Dirichlet's problem for the continuous boundary values , , d r)(s) — /8 ^(s)--arctanj^J— ^, the required solution of the original problem is / \ / N <^ y — ^ u(x. y) = vix, y)-\ — arctan . As the point (x, y) approaches the point (a, /3) along the boundary, from one or the other side, the function u{x, y) approaches one or the other of the values v{a,^)+-6, v{(x, ^) + ^{e - it), where 6 is the angle which the tangent to the curve at (a, /8) makes with the x-axis. If {x, y) approaches (a, /3) from an interior point, the limiting value of v depends on the direction ff of approach, and is equal to v{a,^) + -e', a value which is included between the limiting values approached along the boundary. The fact last stated plays an important part in the method of alternation of Schwarz,* which makes it possible to pass from convex regions to those of more general character. If Dirich- let's problem can be solved for two regions R, R having a part in common, it can be solved for the combined region R + I^- Suppose the region considered can be decomposed into two regions bounded by the convex curves aa, b^. The values of u are given on a and b. Form the solution u, of Aw = which assumes the given values on a, and values on a which join continuously onto the given values at m and n. This function will have certain values (Mj)^ on /8. Form the solu- tion I'j of Aw = which assumes the given values on 6, and the values (m,)^ on /3. This function will take on certain values * See his Oesammelte Abhandlungen, vol. 2, p. 157. 194 MAX MASON. («j)„ on a. Form the solution u^ of Aw = 0, which assumes the given values on a, and the values (v^)^ on a, etc. Two series of functions u^, v^ are thus obtained. The proof is com- pleted by showing that u^ and v^ approach definite limits, as n increases indefinitely, and that these limits are solutions of Am = 0, which coincide in the area bounded by a and /3, and assume the given boundary values on a and b. The solution of Dirichlet's problem is thus obtained for the combined region. Closely allied to Neumann's method of solution of Dirichlet's problem are those of Kiechoff,* RoBiN,t and Stekloff. J In the method of Poincae6§ a series of func- tions is formed, each of which satisfies the boundary condition, and whose limit satisfies the differential equation as well. The most recent method is that of Feedholm, || whereby the functional equation (2) of this section is solved by a new method. This as well as the older methods has been applied to solve the other linear boundary problems of the potential theory.^' § 7. The equation Aw = f; Geeen's functions. On account of the existence of a solution of Dirichlet's prob- lem, there exists a function g which satisfies the potential equa- tion Aw = at all points within the region M and assumes the values 1 - log -^ = log vXx - ^y + {y- vf on the boundary C, where (^, ??) is a point in the interior of a. The function *Aota Mathematlca, 14 (1890), p. 180. tComptu Eendus, 104 (1887), p. 1834. Jlbid., 125 (1897), p. 1026. ? See his Theorie du potential Newtonien, Paris, 1899, p. 260. II Ofversigt of Kongl. Vetenskaps Akad. Forhandlinger, 1900. % For the application of the Feedholm method see in particular Plemklj, Monatshette f iir Mathematik und Phy sik, 15 (1904), p. 337, and 18 (1907), p. 180. For the older methods see, in particular, Korn', Abhand- lungen zur Potentialiheorie, Berlin, 1901. THEORY OF BOUNDARY VALUE PROBLEMS. 195 G{^, y, I V) = log ^ + g{x, y, I rf) is called the Green's function for the region R. It satisfies the equation AG = at all points of B except the point (^, r)) and vanishes identically in (f, rf) when the point {x, y) lies on the boundary C. This function is symmetrical, that is, G{^, y, ^, v) = G(^, 7?, X, y). This is shown by the aid of Green's theorem, (1) jjjv^u - uAv)d.dy = £{''£ -^Zy> where n is the direction of the normal to the curve C, taken as positive when pointing outward. In this equation take u = G{x, y, ^, -q) and v = G{x, y, f, 77'), and apply to the region formed by excluding from R the points (^, rj) and (^', 17') by small circles. In the limit, when the radii of these circles approach zero, the equation G{1 V, r, V) = G{^', v', I v) is obtained. By a similar process an integral representation of the solution of the equation (2) Au=f(x,y) which vanishes on the boundary C is derived. The solution u is substituted in equation (1), and the Green's function 0(x, y, ^, 7?) is taken for v. Equation (1) is applied to the region formed by excluding from R the point (|, 7;) by a small circle. In the limit, when the radius of the circle approaches zero, the equation ^'(1, '') = ~ 27r J J ^^^' ^' ^' '^^■^^^' y)dxdy is obtained, or, since O is symmetrical, (3) u{x, y)=- 2^ J/^(^> y> ^> '?)