? I LECTURES ON MATHEMATICS THE BOSTON COLLOQUIUM Lectures on Mathematics DELIVERED FROM SEPTEMBER 2 TO .">, 1903, BEFORE MEMBERS OF THE AMERICAN MATHEMATICAL SOCIETY IN CONNECTION WITH THE SUMMER MEETING HELD AT THE MASSACHUSETTS INSTITUTE OF TECHNOLOGY BOSTON, MASS. EDWARD BURR VAN VLECK HENRY SEELY WHITE FREDERICK SHENSTONE WOODS Xcto York PUBLISHED FOR THE AMERICAN MATHEMATICAL SOCIETY PA- TH E MAC MI ELAN COMPANY LONDON: MACMILLAN & CO., Ltd. x 95 Copyright, 190o Bv THE MACMILLAN COMPANY lAc5rt. Pa. TO PROFESSOR JOHN MONROE VAN YLECK, LL.D. THESE LECTURES ARE AFFECTIONATELY INSCRIBED BY HIS FORMER PUPILS HENRY SEELY WHITE EDWARD BURR VAN YLECK FREDERICK SIIENSTONE WOODS PREFACE. For a number of years the American Mathematical Society has held a Colloquium in connection with its Summer Meeting at in- tervals of two or three years. In the circular sent out prior to the first Colloquium, in 1896, the purpose and the plan of the under- taking were described as follows: 1 u The objects now attained by the Summer Meeting are two-fold : an opportunity is offered for presenting before discriminating and interested auditors the results of research in special fields, and personal acquaintance and mutual helpfulness are promoted among the members in attend- ance. 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 unprepared, the impression is of too short duration to produce accurate knowledge of cither the content or the method. . . . Positive and exact knowledge, scientific knowledge, is rarely increased in these short and stimulating sessions. "On the other hand, the courses of lectures in our best univer- sities, even with topics changing at intervals of a few weeks, do give exact knowledge and furnish a substantial basis for reading and investigation. . . . 1 Of. Bud. Am. Math. Soc, ser. 2, vol. ?> (1896), p. 49. vi ii PREFACE. "To extend the time of a lecture to two hours, and to multiply 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." As a forerunner of the Colloquia here outlined may be men- tioned the Evanston Colloquium of 1893, which followed the Congress of Mathematics held in connection with the AVorld's Fair in Chicago, Professor Klein, of Gottingen, being the sole speaker. But whereas that Colloquium covered, in a descriptive manner, a variety of topics, it comprised twelve lectures, the Colloquia of the Society have been characterized by close con- tact with the actual analytical development of the topic treated. The following Colloquia have been held : I. The Buffalo Colloquium, 1896. (a) Professor Ma xi: me Booh er, of Harvard University: "Lin- eal' 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.* (h) Professor James Pierpoxt, of Yale University: " Galois's Theory of Equations." This Colloquium was published in the Ainxtl.s of Muthc- mnfirx, sit. 2, vols. 1 and 2 ( 1900). ^ Two of these papers were : "Regular Points of Linear Differential Equa- tions of the Secutnl < )nler"; Harvard 1'niversity, 1S%; " Notes on Some Points in the Theory of Linear Differential Filiations," Annuls of Math., vol. 12, 189S. I'REFAUK. ix IT. The Cambridge Colloquium, 1898. (<7) 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 Foliations of Wave Propagation." III. The Ithaca Colloquium, 1901. (a) Professor Oscar Bolza, of the University of Chicago : " The Simplest Type of Problems in the Calculus of Variations." Published in amplified form under the title : Lecture* on the Calculus of Variations, Chicago, 1904. (6) Professor Erxest W. Brown, of Haverford College : " Mod- ern Methods of Treating Dynamical Problems, and in Par- ticular the Problem of Three Bodies." IV. The Boston Colloquium, 1903. (a) Professor Henry 8. White, of Northwestern University : three lectures on " Linear Systems of Curves on Algebraic Surfaces." (/;) Professor Frederick S. Woods, of the Massachusetts Institute of Technology : three lectures on "Forms of Non-Euclidean Space." (c) Professor Edwaud B. Van Vleck, of Wesleyan University; six lectures on " Selected Topics in the Theory of Divergent Series and Continued Fractions." This colloquium is here published in full. At Commencement, 1903, Professor John Monroe Van Vleck, M.A., LP. I)., completed his fiftieth year of service at Wesleyan University, and retired shortly after from the chair of Mathematics x PREFACE. and Astronomy. All three of the speakers at the Boston Collo- quium were former students of his, one of them being his son and colleague in the department of mathematics. It is fitting that this volume of lectures held at that Colloquium be inscribed to him. Thomas S. Fiske, William F. Osgood, Committee on Publication. CONTENTS. Preface Page. vii Linear Systems of Curves on Algebraic Surfaces By Henry S. White Cremona transformations and the geometry on a curve. Linear series of point sets ......... 1 Rational surfaces defined ........ 4 Linear systems of curves on any algebraic surface. ... 5 Enriques's theorem on the two definitions of linearity ... 7 Hyperelliptic plane curves, two kinds of linear systems. . . 12 Surfaces whose plane sections are hyperelliptic, Castelnuovo's theorem on their rationality . . . . . . .14 Picard's exact linear differentials on a surface; those of first kind exist only on singular surfaces ....... 18 Poincare's and Berry's special quartic surfaces . . . .25 Humbert's hyperelliptic surfaces of sixth order ... 27 Forms of Non-Euclidean Space By Frederick S. Woods 1. The First Two Hypotheses 2. Definitions. ...... 3. The Third Hypothesis .... 4. The Line-Element ..... 5. Geometry in a Restricted Portion of Space 6. The Fourth and Fifth Hypotheses . 7. The Extended Coordinate System 8. The Auxiliary Space 2 xi 32 34 37 . 39 45 51 52 57 xi i CONTENTS. 9. Forms of Space which allow Free Motion as a Whole . . 58 Spaces of Zero Curvature ....... 59 Spaces of Constant Negative Curvature . . . .59 Spaces of Constant Positive Curvature. . . . .60 10. Forms of Space which do not allow Free Motion as a Whole . 61 Spaces of Zero Curvature ....... 63 Spaces of Constant Positive Curvature. . . . .65 Clifford's Surface of Zero Curvature . . . . .69 Spaces of Constant Negative Curvature . . . .71 Selected Topics in the Theory of Divergent Series and of Continued Fractions By Edward B. Van Vleck Part I. Divergent Series. Page. Introduction .......... 75 Lecture 1. Asymptotic Convergence . . . . .77 " 2. The Application of Integrals to Divergent Series. 92 3. On the Determination of the Singularities of Func- tions defined by Power Series .... 107 " 4. On Series of Polynomials and of Rational Fractions. 120 Part II. Algebraic Continued Fractions. Lecture 5. Pade's Table of Approxiinants and its Applications. 134 ' (i. The Generalization of the Continued Fraction . 154 Bibliography 107 LINEAR SYSTEMS OF CURVES OX ALGEBRAIC SURFACES. By IIEXKY S. Will Ti:. Oiiaptki: 1. Transition /rout Plane Curves to Surfaeix. The notion of equivalence as formulated in projective geometry has simplified greatly the study of algebraic curves and surfaces, particularly those of low order. The next step toward a wider survey is the admission of all birational transformations of the plane, or of space of three or more dimensions. In the plane, the theory of Cremona transformations is no longer new, and the elements are familiar to all students of geometry. Not so, how- ever, in space of more than two dimensions ; probably for the reason that nothing is known analogous to the theorem that a plane Cremona transformation is resolvable into a succession of quadric transformations and collineations. And even in plane geometry the intricacies of the transformations themselves have kept most students from the matter of higher importance, the properties of figures that remain invariant under all transforma- tions of the group. Yet there does exist a body of doctrine under the accepted title of " Geometry on an algebraic curve/' and a fair beginning has been made upon a similar theory, the " Geometrv on an algebraic surface." * These titles are intended to cover * Consult, for nn outline of the geometrv upon an algebraic curve, Pascal's Rcpertoriinn 4). For the corresponding theories regard- ing surfaces, the best reference is to the comprehensive summary by Castel- nuovo and Enriques : "Sur quelqties recents resultats dans la theorie des surfaces algebriques," Math. Annalrn, vol. 4S (lSi)t>). Supplementary results are summarized in a later paper by the same authors: "Sopra aleune questioni fotidatnentali nella teoria delle superlieie algebriehe," Anna/i '2p '2. The totality of all sets of J) points corresidual to any one set is termed a group or sa'ie.i, and is denoted by a symbol gj]. Such a series i< called complete. If by any algebraic restrictions a series is separated out from it, of course that would be called incomplete or partial. For example, on a plane nodal cubic a series g- is cut out by all straight lines, incomplete because any three arbi- trary points of the curve are corresidual to any other three. Kverv scries v everv birational transformation, linear ( lr it hc employ iu> auxiliary curves except sueli as are adjoint to that con- taining 1 lie point sets. LIXKAIt SYSTEMS OF (TIIVKS. .'J svstems arc carried over into linear systems. A complete linear svstem is defined most easily by specifying the multiplicity that a curve of the svstem must have in each point of a fundamental set and hv prescribing the order of the curves. Thus (".' ,':';;) can in- dicate that in a 1 every curve is to have a multiple point of order at least *,, etc. If the base points alone, with their respective multiplicities, determine a system under consideration, that system is termed complete. If the base points actually impose, for curves of order m, fewer conditions than would be expected from their several multiplicities, the system is special ; otherwise it is reut the linearity of a svstem ' : Kxtioii:iI c;i of curves is still susceptible of precise definition, and that in two ways whose equivalence constitutes an important theorem. It' on any surface, rational or not, there exists a system of curves doubly infinite, such that two arbitrary points determine one and only one curve containing them, that may be termed a linear net upon the surface in question ; and Enriques proves that the cc 2 curves of such a system can be projectivcly related to the straight lines of a plane. If the series is oc :! , and if three arbi- trary points determine uniquely a curve of the system which shall contain them, then its curves are referable projectivcly to the planes of three-space, etc. Only simply infinite systems escape this far-reaching theorem, and thus give rise to a most interesting unsettled question, indicated by Castelnuovo.* Definitions of residual and corresidual curves upon a surface are those which any one could formulate at once from the use of these terms for sets of points upon a curve; their significance upon a twisted curve is the same as upon its plane projection. 80 of complete systems, both of curves and of surfaces, the latter admitting of course multiple curves as well as base points. For a surface of order ni, the adjoin ts invariantively related are of order m 4, containing as (s l)-fold curve every .s-fold curve of the given surface. If these first adjoint surfaces form a /.'-fold infinite linear system, the number k is an invariant of the surface and is termed its geometric deficiency (/;). Attempting to express this number in terms of the order m of the surface, the order it and deficiency ~ of its double curve (if any), and of the number / of triple points on this double curve, one would find a second number p H = i(,/i - 1) (j,i - 2) O - 3) - <1 (m 4) + 2t + - - 1, called the numerical deficiency of the surface. This number also is an invariant of the surface, as Xocther first proved, and may -"Castelnuovo : "Alcuni risultati sui sistemi lineari
  • 2, the corresponding theorem is not yet fully known. The proof of this theorem I shall give in full. 3. Fpon any algebraic surface f(.v, //, :., /) = a linear differ- ential of first kind is said to exist (Picard), if an expression in- volving four rational functions P v /'.,, P, P , of the coordinates : f [J\ (,', ,,, :;,<) iiperticie alu'ehriche," etc., Lined lirndiennti, January, lS'.t-l. ; "Sui >i-teiiii 1 ineari." etc., Math. Auualrn, vol. l(i ( lS'.lo), pp. 17!' l'. ( .. i For full information, see the second paper, cited ahove, of ( 'astolnuovo and Fnriijtics. I reirret that this [taper had not come to my notice hefore jfiYinjr these lecl Ul'e-. LINEAR SYSTEMS OF CURVES. < when taken upon the surface between any two arbitrary points. It' the surface /= is a cone, such differentials exist, for they are the ubelian differentials of first kind upon its plane sections. Pi curd proves* that if the surface /= have no multiple points or curves, then no such differential can exist upon it. There are however surfaces of all orders above the third which contain (or admit) one such integral ; others, from the sixth order upward, which admit two, and so on. These surfaces and the mode of dis- covering them and of defining them have been the occasion of some of the most interesting studies of Picard and Humbert. The elementary part of Pieard's first paper upon this topic \ shall give in some detail, indicating in conclusion certain points that might prove worth v of further study. Chapter 2. Linear Systems of Curves on an Algebraic Surface. The Two (Geometric Definitions are Concordant. I.v plane geometry a linear system of algebraic curves is defined analytically by an equation containing linearly and homogeneously two or more parameters ; as for example : \ + X^, + \sf> 2 + +\ K K = 0, the A's being parameters, and the cb's a set of polynomials homo- geneous of like degree in the current coordinates. This is called a /c-fold infinite (qc*) linear system. As we restrict our field to include only systems defined by fixed base points, the curves cf).= {) must be supposed all to contain the base points of the system. In a plane such a system may be studied by means of its equation, but for other surfaces one must either assume an analytic representation as definition, or else take such geometric features of a plane linear system as seem most important and transfer them to sets of curves on surfaces in general. \\ c follow * Picard et Si mart: Theorie des functions 1. For if / = 1, the generators of a ruled sur- face would fall under this definition, and one sees immediately the impropriety of calling them a linear system. Notice that a system linear under the first definition must also be linear under the second. For by relating curves to hyper- planes we relate the algebraic surface 1^ to a new surface F' in N, as explained in the preceding chapter; and through / points on A' there will pass one hyperplane, hence through / points in F there will pass one curve of the svstem and no more. The first definition therefore includes the second ; does the second include the first? We shall show that it does, so that the two definitions shall be proven equivalent for all cases except / = 1 , that is, for all except linear sheaves or pencils. The proof is essentially that of Enriques * as presented by Segre.f Two lemmas may well precede the theorem. Lemma 1. I'rojcefivifi/ of tiro flat spaces. Two flat spaces of n dimensions, s' and N', can be projectively related by assigning to anv // -f- - generic hvperplanes or S 's of" the first any n -f '2 generic >' 's of the second, one to one, as corresponding forms. The proof is bv mathematical induction ; to gain a clear idea of it, state it for points instead of hvperplanes, and model the transition from v and N' to N , and S' , niton von Staudt's I transition ' Km - 1 < ( i ; : " I n:i i|iicstmnt' su I In Imc:irit;i del sisU'ini in curve :i| >|:i rtont'ii 1 1 ;i.| hum snpcrlieif ulu'cl.ricii." Koine. Linn-i llcxdirmiti, July. lS'.i.'i. i Si'-rc: /,-.. '.'/. in Aimrtli ili Mnl, nntliru, >,-v. '_'. vmii StMiiil) : <;,. -- % LIXEAK SYSTEMS OF CUKVES. from >'., to >'.,. In >', take any 5 points A, B, (\ J), K, such that no four lit' in a plane, and in S', similarly A', B ' , C , D' , K' , like letters denoting corresponding points. In the plane C'])'K' or S' t call J' the point of intersection with the line A '/>', and in < 'J)E or >'., let /' be on the line All. As J' and P' must correspond, this gives 4 points in >'., corresponding to 4 fixed points of > '.',, and therefore by hypothesis fixes the projectivity between the two planes. The pencil of rays in N, through P corresponds projec- tively to that through P' in the other plane, >'.',. If now Q denotes any point of S to find its corresponding point Q' in N.' let Q be projected from .1 and from B into two points A y and 7^ of the plane >'.,. These are collinear with P, and we can find their corresponding points A[ and B' collinear with P' in S' and so, by using A' and B' as centers of projection, the point Q' desired. Points on the line AB itself have their corresponding j)oints fixed by the assignment of '.'> points A, B, P to the points A', B , P' respectively in the line A' B' . Lkmma 2. In an zc' 1 algebraic system of irreducible curves upon an algebraic surface, if the system is linear according to the second definition, then the points of the surface form sets of n (some finite number), such that if a curve of the system contains one point of any set it must necessarily contain also the other n 1 points of that -ei . The proof rests on the algebraic characters of the system. Call the system (( ') and any curve of the system ('.. Select any point A j of the surface. It does not determine a curve. Let C, and ( ' be any two irreducible curves through .1 . They intersect in n 1 other points .1.,, A v -, A , (n = \\ Since two of these ])oints, './/., A { and A 2 lie on two curves, they must lie on an infinity of curves ; /'. c, it will require at least one additional point to determine a single curve from among those that contain both A { and A.,. If P is a generic point not on all curves that contain J p then by hypothesis the two points .1. and P determine one curve, which shall be denoted by ( [.. Also among the curves that con- tain A { and A.., at least one will contain the additional point /'. 10 THE BOSTON COLLOQUIUM. This can he none oilier than '.,, hence the curve which is deter- mined by J, and the generic point /'will contain also .!. By parity of reasoning it must contain as well A n , , A . But as P was any point, C, was any curve through .1^ consequently every curve of (C) that contains an arbitrary point A x must contain also n 1 other determinate points, as asserted by the lemma. The principal theorem can now he proven if two facts are estab- lished. First the theorem should be found to subsist for the particular case / = 2, so that the base may be provided for a mathematical induction. Then, secondly, the mode of induction employed in Lemma 1 must be shown to be applicable to a sys- tem of curves conforming to the second definition. Particular Thkouem. A doubly Infinite algebraic syst cm of irreducible algebraic curves upon any algebraic .surf act can fie brought into a one-to-one relation vith the system of all fines in a plane by assigning to four arbitrarily chosen curves of the system (no three through one point), four arbitrarily chosen lines of the plant (no three through one point), as corresponding lines, and by rcouiring further that to curves /taring a common point shall cor- respond lines irith a point in common. To prove this, associate every set of m points, such as the A r .1,, , A , of Lemma '2, together as one element A. Then there is upon the surface an oc~ system of O's and a second system of .l's related thus: Two generic C's determine one .1 and two A'> determine one ( '. Now these are precisely the incidence rela- tion- upon which depends the familiar proof that four line- of one plane and four of another determine a projeetivity of the two sys- tem- of line-; here the line- and points of the one plane are replaced bv element- (' and .1. The requisite of continuity is provided for hv the hypothesis that the system is of algebraic character. Therefore the line- of a plane and the curves of the -\ -tern [(') -land in a one to one relation, as asserted by the theorem. This relation is called projective, meaning that it is independent of the particular four pairs, line and curve, that may he selected to determine the correspondence. ()therwi-e -tated : LINEAR SYSTEMS OF CURVES. 11 Ii'tlie lines of two planes are related in the mode above described to the curves of a system, the planes are thereby protectively related to each other. As to the second matter, it is needful to show that the elements used as auxiliaries in Lemma 1 have unique analogues in a sys- tem, triply infinite, of curves conforming to the second definition. What were called points there have become curves here, hence the lines and planes must be replaced by oc 1 and oc 2 systems of curves. We need only examine, accordingly, whether the postulate : a line and a plane intersect in one point, retains its validity. Let a "line" be given by two curves, a "plane" by three ; or to adhere more closely to the definition, consider an S 1 given by two points, a and b, and an S', consisting of all the curves of the oc' system & that pass through a third point c. Then will >', and N have in common one and only one curve. For in the S there is an S' containing the point a ; in this S' 2 there is one curve C that contains the points b and c (and by the explanations of the above theorem we see that it must contain all the intersections of any two curves fixing the >S',). As containing a and b it lies in S, ; as containing; c it lies in S, and as contain- ing these three arbitrary points it is by the definition unique. Therefore, all the constructions of Lemma 1 have their unique analogues in the system #,. We conclude that the transition from an oc 2 system to one oc 3 is possible, and that for / = .*> the first and second definitions are equivalent. Mutatis mutandis, the induction from r=m to / = /// 4- 1 can be made by similar means. Recapitulating we have therefore the theorem : An x r algebraic system <>J irreducible algebraic curves upon any algebraic surface is linear if either (/) its elements ran be put in a one-to-one correspondence, projectively, with the liypcrphuics of an r-fold space ; or {..') if through r generic points of the surface there passes one and only one curve of the. system. For r ]> 1 these two defining properties can be inferred, each from the other. 1- THE BOSTON COLLOQUIUM. Chapter 3. Surface* whose Plane Sections are Hyperelliptic Curves. Plank curves of any deficiency above 1 may be hyperelliptic, and those of deficiency "2 are necessarily so. The specific feature of an hyperelliptic plane curve of order // is this, that its adjoint curves of order n 3, its " -curves," arrange its points in pairs. That is, if a 0-curve contains any one point J* of a hyperelliptic curve C, it will of necessity contain a second determinate point Q of C ; then P and Q form what is called a conjugate pair ; each is the conjur/ate point of the other. It is well known that a (f>- eurve can be found which shall contain p 1 arbitrary points of ( ', where p denotes the deficiency of the curve C. These facts lead to interesting conclusions about any linear system of hyper- elliptic curves in a plane, or in any rational surface. In a plane, a linear system of hyperelliptic curves may be of the first or second kind. In a system of the first kind, a curve passing through any one point is not thereby necessitated to pass through a determinate second point ; in a system of the second kind this compulsion does exist, and all curves of the system that contain a point P contain also Q } its conjugate point. Of the second kind, for example, is a certain family of plane sextics hav- ing double points in seven common points of three cubies : 0=0, cf) : (ij (p = (). The equation E^,,tt = (',*=!, 2,3) gives a linear system of sextics, the (" 7 . being arbitrary. Outside of the seven base points, let anv point I* be on both cubies : (/), = and (p., = 0. Their ninth intersection, O, is determined by the eighth, a familiar theorem ; and sextics of the svstem which pass through /', being given liv the equation (according to Xoether's theorem) nnH contain also the remaining intersection O of y , (f>. (f>, are adjoint <'s of all sextics of the system, so that Q is the conjugate point of i J on every sextic that contains them. We mention systems of this second kind, only in order to ex- elude them from further discussion here. Let (//) be an x; 3 system of the first kind of hyperelliptic plane curves //,, Jf,, etc., of order //, and let ((/>) he the system of adjoint curves of order n 'A, i. c, let the curves < p (f>. have as (/ IV fold points the /-fold base points of the system {II). Consider any point P of the plane. Its conjugate Q on any curve If of the system must lie, by defini- tion, upon every <-curve containing P. Since Q is a variable point, its locus must needs form a part of every <- curve through P, and these (^-curves accordingly must be degenerate. By parity of argument every 0-curve must consist of (p 1) distinct j arts where j> is the common deficiency of curve Jf, and each part must intersect every curve // in only two points, a conjugate pair, out- side the multiple base points of the system (//). For an example of this, let the system (//) consist of all curves of order n having in a fixed point a multiple point of order n 2. Any 6-curve must have in an {n 3)-fokl point, and is itself of order n '>, therefore it will consist of n .'