/(?' '?)^?^'?- 196 MAX MASON. Conversely,* the function u defined by equation (3) vanishes on the boundary : It is continuous together with its first de- rivatives u ,u in the interior of iJ if/ is continuous, and it possesses second derivatives if the function / satisfies the follow- ing condition of so-called " regular " continuity ; there exist, for each point (x, y) of R, a pair of positive constants A, X in- dependent of Ax and Ai/, such that for all values of Ace are Ay sufficiently small, (4) I /(a; + ^x,y + Ay) - J{x, y)\^A \_{Axf + (Ay)^ \ If this condition is satisfied the second derivatives of u are con- tinuous and satisfy the equation (2). The following theorem is an immediate consequence : The solution of the equation (2) Aw=/, where / satisfies condition (4), which assumes given values on the boundary, is given by the equation (5) u{x, y) = ulx, y)--^J J J^i^> y> ^> 'l)fi^> v)d^dr], where ujx, y) is the solution of Am = which assumes the given boundary valves. § 8. Doubly periodic solutions of the equation Am =/.t The only doubly periodic solution f of the potential equation, Am = 0, is m = c, where c is a constant. For if u were such a solution, and v the conjugate potential to u, then u + iv would * The following statements regarding the first derivatives of u are proved in any treatise on potential theory. The statement regarding the second derivatives is generally proved under the assumption that the function /has continuous first derivatives. The statement in the form given here was proved hy Holdke {Dissertation, Tiibingen, 1882). t This section is a reprint, with minor changes, of an article by the writer in the Transactions of the American Mathematical Society, 6 (1905), p. 159. J By a doubly periodic solution is meant a doubly periodic function which is a solution of the equation in the period rectangle, and therefore in the entire plane. Such a function, in particular, has a value less than a fixed finite number for all values of x, y. THEORY OF BOUNDARY VALUE PROBLEMS. 197 be a complex analytic function which has a value under a fixed finite limit for all values of x, y. But, as is well known, such a function is necessarily a constant. It will be shown, how- ever, that the equation (0 ^u=f{x,y), where f{x, y) is periodic in x and in y with the periods a and 6 respectively, has periodic solutions if/ satisfies a certain inte- gral condition. A "doubly periodic Greek's function," O^ will be formed from known functions, and the desired solution of (1) found by quadrature from G and /. The function/ is supposed to be regularly continuous, that is, to satisfy condition (4) of the preceding section. Let 9t denote the real part of the term before which it is written, and consider the function where a-{z) is the Sigma function of Weierstrass, formed with the periods a, ib ; and ^, 7 are two points in the interior of the period rectangle fi bounded by the lines x = 0, y = a, x = 0, y = b. This function is a solution of the potential equation within fi, except at the points (^, 7?) and (a, /S), and has the form ^°^ A^-^r + {y^' ~ ^°^ V(a; - af + (y ^^ + ^' where 5' is a solution of the potential equation throughout fl. Since for any integers m, n, the function o- obeys the law a-{z + ma + inb) = (— 1 )"'"+'"+''e<'"'"+"''»'^^+™+")o-(2), where i]^, rj^ are certain complex constants, * we have ff(z + ma + inb - ?) a{z - ^) dt log —, — ; —T-r ( = Vt log —. ( (2) °'{^ + *"" + ***" — 7) ""(^^ — 7) - mm2v^{^ - 7) - nmvsii - j). Define a real function V by the equation *See e. g., Buekhakdt, ElUptische Fanctionen, p. 53. 198 MAX MASON. Since the last two terms are linear in x and y, V has the form V{<^, y, ^, V, «, /3) = log V, «, /3).* 27ie function G{x, y, I, 1?, a, ^) = F(x, 2/, I, 7?, a, /3) - S{a, /3, |, »?, a, ^) will be called the doubly periodic Green's function for the periods a, b. This function has the following characteristics : 1°. Except at (|, »?) and (a, /8), G is, within XI, a solution of the equation Aw = 0. * In the same manner may be formed a doubly periodic function r{x, y, f„ Vr, ■ ■ ; f„, nj = 2 c.log /(x- f,)^+ (j/-"*)' + S{x, y, fi, Vi, ■ ■ ; f„, >?„) with any number of logarithmic singularities, where 5" is a solution of Laplace's equation in S2 with respect to the variables x, y, provided that "1 + Cj H h c„ = 0. THEORY OF BOUNDAEY VALUE PROBLEMS. 199 2°. (9 has the form 1 O = loe; ^:= — log , + ^(a^j y, lj »?, «, /S)> where i? is a known function, which, with respect to the vari- ables X, y, is a solution of Laplace's equation within il, and which satisfies for all values of f, 77 in fl, the equation i?(«, ^, I V, «, /3) = 0. 3°. (t is doubly periodic in x, y with the periods a, h. The functions G and R obey the following laws of reciprocity : G{., y, i, V, a, ^) + log ^^^_^^.\^^_^^. = G(l, ., ., y, «, /3) R{^, y, l> v, «, /3) = -??(f J ■n, ^, y, «, ^)- To prove these laws apply Green's theorem, I I (?jAu — u^v)dxdy = J (^ ^ - " 5^;: ) '^^^ to the region fi' formed by excluding from O the circles c(f, rj), c(f , V), c((x, yS) of radius r about the points (|, 77), (f , »?'), (a, ^) respectively, and choose u = G^(a;, y, |, 7;, a, ^), v = Q{x, y, f , v, «, /3). Since m and v are solutions of the potential equation within D,', the double integral over Q,' is zero. Furthermore, since u and V are doubly periodic in x, y, each assumes equal values at op- posite points of the bounding lines of the rectangle XI, while at these points the normal derivative of each assumes values numerically equal but opposite in sign. Therefore the line in- tegral over the sides of the rectangle fl is zero, and we have. 200 MAX MASON. replacing ds by rdO, C ^ G{x, y, f , v', «, ^yde - f I G{x, y, f , r^, a, ^)rde + f -{ G{x, y, I, r,, a, /3)- G{x, y, ^', t,', a, ^)}rde+h=0, where lim h = 0. But G{x, y, ^, V, «, 0) - G{x, y, f , r,', a, ^) ^ ^""^ V{x-^y + {y-^' ~ ^""^ V{x-iJ^{y-vy + R(x, y, I, T), a, /3) - R{x, y, f , v', «, ^), and R{a, y8, 1, ,?, a, /3) = 0, R{a, ^, f, v, «, ^) = 0. "We obtain therefore in the limit r = 0, by well known methods, G{^, V, r, V, «, ^) - G^d', '?', I, V, «, ^) or writing a;, y for |', i/'j 1 V)- Furthermore, putting | = a, t; = ^ we have w(«, /3) = 0, S1I1C6 B{x, y, a, /3, a, /3) = i?(«, ^ x, y, a, ^) = 0. From the law of reciprocity for G it follows that 202 MAX MASON. <^{^, y,^ + ma, Tj + nb)- G{x, y, t, V, «, ^) and therefore m(^ + ma,v + nb) — m(^, 97) Therefore, if /(as, y)dxdy = 0, the function u possesses the periods a, b. We have therefore proved the theorems : The necessary and sufficient condition for the existence of a doubly penodic solution {^periods a, b) of the equation (1) Am =fix, y), where f is a regularly continuous doubly periodic function with the periods a, b, is that f satisfy the equation f{x, y)dxdy = 0. If this condition is satisfied, then the doubly periodic solution of (1), with periods a, b, which assumes the value C at x = a, y ^ 13 is uniquely determined, and is given by the formula : u{^, »?) = - 2^ J J Gi'^, y, ^, V, «, S)f{x, y)dxdy + C, where G is a knoum function, expressible in terms of Sigma functions. § 9. The equation Am = cm +/.* The coefficients c and / of the equation (1) Am = cm+/ *SCHWAEZ (1885), see his Gesammelte Abhandlungen 1, p. 241. PiCAKD, Journal de Math^matiques (ser. 4), 6 (1890), p. 145. The following presentation is based on the result of ^ 1 above. THEORY OF BOUNDARY VALUE PROBLEMS. 203 are supposed to be regularly continuous, i. e., to satisfy condi- tion (4) of § 7. Suppose a solution m exists, within the region R, which assumes the values g{s) on the boundary C of R. Let V be the solution of the equation Av=f which assumes the values g(s) on C. Then A(m — v) = cu, and the function u — v vanishes on the boundary. This func- tion may therefore be represented in the form or r (2) u{x, y) = v{x, y)-—jj^ G{x, y, ^, 7;)c(|, ■n)u{^, v)d^dr], where G is the Green's function of § 7. Conversely, a con- tinuous solution u of this functional equation takes on the boundary values g{s), since the integral vanishes on the bound- ary, and it satisfies the differential equation (1) provided that cu is regularly continuous. The function v has continuous