! right lines through O. Kvery constituent right line has with any curve II n 2 intersections in (). and two outside that point ; the latter two are conjugate points on the curve, which is consequently hyperelliptic. Another ex- ample, with the 's compounded of conies, is the system of curves of order 2m ~\- *> with four fixed multiple points of order m + 1. The fact that for these plane systems the points conjugate to a given point fill out a definite locus is the thing to which we shall wish to recur. In space of three dimensions, let a surface /-'have all its plane sections hyperelliptic curves {( ') of deficiency />. Can these be represented by a system of curves all in one plane? 1- the sur- face F rational, I. e., transformable into a plane, point-for-point, rationally? This question again may be approached !>v the aid of conjugate pairs of points. \\ e should expect of course that 14 THE BOSTON COLLOQUIUM. analogues of (^-curves would be, in space, ^-surfaces, and that those that pass through a point P on any curve of the system would contain its conjugate Q ; and further, that all points Q conjugate to /' would lie on some determinate curve of the sur- face. This last supposition can he established by reductio ad ab.surdum. The surface F and its plane sections (C) ai"e algebraic. Each curve C containing a selected point J 1 has, by hypothesis, one par- ticular point Q conjugate to J\ Therefore the Q'a on the oo 2 curves through P must suit one of the following three descriptions. (//) They may be finite in number, Q v Q 2 , Q . But then every plane section of the surface through P would need to con- tain the line PQ V or FQ. , or PQ,.. This is absurd. (//) The Q's may be simply infinite, oo 1 in number, filling one or more algebraic curves on the surface, or lastly (c) The Q's may fill all the oc 2 points of the surface F. We shall reject this after showing that in this case the surface must be rational ; /. c, rationally and reversibly transformable into a plane, whereas on the contrary, in a plane or any other rational surface the ^>'s must be only a simple infinity, oo'. Suppose, therefore, that every point Q of the surface is conju- gate to a given /' upon some one or more curves of the system. It cannot be so upon all plane sections through the secant PQ, for then must every point of any plane section be conjugate to P on that section, contrary to the hyperelliptic hypothesis. Through everv rav PQ there lie then a finite number / of planes in which /' and Q arc conjugate. Any one of these determines all the others, for P and the plane through P fix Q, and the rest follow. Now such :i grouping of the planes through P into sets of / planes, each set being determined by any one of its planes, is called an involu- tion. ( 'astelnuovo * proves the remarkable theorem, that cvcri/ involution o\ lln jiliims (thniit (i /nun/ in span nj t/o'CC (linn iixiniix is riilimi'i/ ; i. i. its groups can be correlated univocallv and rcversiblv to the points of a plane, each group to one point, Mth. .{,,/,. v.. I. II ( is'.i) i, |,,.. rj.v I. v>. LINEAR SYSTEMS OF < TKVES. 1-J Thus through the involution every point Q of the surface can he related to -nme one point of an auxiliary plane, and rice versa. But if the surface /' he transformed algebraically and univocally into a plane, then its plane sections will he transformed into a linear system of hyperelliptie curve- in that plane, conjugate points going into conjugate points : whereas we have seen that in a plane the conjugates Q of /' do not (ill the whole plane, but only an zc ' locus. Supposition (<) is thus dismissed, and (//) alone is tenable. We have then as a starting point this fact, that for a generic point J* of the surface there is a definite curve p containing all its conjugates Q on the curves of the system (/'); and this curve p can meet each curve (each plane) only once outside the point P itself. If then /> is of order s, it must have in P an ( l)-fold point. It must also be a plane curve ; for a plane can be passed through P and two arbitrary points of p, and will contain v 1 4- 1 -f- 1 = .v -j- 1 points of a curve of order s, hence must contain the entire curve //. Ihis curve p can he shown to he either a atraif/ht line or a conic. If jj is not a line or a conic, its order .< must be at least >, whence it must have in Pa double point (2 -=.s 1) or multi- plicity of higher order. As p is a plane curve, this means that its plane is tangent in P to the surface V ; and so that every line joining P to a conjugate Q is a tangent in P to the surface, and by symmetry of the relation between P and Q, tangent also to F in Q. This is not possible unless either the curve p is a curve of plane contact (so that P would be an exceptional point of F), or else the curve p consists wholly of straight lines through P. This alternative is equally impossible, as no ruled surface has through every point three or more generators. Therefore the hypothesis s='.\ leads to absurdity; and we have to examine the two possi- ble eases : .s 1 and s = 2. ,v = I. If j, is a straight line, it does not contain P, since .s 1 = 0. To /' is associated one generator p of the ruled sur- face F, and conversely, to every point Q of p must be associated 1<) THE BOSTON COLLOQUIUM. the generator y passing through P. F is flint a ru/ed surface, of hyperelliptic section, with its generators arranged in pairs cutting conjugate points in every plane. s = 2. If j) is a conic, three cases can be distinguished. First, to every point Q on p may belong a conic 7 containing P but different from p, and these may be in themselves complete plane sections of the surface, It' this were so, the surface would be a quadric. But the conies may not be complete plane sections of the surface, and this possibility it is convenient to divide into two parts, as follows : Secondly, the r/s may be conies distinct from the p's. The surface F will contain in this second case a doubly in fin iff system of decomposable or reducible plane sections. Or thirdly (the only case not trivial), the conic q, while its corre- sponding point Q describes the conic p, may continually coincide with p. There is then only a simple infinity of conies (/>) upon the surface. To show that this system is a rational sheaf, con- sider its section by an arbitrary plane : on the hyperelliptic section curve each conic p cuts two conjugate points P and Q, and either F or Q determines p completely, hence the system ( p) is in one- to-one relation with the series of pairs of conjugate points upon a hyperelliptic plane curve a linear series, and must therefore be rational.* Xow these three alternatives lead to a single conclu- sion, through the application of well known theorems. First if the surface F were quadric, it would be rational ; but then it would be discussed as a surface with all its plane sections rational, i" For the second case we adduce Ivronecker's theorem \ Indeed these |ilancs form the developable, of a twisted cubic curve, since no Miir of them counts twice : ('asU'lnuovo shows that the immediate generalization if tli is remark hold- for hyperspaee. See (taper hv K. I'icard : "Stir les surfaces aljjebriques dont toutes les sec- tions planes sont unieiirsales," < 'rrllr.' .-, Jtniriuit, vol. 100 (18So); and a correlated paper of K. II. Moure: " A l<;ebrai e surfaces of which every plane section is uni- enrsal in the li-ht of //-dimensional geometry,'' Amcr, .Imir. /' Mulh., vol. In i Isss i, p. 17. ; See tic historical note and deinoiist ration hy ( 'asteliniovo. "Sidle superficie altrehriehe idle ammettono tin siMema doppiametite intinito di sezioni piane riduttiliili," /./'///./' II, inlicnnti, Januarv, lSii-1. LINEAR SYSTEMS OF CURVES. 17 that a Mir face haein-curves and g-curves discussed above are component parts of the degenerate ^-curves, and a p coincides with all its 7's. This highly general theorem allows us to study upon plane svstems the geometry upon an extensive family of surfaces in space and in hyperspace as well, since the existence of a triply infinite linear system of hyperelliptic curves in a surface is equivalent to the hypothesis that we have used concerning plane sections in ordi- nary space. And for linear systems of the first kind in a plane reduced types have been found by Castelnuovo, t from which all others are derived by Cremona transformations. It remains to develop to the same extent the theory of svstems of the second kind. This would demand an acquaintance with the work of Bertini on plane involutions of index 2, and of ( 'lebsch and Xoether on rational double planes. An extension in another direction has been given bv (Jastel- t M. Xoether : " Ceber Fliichen, welehe Schaaren rationaler Cnrven besitzen," Mii'n. Anii'tlrn, vol. :5 (1S71). pp. 17. '5-4. The theorem is more general than that here cited. * " Stille superticie algebriehe Ic eui sezioni piane sono curve iperellittiehe." P(ilt'r,> llea/licotiti, vol. 1 (1S90), pp. 73-88. 9 18 THE BOSTON COLLOQUIUM. nuovo,* who discussed under only one specializing restriction the surfaces whose plane sections are of deficiency 3. These are of four kinds, so far as numerated, and not all rational. t In looking for other possible extensions, it should be remembered that there are other classes of highly specialized curves, differing from the hvperelliptic in the degree of the singular series of special groups which occurs upon them. Of such classes, individual curves have received some study, but linear systems little or none. Chapter 4. Linear Exact Differentials of the First Kind on an Algebraic Surface. 1 . The Existence of Integrals on (liven Surface.*. When the theory of integrals upon algebraic curves was ex- tended to surfaces, the first step was the discussion of double integrals. These have been described already (Chapter 1), and attention has been called to two important numbers, characteristic of a surface, to which they give rise, the geometrical and the numer- ical deficiency. Every surface above the lowest orders possesses double integrals of the first kind, everywhere finite, unless its singularities have become too numerous. The increase of singular point- and lines causes a diminution of the geometrical deficiency, p. Double integrals and their classification were introduced by Clebseh and Xoether about 1870. Fifteen years later a different and even more interesting extension of curve theory to surfaces was made by Picard.t The new integrals that he introduced are simple integrals whose path of integration is restricted to ' "Sulle superficie :i ] !_ r '*! ' ii cli t le eili -e/.ioni sono ciin'e di ye ne re ">. " Torino Alii, vol. :T> i ls'.ioj. i It' [lie surface i- of order a hove t lit' fourth, with plane sections nil of deficiency :', it i- rational. See ( asteliniovo and Knrhjnes "Sopni alcnne ipiestioni fon- damentali nella tcoria delle superficie alyehriche," Aitnali
  • S.->). LINEAR SYSTEMS OF CURVES. 19 lie in the surface, while the integrals are further required to be functions of their limit points alone, not of the particular path of integration. The number of linearly independent, everywhere finite, integrals of this kind is a new invariant characteristic of the surface ; and it is found that this number is zero when the surface is non-singular, but increases (according to a law not pre- cisely known) with the multiplication of singularities. This is the theory of which I propose now to give a sketch, following very closely Picard's article cited above.* Upon an algebraic surface f = f(x,y, 2 ) = a linear differential expression in dx, dy, dz can be reduced by the use of the relation : (1) f',dz+f'dy+f',dz-0. By this means any expression of the form Ad. i- + Bdy + Cdz may take on either one of the three aspects : /; dlJ + /; ,- dx, j ij J 1/ dx + ,, dy, Let the first be chosen, and abbreviate it to Qdx - Pdy f] Concerning this expression two things are to be noted. First, if the surface be cut by an arbitrary plane, then by the adjunction of the equation of that plane this must become an abelian differen- tial of the first kind upon the plane curve of section. Secondly, * For details, see also the hook of I'ieanl and Siniart. 2<> THE BOSTON COLLOQUIUM. either x or // could have been taken as dependent variable instead of :. From the first observation it follows that both P and Q must be entire functions of x, ;/, .:, of order m 2 when ui denotes the order of the surface f= 0. From the second, converting the differential into its two equiv- alent forms : -(1x4- id'. ., dti (J-flz we find further that the fractional form ./; must reduce to an integral form upon the surface, /. c, by virtue of the equation of the surface. Let 11 denote this integral form, and A" a suitable polynomial of order m '>, so that we shall have identically : (2) /'/;+ q/; + /.y; + .v-/=o. This gives us for equivalent differential expressions on the sur- face : Q ,/., _ P ,],, 11,1,1 _ Q ,/ 3 /> <>!/\f'J ' ) CX C l Cz \ t I (./,) where I, denotes some integral function of x, //, z, of order 2/// (>. Expanding in part the third term, we distinguish terms which on their face must contain a factor /(.r, y, z) : (DP CQ cR \ v fC X L\ V ex cy cz J V~ ./ ./ 7 = 0. Since the first group of four terms are integral, and of order lower than m, they cannot contain the factor /(.r, //, ~) otherwise than by vanishing identically. Thus we must have for all values of x, y, z the identity () ay j c(> on 1 + J + ~ + A =0. insert again the equivalent of A'/ from (2) : (dP cQ cR The form of this identity invites us to write / homogeneously in (x, //, .:, /), and of course the other functions also, and to employ Euler's identity [f=- r /' +!//', + '/'- + f f' so that equation (7) becomes: (>P + r\)j" j + (m -f //A')/ - ; -f (ml! + zX)f[ + tX-f] = 0. In this it will appear more simple to write , /> + ,r\ = t0 m Q + i/X= td., iii R + -A' = /#. (8) ' A' = ( . To show that 6 0.,, are integral, recall that 22 THE BOSTON COLLOQUIUM. is a total differential nowhere infinite, and that in the plane / it is an abelian integral of the first kind, and so must have the form i/rfx xdi/ (f>(x,y,z)-- ,, ', where cfi is of order m ">. For this reason we must have iden- tical lv : P = x + / Q = y + t 2 , II = # + / 9a , where is homogeneous of order m 8 in x, y, z, and v cf> 2 , 9.., are of order m 3 in x, y, z, and t. Therefore (dP 30 BE V + ty,) + .r(-- i< + ty 4 ) 0, = , = "'9, + -''9, 6, = m 2 + //(/> 0., = ///(/>,, + .:.,, # 4 = <9 -f /<,. Effecting the substitutions (X) in conditions ()^ --/,., ex cy cz ct we have the two relations which the #'s must satisfy : f0, CO, CO., CO, (!) , ' + X- + , " + V s () ' fit.) ,? + "..? + .? +,:-. These conditions are now symmetrical in the four homogeneous variables, and by the aid of four parameters '-,, <.,, <.. <- 4 we can LINEAR SYSTEMS OF CURVES. 23 bring the integral (3) into the symmetrical form used by Berry.* <*1 2 <':>, ( \ 0, (H) -/, 0, e :i *, // .: f hi r/r (It dx This integral is subject of course to the further restriction of re- maining finite in all singular points or lines of the surface. Prima facie, the presence of singular points or lines seems a restriction upon the number of linearly independent sets of polynomials 6. But the fact is, that if the surface have no singularities, it can have no such integrals. For by the identity (10) the (m l) 3 intersections of three first polar surfaces : /; = o, /; = ), /: = o must fall, either on the surface/^ = (>, thus indicating a multiple point of the fundamental surface ; or if not one fall on this fourth polar, then all must needs lie on the surface 6 , = 0. This last, however, is of order m 3, and its equation cannot be a linear combination of three equations of order //( 1. And the conclu- sion cannot be escaped by supposing the polars : f = 0, /' = 0, J[ = to have a curve in common, since then that curve must pierce in a number of points (or else lie wholly in) the fundamen- tal surface. Hence the surface /= must have at least multiple points, as was to be proven. Whether a surface whose equation is given does or does not possess linear differentials of the first kind, and how many linearly independent, this can be determined by first finding the mul- tiple curves and points, and then counting the conditions imposed by them and by the identities (6) and (7), or (9) and (10). Remark. The number and nature of singularities that a sur- face of given order must possess in space of three dimensions in * Twin*. Cambridge Phil. Soc, vol. IS (1900), p. 333-4. '2A T1IK BOSTON COLLOQUIUM. order to admit one, two, , independent exact differentials of this first kind might prove an accessible and profitable subject for fur- ther inquiry. To extend this inquiry to surfaces in hyperspace would require a systematic preliminary study of curves and sur- face- in such a space not yet completed. 2. The Kxixtence of Surface* of Given Character, in particular ITjl per elliptic Surface*. I f a surface in threefold space,/*), possesses exactly two linearlv independent exact differentials of the first kind Qtte - Vth, . . QJx - l\ shows, (/. c) that if .r, //, z are given as fourfold periodic functions of two independent variables, the locus of a point (./, //, .:) is a surface of this sort. For a simplest illus- tration let the functions reduce to elliptic, and in the Weierstrass notation set ' = K") .'/=/>('), '- = !>'(") + ,>'('). This gives an equation between .r, //, and : : * = i !''' .'//' Us + i i// :; //,'/ !/: or |( ir brevit v -. = i 7.'(.'-) + i /'(// s LIXEAK SYSTEMS OF CUUYES. - : > Accordingly tlie surface is of the sixth order : 4 /,'(>) R(>,) - [r - /,(.r) - />>(,,)] 2 = /(,, , y, z) = 0. The two integrals on the surface, '/ and r, arc represented as follows : r <'' f~ 2 - R (- v )+ l *(.v) i " = \ = I -' ' dx, C dy fr + H{x)- //(//) As to double integrals, the one of the first kind belonging to the surface degenerates into / dit-dc, which is evidently finite. The double lines of the sextic surface may be perceived imme- diately, one of them being obviously the straight line x = //, - = 0; another a conic in the (x, if) plane; and three lines at infinity. One linear differential of the first kind can exist on a surface of order as low as the fourth. There are five types of such quartic surfaces, found by Poincare,* Berry t and de Franchis ; the five types are projectively distinct, that is, collineations cannot trans- form one into another; but Berry has found that under birational transformations all five are equivalent to a cubic cone devoid of double line. Of these five types, perhaps the easiest of derivation is the fol- lowing. The quantities 0, being of order //) 3, are linear. Let their planes coincide with those of the tetrahedron of reference, viz.: e l = .r, e 2 = ih e :i =-~, o^-f, thus satisfying the condition (!). It remains to satisfy (10), ex dij ' cz ti *('mnpU>* llemhix, vol. 99 (Dec. 29, 1884). t Ibi'lem, Sept. '_', 1899. See also his papers, cited above, in the Trans. C'mn- brhlye Phil. Sor. 26 THE BOSTON COLLOQUIUM. Since also by Kuler's identity for homogeneous functions df Of of of x : + y :' + z 1 + ti, a if, ox oy oz of ; it follows that Of Of df df x ; + ?/ ; =s; + r^ = 2/ t.r c// c,~ (7 Hence the form is homogeneous of order 2 in x and ?/, also in s and /. Symbolically / = (V/ r r + 2< W + ; /)(^r 2 + 2M + 6/), each product a.b k denoting an arbitrary real quantity. This is a familiar ruled quartic surface with two double lines (x = y =0 and z = t = 0). It is generated by taking for directrices these two double lines and any plane quartic which has nodes upon the two lines. This suggests the interpretation of conditions (9) and (10) by a complex of lines. The connex u l l + u 2 2 + fr 3 3 + u i i ~ 0=0 gives rise to a complex when every point (.<) is joined to its cor- responding point, and condition (10) is the requirement that the complex line originating in a point (x) of the surface /= 0, shall be tangent to that surface. Speaking of the line joining a point (.?) to its corresponding point in the connex : = as a trajectory of that connex, we say : A surface f = of order in will possess a linear exact differential of the first kind if a coinj)Iex (m ', 1) exists . siic/i that the trajectories oj all points on the snrfu'i a re ta iKjent to tin surface. Hi urn rl:. When one linear exact differential exists on the sur- face, and only onc.it is invariantively related to the surface under ;i much larger group than that of the eollineations and a fortiori under the latter group. Instead of seeking the integral when the surface is given, and finding it as an irrational covariant of the surface, one might attempt to determine the surface as a rational LINEAR SYSTEMS OF C.TRVES. 27 eovariant of the forms V 0.,, V V occurring in the integral. But the surface may be not determinate. (hi the above example it had still 8 free parameters.) Also the 0's depend on the choice of planes of reference. Hence more precisely one should seek to determine the mixed form f(x, u) (connex) eovariant with the connex = u 1 l + ufi 2 4 /, + "A such that every set of values (u) makes the surface f(x, u) = a surface possessing the integral of the first kind represented in (11). In other words the connex is to satisfy the relation 3 2 3-0 d 2 6 c-6 -, -, + -, 1 - 4- -, -, 4- ^ ~r = > ou 1 ox 1 cu.,cx cn^cx^ cu 4 cx i while the eovariant /(.r, u), or/, is to be of order in the (.r) higher by 3 than 0, and shall satisfy also identically the equation : 30 3f 30 3f 30 Of 30 3f . : + 1 : 4 . / 4- , / a o. an ox at., ex,, ou ex ou i cx 4 Of course the chief interest in this problem would be found in the lower orders, 4, 5, . If a surface have two such differentials, its plane sections are curves of deficiency at least p = 2, and its geometrical deficiency is I) i^ 1. Picard establishes directly the existence of a class of surfaces with two differentials, in brief as follows : Let the Cartesian coordinates of a point be given as uniform functions, quadruply periodic, of two independent variables. Let the relation lie such that to every point (x, //, ~) of the surface there corresponds one and only one pair of values of the two independent variables z> THE BOSTON COLLOQUIUM. //, r. Then the surface has exactly two linear differentials of the first kind.* For if the surface equation /(>', V, ~) = is satisfied identically by three uniform functions ' = J \(", '"), y = ^,(", f), - = F 3 (u, ') and the functions F v F,, F., have four simultaneous systems of periods, then since dF t BF t d.r = ' (In -f- ' f fr Oil cv cF r/-:, at/ = _, " (J a -f- , " .. ,iA i () , ,/,- - /', th \ l c// 'c// y \ ' " ( / ( ;/ = (/)(.', //, :)'/'/ -f \f/{.<\ ii, :)'/r, /. .m rill,, >T. 1. vol. I I ISS.-1 . LINEAR SYSTEMS ( >F CURVES. 29 then as the functions cf) and -v^ arc assumed to remain finite throughout the surface /= 0, and are seen by the foregoing to be rational in x, //, z, they can be nothing but constants, as was to be proven. The double integral of first kind on the surface is J f (he dv; the proof that it is unique is closely similar to the above. Functions of the properties required for /-',, F F,, are readily expressed by quotients of theta-funetions of two variables. Surfaces of this sort are called by Picard and Humbert hi/per- elfiptic .surface*. They are to be distinguished carefully from sur- faces whose plane sections are hyperelliptic, or which have a linear net of hyperelliptic curves upon them, for those we have seen to be rational (y> = **j; while these, possessing one double integral everywhere finite, have p = 1. Hyperelliptic surfaces of order lower than the sixth do not exist, as was said. This evokes recollections of Kummer's sur- face of the fourth order; but that, as Picard shows, is not of this class, because it has tiro .sets of values (u, r) for every point (x, tj, z). lFunibci't has discussed hyperelliptic surfaces in extenso,* in par- ticular those of sixth order. He extends this mode of establishing their existence by theta-formuke, so as to employ the next higher class of thetas, those in three independent variables. In this way he reaches surfaces containing three linearly independent exact linear differentials of the first kind and proves that their order must be higher than six. An example is given of the eighth order, but the order seven is left in doubt. Of such representation of these sur- faces, the chief advantage is that every algebraic curve lying in the surface is given by the vanishing of some theta function, s< > that by the use of theorems more or less familiar in the theory of thetas, one obtains an exhaustive treatment of geo'metrv upon a surface. It is apparent that this line of investigation opens a prospect of a classification of surfaces based on properties much more gen- eral than those merely projective. As was indicated in a remark upon quartics, this calls for the projective study (for the sake of * I/nmrilli sit. I, vol. ") (lS.S'.t), vol. ( J (1K93), and ser. ">, vol. 2 llS'Hi). '<> THE BOSTON" COLLOQUIUM. models) of surfaces which become interesting under this more searching light. And the special classes as those related to point-pairs on one curve, on two curves, those in which the periods of the arguments fall into some integral relation, etc. those offer a field most inviting: and likelv to vield rich fullness of even the simpler geometric forms. FORMS OF XOX-EUCLIDEAX SPACE. By FREDERICK S. WOODS. By a non-euclidean geometry we shall mean any system of geometry which, while differing in essential particulars from that of Euclid, is nevertheless in accord with the facts of experience within the limits of the errors of observation. The space in which such a geometry is valid is a non-euclidean space. It is clear that the test of experience can be applied only within a restricted por- tion of space, so that non-euclidean spaces, while having essen- tially the same properties in such a restricted region, may differ widely when considered in their entirety. It is the purpose of the present lectures to present especially those non-enclidean spaces, investigated by Clifford, Klein and Killing, which have been named by the last author the Clijfbrd-Kh'hi'sehe Itdumfonnen.* For the sake of clearness it is necessary to begin with the geometry of a restricted portion of space. Here the author has followed the development of his own article on " Space of Con- stant Curvature/'t to which the reader is referred for references to the literature and for fuller handling of some of the subject matter of the first five paragraphs of these lectures. The point of view adopted is that objective space presents cer- tain phenomena of form, position and magnitude, which demand explanation as do other physical phenomena. This explanation the geometrician gives by the assumption of certain hypotheses, * Clifford, \V. I\., "A Preliminary Sketch of Biquaternions," Mathematical Paper*. No. XX. Klein. P., "Zur Nicht-Euklidischen Geonietrie," Math. Annalen, vol. 37 (1890). ]. 344. Lecture* on Mathematics, Lecture XI, New York. 1894. " Zur ersten Vertheilung der Lobatchewsky Preise," Math. Annaleu, vol. ">9 (1S9S), especially pp. o91-592. Killing, \\\, " I'eber ClilTord-Klein'sche Rauniforinen," Math. Annaleu, vol. 39 (1891). Eiiifdhrunrj in die Grundlanen tier Geonietrie, vol. 1, Chap. 1 ; Pader- born, 1893. j Annals of Mathematics, ser. 2, vol. 3 (1902), p. 71. 31 V2 T1IK BOSTON COLLOQUIUM. which he is free to make as he pleases, provided that they are self-consistent. The test of the validity of the hypotheses lies in their results. We make at tirst hypotheses which follow the ideas of Niemann's famous Ilabilitationsschrift.* It is admitted that questions may be raised which lie back of these hypotheses, as, for example, the possibility of reducing them to simpler axioms, but the discussion of such questions lies outside our present province. The Riemann method has for us the double advantage of allowing the immediate use of analytic methods and of restricting the discussion at the outset to a small region of space. A geometry having thus been developed in a restricted portion of space is extended to all space by means of new hypotheses, which are essentially those used by Killing in his Grundlagen der Geometric In the further development the ideas of the last named treatise have been largely followed. 1. Tin-: First Two Hypotheses. As already said, we adopt in our investigations the method of Riemann by which our objective space is assumed to be an example of an extent CMannif/faltikvif) of three dimensions in which an element may be determined by means of coordinates. We assert tin- explicitly in the following words : Knist Hypothesis. Space in a continuum of three dimension* in which a point nidi/ In determined !>ij three independent real coor- dinate* (- ,:.,, :.,). // a proper/// restricted portion of space is con- sidered, the correspojidence between point and coordtnalt is one-to-one and coiifin nous. \\ ithin our space, we mav pick out at pleasure one-dimensional extents or lines. We shall restrict ourselves to lines which mav be expressed bv t he equations :,=/,(/), -..=/M), -,=./; ; (0> where / is an arbitrary parameter and /.,./*, and/', are continuous I ,' h mil mi, I'>. , "I din 1 1 if 1 1 ypotlirseii, wclclic dcr ( it'iiinctrio zu (irmide li.-ir.-ii." '.>, ,,,,,. 7, If.,/., 1st i-il. p. ll.Vl ; 2d cd. p. SJ]>. FORMS OF NON-EUCLIDEAN SPACE. ->> functions possessing continuous first derivates, nowhere vanishing simultaneously. For such a line we may introduce the concep- tion of length as follows. Consider a portion of the line corre- sponding to values of t lying between the values / and T inclu- sive, and let this portion be divided into n segments to the extremities of which correspond the values t v /. t v t n _ v T. Let further (~ p z. s 3 ) and (~, + Bz { , z., -f 8z 2 , z 3 -f Bz.,) be the coordi- nates of the extremities of any segment, corresponding respect- ively' to /. and t._ .. We may then assume arbitrarily a function 4>(h> z 2> ~ 3 ; g -P ^ 8 h) which has the following two properties : First, it shall become infinitesimal with Bz v Bz 2 , 8:,, and consequently with t i+l t.; and secondly, the sum of the n values of this function, computed for the n segments of the line, shall approach a limit as n is indefinitely increased and each of the n quantities t. +1 t. approaches zero, this limit to be independent of the manner in which the segments of the line are taken. This limit is defined as the length of t/ie line. If in particular we take < = l 2 A & * (i, A =* 1 , 2, 3 ; a k . = a. k ) the length of the line is expressed by the integral I d.r. d.r, ,, a 2a ' (lt - \ '> dt dt The differential of this integral, namely, (7s = i 1a. k dxxlx k , we call the line-element of the space. We express these conventions in a new hypothesis as follows : Second Hypothesis. The length, of a line shall he determined by means of a line-element given by the equation ds= , 2^.r/^ (,,- = ,.,; i,* -1,2, 3) where the a {k are functions of z v ~, z v possessing continuous deriva- tive* of the first four orders, the determinant \a ih | does not vanish identically, and the expression under the radical sign is positive for 34 THE BOSTON COLLOQUIUM. all value* ~,, z. ~,, dz v dz. z , dz 3 , provided that (z v z.,, z 3 ) is a point of space and that dz v dz. dz 3 are not all zero. 2. Definitions. We proceed now to develop the conceptions of a geodesic sur- face, a geodesic line, an angle and a direction, which shall corre- spond to the conceptions of a straight line, a plane, an angle and a direction in Euclidean space. 1. Geodesic. Line. A geodesic line is defined roughly as the shortest distance between two points. To determine its equa- tions, we have to find the conditions that the integral X \ 2 "* dt a l \l dt shall be a minimum. The Calculus of Variations gives as neces- sary conditions, the three equations d 1 / dz x dz 2 i + a n i \ = >~ n ;' I > ds l_ as - ds '' ds ^ - cz x ds rf.s d\ dz. dz., dz~] da dz t dz,. dV dz, dz, dz,l _ da., dz. dz, ^/x[_ '> r/.s -' c/.s " ff.s J - c ri. { f/.s rif.s' which must be considered in connection with the identity dz. dz, "' ^fx rr.s Conversely these conditions are sufficient if s is not too great. FORMS OF NON-EUCLIDEAN SPACE. H~) More precisely : Let (z^>, z.'*', .:\j' ') be any point point of space, and (:,, . 2 ) any second point such that j 2. .:!' j does not exceed a suitably chosen positive quantity, h. Then the above equations admit one and only one solution which passes through the points (.: :'" ) and (;) and has all its points lying in the region j f !'' 2,-1 < h ; and for the corresponding curve the integral s has a smaller value than for any other curve joining the points (z" ] ) and (2). We take the equations accordingly as the defining equations of the geodesic lines and shall apply this name to the curves satis- fying these equations, even if the curves have been so prolonged that the minimum property no longer holds. 2. Direction. In accordance with the theory of differential equations it is always possible to find one and only one solution of the above equations which takes on at an arbitrary point (z v z. z ,) any arbitrary values (not all zero) of the differential coefficients dz } dz 3 r/2, ds' da' dx' If these differential coefficients satisfy initially the condition dz. dz, '* (Is dx this relation will be fulfilled for all values of x. The geodesic lines which radiate from a point are hence dis- tinguished from each other by the ratios of the values of the dif- ferential coefficients, which may consequently be regarded as fixing the direction of the line; the direction being, broadly, that property of the line which distinguishes it from all others through the same point. It will be convenient to denote dz.jds by . and to speak shortly of the direction (",, ^), or . These quantities satisfy the relation 3. Anyle. The angle 6 between two intersecting curves with the directions " and "' is defined by the equation cos e = s.^r;. 30 THE BOSTON COLLOQUIUM. In particular two intersecting curves ure perpendicular if 4. Geodesic Surface. A geodesic surface is defined as a pencil of geodesic lines. More precisely : Take any two geodesic lines OA and OB intersecting at 0, having at that point the directions a. and /3 respectively, and making the angle w with each other. Consider any other geodesic line OM with the direction Si = *<*< + f*Pii where X and /x are parameters subject only to the condition X 2 -j- fx 2 -f 2\fx cos to = 1, which arises from substitution in As X, /x take all possible values, OM generates a pencil of lines, which is defined as a geodesic surface. It may be shown without difficulty that in this pencil there is one and only one line per- pendicular to OA and that this may replace OJ> in defining the pencil. We shall then have f. = a. cos 6 -f /3. sin 0, where is the angle between OA and OM. If now /' is any point on O.l/ and / is the length of OP, the coordinates z. of /'are determined by integrating the equations of the geodesic lines, choosing the solution which has at the direc- tion , and substituting / for .v. We have then , = /;.( ,, t : r)= .(0, /'), the functions being continuous together with their partial derivatives of the first and second orders. l>v taking and e as independent parameters, we have the equations of the geodesic surfaces. FORMS OF XOX-EUCLIDEAN SPACE. 37 3. The Third Hypothesis. The method of superposition, involving the assumption that a geometric figure may be moved from one position to another without altering its size or properties, is fundamental in the Euclidean geometry and would seem to be a necessity in any ex- planation of spatial phenomena. The hypotheses thus far made do not carry with them the necessity of any such superposition. This may be clearly seen by examples from the Euclidean geom- etry of a kind which we shall frequently employ in the follow- ing pages. In thus using the Euclidean geomety, we do not as- sume that it is objectively true, but that it is a self-consistent system which explains experience. Consider any surface on which a sys- tem of curvilinear coordinates (u, v) have been established. This surface is a two-dimensional space satisfying the first two hypoth- eses, the line element being of the form ds 2 = Eda 2 + 2Fdudv + Gdv\ Such surfaces, however, offer various possibilities in the matter of superposing one portion upon another. One needs only to con- sider the ellipsoid, the right circular cylinder, and the sphere as examples. To bring the principle of superposition into our present discus- sion, we shall define a displacement as a transformation by which a continuous portion of space is brought into a continuous point for point correspondence either with itself or with another portion of space in such a manner that the lengths of corresponding por- tions of lines are the same. Let S be a portion of space in which the coordinates of a point P are (z v z. z,), and let S' be a portion of space in which the coordinates at a point P ' are (z[, z', '). Let the line-element in She denoted by ds = ] Ha.,dz.dz, ik i k and the line element in H ' by ds' = i /y 2.a'. k dz'.dz l; , 38 THE BOSTON COLLOQUIUM. where a' ik denotes the value of a ik for (z[, z' 2 , z' 3 ). In order that S may be displaced into . 40 THE BOSTON COLLOQUIUM. a region of space such that any point P of the region can be joined to by one and only one geodesic line lying in the region. We shall call this region T. Through take in T three mutually perpendicular geodesic lines OA, OB, 00. This can be done by taking directions (a v a a.^), (/3 1? /3. /3,), (7^ 7.,, y 3 ) so as to satisfy the relations 2<,V 4 = 1, 2^/3^=1, S^ 7i7i =l, 2^ = > ^TA%, = 0, 2. <(.,, r) as the coordinates of P. Then to any set of values of the coordinates corresponds only one point P, and to any point P correspond only the coordinates (7/,, ",, c/.j, /) or ( " v ":> "v r )> where the functions F. are continuous and possess continuous derivatives of the first two orders since they arc the solutions of the differential equations of the geodesic lines. By the substitution in the form of the line-element is obtained as FORMS OF NOX-EUCLIDEAN SPACE. 41 f/ , 2 = Y:A ik dcula k + ^.A^hi/h- + A u d,- 2 , 0,1; 1=1, 2, 3) i/c I where cz. c The direct calculation of the values of the coefficients is difficult : hut we shall prove by an indirect method that the proper form is (Is 2 == , (-=uo- If P tl is any fixed point on C and 6 is the angle between the geodesic lines OP t) and OP, 6 is a function of t and hence ^ is a function of 6, which for small portions of C is one valued. We may consequently write for the equations of C a^^lO), a 2 =< 2 (0), 3 = 3 (0), r = 4>ld). \i' from these four equations we omit the fourth, thus allowing r to take any value, we have the equations of a surface, which passes through the curve C, as is evident, and also contains the point since the equations are satisfied by / = 0. The surface is analogous to a cone of the Euclidean geometry, for the lines = const, are geodesic lines radiating from to the points of C. These lines form one of the systems of coordinate curves on the surface ; the other system is composed of the lines / = const., each of which is the locus of points equally distant from 0. If we refer to the general form of the line-clement of a surface ds 2 = Edr + 2Fdrdd + <;., Calculus <*/ Variations, p. 164. 42 THE BOSTON COLLOQUIUM. and we proceed next to replace d6 by its value in terms of a { . For that, we call 89 the angle between two neighboring geodesic lines OP and OQ, with directions a and a 4 &/i, where 'i 4- a\ + l = 1, (, + 8J 2 + (, + 2 ) 2 + (a, + 8a 3 f = 1. Then cos S# = a l (a l + 8j) 4- a 2 (a 2 + 8a 2 ) + ^ 3 (" 3 +5 3 ) = 1 -)- a l Sa l 4- 2 ^ 2 + a J* a s so that sin 2 9 = \{8a 2 + 8a 2 -f 8a 2 ). From this follows in the differential notation dd 2 = da 2 + da\ 4. f?a^, So that the line-element of the snface is (Is 2 = G(ddl + da:, 4- efo 2 ) 4 dr. This is in particular the element of the length of the curve (\ since C is on the surface. But C is any curve in space and hence the above expression is the line-element of the space. We seek now to determine (r. For that purpose consider where (see p. 40) r -,= ^'i('"p "j> (l v '*) = s i + K a i + "^- + ":;%)'' + Hence (7 = i'~*,' 2 with the initial conditions I lonce I sin // (1 I !' /.- i> real this determination of/; is final 1 (d = FORMS OF NON-EUCLIDFAX SPACE. 45 If I; is pure imaginary ,we may place /; = ik' and sinh /// /:' ~ ' If /; is zero, we may place sin />/ |/ (V = Lini - j - = r. It will he more convenient to retain the general form for /;, since the above changes are readily made. We have accordingly the line-element in the desired form, 'Is 2 = " ,, --' {da\ -f da\ + d1 + k\B\ + B\ + Bl) = 1, (6) A {) B + k\A,B x + A 2 B 2 + J 3 5 3 ) = 0, J which are necessary and sufficient in order that the conditions (2) and (3) may be satisfied. In fact the constants jB. are the coordi- nates of the point from which s is measured and the constants kA. are the values of dx.jds at that point and consequently fix the direction of the line. We may write the equations of a geodesic line in terms of any two points upon it. Let y i and z. be the two points, and let I be the distance between them measured on the geodesic line. If we measure .s from z., we have from (5), z. = B ff Vi = A. sin kl + B. cos kl (7) From these follow, with aid of the relation (6), .o + k2 0h z i + V-ri + y&) = cos m> ( 8 ) an important formula which gives the distance between two points in T, Tf x t is any other point on the geodesic line, we have from (o) and (7) ', = \'/, + fl3 {i (9) where sin k.s sin k(l s) X = sin/;/' ^ = sin kl ' or, otherwise written, X sin // = sin ks, X cos kl -f /x = cos k>t. Hence X and fx must satisfy the condition X 2 4- ^ -f 2X/x cos //= 1, (10) FOKMS OF NON-EUCLIDEAN SPACE. 47 which is also the necessary and sufficient condition that ?.'. may satisfy relation (2). Conversely any equations of the form (9) for which conditions (10) and (S) hold represent a geodesic line, provided they are sat- isfied by points in T. For it is always possible to find an angle v, such that sin kv = X sin hi, cos l;v = X cos hi -\- fx. From the condition (3) it follows that dtr = dp 2 . It can then be verified that the functions x i = x !f, + f- z i satisfy the differential equations (4). We collect these important results in the following theorem : Any geodesic line may be represented by the equation* x i =Xy i + ixz., (i= 0, 1, 2, 3) where y i and %. are any two points on the line, and. X and fx are parameters satisfying the relation X 1 -f p* -f 2X/x cos hi = 1 , / being the distance between the two points y. and z.. Conversely any equations of the above form represent a geodesic line if they are satisfied by points of T. From this follows immediately : Any tivo linear homogeneous equations in ,e { represent a geodesic Une if satisfied by coordinates of points in T ; and conversely any geodesic tine may be represented by two such equations. As to the geodesic surfaces we have the theorem : Any geodesic surface is represented by a linear homogeneous equa- tion in ;e. ; and conversely any such equation represents a geodesic surface if it is satisfied by points in T. To prove the last theorem, consider a pencil of geodesic lines determined by two lines through B. with the directions A', and A" x respectively. It has the equations x. = (XA'. -f pA'l) cos hs -f B i sin hs, 48 THE BOSTON COLLOQUIUM. where A. 2 -f- /x 2 -f-2X/Lt cos 6 = 1 # being the angle between the two lines A', and y1'.'. From this it readily follows that the coordinates of any point on the pencil satisfy an equation of the form ex -4- c x 4- c x -4- c x =0. Vo ' 11 ' 2 2 ' 3 3 Conversely if this equation is given and y. and z { are any two points satisfying it, the point x. = \y. + fiz it (X 2 + yU 2 + 2X/H cos H = 1) will also satisfy it. Hence any points on the locus of the equa- tion may be connected by a geodesic line lying wholly on the locus. The locus may therefore be considered as a pencil of geo- desic lines and is therefore a geodesic surface. Explicit formulas for the displacements in T may now be writ- ten. Since these displacements are continuous, one-to-one point transformations by which a geodesic line is transformed into a geo- desic line and the expression for cos kl is invariant they will have the form : x\ = a v c x + a 2 x 2 + a 3 x 3 + a^ , < = 0,-r, + Pp 2 + /3,r 3 + Pfa x s = 'V'i + ^2 + V:>, + % x o> < = Vi + *& + S 3" T 3 + V*o> where S 2 + /r(a 2 + /3 2 + 7 2 ) =1, 8;' f /r( 2 + /9= + 7 2 ) =lr, (k = 1,2,3) 5-5,, -f ^(ot.^ + /3./3 A + 7,7.) = 0. (/, /< = 0, 1, 2, 3 ; / =J= //) From these conditions it follows that determinant |a 1 /3./y8 () j = 1. If we add to our definition of a displacement the condition that it may be reduced to the identical substitution by a continuous change of the coefficients, we shall have the new condition FORMS OF XOX-EUCLIDEAX SPACE. 41) Conversely, any linear substitution in which the coefficients sat- isfy the above conditions represents a displacement in T, provided that it is satisfied by at least one pair of corresponding points in T. We have now the full data for constructing a system of geom- etry in T. The following are some of the fundamental theorems which are readily proved.'* In fact some have already been proved in the preceding discussions and the theorems are repeated here for completeness. 1. A geodesic line is completely and uniquely determined by any two points. 2. A geodesic surface is completely and uniquely determined by any three points not in the same geodesic line. 3. If two points on a geodesic surface are connected by a geo- desic line, the line lies wholly on the surface. -4. Two geodesic lines, or a geodesic line and a geodesic surface, intersect in at most one point. 5. Two geodesic surfaces intersect in a geodesic line, if they intersect at all. 6. On a given geodesic surface, one and only one geodesic line can be drawn perpendicular to a given geodesic line at a given point. 7. If a geodesic line is perpendicular to each of two intersecting geodesic lines at their point of intersection, it is perpendicular to every line of the pencil defined by the two intersecting lines. Such a line is said to be perpendicular to the geodesic surface defined by the pencil. 8. Through any point of a geodesic surface, one and only one geodesic line can be drawn perpendicular to the surface. 9. Through a given point on a geodesic surface, one geodesic line can in general be drawn perpendicular to a given geodesic line on the surface not passing through the given point, and never more than one. 10. Through a given point not on a geodesic surface, one * Proofs of all these theorems may be found in the Anmih article already cited. 1 50 THE BOSTON COLLOQUIUM. geodesic line can in general be drawn perpendicular to the surface, and never more than one. 11. The sum of the angles of a triangle formed by three inter- secting geodesic lines is equal to, greater than, or less than, ir, according as k is zero, real, or pure imaginary. It appears that the geodesic lines in T have all the properties of the straight lines of practical life or of the Euclidean geometry. In the endeavor to construct a material line which shall be " straight," we may proceed by attempting to realize the shortest distance between two points by stretching a string or otherwise. The result is simply a geodesic line by definition. Or we may look fur a line which may be revolved upon itself when two points are fixed. This is also a property of the geodesic lines. A geodesic surface has the properties of a plane. The practical testing of a plane surface by the application of a straight edge has its full significance in T. The practical measurement of length and angle by the application of an assumed unit is also possible in T. We see then that the groundwork of experimental geometry is the same for all spaces which satisfy our three hypotheses. These spaces agree also in the first ten theorems above stated. A distinction appears first in the eleventh theorem, which appears to present a means for determining the curvature of our objective space. The test fails, however, owing to the impossibility of exact measurements. All we can discover is that the sum of the angles of a triangle does not differ very much from it and it is possible to show that if the sides of a triangle are sufficiently large compared with / the divergence of the sum of its angles from it is within the limits of the errors of observation.* \\ e may sav then : Ainj space irhich satisfies the three hypotliescs is f ax jar ax 1: ,. __ cos />, where <' in around O corresponds to a region '/', around l in such a manner that the portion of the geodesic line OQ which lies in 7' ( corresponds to the portion of the same line which lies in 1\ and extends in the same direction. Here 7', and 1\ are both contained in S a , but by vir- tue; of the fifth hypothesis this displacement of T into J\ determines a displacement of S into a new position S { . The line OQ of length P goes then into a line O l Q ] of the same length ; that is, the line OQ { has the length P -f I. Now we can repeat this operation with the region .S', by selecting on O l Q ] a point O., such that 0,0.,= I, and displacing 0, into O., in the proper manner. Jn this way the line OQ is extended indefinitely, but it is of course consistent with the theorem that the line should be a closed line. Any point in space may be joined to by a geodesic line. A rigorous proof of this statement may be given by means of the method introduced by Hilbert into the Calculus of Variations under the name of the " Huufungsverfahren."