second derivatives inside of R, and c is regularly continuous by assumption. Now the function fl. G{^> y, ^, ■n){^, v)dtdv, where + S'+ ■■■ is uniformly convergent in R. The operator S is therefore convergent, and there exists one and only one solution of the boundary value problem if the region B be taken sufficiently small in area. This theorem regarding the existence of a unique solution of the boundary value problem for the differential equation 206 MAX MASON. Am = cm + / holds true also for the most general equation of elliptic type, Aw == au + 6m. + cm + 6. The proof is however complicated by the presence of the terms in u^ and u^ and an investigation is necessitated regarding the behavior of the derivatives of a potential function in approach- ing the boundary of its region of definition.* § 10. The analytical character of solutions. It was Du Bois Reymond who characterized the three types of differential equations — elliptic, hyperbolic and parabolic, as " himmelweit verschieden." Their difference appears the more striking in view of later knowledge regarding the analytical character of their solutions.f To PicardJ is due the remark- able theorem that every solution of a differential equation of elliptic type whose coefficients are analytic is itself analytic. Even though the solution be required to assume non-analytic values on the boundary of a closed region it remains analytic in the interior of the region. In an important article, cited in § 4, Serge Bernstein has derived far-reaching results along these lines. He has shown that every differential equation of hyper- bolic type possesses non-analytic solutions. That the solutions of the dif- ferential equation of parabolic type ^.. = F{^, y> u> "J. *In PiCABD's first proof of the theorem [Journal de Math^ma- tiqnes, ser. 4, 6 (1890), p. 145] this investigation was omitted, and the proof was criticised on this account by DiNl [Acta Mathematica, 25 (1902), p. 185] who supplied a discussion of the point in question. In a later article [Journal de Math^matiques, ser. 5, 6 (1906), p. 129] Picaed revises his former proof, restricting the nature of the bounding curve to be analytic. A more recent discussion of the behavior of the derivatives of a potential function in approaching the boundary of its region of definition is given by Kellogg [Transactions of the American Mathematical Society, 9 (1908), p. 39]. t A function is analytic in a region if it may be expressed as a converging power series about every point of the region. t Journal de I'Eoole Polyteohnique, cab. 60 (1890), U 1-4. THEORY OP BOUNDARY VALUE PROBLEMS. 207 in which the first derivative of u with respect to y does not occur, are analytic with respect to x. Finally that the general non-linear equation F{x, y, u, u^, u^, u^^, u^^, uj = possesses only analytic solutions, provided that 8F OF 5m.^ du The coefficients of these equations are of course supposed to be analytic. §11. The boundary value problem for the equation y" + {x)y=f(x) by a change in the dependent variable. For simplicity this form will be used. In connection with the existence proof of § 2 it was remarked that the general solution of this equation has the form (2) y = Cj7?j + cjq^ + Vi I vjdo' + V2 I vjdx where e, and c^ are arbitrary constants, t;^ and rj^ are linearly independent solutions of the homogeneous equation (lo) f + 'i>{^)y = 0. and are connected by the relation (3) '?i'72 — ■^2'?i= 1- It is required to determine a solution of the equation (1) 2/" + <^(%=/(^) in the interval {a, b) which satisfies the boundary conditions (I) 2/(«) = A y{i>) = B. On substituting the general solution (2) in equations (I) the a = 208 MAX MASON. following equations for c^ and c^ are obtained : Cj»?j(a) + G^v^iia) = A~ 7)Ja) i -njdx, c.Viib) + c^Tj/i) = B- j?,(6) J »?j/rfa3. ^ unique solution of the boundary value problem therefore exists if the determinant Vi{a) -nla) is not zero. The equation 8=0 is the necessary and sufficient condition for the existence of a solution other than zero of the homogeneous problem (1„) y" + {x)y = 0, (I„) y{a) = 0, y{b) = 0. Suppose S = and choose i)^ as the solution of the homo- geneous problem. Then a solution of the non-homogeneous problem (1), (I) coexists with this solution when and only when ^^nip) = -^ — Viia) \ vjdx, c,vlb) = B. On account of equation (3) and the equations r)^{a)=0, T]^(b)=0 these equations may be given the form c^ = Ar][(a) — j njdx, c, = Bv[ib). The equation Av[{a) — Bv[{b) = I Vifdx is therefore the necessary and sufficient condition for the existence of a solution of the non-homogeneous problem (1), (I) when a solution 7]^ of the homogeneous problem (1,,), (I^) exists, L e,, when S = 0. THEOKY OF BOUNDARY VALUE PEOBLEMS. 