* The details are too involved to be presented here. We content ourselves with noticing that since space is a continuum by our first hypothesis, any point P may be connected with by a continuous curve. Now the Hilbert method consists in showing that among all the curves that can be drawn between and P there is one such that no other has a greater length, and that this curve in sufficiently small portions is a geodesic line as we have defined it. By virtue of the two theorems just proved, we may write sin hi '< = . /. > 0=1,2,3) ;r = eos/;s, {a\ + a: + a* = 1), where -s is unrestricted, with the assurance that all values of x. thus determined represent a point of space and that any point of space may be represented in this way. This is our generalized coordinate system. Let us take now any point P. By the fourth hypothesis, the * Consult for example the dissertation of ('has. A. Noble, " Fine neue Methode in der Variationsrechung," Gottingen, 1901. r >4 THE BOSTON COLLOQUIUM. region 8 may be so displaced that corresponds with P and 8 Q with a congruent region 8 n . There exist then relations between the coordinates of points in 8 n and the coordinates of points in S . We shall show that these relations have the same form as those which define a displacement in S . For that purpose connect and P with a geodesic line and take on this line the points O, O v 2 , , 0,P, such that the distance 0.0. +1 is less than II. If then is displaced so as to coincide in succession with O v 2 , , P, there is determined a chain of congruent regions 8 , S l} 8 2 , , S n , eacli of which has points in common with the preceding one. The displacement of S Q into S t however is fully determined by the fact that a region around is dis- placed into a region around O v both regions lying in 8 . Hence all coordinates of all the points in 8 { are connected with those of 8 by relations of the form given in paragraph 5. It follows that in *S'j the line element is the same as in 8 , that a linear equation represents a geodesic surface, that two such equations represent a geodesic line, and that a displacement of a portion of 8 is repre- sented by equations of the same form as in 8 . In like manner we can proceed from 8 l to 8, and hence eventually to 8 n , thus establishing the fact to be proved. It is clear that if more than one geodesic line can be drawn from to P, P will have more than one set of coordinates and more than one set of equations will connect the coordinates of N and N . Let now any displacement be imparted to X (| . By the fifth hypothesis, a displacement is then imparted to 8 n through the chain N (| , 8 V N, , S' . It is easy to see that the analytic ex- pre.ssion of this displacement of 8 will be found by substituting in the displacement defined for S the co()rdinates of the points of N determined bv the chain 8 ()f 8 , 8 . We may now establish the imj)ortant proj)osition : If I: is a real (ptfiiifdif, frr/ 1 )/ (jvodesic line ts closed and has a len an( l each region being ob- tained from the preceding one by the substitution sin /.V ''i = ''. ( ' ()S + x o /. ' X 3 = ''";' x' = xj: sin M + cos /, where / < R. Apply now to 13 the displacement x[ = x } cos x 2 sin <, a:.', = Xj sin < -f x., cos <, k! = x.,, 0* This displacement will be transmitted to S n through the chain S ti , S v , S n . The distance I) between the new and the original position of a point is given by cos kB = x x' -f /;-(.t 1 p" 1 ' -f x 2 x' 2 + x.jjc' s ) = x 2 -f k 2 x 2 3 -f k 2 (x 2 + &l) COS varies from to 27r, the point Q is moved on x = 0, .r, = through a distance 'lirl;. But a continuous variation of < from to 2tt restores S and hence .S' /t to its original position. Hence the geodesic line x Q = 0, .? 3 cannot have a length greater than 2-7T /;. The theorem is thus proved for a particular geodesic 56 THE BOSTON COLLOQUIUM. line; but by proper choice of the origin and coordinate axes, any geodesic line may be given the equations x = 0, .r., = and hence the theorem holds universally. It may be explicitly noted that we have not proved that a geodesic line may not have a length less than '2tt k, nor that all geodesic lines have the same length. We are now prepared to prove the proposition : To . Forms of Space Which Allow Free Motion as a Whole. We are to examine in this paragraph the results of assuming that the displacement of S n caused by a displacement of ', and let ns assume that /' corresponds to two points J I and II'. As shown in the last paragraph, if II and IT are connected by a straight line 7, there will correspond in S a line g which starts from P and returns to the same point. Along this line we may construct a chain of congruent regions S , >>',,. S'. -, S n , where S n is the same region as S' n must be the same as that of S , if sj>ace is movable as a whole. If, for example, # is so moved that all points on a geodesic line I are fixed, S n must be moved in the same manner. Correspondingly, we must have in 2 a dis- placement by which two straight lines X and X', one lying in 2 (J , the other lying in 2 n are each point for point fixed. This, how- ever, is impossible unless 2 n coincides with 2 . Hence the as- sumption that P corresponds to two points II and II' is untenable. Spaces of Zero Curvature. If k = 0, the relation between points ofS and those of 2 is one to one. In other words, to each point of S' corresponds one and only one set of coordinates x { and conversely. We have there- fore a geometry in which the theorems of paragraph 5 hold univer- sally. In addition all geodesic lines are infinite in length. We may consequently introduce the conception of parallel lines by the following definition: A line AB is parallel to CD when AP> is the limit approached by a line AG intersecting CD, as the point of intersection recedes indefinitely. It may then be shown that through any point of space there goes one and only one geodesic line which is parallel to a given geodesic line not passing through the given point. The resulting geometry is the Euclidean Geometry. Space* of Constant Xee/ativc Curvature. If/; is pure imaginary, again the relation between the points of S and those of 2 is one to one. We have again a space in which 60 THE BOSTON COLLOQUIUM. the theorems of paragraph 5 hold universally and in which all geodesic lines are infinite in length. If parallel geodesic lines are defined as for /. = 0, then through a given point there go two and only two geodesic lines parallel to a given geodesic line not pass- ing through the given point. All other geodesic lines through the point and lying on the geodesic surface determined by the given point and the given geodesic line are separated by the parallel lines into two classes, consisting respectively of the lines which do, and of the lines which do not, intersect the given geodesic line. The geometry is the Lobaehevskian Geometry. Spaces of Constant Positive Curvature. If/,' is real, two cases present themselves. In the first ease, the relation between the points of S and those of S is two-to-one. Then to each point of & corresponds only one set of coordinates and conversely. In particular, the coordinates .r. and . belong to different points of space. The theorems of paragraph 5 hold only in a restricted portion of space in which the greatest geodesic distance is nr/k. All geodesic lines are closed and of length equal to 2ir/k. Two intersecting geodesic lines intersect again at a dis- tance irJJ: on each of them from the first point of intersection. There are no parallel lines in the sense of the definition given for k = 0. In fact any two geodesic lines on the same geodesic surface intersect. All geodesic lines perpendicular to the same geodesic surface intersect in two points which arc distant irj'lh from the sur- face. The geometry is that called by Klein the Spherical Ceometry. In the second ease, the relation between the points of S and those of ! is one-to-one, in the sense; that to each point of N belongs the two sets of coordinates f and ;>'.. The theorems of paragraph o hold for a portion of space in which the greatest geodesic dis- tance is it Ik. All geodesic lines are closed and of a length it Ik and any two intersecting geodesic lines return to the point of in- tersection without previously meeting. All geodesic lines per- pendicular to the same geodesic surface meet in a point at a dis- tance it J '11; from the surface. The geometry is called by Klein the I'll i [>l a- Ceomefri/, FORMS OF XOX-EUCLIDEAX SPACE. 01 \\ r e may sum up as follows : The only spaces satisfying our fire hypotheses and allowing free motion as a whole arc the Euclidean, Lobachevskian, Spherical and Elliptic spaces. 10. Forms of Space Which do Not Allow Free Motion as a Whole. We consider next spaces in which the displacement of & n caused by the displacement of S is dependent upon the manner in which S n is connected with N () . These are called by Killing the Clifford- Klein space. They have been illustrated in paragraph 6. From what has preceded, it is clear that in the Clifford-Klein spaces a point must have more than one set of .^-coordinates. Consider then the region >' and let a*, be one set of coordinates of its points. Then if x'. are also the coordinates of its points, x[ may be obtained from x., as we have seen, by following out a chain of displacements by which # takes in succession the positions S , S y , S. S it = N (l . That is x'. and x. are connected by relations which have the form of the displacement formulas. Suppose these relations denoted by I) v Let now y. be the coordinates of a point P lying outside of N,. It may be connected with # by a geodesic line and a chain of regions S S[, S'.',, , S' constructed along this line. If the displacement />, is imposed upon S' , it will be trans- mitted to S' ; and since S returns to its original position the same is true of #', by the fifth hypothesis. That is the transformation I\ gives a relation between two sets of coordinates of any point of space. Such a transformation is said by Killing to represent the coincidence of points. It is clear that the inverse transformation T)~ x also represents the coincidence of points, and if D l and J)., each represents the coincidence of points, the transformation D.D., dot's also, and this is true when ]) is the same as J> r That is, the transformations which represent the coincidence of points in space form a group. This group we shall call the group of the space. The group of the space interpreted in !l is a group of collinea- 62 THE BOSTON COLLOQUIUM. tions by which the fundamental quaclrie is invariant and by which points that correspond to the same point in S are transformed into each other. Because of the theorems established in paragraph 8 it follows that the group of the space interpreted in 2 must not only be properly discontinuous but must be subject to the condition that the distance between corresponding points shall never be less than a certain finite quantity. In particular, no transformation of the group maij hare a real fixed point. If the region of discon- tinuity of the group in 2 is obtained, this region will correspond in a one-to-one manner to S', when 1: is zero or pure imaginary, and in either a one-to-one manner or a one-to-two manner to *S' when /, is real. Conversely, the region of discontinuity of any properly discontinuous group in 2, by which the distance between two corresponding points is never less than a finite quantity, will furnish an example of a space satisfying the five hypotheses. Hence the problem to determine the Clifford-Klein space is reduced to the problem to determine all groups with the required properties. Before proceeding to the nearer discussion of the problem, we may note that our derivation of the group of the space is based upon the consideration of a three-dimensional region S' in which each point has different sets of coordinates. This region gives opportunity to apply the fifth hypothesis. There is still the pos- sibility therefore that certain exceptional one-dimensional or two- dimensional regions may exist, upon which the same point may have sets of coordinates not connected by transformations of the group. The following two examples are given by Killing of a two-dimensional space of zero curvature having an exceptional line. 1. Consider a cylinder in Kuclidian space standing upon a cubic curve with a double point. The geometry of the cylinder is that of the Kuclidean plane except for the presence of the double line. We call 'la the length of the loii)) of the cubic, and take as the origin of coordinates the point on the loiip equidistant from the double point in each direction. Then if we take for one coordinate FORMS OF NON-EUCLIDEAN SPACE. 63 the length s of the cubic and for the other the length // of an element of the cylinder, the coordinates (s, //) correspond in a one-to-one manner to the points of the surface, except that the coordinates (a, h) and ( a, li) correspond to the same point of the surface. '1. ( 'onsider a cylinder in Euclidean space standing on a leninis- cate. Its geometry is the same as that of the Euclidean plane for restricted portions. We will take the origin at the double point of the lemniscate, define s as the length of the curve and h as the length of an element of the cylinder. Then if '2a is the entire length of the lemniscate, the group of the surface is s' = s + 2na, h' = h, where n is an integer; that is, the coordinates (s, It) and (s -f 2na, A) refer to the same point of the surface. But the coordinates (0, h) and (ita, h) also refer to the same point of the surface, since they give points on the double line. Examples of a similar kind may be formed for three dimensional spaces without difficulty as far as the analytic work is concerned. How far they are conceivable as an explanation of physical space, involving as they do the passing of space through itself without break in the continuity of each of the intersecting portions may be open to question. They have been examined by no one in detail and we shall rule them out of the following discussion. We pass now to the special consideration of the three kinds of space. Spaces of Zero Curvature. If /; = 0, 2 is the Euclidean space and its movements arc the Euclidean movements. A rotation around an axis cannot be a transformation of the group of the space S since, as we have seen, no transformation of the group can have a real fixed point. We must form the group therefore by the use of translations and screw motions. The use of translations alone lead to three and onlv three 04 THE BOSTON COLLOQUIUM. properly discontinuous groups, having for regions of discontinuity respectively : () A parallelopiped with three finite edges. (/>) The limiting figure of a parallelopiped when one edge he- comes infinite. (c) The limiting figure of a parallelopiped when two edges be- come infinite. The geometry in ) The group of the space is ar; = (-l)'a! 1 + wia, V'., = (~1)'X 2 + III), .-.; = x 3 + ic, where a, b, c, are constants and /, m, n, are arbitrary parameters. AVe give without proof some of the striking peculiarities of S in the case (). There is a unique geodesic line of length // which we shall call the axis of the space. If a and it are incommensurable, this is the only closed geodesic line ; if a and it are commensurable, all geodesic lines parallel to the axis are closed and of lengths equal to multiples of //. For all values of a there are geodesic lines with double points. Through any point of space there goes an infinite number of such geodesic lines having the given point for a double point; and for a given direction, not parallel or perpen- dicular to the axis, there exist an infinite number of geodesic lines with double points. Geodesic surfaces are of three kinds : (1) those perpendicular to the axis, (2) those parallel to or contain- ing the axis, ('>) those which have neither of these relations to the axis. On geodesic surfaces of the first kind, all geodesic lines are infinite in length and the geometry is that of the Euclidean plane. On geodesic surfaces of the second kind, there are no closed geodesic lines but a geodesic; line may have a double point. On geodesic surfaces of the third kind, all kinds of geodesic lines lie. The last two kinds of surfaces present the peculiarities of cylinders with double lines mentioned on pp. 623. Spaces of (bus/ant Poultice Curvature. If I; is real, there is a fundamental difference between spaces of an even and those of an odd number of dimensions. It is a simple matter to apply our foregoing discussion to space of two 6(> THE BOSTON COLLOQUIUM. dimensions by dropping the coordinate .*' and making necessary changes. It appears that any displacement has a real fixed point and consequently there can be no group of the space. If we rule out such special lines and points as occur on cylinders with double lines, we are then led to the discussion of paragraph 9. Hence the theorem : A non-euclidean space of tiro dimension* and of constant positive curvature for which our hypotheses hold and, in which no special point* exist has the connectivity and the geometry of either the spherical surface or the elliptic plane. Consider now a space of three dimensions. The study of the collineations which leave invariant the quadric S + * 2 (; + g + 8) = o, and which we call the movements of 2, lead to the following results.* By any real movement in 2 two real lines G and II, reciprocal polars with respect to the fundamental quadric, are unaltered as a whole, each point on each of the lines being dis- placed through a distance which is constant for that line. If the displacement is different for the two lines G and If, these are the only fixed lines. If however the displacement is the same for G and JF, then all lines of a certain line congruence are fixed, this congruence being made up of all lines which intersect the same two conjugate imaginary generators of the fundamental quadric. Any point of 2l is then displaced a constant distance 1 along the line of the congruence which contains the point. Such a transformation is the nearest analogy in a space of con- stant positive curvature to a translation in Euclidean space. It is accordingly called a translation, and the congruence of fixed line- are called Clifford parrdle/s. The name parallels is sug- gested bv the relation of these lines to a translation, but they have other properties analogous to those of the Euclidean par- allels. For example, from any point in either of two Clifford ('niisuh tor tlic details of the geometry of this paragraph: Klein, " Zur Nieht-Kuklidischeii < ieometrie," Mnlfi. Annul,,,, vol. :;7 (1SK)), p. oil. FORMS OF XOX-EUCLIDEAN SPACE. 67 parallels a common perpendicular can be drawn to the two, and the portion of the perpendicular included between the two has always the same length. Again, if a line cut two Clifford par- allels the corresponding angles are equal. The Clifford parallels are of two kinds, according as the gener- ators of the fundamental quadric which determine them belong to one or the other of the two sets of generators of the quadric. Similarly we must distinguish between two kinds of translations. Two translations of the same kind carried out in succession are equivalent to a translation of the same kind, but two translations of different kinds are not equivalent to a translation. Hence the translations of each kind form by themselves a group. Let us consider first non-euclidean spaces whose groups are formed by translations alone. These translations must all be of the same kind. If we place /. = 1, for convenience, and intro- duce \ and fx as the parameters of a point on the fundamental quadric, whereby + %- fc'-iV then any translation of the one kind causes a substitution of the form ,_(<*+ ie)\ - (b - in) A ~ (6 + ia)\ + (7/ - iV) ' ^ _ fX and conversely. On the other hand, if we interpret \ in the usual manner as a complex variable upon the unit sphere, the above substitution represents a rotation of the sphere. To any translation of the one kind in 2 corresponds then a rotation of the sphere, and in fact the angle of rotation of the sphere is equal to the distance by which the points of 2 are displaced along a system of Clifford par- allels. The group of the space corresponds to a group of rotations of the sphere, and since the amount of displacement by any trans- formation of the group is never less than a finite quantity R it 68 THE BOSTON COLLOQUIUM. follows that the group of rotations can contain no infinitesimal rotation. This condition is met only by the groups of rotations by which a regular polyedron concentric with the sphere is trans- formed into itself. We have accordingly the theorem :* If a Clifford-Klein space of constant positive curvature is trans- formed into itself by a (/roup of translations, this group must be holoedric-isomorph with a group of tlie regular polyedra ; and con- versely, to any group of the regular polyedra correspond four spaces of constant positive curvature, according as the coordinates x. and .r. represent the same or different points of space and as the group of the space is made up of translations of one or the other kind. It remains to ask if groups of the space may contain displace- ments which are not translations. This question is answered in the negative by Killing (/. c.) but his proof is not satisfactory. Pie shows conclusively that if I) is a displacement belonging to the group of the space and if G and JT are the two tixed lines, then the smallest displacement along either line caused by the repeti- tion of D must be tt\o, where q is an integer, the same for both lines. But he errs in assuming that this minimum displacement is caused in both lines by the same transformation. For example, consider the displacement I) TT TT 7T IT in _ + ''., cos . >7T . o7T _ .v.. sin . .v = .<., sin _ 4- '' cos . > o ) and the group 7), I >\ I )\ D\ //' = 1. The two fixed lines are '"This theorem is new as far as (lie author knows. Killing ((riniilliii/rn rfrr (,'.',i,ii irir, vol. I, p. .". 11 ) not ires that if the fjroiip of a space of /.' 1 contains a translation, the amount of the translation must he an aliquot part of ~. hut he leaves the impression that any three such translations may he comhim-d at pleas- ure to form a <;roup of a space. FORMS OF XOX-EUCLIDEAX SPACE. 69 G (.r, = 0, .r, = 0) and 7/"(.r, = 0, x = 0) and the smallest displace- ment along each is it lb. But this displacement is produced along G by 7) and D 4 and is produced along 11 by D 2 and Z>\ By no substitution of the group, however, can the distance between two corresponding points fall below a definite finite quantity. Hence the group, which is not composed of translations, is allowable as the group of a non-euclidean space. The investigation of such groups is yet to be made. Clifford's Surface of Zero Curvature. It is of interest at this point to mention Clifford's surface of zero curvature and finite extent which first led to the conception of the Clifford-Klein spaces. This surface may be obtained by choosing on the fundamental quadric of the above space of con- stant positive curvature two conjugate imaginary lines from each set of generators. The quadric surface which passes through the quadrilateral thus formed is the surface required. It is clear that the surface contains two sets of Clifford parallels and is transformed into itself by two translations. If we take the four lines on the fundamental quadric as corresponding respectively to A. = 0, A. dc, jx = 0, and \i = oo in our previous notation, the equation of the surface is R + B- 8 (e + g)-=o, where a is a real constant. We may define the two sets of Clifford parallels on the surface by the parameters u and r, where _ i + <, = t> - = , t z af ^ ". To obtain the line-element of the surface, we write first f . = px., where p~ = Sf 2 . Then for the space p*ld$ - (p X=_, 7*- + S 7\ + S where the determinants aS /3y and a /3y are not zero, de- termines such a collineation.* These collineations are the move- ments of . A real movement occurs when and only when a, /3, 7 and S, are conjugate imaginary to a, /3, 7, 8, respect- ively. A real movement may consequently be determined by the single substitution , aX + /3 X =yX + o- Let us suppose first that the substitution in X leaves two dis- tinct values of X unaltered. There correspond two fixed points on the fundamental quadric, and we may without loss of general- ity assume the coordinate system in such a way that these cor- respond to the values X= 1. The substitution may then be written * Consult for proof and historical references: Fricke-Klein, Vorlesungen iiber die Thw ><'<>' dcr antumorphen Faactwn-ni, vol. 1, pp. 44-59. THE BOSTON COLLOQUIUM. X' - 1 \' 4- 1 \- 1 X + 1 and is a loxodromic substitution when a =f= 0, (3 4= 0, an elliptic substitution when a = 0, /3 =)= 0, and a hyperbolic substitution when 1 4= 0, /3 = 0. The corresponding substitution of .r. is readily computed to be x\ = x x eosli a x sinh a, .'".', = .''., cos /3 x., sin /3, x = x sin /3 4- >', cos /3, ;/ = x. sinh a -f 33 cosh a, and the distance / between two corresponding points is determined by the equation cosh / = - ./.', - ;-> 2 - ;*y? 3 + x' x = (x: -f ^) cos p 4- (.cj; .';) cosh a = cosh a 4- (.'':! 4- .''3) (cosh a cos /3). If a = (), every point on the line x, = 0, x., = is fixed. Hence an elliptic substitution can not occur in the group of the space. It' 7. 4= 0, I zr~ a. Hence hyperbolic or loxodromic substitutions may occur in the group. Consider next a parabolic substitution of A. by which onlv one value of X is unaltered. By proper choice of the coordinate sys- tem this substitution may take the form \' = \ + a and the corresponding substitution of x is = '., 4 1 (a a) FORMS OF NON-EUCLIDEAN SPACE. 73 a + a i(d a) ad >', = -''a + " ., i + 9 **+ 9 ( x o~**)> + a i(a ) ad ' = ''"o + o ! + - ' o ^ + 9 ('''o .('., .T, . The distance / between two corresponding points is given by the equation (l( l , There is no fixed point in finite space, for the assumption ,r = x 3 carries with it the equality x\ + x\ = - 1. We may however find corresponding points whose distance apart is less than any assigned quantity. For if we take y { to represent any point, the coordinates x i = x l/v ''> = X y x s = X .";> + P> ' r o = X Vo + ^ represent a point for all values of X and /* which satisfy the relation x 2 -f 2 v(i/ - .'/,) = 1 The displacement / of the point x. is determined by ad , cosh/== 1 + - 2 \-(i/ - jj 3 )-, and / can be made as small as we please by taking X sufficiently small. Hence a parabolic substitution can not occur in the group of the space. We may have, then as allowable groups of a Clifford- Klein space of constant negative curvature only those which correspond to groups of linear substitutions of X which are properly discontinuous when interpreted in S and contain only hyperbolic and loxodromic sub- stitutions. The more minute discussion of the Clifford-Klein space depends therefore upon the knowledge of the groups called by Poincare 74 THE BOSTON COLLOQUIUM. the Kleinian groups. It is worth noticing that whereas in the group theory the greatest attention has been paid to Kleinian groups with elliptic and parabolic substitutions, it is exactly these groups which are of no interest in the geometric problem before us. Geometry here waits for the development of the theory of groups. SELECTED TOPICS IX THE THEORY OF DIVER- GENT SERIES AXD OF CONTINUED FRACTIONS. By EDWARD B. VAX VLECK. Part I. Lectures 1-4. Divergent Series. It may not be inappropriate for mo to preface the first four lectures with a few words of a general character concerning diver- gent series. These will serve the double purpose of indicating the nature of the problems to be treated and of binding together the separate lectures. The problem presented by any divergent series is essentially a functional one. When a divergent series of numbers is given, its genesis is usually to be found in some known or unknown func- tion. The value which we attach to it is defined as the limit of a suitably chosen convergent process, and the elements of the proc- ess are the terms of the given series or are functions having these terms for their individual limits. Most commonly the given numerical series a Q -f a, -f (i., + is connected with the power series (1) a -j- a x x + a 2 x 2 + . . , and the question thus reduces to that of determining under what conditions or restrictions a value may be assigned to the latter scries when x approaches 1. The primary topic therefore is the divergent power series, and to this we shall confine our attention exclusively. This topic, if broadly considered, presents itself under at least four very different aspects. What is given is in every case a power series with a radius of convergence which is not infinite. Suppose first that the radius is greater than zero and that the 7(5 THE BOSTON COLLOQUIUM. circle of convergence is not a natural boundary. Then the series defines within this circle an analytic function. In the region of divergence without the circle the value of the function may be obtained by the familiar process of analytic continuation. The- oretically the determination of the function is a satisfactory one, for Poincare * has shown that the function throughout the domain in which it is regular can be obtained by means of an enume- rable set of elements, P n (x ~ " ') Practically, however, when WeierdraHx process is employed for analytic continuation, the labor is so excessive as to render the process nearly valueless except for purposes of definition. Hence to-day a search is being made for a workable substitute. I may refer particularly in this connection to the investigations by Borel and Mittag-Leffier. As I consider the work of the former to be both suggestive and practical, I have taken it as the basis of my second lecture. A second aspect of our topic, intimately connected with the continuation of the function defined by (1), is the determination of the position and character of its singularities in the region where the series diverges. This subject is treated in Lecture 3. When the circle of convergence is a natural boundary, it does not appear to be impossible, despite the earlier view of Poincare to the contrary, t to discover, at least in a certain class of cases, an appropriate, although a non-analvtic mode of continuing the function across the boundary into other regions where it will be; again analytic. The thesis of Borel i\iu\ its recent continuation in the Aria }Ftitlien\tlv, we have the conundrum of the truly divergent power series the scries which converges onlv when ./ = (). It is upon ' l^u.l.mntl ,1,1 Cnrn'n M, I tr , I in , ,li I'f,;;,l, Vol. 2 (1SSS), p. |<7, or woe IJori-1 - Th.'uri, ,/... 1. 78 THE BOSTON COLLOQUIUM, which is a solution of the equation (2) x 2 ; + x ' + .>' 2 - n 2 );/ = 0. v ' ax dx, This is convergent for all values of .r, but when x is very large the series is worthless for computation owing to the rapid and long-continued increase of the terms before the convergence finally sets in. The astronomer and physicist therefore have been driven to use for large values of x an expansion which is of the form * 4sina;( A A, . A, Ax~Hmx[ A n + '+ .;-f X X" p p + Bx~l cos x ( />' + ' + 2 + J X X or, what is the same thing, (3) rv-*>-i C+ + + /> /> Here the multipliers of ('and I) are only formal solutions of the differential equation (2). Tn respect to convergence they have a character exactly opposite to that of J n , since for very large values of x the terms at first decrease rapidly but finally an increase begins. At this point the computer stops and obtains a good ap- proximate value of J . What is the significance of this ? It is strange indeed that no attempt was made to study the question until 1886, when Poin- i-tirfi and Sfic/tjcs'l simultaneously took it up. That so evident and important a problem should have been so long ignored by the mathematician emphasizes strongly the need of closer touch between him and the astronomer and the physicist. Both Pohieare and Sliiltjts regarded the series as the asymptotic representation ' Sec, for example, (iray and Mathew's Treatise cm Bessel Functions, chap. 4. i An,, Math., vol. S, p. L".t:> IT. ; Tin-si*, Ann. >lc I'Kc So,:, ser. 3, vol. .'!, p. 201. DIVERGENT SERIES AND CONTINUED FRACTIONS. 7! of one or more functions. While the latter writer studied care- fully certain divergent series of special importance with the object of obtaining from the series a yet closer approximation to the function by a species of interpolation, Poincave developed the idea of asymptotic representation into a general theory. To explain this theory ::: and at the same time to develop certain aspects scarcely considered by Poineare, I shall start with the genesis of a Taylor's series. Take an interval (0, a) of the posi- tive real axis, and denote by /(.<) any real function which is con- tinuous and has u + 1 successive derivatives at every point within the interval. Xo hypothesis need be made concerning the char- acter of the function at the extremities of the interval except to suppose that _/'(.'.'), /'(.r), , f'"(j')/n ! have limiting values a , a v , a when x approaches the origin. Thus the function at any point within the interval will be represented by Taylor's formula : (O<0<1). If the function is unlimitedlv differentiable and limiting: values of/ " (')/" ! exist for all values of n when ,v approaches 0, the number of terms in the formula can be increased to anv assigned value. Thus the function gives rise formally to a scries (1) ff -f a x x + <> 2 .r 2 + , uniquely determined by the limiting values of the function and its derivatives. The converse conclusion, that the series determines uniquely a function fulfilling the conditions above imposed in some small in- terval ending in the origin, can not, however, be drawn. This is not even the case when the series is convergent. Suppose, for example, that c for all values of n. Then in addition to * Cf. Feano, Atti della R. Accari. delle Scicnze di Torino, vol. 27 (1891 ), p. -10 ; reproduced a- A n hong II I ("Leber die Taylor' sclie Forniel") in Genocchi- I'eano's Differential- mid Tntegral-RecJtnunr/, p. 3-7.). 80 THE BOSTON COLLOQUIUM. f(x) = we have the functions e~ Vr , f~ ] x ~, , which fulfill the assigned conditions. They are, namely, unlimitedly differentiable within a positive interval terminating in the origin, and when x approaches the origin from within this interval, the functions and their derivatives have the limit 0. From this it follows imme- diately that if values other than zero be prescribed for the a n , the function will not be uniquely determined, since to any one deter- mination we may add constant multiples of e~ Vx , e~ i/x ', Inasmuch as the correspondence between the function and the series is not reversibly unique, the series can not be used, in general, for the computation of the value of the generating func- tion. Nevertheless, although this is the case, the series is not without its value. For consider the first //( terms, m being a fixed integer. If x is sufficiently diminished in value, each of these terms can be made as small as we choose in comparison with the one which precedes it, and the series therefore at the begin- ning has the appearance of being rapidly convergent, even though it be really divergent. Evidently also as x is decreased, it has this appearance for a greater and greater number of terms, if not throughout its entire extent. Xow by hypothesis the generating function was unlimitedly differentiable within the interval, and the successive derivatives are consequently continuous within (0, a). Hence if the interval is sufficiently contracted,/'" rl (a?) /(in -f ])! can be made as nearly equal to rH throughout the interval as is desired. We have then for the remainder in Taylor's formula : f "(0.10 in which e is an arbitrarily small positive quantity. Consequently if the first //; -J- 1 terms of the series should be used to compute the value of the generating function, the error committed would be approximately equal to the next term, provided x betaken suf- lieientlv small. In the^e considerations there is, of course, nothing to indicate when .- is sufficiently small for the purpose. If the result holds DIVERGENT SERIES AND CONTINUED FRACTIONS. 81 simultaneously for a large number of consecutive values of m, the best possible value for the function consistent with our informa- tion would evidently be obtained by carrying the computation until the term of least absolute value is reached and then stopping. Herein is probably the justification for the practice of the com- puter in so doing. Equation (4) which gave a limit to the error in stopping with the (m -j- 1)/// term shows also that tins limit grows smaller as x diminishes. Since, furthermore, by increasing m sufficiently the (//* + 2)th term of (1) may be made small in comparison with the (ni -f l)th term, it is clear that on the whole, as x diminishes, v:e must take a greater and greater number of terms to secure the best approximation to the function. These two facts may be comprised into a single statement by saying that the approximation given by the series is of an asymptotic character. This will hold whether the series is convergent or divergent. This notion can be at once embodied in an equation. From (4) we have ]ini . ,,,)_, ,._, ,,, = lim / '":;,: (: ' >) = 0=1,2,...). This equation is an exact equivalent of the two properties just mentioned and is adopted by Poineare'^ as the definition of asymp- totic convergence. More explicitly stated, the series (1) is said by him to represent a function f(x) asymptotically when equation (o) holds for all values of in. It will be noticed that this definition omits altogether the assumptions concerning the nature of the function with which we started in deriving the series. Not only has the requirement of unlimited differentiability within an interval been omitted but the existence of right-hand limits for the derivatives as .' approaches the origin is not even postulated. If the value a tl be assigned to * Loc. cit. i; 82 THE BOSTON COLLOQUIUM. the function at the origin, it will have a first derivative, a v at this point but it need not have derivatives of higher order.* The exclusion of the requirement of differentiability has un- doubtedly its advantages. It enlarges the class of functions which can be represented asymptotically by the same series. It also simplifies the application of the theory of asymptotic representa- tion, and this is perhaps the chief gain. The results of Poincar&s theory can readily be surmised. The sum and product of two functions represented asymptotically by two given series are represented asymptotically by the sum- and product-series respec- tively, and the quotient of the two functions will be represented correspondingly, provided the constant term of the divisor is not 0. Also if /(.'.*) is any function represented by the series (1), whether convergent or divergent, and ('') = h v + V- + V 2 + is a second series having a radius of convergence greater than \a.'\, the asymptotic representation of $[/(*)] will be the series which is obtained from K + &iK + "!' + ) + 6 2 (o + a i x + ) 2 + by rearranging the terms in ascending powers of :>'. Lastly, the integral of /(a*) will have for its asymptotic representation the term by term integral of (1 ). J>ut the correspondence of the func- tion and series may be lost in differentiation, for even if the function permits of differentiation, its derivative will not neces- sarily be a function having an asymptotic power series. Examples of this kind can be readily given. t * The ordinary definition of an //th derivative is here assumed. If, however, we define the second derivative by the expression /"(<>) 1 1 in' , I .'* and the higher derivatives in similar fash inn. the f unit ion niu^t have derivatives of all orders. i < f. Morel, !.<< .SVnV.1 ,/,'n ,-(.*) ; a t) -f p. S'.l f x2 of Morel's article, Ann. . Snr., ser. 3, vol. !; i Mm i. DIVERGENT SERIES AND CONTINUED FRACTIONS. 85 when the series in parenthesis gives such a representation of The applications of Poineares theory have been made chiefly in the province of differential equations* where divergent series are of very common occurrence. We will take for examination the class of equations, of which the theory is perhaps the most widely known, the homogeneous linear differential equation with polynomial coefficients : (6) P n ) 2' + 7> --^) IP' + + P v% = * ' This is, in fact, the class of equations to which Poincare first applied his theory, t but his discussion of the asymptotic repre- sentation of the integrals was limited to a single rectilinear mode of approach to the singular point under consideration. The de- termination of the sectors of validity for the asymptotic series has been made by Hon\,\ who in a number of memoirs has care- fully studied the application of the theory to ordinary differential equations. As is well known, the only singular points of (6) are the roots of P n (x) and the point x = oc. For a regular singular point jj we have the familiar convergent expressions for the integrals given by Fuchs. Consider now an irregular singular point. By a linear transformation this point maybe thrown to oc, the equation being still kept in the form ((>). Suppose then that this has been done. If P n is of the pth degree, the condition that x= oc shall be a regular singular point is that the degrees of P H _ V l ) ,^o, , P shall be at most equal to p 1, p 2, ,[> n, respectively. For an irregular singular point some one or more of the degrees must be greater. Let It be the smallest positive integer for which the degrees will not exceed successively * In addition to the memoirs cited below Polncar^'s Les methodes nonre/les d- la mecanique celeste and various memoirs by Kneser may be consulted. fActa Math., vol. 8 (1886), p. 303. See also Amer. Jour., vol. 7 (1885), p. 203. tMath. Ann., vol. 50 (1898), p. 525. \ See various articles in Crelles Journal and t lie Mathematische Annalen. 1 Stelle der Bexttmmtheit. 8G THE BOSTON COLLOQUIUM. p+(h-l), /) + 2(A-l), p-h 3(A-1),.... The number /< is called the rank of the singular point oc, and the differential equation can be satisfied formally by the series of Thomae or the so-called normal series : a i x ' : > 1.1. / PC 1 \ (0 $ = " ' 1 ^^c i + ;; ! + -3- + ...J (i-1,2, ...,n). Unless certain exceptional conditions are fulfilled, there are n of these expansions, and in general they are divergent. To simplify the presentation let us confine ourselves to the case for which h1. Then at least one of the polynomials succeeding P will be of the pth degree, and none of higher degree. Place P =Aa*+ B ,^- 1 + , n n ' h ' > and construct the equation (8) A n a + A n _^'~ l + ...+ A Q =0. The ?? roots of this equation are the n quantities a. which appear in the exponential components of the S.. As a particular illustration of the class of equations under con- sideration, fiessel'x equation (Eq. (2)) may be cited. Here the point oc is of rank 1, the characteristic equation is A a 2 -f A ,/r -f A., == or -f 1 = 0, with the roots *, = ', ^ = 4- ', and the two Tlmmacan integrals are th * Cf. Picard's Traitc d' Analyse, vol. 3, p. 383 ff., or Poincar, Amer. Jour., vol. 7 (1885), p. 217 ff. 88 THE BOSTON COLLOQUIUM. degree, the transform is of the ^>th order with coefficients of the nth degree. Its singular points in the finite plane are the roots of the first coefficient of (11), which is identical with the left hand mem- ber of (8). Furthermore, an inspection of (11) shows immedi- ately that each of these singular points a. is regular, and the exponents which belong to it are o, 1,2, ..., i >-2,/3 i = -(/) i +l) (/ = 1, 2, ;), in which p. is the exponent of x, hitherto undetermined in (7). Hence if (3. is not an integer, there is an integral of (11) having the form which, when continued analytically, can be taken as the function v.. Thus for the solution of (6) we obtain >?<= j. The asymptotic character of the series is then argued and, finally, the sector within which this representation is valid is determined. The status of the theory thus exhibited seems to me an unsat- isfactory and transitional one. It is to be hoped that ultimately the theory will be so developed that the mere existence of a diver- gent power series as a formal solution of the differential equation will be sufficient for the immediate affirmation of the existence of one or more solutions which are analytic functions with certain specified properties. * Math. Aim., vol. .",1 (1898), p. 34t>. In Crellex Journal, vol. US (1807), still another method is used for obtaining the solutions. 90 THE BOSTON COLLOQUIUM. Tt remains yet to fix the sectors within which the solutions r). can be represented asymptotically by the normal integrals. These sectors have been specified by Horn * in the following manner. Let straight lines be drawn from each singular point a. to every other point and produce each joining line to infinity in both direc- tions. A set of lines will be thus fixed, radiating from the point oo. Let their arguments, taken in the order of decreasing magnitude, be denoted by &) p &).,, , ft) , ft) = CO l 7T, , CO., = CO IT. Suppose now that the argument of the rectilinear part of the path of integration for ?;. in the plane of z lies between w _ 1 and co p . Then ?;. is represented asymptotically by S { for values of the argument of a; between irj'l co and 7r/2 &> p+/ ..t To the general solution of (6), c l rj l -f c 2 rj 2 -f- -f- c n V n > there corresponds the divergent expansion (13) C (' -!;,' + v + ( ' c + c e a " x x p [ C -f "'4- "', 2 " X X" Here the real parts of two exponents, a.x and ax, are equal only when arc/ (a. a.)x is an odd multiple of irj 2; that is, when argx is equal to 7r/2 &>.(/ = 1, , 2/'). Suppose then that for w/2 %_, < wff x < tt/2 co p r we so assign subscripts to the a. that /.'(a,./-) > /.'(a,*) > > /?(a^). Then all the integrals for which o =j= ^ have in common the asymptotic series ',#,, while those for which c = c = = c. ., *Horn, .!///,. .!,/., vol. 50 (1K<)K), p. 531. 1 In certain cases the asymptotic representation may be valid fur a greater range of values of tlie argument of x, as in the case of Bessel's. equation discussed below. DIVERGENT SERIES AND CONTINUED FRACTIONS. 01 c. ={= 0, are represented by <\S r Thus it appears that between the arguments considered S n is the only one of the n asymptotic series S { which defines a solution of the differential equation (6) uniquely. Changes in the asymptotic series representing a solution may occur from two causes, either because x passes through one of the critical values above mentioned for which there is a change in the dominant exponential in (13), or because of a sudden alteration in the values of the constants c. for certain values of the argument. This can be made clear, in conclusion, by illustrating with BessePs equation.* For this equation, as we saw, a x = i, a,, = + i, and hence 'j7T 7T Also since Laplace's transform for the particular case before us is f (* 2 + 1) \& + 3, -^ + (> = 0, the exponent p { for either of the two singular points z = i has the value h Accordingly the series (13) for c l r) l -f- ct/., may be written / C C \ x x + De~^x-^D + ^ + ~l + ) = CU(x) + DV(x), as previously given in (3). If the imaginary part of x is nega- tive, CU(x) is the dominant term in (3) and gives the asymptotic representation of the general solution, c 1 t] l -f c.,77.,. On the other hand, if the imaginary part is positive, the dominant term is * A brief but very interesting discussion is given in a letter of Stokes in the Acta Math., vol. 26 (1902), pp. 393-397. Compare also J. 3 of Horn's article, Math. Ann., vol. 50 (1898), p. 52."). fMath. Ann., vol. 50, p. 539, Eq. IV. 02 T1IK BOSTON COLLOQUIUM. 1> !'(.'') The changes in the values of (' and I) take place only when arg x passes through the values (2/* -f- l)7r/2. Then the coefficient of the dominant term remains unaltered, while the coeffic- ient of the inferior term is altered by an amount proportional to the coefficient of the dominant term.t We conclude, therefore, that in general the asymptotic series for any solution of BessePs equa- tion changes abruptly for values of the argument congruent with (mod 7r). Furthermore, the series can not he valid for a greater range of values of the argument unless when arg x = 0, either I) = or (' = 0. In the former case we have a particular solution G'r;, which is represented by the series CU(x) for 7r <; arg x < arg .' <; 7r. Thus from the infinitely many solutions of BessePs equation having the common asymptotic representation Cl\x) and I)V(x\ respec- tively, these two solutions can be singled out by the requirement that the asymptotic representation shall have the maximum sector of validity. Lkcti'RK 2. The Application of Intermix to Dicerr/eyif Series, In the first lecture a divergent series was connected with a group of functions, for which it afforded a common asymptotic represen- tation. In the present lecture I shall treat of methods which have been used to derive a function uniquely from the series. To establish, whenever possible, such a unique connection, to develop the properties of the function, and to determine the laws and conditions under which the series can be manipulated as a sub- stitute for the function this may be said to be the ultimate aim of the theory of divergent series. Up to the present time this goal has been reached only for a restricted class of divergent series. Furthermore, the uniqueness I Stokes /..,-. ril. DIVERGENT SERIES AND CONTINUED FRACTIONS. ( J3 of correspondence between the function and the scries has been attained, not by a specification of the properties of the function, but by means of some algorithm which, when applied tothe series, yields a single function. Unquestionably the instrument by which the greatest progress has been made thus far is the integral. The first successes, however, were reached by Laguerre * and Stieffjex t through the use of continued fractions, and very possibly in the end the continued fraction will prove to be the best, as it was the earliest tool. But as yet it has been applied only in cases in which the function can be represented under the form of an integral as well as of a continued fraction, although with greater difficulty. To explain the use of integrals let us consider the familiar divergent series treated by Laguerre, (1) 1 -f .,+ 2!,- + :}!,-+.... This is, I believe, historically the first divergent series from which a functional equivalent was derived. % Since *See No. 20 of the bibliography at the end of lecture 6. t Bibliography, No. 2Ga. i Laguerre [loc. cit.) gives the function first in the form of a continued fraction and later proves its identity with the integral which gives rise to the divergent series. Borel at the opening of the second chapter of Les Series divergenles remarks that " Laguerre parait avoir le premier niontre nettement l'utilite qu'il pent y avoir a transformer une serie divergente ... en une fraction continue conver- gente." It seems almost to have escaped notice (see, however, p. 110 of Prings- heim's report, Eucyklopadie der Math. Wiaxenschaften, I A 3), that Euler (Biblio- graphy, Xo. 40) derived a continued fraction from the divergent series 1 -j- nix -f- m( in -j- n)-'' 2 -';- m(//i -f- ) (m -- 2n)x 3 -\- , of which Laguerre' s series is a special case, and clearly realizes the utility of the continued fraction. Moreover, a close parallel to the course followed by Laguerre is found in the work of Laplace who derives from the expression e x- j e-*-dx a divergent series and from this in turn a continued fraction, the convergents of which were stated by him and proved hyJacobi to be alternately greater and less than the expression. Had Jaeobi proved also the convergence of the continued fraction, the work of Laguerre would have had an exact parallel for real values of . Cf. No. 47 of the bibliography. 04 THE BOSTON COLLOQUIUM. "> will represent an analytic function of x over any closed region of the .''