209 If this condition is satisfied the solution of (]), (I) is not uniquely determined, but has the form y = y + cv^, where y is one such solution and c is an arbitrary constant. 1£ A = 0, B = the above condition becomes f %J a. ■njdx = 0. This case is of special importance for the developments of § 13. Similar theorems hold for the general equation y" + p{x)y' + q{x)y =f{x), and the boundary conditions a^yix^) + a^{x^ + a^y'{x^) + a,y'{x^) = A, • b^y{x;) + b^y{x^) + b^'{x^) + b^\x^) = B, provided that certain relations are satisfied by the coefficients a., 6..* § 12. The teanb verse vibrations of a coed. The small transverse vibrations of a stretched cord are to be determined as the solution u of the differential equation where t is the time, x the distance along the cord in its position of equilibrium, and A(x) a function depending on the nature of the cord and on the tension. Suppose the cord is fixed at its ends, cc = a, X = 6, is initially distorted so that u{x, 0) = f{x) is given, and is then allowed to vibrate freely from rest. Then the function u must satisfy the further conditions (2) u{x,Q)=f(:x), (3) ulx,0) = 0, (4) u{a, t) = 0, (5) u{b, t) = 0. In accordance with the usual method, assume that u is the *See an article by the writer in the Transactions of the American Mathematical Society, 7 (1906), p. 337. 210 MAX MASON. product of a function of x alone and a function of t alone, u{x, t) = y{x)T{t). Then the differential equation becomes After separating the variables by division each member of the equation must be a constant, say — X, and the equations T" + Xr = , 2/" + \A{x)y = result. On account of (3 the solution T=cos 0, 212 MAX MASON. and by assumption, X^ > 0. Furthermore, from equation (4) and the similar equation for the index h, the equations Ay.y^dx = J {y',:y. - y'^y^dx = J {y'kv'i — y'iy'k)dx = o are deduced, and therefore the equation (5) j^ Ay,y,dx = holds for unequal v^alues oft and k. The existence of a new normal function y^ will now be proved in the following manner : * Let X^ be defined as the lower limit of the values of the integral J — I y' dx formed for functions y which satisfy the following conditions: K) {a) ("^) y{a) = 0, y{b) = 0, £ Afdx = 1, (a,) [ I Ay^dx = (i = 1, 2, •• ■, n-1). Since the values of / are all positive such a lower limit must exist, even though it is not a priori evident that a function exists which satisfies condition (a) and gives to the integral J the value X . It will be shown that for X = X a solution of the boundary problem exists which is independent of the known solution y., i. e., that X^ is a normal parameter value, a root of S(X) = 0. It will also appear that this new normal function Vn gives the value X^ to the integral J and satisfies conditions (a), so that it is the solution of a minimum problem. The proof of the former statement will consist in showing that on * Based upon an article by H. Webke [Mathematisohe Annalen, 1 (1869), p. 1], on the partial differential equation Au-^XAu=^0, where reasoning similar to that of Dieichlet's principle is employed. THEORY OP BOUNDARY VALUE PROBLEMS. 213 assumption of the contrary a function may be found which satisfies conditions (a) and gives to /a value which is less than \, its least possible value consistent with these conditions. Let be an infinite series of approximating functions for the problem of giving / its least value consistent with conditions (a), i. e., the u. form a set of functions which satisfy conditions (a), and for which* (8) limit J{u.) = X^. Define a set of functions f. by the equations Multiply these equations by u^ and integrate from a to b. On account of conditions (a) the result is *J a It therefore follows from (8), that (10) limit I f.udx = 0. Suppose now that X^ is not a normal value distinct from Xj, Xj, ■ • •, X^_j. Then either it is not any normal value, or else it coincides with one of the known set. In the first case a solution of the equation 2/" + \Ay=f under the boundary condition (aj exists, whatever be/; in the second case such a solution exists, according to § 2, if/ satisfies the condition where h is the index of the normal value with which X^ coin- * It may be assumed without restriction that the functions ut have con- tinuous secoud derivatives. 