-plane which excludes the positive real axis. If, now, t passes through any indefinitely increasing set of values, 'i < * " ; we have in a series of analytic functions which is seen at once to converge uniformly over the region considered, since W ,-) - fj (x) |- J - for sufficiently great values of / and/. The limit (2) is therefore analytic. By deforming the path of integration the same conclusion con- cerning the analytic character of the function (2) can be extended < i >. to all values of x upon the posi- tive real axis excepting and oc, and when the deformation is made on opposite sides of a fixed point x, the two values of the integral will be found to differ by 1 i (3) 2/tt -,>~-. x The integral accordingly represents a multiple-valued function with the singular points and x>, the various branches of which differ from one another by multiples of the period (3). For the initial branch which was given in (2) the limit of /"(/')//i -j- l)th coefficient of (1) if x approaches the origin along any rectilinear path except the positive real axis. Let the process which has been adopted for the series of La- guerre be applied next to any other series (I) + 'V + y + !H] THE BOSTON* COLLOQUIUM. having a finite radius of convergence. If we write the series in the form ,-,)* + + '.{',)* + , then replace the factor n ! by its expression as a P-integral, and finally, by a step having in general only formal significance, bring all the terms under a common integral sign, we shall obtain j", (: +,,,,+ :=,,v + ...v or (4) f e-F(zx)" +/ ( = -'') This integral is the expression upon which Borel builds his theory of divergent scries, and may be regarded as a generalization of a very interesting theorem of Caesar o* The series (5) is called the associated scries of (I). Two cases are now to be distinguished according as the funda- mental series (I) has, or has not, a radius of convergence II which is greater than 0. If the radius is not zero, the associated series has an infinite radius since lim \\ = Inn \| = 0, and it accordingly represents an entire function. It is a simple matter to prove that the integral ( I) will have a sense if x lies within the circle of convergence of (I), and that the values oi the integral and series arc identical. I>ut the integral mav also have a sense for values of , which lie without t he circle, and in this case the integral mav be used to get the analvtie continuation of ( I ). ' < f. r.ul'cl, /.< "'.', i'i tlin ,-,/, /. -. p|i. SS !tS. DIVERGENT SERIES AND CONTINUED FRACTIONS. 7 The scries is said by Borel to be mmmable* at a point .' when the integral (4) has a meaning at this point. The second case is that in which the fundamental series is divergent. The associated series in this case may be either con- vergent or divergent. If it is convergent only over a portion of the plane of u = zx, we are to understand by /'"(") not merely the value of the associated series but of its analytic continuation. Let x for an instant be given a fixed value. Then when z describes the positive real axis, u in its plane describes the ray from the origin passing through the point x. 1 t\u) is holomor- phic along this ray, it is possible that the integral (4) will have a sense. Suppose that this holds good as long as x lies within a certain specified region of its plane. Then for this region a func- tion will be obtained uniquely from the divergent series by the use of the integral, precisely as in the case of the series of Laguerre. This method of treatment is obviously restricted to divergent series for which the associated series are convergent, and it will not always be applicable even to these. A divergent scries in which there is an infinite number of coefficients of the same order of mag- nitude as the corresponding coefficients of (6) 1 + x + (2 !)V + (3 !) 2 .r +... + ( !)V + . can not be summed in this manner. It will be noticed, however, that the scries just given is one whose first associated series is the series of Laguerre, and whose second associated series is conse- quently convergent. The method of Borel can be readily extended so as to take account also of such series, or, more generally, of scries that have an associated series of the uth order which is convergent. One mode of doing this is by the introduction of an /(-fold integral. Suppose, for example, that in (6) one of the two factorials n ! is replaced by e~~z n dz f. * Some other term would be preferable since his definition refers only to one of many possible modes of summation. A series may be simultaneously " sum- mable" at a point x by one method, and non-summable by another. 98 THE BOSTON' COLLOQUIUM. and the other l>v e-'t H (ft. ./; The (// -f l)th term of the series becomes x" I | e-'- s z H t H dz(ft, Jo Jo and we obtain the two-fold integral l Jo 1 e 1 dzdt tzx for the functional equivalent of the series. This is a function, the initial branch of which is analytic over the entire plane of x except at the points and dc. We turn now to the consideration of the region of sumniability, in which x must lie in order that the integral shall have a sense. Borel has determined the shape of this region when the funda- mental scries (I) is convergent, but in so doing he restricts him- self to what he calls the absolutely summable series. The series (1) is said to be absolutely summable for any value of x when the integral (4) is absolutely convergent and when, furthermore, the successive integrals r d K F(zx\ , (<) ' (h , 'k (*= >,2, ) have also a sense.* To fix the shape of the region Borel shows first that if a func- tion defined by a convergent series (I) is absolutely summable at a point P, it is analytic within the circle described upon the line ()P as diameter, connecting P with the origin O ; conversely, if it is analvtie within and upon a circle having OP as diameter, it must be absolutely summable along OP } inclusive of the point * The condition 7] was not originally included in Borel's definition of abso- lute suinruahility {Ann. tie /' J'.'r. .W., >er. ''>, vol. Id, IKD9), and is superfluous in lixin the shape of the region. < f. Math. Ann., vol. 5o (l'.t02), p. 74. The modification of the definition was introduced in the St'rirs divert/rnte* and is needed for the developments explained helow, p. 102. Chapters 3 and 1 of this treatise can lie read in connection with the present lecture. DIVERGENT SERIES AND CONTINUED FRACTIONS. 09 P. As P moves outward from the origin along any ray, the lim- iting position for the circle is one in which it first passes through a singular point S, and at this point SP and OS subtend a right angle. The region of absolute summability can therefore be obtained as follows : Mark on each ray from the origin the nearest singular point of the function defined by (I), if there is such a point in the finite plane. Then through this point draw a perpendicular to the line. Some or all of these per- pendiculars will bound a polygon, the interior of which con- tains the origin and is not penetrated by any one of the perpen- diculars. This region is called the polygon of summability. If the singularities of the function are a set of isolated points, the polygon will be rectilinear. For the extreme case in which the circle of convergence is a natural boundary, the polygon and circle coincide. In every other case the circle is included in the polygon. Thus by the use of (4) Borel effects an analytic con- tinuation of the series over a perfectly definite region whenever an analytic continuation exists. On passing to the exterior of the polygon the series ceases to be absolutely summable. As an example of this result, take the series x 3 x 5 which is the familiar expansion of J log (1 + x)Ul x). The singular points of the function are -f 1 and 1, the circle of convergence is the unit circle, and the polygon of summability is a strip of the plane included between two perpendiculars to the real axis through the points =b 1. When the given series is divergent, the form of the domain of summability has not been determined with such precision. The only information which we have upon the subject is contained in a brief but important communication by Phragmen in the ( 'omjj- trs Rendu* ,* published since the appearance of JtoreF* work. Phragmen considers here the domain, not of absolute, but of sim- ple summability for Laplace's integral x Vol. 132, p. 139(5 ; June, 1901. 100 THE BOSTON COLLOQUIUM. (8) fe-i{zx)ch, ill which f(z%) denotes an arbitrary function. To adopt a term of Mittag-Lefjier, the domain is a "star," which is derived as follows : Draw any ray from the origin. If the series is suminable at a point x {) of tins line, Phragmen shows that it is snmmable at every point between x and the origin 0. There is therefore some point P of the line which separates the interval of summability from the interval of non-summability. If the function is snmmable for the entire extent of the ray, P lies at infinity. In any case let the segment OP be obliterated and then make a cut along the remainder of the line. When the same thing is done for every ray which terminates at the origin, there is left a region called a star, bounded by a set of lines radi- ating from a common center, the point at infinity. Phragmen says that the proof of this result is so simple that it can be given "en deux moU" For this reason I shall repro- duce it here. We are to show that if the integral converges for any value x = x , it will also converge for x = dx^, if <; 6 <; 1. Place f(zx ) = 4>{z) + !f(z). For x = x (t the real and imaginary components of the integrals, (9) C^e-.h, i r ^(z)e-'l: DIVERGENT SERIES AND CONTINUED TRACTIONS. 101 By the change of variable w = 6z this becomes /=a| ("'>' * dw = n\ e d 4>{w)e- w dic. " Je{ic)e-dic, u Je<'i in which a designates an appropriate value between x and 2 . This, as Phragmen says, proves the theorem, but a word or two of explanation additional to his " deux mot* " may not be unac- ceptable to some of my hearers. The necessary and sufficient condition for the existence of the first of the two integrals given in (10) is that by taking two values a, and a sufficiently small or two values sufficiently large, the integral ./ may be made as small as we choose. Now this is true of X(ic)e "d>r since the integrals (9) exist, and equation (11) show then that it must be true likewise of J because the factor ( -' ll(l ~ e) /0 has an upper limit for < 6 X <; < 1 and Math. Wins., I] A 2, \ :>5. 102 THE BOSTON COLLOQUIUM. It might be thought that the result of Plwagmen makes the con- cept of absolute summability useless. This is, however, in no wise the case. At any rate, Borel employs the concept to estab- lish the important conclusion that a divergent series, if absolutely summable, can be manipulated precisely as a convergent series. Thus if two absolutely summable series, whether convergent or divergent, are multiplied together, the resultant series will also be absolutely summable, and the function which it defines will be the sum or product of the functions defined by the two former series. Or, again, if an absolutely summable series is differen- tiated term by term, another such series is obtained, and the latter yields a function which is the derivative of the one defined by the former series. Lastly, the function determined by an abso- lutely summable series can not be identically zero, unless all the coefficients of the series vanish. These facts make possible the immediate application of Borer s theory to differential equations. If, in short, P(x, .'/,//',-, 2/ (n) )= is a differential equation which is holomorphic in x at the origin and is algebraic in // and its derivatives, any absolutely summable series (I), which satisfies formally the equation, defines an analytic function that is a solution of the equation. For example, it will be found that the series of Laguerre satisfies formally the equation and hence the function ./( '-.[ must be a solution of the equation. These conclusions of fiord should be strongly emphasized. In auv complete theorv of divergent series it is an ultimatum that thev shall in all essential points* permit of manipulation In an absolutely tiiimmnhle scries it is not always legitimate to change t lie order of an infinite nuiiilier of terms. < '!'. Morel, Jnurn. , vol. '1 ( Is'tt',). ,,. 111. DIVERGENT SERIES AND CONTINUED FRACTIONS. KKj precisely as convergent series, this property being a requisite for satisfactory application to differential equations. In our preceding exposition of BorePtt theory, we have intro- duced his chief integral by a method which permits of expansion in various directions. Lc Hoy in his very excellent thesis* suggests a change of the function in Laplace* integral which greatly en- larges the applicability t of BorePs method without essentially changing its character. Let the initial series (I) be first written . + ?\p + ] ) r(/+ i) + "; r< -'' + V T(2/h i) + ' ' + "'>'' + , )r ( ,v" + T)+-"' and then replace the second factor in each term by This gives for the formal equivalent of the scries the integral 12) I e-'"^"- 1 F(z.r)(h in which the associated function is now The number /> remains to be fixed. If the series (I) is divergent, there is a critical value of j> such that any smaller value of \> gives an associated scries having a zero radius of convergence, while a larger value gives one with an infinite radius of conver- gence. This critical value // may be stud to gauge 1 or measure * Annates de Toulouse, ser. 2, vol. 2 (1900), p. 41(5. t Since this was written, a very interesting application of Le Roy's idea to differential equations has been made by Maillot, Ann. de /' Ec. Xor., ser. 3, vol. 20 (is!).",), p. 487 ft". 104 THE BOSTON COLLOQUIUM. the degree of divergence of the series. For the divergent series treated by Borel, p' = l. If p' = 0, the series (I) has a finite radius of convergence. On the other hand, when ;/ = oc, Lc Roy's integral can not be applied, but it may be conjectured that such eases will be of very rare occurrence. Le Hoy proposes to employ the integral when the associated series is convergent for p = p' and when also its circle of convergence has a finite radius and is not a natural boundary. The function obtained from (12) will be unique, and he shows that the series which are summable by its use like the series of Borel, can be manipulated as convergent series. One might also inquire whether, in case (13) diverges for j) = p and we take p >//, we shall not get a unique result irre- spective of the value of/>. Other forms of integrals may also be selected for the summa- tion of the series, as for example, :;: in which f J(z)F{Z.r) so that (14) ,= f f{z)-zdz. At first sight this choice of functions would seem to be a very desir- able one, for the function defined by the divergent series is obtained in the familiar form [z)dz zx (.8) *(.)-ff Upon examination, however, it turns out to be otherwise. For suppose the divergent series to be given and f(z) is to be found. The problem is then a very difficult one, that of the inversion of the integral (14) when a n is given for all values of n. This is what Stieltjes terms " the problem of the moments." It does not admit of a unique solution, for Stieltjes himself* gives a function, f(z) = e~*" - s i n ,' Zj which will make a n = for all values of n. If tlie supplementary condition is imposed that f(z) shall not be negative between the limits of integration, only a single solution f(z) is possible, but the divergent series is thereby restricted to belong to that class which Stieltjes derives naturally and elegantly by the considera- tion of his continued fraction. Thus far our attention has been confined exclusively to integrals in which one of the limits of integration is infinite. There are, however, advantages in using appropriate integrals having both limits finite, at least if the given series is convergent and the integral is used for the purpose of analytic continuation. In particular, the integral (16) f(x)= f V(z)F(zx)dz Jo should be noted, to which Iladamard has drawn attention in his thesis. t This falls under Vdllee-Poussi it's theorem when V(z) is *Lor. cit., I ")'). fJourn. de Math., ser. 4, vol. 8 (1892), pp. 158-1(50. iog tup: boston colloquium. continuous along the path of integration and when also F(u) is analytic in it = zx for all values of z upon the path of integration and for values of x in some specified region of the r-plane. If, as we suppose, the path is rectilinear, the values of x to be ex- cluded are evidently those which lie on the prolongations of the vectors from the origin to the singular points of F(x). The region of convergence of (1(5) is consequently a star, whose boun- dary consists of prolongation of these vectors.* Thus Hadamard's integral, when applied to the analytic continuation of a function, is superior to Bard's in the extent of its "region of summability." This is illustrated in Le Hoy's thesis f with the very familiar series : - + , 1:;".. (2 " s -. ,) Here the coefficient 01 .c is 1 r *** . , 7T , 1 |.(1 -z) so that ./'(') <\ (h Since /'('-<') - 1/(1- ~ *'), the region of summa y is the entire plane of .r with the exception of the part of the real axis between :r = 1 and ,/== oc. Hard's polygon of summabilitv for the series, on the other hand, is only the half plane lying to the left of a perpendicular to the real axis through the point .' = 1. Much, it seems to me, can vet be done in following up the use of IJuihimttt'iVa integral. ( )ne special case has been studied already by l.i Hay, in which the (n f 1 )th coefficient of (I) has the form [ --'^(:)~ The series therefore defines a function 1 1 - zx whieh is analytic over the entire plane except along the real axis between x = 1 and x= cc. The path of integration may also permit of deformation so as to show that the cut between the points is not an essential cut. It is interesting to note that if (j>(z) is positive between and 1, the primary branch of the func- tion has only real roots whieh are, moreover, greater than 1.* Lecture 3. On the Determination of the Singularities of Func- tions Defined by Dover Series. Up to the present time comparatively little successful work has been done in determining the singularities of functions defined by power series, and the little which has been done relates mostly to singularities upon the circle of convergence. "Work of this special nature I shall omit from consideration here, thus passing over the memoirs of Fabry, and I shall call your attention to the literature which treats of the singularities in a wider domain. The most fundamental and practical result yet obtained is undoubtedly a brilliant theorem of Iladamard,^ in the wake of whieh a number of other interesting memoirs have followed. This theorem is as follows : If two analytic functions are defined by the convergent power scries (!) (2) +(x) = b i] + b l x + b.^+..., the only singularities of the function (3) f{x) = aj, + aj> x x + a 2 b 2 x 2 + io ill be points whose affixes y r arc the product of (((fixes of the singu- lar points a. and f3. of the first tiro functions. * Le Roy, Joe,, ci!.. pp. :::'>( > -."."> 1. f Acta MdJl,., vol. 22 (lHiKS), p. '>'). 108 THE BOSTON COLLOQUIUM. The possibility that x should, in addition, be a singular point has been pointed out since by Lindelof. Although HadamarcVs proof of the theorem is not a compli- cated one, I shall present here a still simpler proof given by BoreL* Let II and R' be the radii of convergence of (1) and (2) respec- tively, and take a number p such that 7i/p > 1 ft'. If then | zx | = | px | l/R' } the product of + afo + "A"- 2 + Consider now the integral (>) 'llTT , in which c is a closed path surrounding the origin and contained within the circular ring. As long as z in its plane lies within a circle of radius p <; R R', having its center in the origin, the integral will surely define a function of z, and this function is evidently equal to the residue of the integrand for x = 0, which is /(,->). We shall now seek to extend this function by varying z and at the same time deforming appropriately the path of integration. By the theorem of \'//. ,/, hi Sue. Math. y Piii.-li.Tlc iii tlit- H.-i,ilirhi ih-lln /,'. Acrutl. ih-llt Scicnznli liulwpw, new ser., vol. :; i -.. . p,,. iiT-T l. DIVERGENT SERIES AND CONTINUED FRACTIONS. 109 1 ,-i ' g The points x = 1 /5. lie within the circle (1 7/') which is the inner circumference of the ring, while the points x = a.jz before the variation of z lie without the outer circumference (R/p). For simplicity of presentation it may be convenient to assume at first that these points form an aggregate of isolated points. Suppose then that z follows any path in its plane emerging from the eircle (p). Then the points a. z describe certain cor- responding paths which we will mark in the aj-plane. At the same time the contour < may be deformed continuously so as to recede before the points aJz without sweeping over any point 1//S., provided merely that aJz never collides with a point 1 /3. ; that is, z must never pass through a point a./3.. Now when z is held fixed, a deformation in the contour c, subject of course to the condition indicated, produces no change in the value of the integral ,/(), since the integrand is holomorphic between the initial and deformed paths. On the other hand, when the path is kept fixed and z is varied, we have the analytic continu- ation of f(z) in accordance with the theorem of Vallie Poussin. By the two changes together f(z) maybe continued over the entire plane of z with the exception of the points a.j3. = y. . To these should, of course, be added z = oc, also z = as a possible singular point for any branch of f(z) except the initial branch. It should be observed that 7.. is shown to be a potential rather than an actual singular point. When, however, it is such a point, the character of the point depends in general solely upon the nature of the singularities a. and (3. for (1 ) and (2) respectively. This fact w r as noticed by Bore/ and demonstrated in the following manner. Let 11" THE BOSTON COLLOQUIUM. be any convergent series defining a function $ v (x) which is regu- lar at a.. Then tf> 2 (x) = ^(a?) -f (v) is a function which lias at a. the same singularity as (f)(x). The combination of the series for <,(.<') and for ^(.'') by Ilculamard's process gives the function ./>) = ( + <\)K + K+OV; + K>+<- 2 )& 2 2 + =/0>0 +./, in which ./ = <\, ft o + ( 'A X + ^^ + Now since <,(.'') is regular at a., when compounded with i/r(.?) it must give a function ,/",(?) which is regular at 7... It follows that /,(.'') and ,/'(.') have the same singularity at 7... Thus the nature of this singula]' point is not altered by any change in (x) or ^(.c) which does not affect the character of the points a. and /?.. It depends therefore solely upon the character of the singularities compounded. Complications arise only when there is a second pair of singu- larities 7,, ft, such that 7, 7 = *& = A Clearly the resultant singularity is then dependent upon both pairs. Their effects may be so superimposed as to create an ugly singularity, or they may, on the other hand, so neutralize each other that 7, is a regular point. Very simple examples of the latter occurrence can be easily given. It seems probable that when 7 is but once a product of an 1. by a ft, it must always be a singular point, but this has not yet been proved. Its demon- stration will greatly enhance the value and applicability of llada- marrf's theorem, for then it can be stated in numerous cases, not what the singular points of /(.>) may be, but what they actually are. A detailed study of the nature of the dependence of the singu- larity 7 ( upon t. and (3. would probably be both interesting and profitable. Ilorcl examines the case in which oj.and/3 are poles of auv order-, j> and 7, and shows that 7., is then a pole of order l> -f 7 1 . It can. furthermore, be easily shown that whenever a is a pole of the first order, 7. is the same kind of singular point DIVERGENT SERIES AND CONTINUED FRACTIONS. 1 1 1 as ft.. For suppose that we put a, .= 1 , which may be done without loss of generality. The principal part of (/>(.') at the pole t. is then :' 1 and the composition of this with yfr(x) gives for the corresponding component of ,/*(#) Hence the singularities 7.. and /3. differ bv a multiplicative constant. Only one other general fact concerning the composition of sin- gularities seems to be known. Bore/ proves that if the functions (.>) and yjr(x) are one-valued at a. and IS. respectively, f(x) is also one-valued at 7... Thus when two one-valued functions are compounded, the resultant function is also one-valued. But this statement, as he himself points out, must be correctly con- strued and will not necessarily hold true when the singular points of the two given functions are not sets of isolated points but con- dense in infinite number along curves. To construct an example in which /(;) in not one-valued, Borel makes use of the fact, now so well known, that the decision whether the circle of con- vergence is or is not a natural boundary of a given series depends upon the arguments of its coefficients. If, for instance, we take the scries 1 + e**x -f e^ 2 + ..., which has a radius of convergence equal to 1, by a proper choice of the arguments 6 n the circle of convergence can be made a natu- ral boundary. Put now (<>) v 1 -.r = r n + Cl x + 'V'"+ ..., in which the coefficients are necessarily real. Clearly the unit circle will be a natural boundary for <(') = '', + c/ 9l X + r/*x 2 + ... 112 THE BOSTON COLLOQUIUM. and for f(,') = 1 -f ,.-<*.,. + ,.-<,v,*+ .... Yet the function fM which is derived from these two one- valued functions by Hadamard's process is the two-valued func- tion ((3) which exists over the entire plane of x. I have dwelt at some length upon Flarfamard's theorem and its consequences because of their evident interest and importance. It is worthy of note that for analytic functions defined by power series the first great advance in the determination of the singu- larities over their entire domain has been made by methods that are roughly parallel to those currently employed in the considera- tion of their convergence. The convergence of series is indeed too difficult a question to be settled by any one rule or by any finite set of rules, but the methods of comparison with scries known to be convergent have been found to be not only most efficient but also adequate for most practical purposes. In somewhat similar fashion Iladamard'.s theorem will determine the singular points of numerous functions by linking them with other series, of which the singularities are known. One of the simplest applications of this theorem is obtained by compounding a given scries (") " + ",.' 4- , are included among the points obtained by multiplying / affixes of the singular points of (7) among themselves in all possible ways. If the n> scries (X) are multiplied each by a constant /; and are then added, a new series (U) fr'O'J + '''(",)'+ '''("::)''"+ is obtained, in which <'{") denotes the polynomial /."," + -}- /',""' DIVERGENT SERIES AND CONTINUED FRACTIONS. 1 1 3 This function lias no singular points other than those which are possible for the m series from which it was derived. "When r different series o + a x x + a.y- + , K + h i x + b ^ + > '"(J + r i X + ''2 X "~ + t are used, a similar conclusion is readied for the series #K> K '"> r o) + G X a v \ ; "> + G{ (l v K ' r 2 K + > where G denotes a polynomial in which the constant term is lacking. These results are of particular interest when applied to the series (10) 1 + x + 2x 2 -f -f nx n -f and (11) x l 1 4- x + s- + x n 4- 4- , 71 which are the expansions of 1 -f xj{\ x) 2 and log (1 4- x). Since these functions have only one singular point, x = 1, in the finite plane, the only possible singularities of 26 '(">,!>" are x0, 1, oc.* The continued repetition of the above process for combining series leads naturally to a consideration of series of the form (12) 2 J(>" in which a convergent power series P{u) appears in place of the polynomial G(u). Various theorems concerning cases of this * Obviously a constant term can be included now in the polynomial G(n, 1/n). 11 1 THE BOSTON COLLOQUIUM. series have been given recently by Lean,* Le Iioy } i l)e*aint,\ LindeloJ,^ Ford and Faber,\ though the proof of some of these theorems has no direct relation to Iladamai'd'a theorem. The importance of such work is, however, apparent, inasmuch as nu- merous series which occur in analysis can he put into the form under consideration, as for example 2(sin tr n)x n . Three cases must be distinguished according as the radius of convergence of the initial series (7) is less than, equal to, or greater than 1. If the radius is less than 1, the singular point nearest to the origin has a modulus less than l,and the continued multiplication of the affix of the point by itself gives a series of points which approach indefinitely close to the origin. The pre- sumption, therefore, would naturally be that the series (12) is then divergent, but this is very far from being always true, as will be seen at once by referring to the series 3L(x" sin r/Jand 2(.v" cos n ) in which c n is real. The applicability of Iladamard's theorem consequently ceases. The case in which the radius of convergence of (7) is greater than 1 has been investigated very recently by Desa.iat. In this case the expected theorem is obtained. If, namely, P(v) is a conver- gent series without a constant term, 2-P( n ).r" defines a function which can have no singular points, besides x = and .r = oc. than those which result from the multiplication of the affixes of the singular points among themselves in all possible ways and to any number of times.'* Demhd'.s proof is based upon the fact that 1P((i n )z", after the omission of a suitable number of terms, can l>e expressed in the form *. Jouni. (h Math., Moth., ser. ">, vol. 8 (19<)2), p- 433. \ Acta Sor'ftntU Srientinrum Fumirir, vol. 31 (1902). J, .urn. th Moth., ser. ">, vol. 9 I 1903 . p. 223. Moth. A i< it., vol. ">7 ( 1903 . p. 369. ** This is ;i somewhat sharper statement of the result than that given by De- saint. In his theorem j 1 is given as a possible singular point, but this, as appears from the proof to Le given here, is due solely to the admission of a con- stant term into /'(;. He also fails to note that r may be a singular point. DIVERGENT SERIES AND CONTINUED FRACTIONS. 115 /(M/(0 /(/,.) (27ir)*(V f ...y* dtjlt, tlt k , ^^ in which f(t) is the function defined by (7) for x = t, c' is an ap- propriately chosen contour, and c denotes the nth coefficient of P(u) = Cl u -f c.y + . Although his proof is essentially simple in character, I shall give here a new and simpler proof, based directly upon Hadamard's theorem. Place first f.(x) = a<+ a[x + atf + . . . (/ = 2, 3, . . ), and consider the expression in which n denotes some fixed integer. If r > 1 denotes the radius of convergence of the fundamental series (7), the radius of /.(.r) will be >*. Describe about the origin a circle (r) having a radius r' < r n . If a sufficient number of initial terms be cut oif in each of the series, the maximum absolute values of the remainders within or upon the circle (/') can be made as small as is desired. Suppose then that after m terms of each have been removed, the remainders ( 13 ) r n ( X )> ''n-rl('<*)> > >'2n( X ) do not exceed respectively, in which is some small positive number. Let us now substitute in Hadamard's integral 116 TILE BOSTON COLLOQUIUM. v ' l\ir J e x any two of the functions (13) for $ and ^r. Put for example <}>(zx) = r n+i (zx), +(l) = ntj and choose the unit circle as the path of integration. Then if \z\=r', the absolute values of the arguments of the series 1. The conditions for the existence of Hada- mard's integral are therefore fulfilled. Since also +i (zx)' X we have r" - ''+>' r ! di ft+t+j r ( ix\ But by Iladamard's theorem F(z) = ?' 2n+i+ .(z), and hence (i4) h (*)!<*< (1*1^'), for all values of / from 2n to 4/; inclusive. The reasoning can now be repeated with 'In in place of n, and so on ; therefore (14) is true for all values of i= n. Tims far the value of e has remained arbitrary. Let its value now be taken less than the radius of convergence of P (w). Then by (1 4) the series 5 ) vi-'O + 'v,, '-,,+,(*) + will be uniformly convergent in (?'). Since, furthermore, all the component series >',,.,(' = 0, 1, 2, ) are likewise so convergent, by a fundamental and familiar theorem of Weieratrdss* the terms of the collective series (15) may be rearranged into an ordinary series in ascending powers of x. Jmt this rearrangement gives ' IliirkiiL'ssand Murley's IntnidiicAion t<> the Theory nf Analytic Functions, p. 134. DIVERGENT SERIES AND CONTINUED FRACTIONS. 117 jm \ i=n / or the remainder after the (in l)th power of x in (16) l\y - cj\r) - cj 2 (x) o n _J n _ x {x). Now the series (15) before its rearrangement was a uniformly convergent series of analytic functions and defined a function which was analytic within (/'). It follows that (16) is also analytic within this circle, and hence has no singularities within this circle except those of But the radius of (>') was any quantity short of /'", and this con- clusion therefore holds within a circle having its center in the origin and a radius equal to /". By increasing n indefinitely, the theorem of Desaint results. It is evident also that if f x (.r), and therefore f { (x), represents a one-valued function, SP(a i )?;" must also he such a function. There remains yet for consideration the third class of cases in which the radius of convergence of the fundamental series is 1. If upon the circle of convergence there is any singular point with an incommensurable argument, the continued multiplication of its affix by itself gives a set of points everywhere dense upon the cirele of convergence. It is therefore to be expected that this circle will be, in general, a natural boundary for S7 J (^. ),'", and accordingly the cases which will be of chief interest will be those in which all the singular points upon the circle have commensur- able arguments. A simple case of this character is obtained when either (10) or (11) is chosen as the generating series. If the former be selected, the resulting series has the form 2P()a3 n . This has a special interest inasmuch as its study has proved to 118 THE BOSTON COLLOQUIUM. be of profit both for the theory of analytic continuation and of divergent series. The reason becomes apparent when the state- ment is made that it is possible to throw any Taylor's series, 2a as", whether convergent or divergent, into the particular form 2P(?*)as", and in an infinite number of ways. This fact follows as a corollary of a very general theorem of Mittag-Leffler,* which, when restricted to the special case before us, establishes the exist- ence of a function P(x), which is holomorphic over the entire finite plane and assumes the pre-assigned values a , a v a 2 , in the points x = 0, 1, 2, . Consequently the character of the function defined by 2jP(w)os" is made to depend upon the behavior of -P() as x approaches oc. Inasmuch as 2.P(??)os n is perfectly general, limitations must be imposed upon jP(it) in any attempt to extend Hadamarcl's theorem to this series. But whenever the theorem is applicable, the only possible singularities of 2P(n)as" are x = 0, 1, oc. Lean t estab- lishes the correctness of this result when JP(m) is an entire function of order less than 1,| giving also a more general theorem concern- ing SP(J.c" of which this is a special case. The like conclusion holds concerning the singular points of 2J 3 (l/n)as", provided only that P(x) is holomorphic at the origin. || A T ery recently these results of Lean have been proved more simply by Faber, but in a more restricted form, an artificial cut being drawn from x = 1 to x = oc to obtain a one valued func- tion. In addition, Faber shows that if for any prescribed e and for a sufficiently large / the inequality (17) |P(rc'*)| R * I.e Roy three years earlier had noted this conclusion when P{x) is an entire function whose "apparent order" is less than 1 ; he. cif., p. 34S, footnote. Faher does not seem to be aware of Le Roy's statement. The difference between the two statements is slight but becomes important in formulating the new and interesting converse which Faber adds. r Lnr. rit., VA. 120 THE BOSTON COLLOQUIUM. then the principal branch of the function '2,JF > (n)z n will be holo- morphic throughout the complex plane excepting possibly on the segment (1, -f oo ) of the real axis. Furthermore, the function approaches as a limit when x tends toward the point at infinity along any ray having an argument between and 2tt. Lecture IV. On Scries of Polynomials and of Rational Fractions. In the last two lectures I have spoken of the use of integrals for the study of analytic extension and of divergent series. The topic of to-day's lecture is the representation of functions by means of series of polynomials and of rational fractions. This subject forms a very natural transition to the succeeding lecture upon continued fractions, since an algebraic continued fraction is in reality noth- ing but a series of rational fractions advantageously chosen for the study of a corresponding function which, when known, is com- monly given in the form of a power series. The literature relating directly or indirectly to series of poly- nomials and of rational fractions is a vast one, with many ramifi- cations. Thus in one direction there are various researches of importance upon the non-uniform convergence of series of contin- uous functions, and in this connection I may refer particularly to the recent work of Osgood and Bairc, an excellent report of which is contained in Schonjlies' Bcricht i'tber die Mengenlehre.* An- other part of the field comprises numerous memoirs devoted to special series of polynomials and rational fractions. Quite re- cently a more systematic and general study has been begun by /lore/, Mittag-LeJIer, and others, and it is to this that I am to call your especial attention. Two very familiar facts, both discovered by Weiersfrass, may be said to be the origin of this stud v. I refer, of course, to the theorem that any function which is continuous in a given finite interval of the real axis can be expressed in that interval as an ' Jahre.;; 2) ( ) + + z^f ] ^ \x aj ~ x \x~a 2 J ~ \a> - a in which a x , , a are points arbitrarily chosen in the separate pieces. In the familiar case in which only a single analytic function (1) a Q + a x (x a)-\- a 2 (x a) 2 + is given, it is natural to seek a series of polynomials having the greatest possible domain of convergence. Unless the function is one-valued, the most convenient domain is in general the star of Mittag-Lejjler. This is constructed for the series (1 ) by first marking on each ray which terminates in a the nearest singular point and then obliterating the portion of the ray beyond this point. The region which remains when this has been done is a star having a for its center. Mittag-Lefflcr* shows that within the star the given ana- lytic function can be represented by a series of polynomials in which the coefficients of the polynomials depend only upon the value of the function and its derivatives at a, f or, in other words, upon the coefficients of (1). li\ in short, we put .^) = EE'-E ( X l + X 2 + " ' ' + \.) ! (l n X, !X. ' \ ! V " ^' = .7-.7 Jl -,0), ' Ann Math., vol. '_':'. ( ls'.i'.o, p. !:'>; vol. 24, pp. is:;, 20o ; vol. 2f>, p. :>">.".. A pood summary is found in the Prof, of I lie London Moth. Soc, vol. ''>- ( l'.HK)), pp. i In this respect his work is superior to that of Runge and others. Kun^e, for example, presupposes a knowledge of the function at an infinite numher of points. DIVERGENT SERIES AND CONTINUED FRACTIONS. 128 then ^ &,( x ) i s a series which converges uniformly in any region lying, with its boundary, entirely in the interior of the star. The series may also converge outside the star. Borel * has shown, furthermore, that the series of Mittag-Leffier is not the only possi- ble one, but there is an infinity of polynomial series sharing the same property within the star. It will be noticed that the construction of the series of Mittag- Leffier is in no wise dependent upon the convergence of the initial power series. In certain cases, at least, the polynomial series con- verges when the given series (1) is itself divergent. It is natural therefore to look for a theory of divergent series based upon con- vergent series of polynomials. As yet, however, no such theory has been invented. One of the chief difficulties in the way is that the polynomial series do not afford a unique mode of representing an analytic function. Xow the difference between any two series of polynomials for the same function in an assigned area is a third series which vanishes at every point of the area, though the sep- arate terms do not. This is a decidedly awkward point, and occasions difficulty in proving or disproving the identity of two functions expressed by polynomial series. It is true, indeed, that this difficulty will scarcely present itself when we start with a con- vergent power series which is to be continued analytically, the polynomial series then giving continuations of a common function. But when the series (1) is divergent and there is no known func- tion which it represents, it is an open question whether the differ- ent series of polynomials which are obtained from (1) by applica- tion of diverse laws will furnish the same or different functions. If different functions, is there any ground for preferring one series of polynomials to another '! Up to the present time two essentially different principles seem to have been followed in the formation of series of polynomials. In the work of Iiunge, Borel, Vainleve and Jlittag-Lejjter the co- efficients in the polynomials vary with the character of the ana- *Ann. de /'/>. Xor., sct. 2, vol. 1(1 (1899), p. 132, or Les Series divergentes, P . 171. 124 THE BOSTON COLLOQUIUM. lytic function to be represented ; for example, in the polynomials of Mittag-Leffler they are functions of the coefficients of the given element, yYt x n . By appropriately choosing the coefficients of the polynomials these writers obtain a very large region of conver- gence and at the same time are able to greatly vary its shape. On the other hand, the series which are met in the practical branches of mathematics for instance, in the theory of zonal harmonics have the form (2) %&o(*) + *&(*) + *&*) + '' in which the polynomials G n (x) are entirely independent of the function represented, while the c. vary. The polynomials them- selves arc selected according to the shape of the region of con- vergence. Thus if the region is a circle, we may put G n (x) = (x - ay, and we have then the ordinary Taylor's series. Or if it be an ellipse having the foci 1, we may take for our polynomials either the successive zonal harmonics or a second succession of polynomials (also called Legendre's polynomials) which are con- nected by the recurrent relation : (3) 0' .,(.') - 2sb(2i + 2)G n+l (x) + 4(n -f 1) 2 = 0. In a recent number of the Mathematischr Annalen (July, 1903) Fohi'r has considered this second class of polynomials from a some- what general point of view and has demonstrated that any function which is holomorphic within a closed branch of a single analytic curve, as for example an ellipse or a lemniscate of one oval, can be expressed by a series of the form (2). The properties of his -erics are similar to those of Taylor's series. In the case of the latter, to ascertain whether -j >, ). If it is exactlv equal to 1 /.'. the circle ( /.') is the circle of con- DIVERGENT SERIES AND CONTINUED FRACTIONS. 125 vergence, and there is at least one singularity upon its circumfer- ence. If, on the other hand, it is greater or less than 1 R, the series will have a smaller or a larger circle of convergence. So also to the given branch of the analytic curve there corresponds a certain critical value. When this is exactly equal to the upper limit of | i c\ in Faber's series, the given analytic branch is the curve of convergence. At every point within, the series converges, while it diverges at every exterior point, and upon the curve there must lie at least one singular point of the function defined by (2). If, however, the upper limit is greater or less than the critical value, we consider a certain series of simple, closed analytic curves, (as for example a series of confocal ellipses), among which the given analytic branch must, of course, be included. The curve of con- vergence is then fixed by the reciprocal of the upper limit of j | ej provided this limit is not too large. Moreover, as in the case of Taylor's series, the function cannot vanish identically un- less every c n = 0, and in consequence the series vanishes identi- cally. It is therefore impossible that the same function shall be represented by two different series of the given form. In view of the last mentioned fact it might be of especial inter- est to apply this class of polynomial series to the study of diver- gent series. In the most familiar and useful polynomial series the successive polynomials are connected by a linear law of recurrence, (4) LG , (x) + k G . ,(.r) + ... + k G (x) = 0, \ / u+m\ J ' 1 n + m 1\ / m n\ / > in which the coefficients k. are polynomials in x and /(. Thus the zonal harmonics have as their law of recurrence + l)G u+l (x) - (2n + \)xG n {x) + nG n _ x {x) = 0. Many series of this nature are also included in the class con- sidered by Faber. The form of the region of convergence has been determined by Poincare * upon the hypothesis that equation * Amer. Joum. of Math., vol. 7 (1885), p. 243. 126 THE BOSTON COLLOQUIUM. (4) lias a limiting' form for n = oo. Let the equation be first divided through by k Q) and then denote the limits of the successive coefficients for n = oc by l\{x), kjx), /",(*") Construct next the auxiliary equation (5) " 4- fc l ( aJ >- 1 + \{x):^ + + *(*) = 0. Except for particular values of x there will be one root of this equation which has a larger modulus than any other. Let r(x) be that root. Poinoare* shows that with increasing n the ratio G(x)jG n _ 1 (x^ will approach, in general, this root as its limit. The region of convergence is therefore confined by a curve of the form C= \i'(x)\, and the value of C for the series (2) is to be taken equal to the radius of convergence of Scj/'.f By way of illustration let us take the series ~2c h G h (x) in which the polynomial obeys the law *More specifically, Poincare proves that if no two roots of (5) are of equal modulus, G n {x)jG n i(x) has always a limit, and this limit is equal to some root of (5), usually the one of greatest modulus. t Poincare has given no proof that the series (2) will converge at those points within the curve | r(x) | = C, for which there are two or more distinct roots of (5) having a common modulus greater than the moduli of the remaining roots. Thus in the example which is quoted below (p. 127), these are the points of the real axis which are included between -\- 1 and 1. This gap in Poincare's theory can lie filled in by the following theorem which I have given in the Transaction* of the Amer. Math. Soc, vol. 1 (1900), p. 29S : Jf the coeflicients in the series ~.1V" are connected by a recurrent relation having the limiting form the series will converge at t lie worst within a circle whose radius is the recipro- cal of the greatest modulus of any root of the auxiliary equation ... + /--! .;.... .{.* w== 0. Denote this maximum by r, irrespective of the number of root. < fairing this maximum modulus. Then \A [< M(r I 0" ( = 1, 2, ... ). Hence if ( 'is the radius of convergence of ->//", the series 2,c n A will converge when (' . Suppose now th;it A depends upon x and put A n = G(x). It follows thru from my theorem that !>,/,'(., ) will ahraijs converge when C^> r. Hut this i- what was to he proved. At the time of the publication of my work I was not aware of Poincare's article, and 1 therefore tailed to point out the relation of the two memoirs. DIVERGENT SERIES AND CONTINUED FRACTIONS. 127 (,r + 1 )C ir J.r) - 2n*xG n _ ,(,) + (>r + -'' 2 )^ = 0. For // = x the limiting form of this equation is G H+ Jx) - 2xG n+l (x) + G H (x) = 0, or the same as the limiting form for the zonal harmonic. The auxiliary equation is ?} 2xz +1=0, of which the roots are % = X dz V X 2 1 . The curves ! x d= i x 2 1 j = C are easily seen to be ellipses having the foci 1. Hence if i? is the radius of convergence of Sc ?/", the region of convergence of (2) is the interior of an ellipse, S x i/.^ri | = #. Poincare also examines such exceptional cases as that which is specified by relation (3), which has no proper limiting form. But upon this work we can not longer dwell. I wish, however, to emphasize its fundamental character, inasmuch as many previous, and even subsequent conclusions concerning the convergence of series of the form (2) are comprised in Poincare's result. Somewhat earlier in the lecture I set forth the arbitrary charac- ter of the function which could be represented by series of poly- nomials and rational fractions. We have seen also how this arbi- trary element was entirely eradicated by confining ourselves to polynomials which obey a linear law of recurrence. In the remain- der of this lecture I wish to develop the consequences of restrict- ing a series of rational fractions in the manner supposed by Borel in his thesis ;;: and its recent continuation in the Acta Mathematical Borel seeks to so restrict a series of rational fractions, 2P(.t')/i? B (.T), as to ensure a connection between the position of the poles of its sep- arate terms and the position of the singular points of the function which the series collectively represents. On this account he assigns * Ann. de l' Ec. Xor., scr. 3, vol. 12 (1895), p. 1. t Vol. 24 (1900), p. 309. 128 THE BOSTON COLLOQUIUM. an upper limit to the degrees of P n (.<) and JtJ-r). But this is not enough, and he proceeds therefore to limit the magnitude of the co- efficients in the numerators. On the other hand, he allows any dis- tribution whatsoever for the roots of the denominators, thus leaving himself at liberty to vary greatly the nature of the function rep- resented. In his thesis he develops the case 1=1 V" n) which had been previously considered by Poincari * and Goursat.if To avoid semi-convergent series or, in other words, functions, of which the character depends not merely upon the position of the poles a n and the values of A n but also upon the order of summation, the condition is imposed that *SLA n shall be absolutely convergent. Then if there is any area of the z plane which contains no poles, the series (G) must converge within this region. Since further- more it is uniformly convergent in any interior sub-region, it defines an analytic function within the area. There may be several such areas separated by lines or regions in which the poles are everywhere dense. This is precisely the case to be considered now. To simplify matters, let us suppose that the poles are every- where dense along certain closed curves of ordinary character, but nowhere inside the curves. Poincari and Goursat show that each curve is a natural boundary for the analytic function cf>(z) defined by (G) in its interior. BorePs proof is as follows. De- note the component of (G) which corresponds to a n by *> CO * Acta Societal L-t Faniicu, vol. 12 (188H), p. 341, and Amer. Journ. of Math. 1. M ( 18'.i2), ). 