214 MAX MASON. cides. Now on account of condition (aj) the equations (11) j AyM^dx = (i = l, 2, •••,«— 1) hold for all values of h. There therefore exists a solution of the equation (12) < + KAv, = Au^ under conditions (a^) for each value of h. It is these solutions which, when combined with the approximating functions m^ of the minimum problem, show the impossibility of the assump- tion that \^ is not a new normal parameter, value. From (12) and the equations y'^ + X; Jy, = it follows, on account of (11) and the boundary conditions, that {\ - \) I ^''^nvfi^ = I iy>H - \_.^ > \_^ ~> ■ • ■, decreasing vnthout limit. The function ?/„ satisfies the conditions * f PAyMx = ± I , r PAysydx = [ J{y) =J 2/'' [i=-±l, ±2, ■•■, ±(n-l)], and gives to the integral ' dx * The upper or lower signs are to be taken according as n is positive or negative. 218 MAX MASON. its least possible value consistent vnth these conditions and with equations (2). This minimum value of J is ± \^. § 14. The expansion of a function in teems of normal functions.* On the basis of the theorem stated at the end of the preced- ing section the following theorem will be proved : A function fix) which vanishes at a and b, is continuous in (a, 6) and has a derivative which is continuous except at a finite number of points, may be expanded in a uniformly and absolutely converging series of the form X ".y.' where %J a A lemma regarding the constants c^ will first be proved, namely, that the series of numbers are convergent.f The following equation is a result of the * Kecent articles of special importance on this subject are those of Hil- BERT, Orundziige einer allgemeinen Theorie der linearen Integralgleichungen, Naohrichten der K. Gesellschaft der Wissenschaften zn Got- tingen, 1904, 1905, 1906, and Kneseb, Mathematische Annalen, vol. 63 (1907), p. 477. These articles are based on the theory of integral equations, as developed by Hilbeet and Schmidt. Kneser removes the restrictions imposed by Hilbebt, that the function be continuous with its first and second derivatives, and satisfy the same boundary conditions as the normal functions. In all work on this subject the boundary conditions for the normal functions are special cases of equations (2), above, except in the important case of singularities of the differential equation. The function A is supposed positive in all previous work, except in the fifth memoir of Hil- bert'b series, in which, by considering a new class of integral equations, the restriction is removed. This restriction will not be made in the following treatment, which is a special case of the discussion given by the writer in the Transactions of the American Mathematical Society, vol. 8 (1907), p. 427. A very general expansion problem is treated by Biekhoff, ibid., vol. 9, (1908), p. 373. tCt. Schmidt, Mathematische Annalen, vol. 63 (1907), pp. 439, 440. THEORY OF BOUNDARY VALUE PROBLEMS. 219 properties of the functions y^ : / - Z o,y, )dx= \ fdx + ^ c^\ -2c,E fyldx+ X; I y'iv'^dx. {» + *) From the differential equation for y. and the boundary condi- tions it follows that fy!dx=- I /2/."dx = X. fAydx=\c., \) a. ^J a Xh nb /%b y[y'idx= - \ y,y"dx=\.j Ay^y.dx = 0. Since the first member of equation (3) is positive, it follows from this and the above equations, that frdx^tc^^x^. This holds for all values of n, and since the terms of the series forming the second member are all positive, this series is con- vergent. The convergence of the series with negative sub- scripts is proved in the same way. The convergence of the series of functions may now be easily shown. Consider the terms with positive subscripts, and write Then i=n i=n \ t/a I m / /*x \ 2 {n.j^^i:cH I y-dx), since a^ -|- /3'' = 2a/3 for any numbers a, /3. Furthermore, since the value of I (/* + y'ifdx = (x - a)^' + 2/i j y'idx + j yfdx is positive for all values of p., the discriminant is not negative, 220 MAX MASON. and therefore ( J y'i