201. f Cnnpt. Rmd., vol. 91 (1882), p. 715. -i+ +;t g part by DIVERGENT SERIES AND CONTINUED FRACTIONS. 12 ( J It is evident that if a lies within any one of the curves considered, a r is a pole of (z). Now when these interior poles condense in in- finite number in the vicinity of any point of the curve, it must, of course, be a singularity of (z). Consider next any one of the points a n which lies upon the boundary but is not a point of con- densation of the interior poles, and let z approach this point along the normal. Describe a circle upon the line z a as diameter. If z is sufficiently near to a n , the circle will exclude every one of the points a., excepting a n which lies upon its boundary. Since also S.i is absolutely convergent, by increasing / the second component of .,("-) may be made less in absolute value than ej\z a n \ m , in which is an arbitrarily small prescribed quantity. If, then, // denotes the maximum of the first component of 2 (^) as z now moves up to a n , we have Consequently, lira 00) (z - a J" = lira 4> x (z) (z - a n ) m + lira 2 (z) (z - a) m =B m . This shows that |<(s)| increases indefinitely when?: approaches any pole a n of the //ith order along a normal, and removes the pos- sibility that the poles, because they are infinitely thick upon the curve, may so neutralize one another that the function can be car- ried analytically across the curve at a n . As, moreover, we sup- pose the points a n of order m to be everywhere dense upon the curve, it must be a natural boundary. It is apparent now that the expression (6) continues the initial function (z) across a natural boundary into other regions where it defines in similar manner other analytic functions with natural boundaries. But, it may be asked, is there any proper sense in which these analytic functions may be regarded as a continuation of one another? Just here Borel steps in and, after imposing 130 THE BOSTON COLLOQUIUM. further conditions, shows that when the function defined by (6) within some one of the curves is zero, the functions defined within the other curves must also vanish.* Take m = 1, so that (7) #0 = 2 A " v ' r ~ ' z a i By a linear transformation , (z) into a Taylor's series are the negative of <8) s A - z A - z A ... while those in the expansion for z = x> are (9) 2J tt , ZAa n , 2A H al, . Floret proves that when lim i" A = 0, n n the coefficients (!)) must vanish if those given in (8) do. Any one of the analytic functions under discussion is therefore completely determined by any other, the expression (7) being the intermediary bv which we pass from one to the other. So far as yet appears, this method of continuing an analytic function across a natural boundary is of very limited applicability. Its significance has been made clearer by /Jo/v/'.v later memoir in the . 1 eld Muthciiudit'd. 1 Iere t he rational fractions are of a less highly specialized character, but the essential nature of the investigation can still be exhibited without abandoning the expression (<)). Let A -^ " ', where n denotes the th term of a convergent series v C\~. pp. ''<- ''<'> nf hi> tlu'sis or pp. ! l-'.)s of his Tlu'nrie ). At c erect a perpen- dicular to ab. This will be a line parallel to the assigned direc- tion which throughout its entire extent lies without all the circles, excepting possibly the first X. Hence the series (6) will con- verge absolutely and uniformly along the line, even though the line lie inlinitesimally close to some set of poles in the system. Lastly, because ab was an interval of arbitrary length, these lines of convergence must be everywhere dense throughout the plane, obviously forming a non-enumerable aggregate. 132 THE BOSTON COLLOQUIUM. Since the series is uniformly convergent, it can be integrated term by term. Clearly also the numerators A. in (6) can be so conditioned that the term-by-term derivative of (6) shall be uniformly convergent. Then the derivative of {z) is coincident with the derivative of the series. It is even possible to so choose the A. that the series will be unlimitedly differentiable. I may add that in any region of the plane there will be an infinite or, more specifically, a non-enumerable set of points, through each of which passes an infinite number of lines of con- vergence. If a closed curve is given it will be possible to approximate as closely as desired to this curve by a rectilinear polygon, along whose entire length the series converges and defines a continuous function. Integration around such a polygon gives for the value of the integral the product of 2c7r into the sum of the residues of those fractions whose poles lie in the interior of the polygon. Finally, if we take for axes of x and y two perpen- dicular lines of continuity of (f>(z), all the lines of uniform continuity which meet at their intersection will give a common value for (f>'(z) } and the real and imaginary parts of (f>(z) will satisfy Laplace's equation : ex cif Thus we have in (f>(z) a species of quasi-monogenic function. One question Borel has as yet found himself unable to resolve. If (f)(z) = along a finite portion of any line, will the series in consequence vanish identically? If this question be answered in the affirmative, the analogy with an ordinary analytic function will be still more complete. Let us now return to the case in which two or more functions with natural boundaries are defined by (7). The lines of con- tinuity just described form an infinitely thick mesh-work along which (j)(z) can be carried continuously from the one analytic function into the others. Suppose again that the origin is not a point of condensation of the poles a so that cf)(z) can be expanded DIVERGENT SERIES AND CONTINUED FRACTIONS. 1^3 at the origin into a Maclaurin's scries 1>cz\ Now if a ray is drawn from the origin through the pole a n and the portion of the ray between a n and oo is retained as a cut, the mth term of (7) can be expanded into a series of polynomials a ~, " \a J m 'I ' \ ms which converges over the plane so cut. The series (7) can there- fore be resolved into a double series m n m \ ,/ and this expression will be valid on an infinity of rays from the origin which do not pass through any of the poles. Since, moreover, the poles are an enumerable set of points, these rays will be infinitely dense between any two arguments which may be taken. By fur- ther conditioning the A n , Borel is able to rearrange the terms of the double series so as to form a series of polynomials y^ Q.(z), n in which m=l u m \ w ,a/ and in this way he obtains a series of polynomials which is con- vergent on a dense set of rays through the origin. It also appears that the polynomial series 2 Q n (z) can be formed directly from ILcz 1 without the intervention of (7). When, there- fore a Maclaurin's series is given which corresponds to such an expression (7) as is now under discussion, the continuation of the function can be made along the above set of rays. Now the rays cut any curve upon which either (7) or H,Q n (z) defines a continuous function in a set of points everywhere dense. The value of the function along the entire curve therefore depends only upon the coefficients c. ; i. e., upon the value of the function and its deriva- tives at the origin. It is shown, moreover, that any point of the plane which is not a point of condensation of the poles a n may 134 THE BOSTON COLLOQUIUM. be converted by transformation of axes into such an origin. Finally, Borel gives a case in which the poles may be everywhere dense over the entire plane, so that the function defined by (7) is nowhere analytic, and yet its value is determined along the lines of continuity by the value of the function and its derivatives at the origin. Here then is a class of non-analytic functions sharing a most fundamental property in common with the analytic functions! Is it not then possible, as Borel surmises, that there is a wider theory of functions, similar in its outlines to the theory of ana- lytic functions and embracing this as a special case? If so, the con- ceptions of Weierstrass and of Meray are capable of generalization. Part II. Ox Algebraic Continued Fractions. Lecture 5. Pade's Table of Approximanfs and its Applications. Both historically and prospectively one of the most suggestive and important methods of investigating divergent power series is by the instrumentality of algebraic continued fractions. It is for this reason that I have ventured to combine in a single course of lectures two subjects apparently so unrelated as divergent series and continued fractions. I shall not, however, confine myself to the consideration of the latter subject solely with reference to the theory of divergent series. It is rather my purpose to give some account of the present status of the theory of algebraic continued fractions. At the close of the next lecture a bibliography of memoirs connected with the subject is appended, to which refer- ence is made throughout this lecture and the next by means of numbers enclosed in square brackets. By the term ah/chraiv continued fraction is understood, in dis- tinction from a continued fraction with numerical elements, one in which the elements i, e., the partial numerators and denomi- nators are functions of a single variable x or of several varia- bles [Hi, ft, p. 4]. All hough the term algebraic does not seem to DIVERGENT SERIES AND CONTINUED FRACTIONS. l3o me to bo fortunately chosen, I shall nevertheless accept it and use it to indicate the class of continued fractions which it is proposed to consider here. The first foundations of a theory of continued fractions were laid by Enter, who early employed them [1, > ^] hi his celebrated continued fraction for F(a, fi, y, x)jF(a, /3 + 1,7+ 1, *'). From this time on still other forms were discovered so that it became impossible to speak of a unique development of a function into a continued fraction. Among these forms may be especially mentioned the continued fraction 111 a y r + 6, + a 2 ,r + b., -f a~p + b\ + " '' used by Heine, Tchebychef, and others in approximating to series in descending powers of x. By the substitution of l/.r for x and a simple reduction this can be transformed, after the omission of a factor x, into x~ r~ (3) - ... ". + V + " 2 + b 2 x + a. i x+ b. s + The reason for this variety of form and for the occurrence, in * I'ade in his thesis (p. 38) traces it back to Lambert [2, (.r) denotes an arbitrary rational fraction in which the numerator and denominator are of the ^>th and ^th degrees respec- tively, there will be p -}- q + 1 parameters which can be made to satisfy an equal number of conditions. Let them be so determined that the expansion of N t D in ascending powers of x shall agree with (4) for as great a number of terms as possible. In general, Ave can equate to zero the first p -f- q -f 1 coefficients of the expan- sion of J> S(x) X in ascending powers of x, and no more. Hence, unless X and I) have a common divisor, the series for X D agrees with (4) for an equal number of terms, and the approximation is said to be of the (p -f q -f l)th order. In excep- tional cases the order of the approximation may be either greater or less. Po.de examines these exceptional cases and proves strictly that among all the rational fractions in which the degrees of numer- ator and denominator do not exceed p and q respectively, there is, taken in its lowest terms, one and only one, the expansion of which in a series will agree with (4) for a greater number of terms than any other. Such a rational fraction I shall term an approxl- munf of the given series. The existence of approximants was, of course, well known before Pode, but no systematic examination of them had been made except by Frobeniux [13], who determined the important relations which normally exist between them. Pad*' goes further, and arranges the approximants, expressed each in its lowest terms, into a table of double cut l*v : DIVERGENT SERIES AND CONTINUED FRACTIONS. 137 p=0 p=1 )> 2 7=1 7 = 2 02 II, *u " X, A, , ^,2 a:,, 7> >> a", 1 J "* <1 When the order of approximation of a rational fraction, taken in its lowest terms, is exactly equal to the sum of the degrees of numerator and denominator, increased by 1, the fraction will be found once and only once in the table. If, conversely, a fraction N ID occurs but once in the table, the numerator and denomi- nator are of degree p and q respectively, and the order of the approximation which the fraction affords is exactly p -f q -f 1 . The approximant is then said by PaiU to be normal. We shall also call the table normal when it consists only of normal fractions, or, in other words, when no approximant occurs more than once in the table. Obviously all approximants which lie upon a line perpendicular to the principal diagonal of the table correspond >to the same value of p -f q + 1. Hence in a normal table they approximate to (4) in equal degree, and accordingly may be said to be equally advanced in the table. If p -f q + 1 increases in passing from one fraction to another, the latter is the more advanced. Two approximants will be called contiguous if the squares of the table in which they are contained have either an edge or a vertex in common. Consider now a normal table, and take any succession of approx- imants, beginning with one upon the border of the table and pass- ing always from one approximant to another which is contiguous to it but more advanced. Pade shows that any such sequence of approximants makes a continued fraction of which the approxi- 138 THE BOSTOX COLLOQUIUM. mants are the successive convergents. * Thus a countless manifold of continued fractions can be formed, any one of which through its convergents gives the initial series to any required number of terms and hence defines the series and table uniquely. In all of Pade's continued fractions the partial numerators are monomials in x. The continued fraction is called regular when its partial numer- ators are all of the same degree and likewise its denominators, certain specified irregularities being admitted in the first one or two partial fractions. These irregularities disappear when the continued fraction, as is most usual, commences with the corner element of the table. (Cf. the continued fractions (2) and (3).) In a normal table a regular continued fraction can be obtained in any one of three ways. If we take for the convergents the approximants which fill a horizontal or vertical line, a continued fraction is obtained which except for the irregularity permitted at the outset is of the form (1) given above. If the approxi- mants lie upon the principal diagonal or any parallel line, the con- tinued fraction is of type (3). Lastly, if the convergents lie upon a stair-like line, proceeding alternately one term horizontally to the right and one term vertically downward, the continued fraction is of the familiar form (2). When a table is not normal, the approximants which are iden- tical with one another arc shown by P.de to fill always a square, the edges of which are parallel to the borders of the table. When the square contains (n -f I) 2 elements, the irregularity may be said to be of the th order. The vertical, horizontal, diagonal and stair-like lines give regular continued fractions as before, unless they cut into one or more of these square blocks of equal approxi- mants. When this happens, certain irregularities appear in the continued fraction which give rise to various difficulties in the consideration of matters of convergence 1 and other questions. ( )n this account it is natural to inquire first whether the con- tinued fraction lias or has not a normal character. If it has, the * Tliis is iilso tacitly implied in the relations jjiven by Krobenius [l.'i, p. ">]. DIVERGENT SERIES AND CONTINUED FRACTIONS. 130 existence of the three regular types of continued fractions is as- sured. The necessary and sufficient condition that the table shall be normal is that no one of the determinants a a 'a-S-M ( 'a-fi:Z 'a-B-^2 ( 'a-a-r, c. = if i < 0) <' c shall vanish [16, , p. 35]. It will be noticed that the determi- nants are of the same sort as those which play so conspicuous a role in Hadamard's discussion of scries representing functions with polar singularities. So far as I am aware, the normal character of the table has been established as yet only in the following cases : (1) for the exponential series [37] and for (1 + x) m when m is not an integer [35, e/] ,+ by Fade ; and (2) for the series of Stieltjes, by myself [45] . The construction of PacU's table leads at once to a number of new and important questions. The numerators and the denominators of the approximants constitute groups of polynomials which it is only natural to expect will be characterized by common or kindred properties. The table then affords a suitable basis for the classifi- cation of polynomials. Thus, for example, the polynomials of t At least half of the table for F\ , 1, y, /) lias a normal character. This was proved incidentally in my thesis [70] by showing that the remainders corre- sponding to approximants on or above the diagonal of the table were all distinct. The method of conformal representation was there employed, but the same fact can also be demonstrated very simply by means of Gauss' relntiones inter contiguas (formulas (19) and (20) of [34]). The approximants in the other half of my table (Cf. [76], p. 44) were constructed on different principles from Fade's, the approximation being made simultaneously with reference to tv:o points, = 0and r=oo, but the resulting continued fractions were of the same form as Fade's. It is noteworthy that the relation's inter contiguas lead to such a table rather than to the one of Fade's construction. In the case of F( m, 1, 1, x) = (1 --'')'" the half of Fade's table below the diagonal is also normal, since the reciprocal of the approximants in the lower half are the approximants in the upper half of the table for FKl,l,-a:) = (l + z)-. The normal character of the table for e r then follows since e r lim F(g, 1,1, xfg). 140 THE BOSTON COLLOQUIUM. Legendre and similar polynomials are obtained from the series for log (1 while the numerators and denominators of the approximants for (1 + ?')'" arc the hypergeo metric polynomials /*'( /x, v d= m, fi + v, x), in which /x and v are integers, or the so-called polynomials of Jacobi [65] . In these, as in numerous other cases, the denominators of the convergents and the remainder- functions,* formed by multiplying each denominator into the cor- responding remainder, are solutions of homogeneous linear dif- ferential equations of the 2nd order which have a common group, and the relations of recurrence between three successive denomi- nators or remainder-functions are the relatione* infer contiffuas of Gauss and Riemann. (See in particular, [To, <7] and [~6].) The further study of such groups of polynomials will probably bring to light new and important properties. The position of the roots of the denominators should especially be ascertained, be- cause the distribution of these roots has an intimate connection with the form of the region of convergence of the continued fraction and oftentimes also with the position and character of the function which the continued fraction defines. Probably the most fundamental question concerning Pade's table is that of the convergence of the various classes of continued fractions or lines of approximants. The first investigation of the convergence of an algebraic continued fraction was made by Rie- mann [18] in 1863, followed by Thome [19] a few years later. f Both writers investigated the continued fraction of Gauss by rather painful methods, not based absolutely upon the algo- rithm of the continued fraction but upon extraneous considera- tions. This is not surprising, for there were at that time no gen- eral criteria for the convergence of continued fractions with complex elements, and even now the number is astonishingly small. ' In :it 1 ;i - 1 half of the table. See the preceding footnote. t As Niemann's work appeared posthumously, Thome's lias the priority of publication ( lHfJfJ) hut was itself preceded by Worpitzky's dissertation, to which reference is made in a subsequent footnote. DIVERGENT SERIES AND CONTINUED FRACTIONS. 141 The two principal criteria for convergence correspond to the familiar tests for the convergence of a real continued fraction (5) * % ^ \ + \ + \ + ' in which either (1) all the elements are positive or (2) the partial denominators X. are positive and the partial numerators fi i are negative. The latter class of real continued fractions is known to converge if X. 1 /m { . Pmigsheim [29] has shown that when the elements are complex, the condition I \. | = 1 -f I /**< | is still sufficient for convergence. If, furthermore, the continued frac- tion has the customary normal form in which \x n = 1 , the condition mav be replaced In* the less restrictive one [29, p. 320], The necessary and sufficient condition for the convergence of the first class of real continued fractions can be most easily expressed after it has been reduced to the form 111 , x;+x: + x; + "- . K>)- If then 2X'_ is divergent, the continued fraction converges, while it diverges if 2X' is convergent.* But in the latter case limits exist for the even and the odd convergents when considered separately. This result is included in the following theorem which I gave in the Transactions of 1901 for continued fractions with complex elements [31 ] : If in 1 1 1 *,"+{, + *., + i8 2 + i, + ijSJ + " " the elements a. have all the same sitni and the 8. are alternately positive and negative, t the continued fraction will converge if 2 j a n 4- I8 n ' is divergent ; on the other hand, if 2 I a 4- i/3 ; is *Seidel, Ilabilitationsseliri/t, 1 840, and Stem, Joum. fir Math., vol. 37 (1848), p. 269. t Zero values are permissible for either ; or ?,-. 142 THE BOSTON COLLOQUIUM. convergent and either the a. or the {3. fulfill the condition just stated concerning their signs, the even and the odd convergents have separate limits. The most general criterion for the convergence of 6, b 2 6, 1 + 1+ 1 + ' * ' (6. real or complex) seems to be the one which I gave in October, 1901 [32, b, 5]. Two remarks of a general nature concerning the convergence of algebraic continued fractions may be of interest. In the con- sideration of numerical continued fractions a difficulty frequently encountered is that the removal of a finite number of partial fractions ^J\ i at the beginning of (5) may affect its convergence or divergence. The convergence is therefore not determined solely by the ultimate character of the continued fraction, as is true of a series. Pringsheim [29] has proposed to call the con- vergence unconditional when it is not destroyed by the removal of the first n partial fractions of (5). The difficulties due to con- ditional convergence usually disappear from consideration in treat- ing algebraic continued fractions. For let N n !D n now denote the th convergent. If after the removal of the first n partial fractions the continued fraction converges uniformly in a given region and accordingly represents a function F(s) which is holo- morphic within the region, then after the restoration of the initial terms the continued fraction will define the function which must be either holoinorphie or meromorphic within the given region [.'52, ft or c] . An exception occurs only if the denominator of ('<>) vanishes identically in the region. This is impossible for the second and third tvpes of continued fractions, since the de- velopment of a rational fraction I) /> in either type (2) or {')} consists of a finite number of terms, whereas the develop- ment of /'--), by hypothesis, continues indefinitely. DIVERGENT SERIES AND CONTINUED FRACTIONS. 143 The second remark relating to convergence is that its discus- sion for a continued fraction is usually reduced to the correspond- ing question for an infinite series. The succession of convergents is, in fact, obviously equivalent to the series n \ n1 n/ \ )it2 n+l/ But the latter by means of the familiar relations connecting the denominators or the numerators of three consecutive convergents may be reduced to the form : n \ n n+\ iitI r2 J> J> 3 n+2 n+o We turn now from these general considerations to the questions of convergence connected with Pade's table. Untjer what con- ditions will the various lines of approximants converge; in par- ticular, the three standard types of continued fractions obtained by following (1) the horizontal or vertical lines, ('2) the stair-like lines, and (3) the diagonal lines? When they converge simul- taneously, have they a common limit? If not, what are the mutual relations between the functions which they define? What is the form of the region of convergence ? These and other questions press upon us, and are of great in- terest. A complete investigation has been made only for the exponential series. Pade [37, a] finds that when j> 7 for any suc- cession of approximants X D converges to a value w, the an- proximants converge toward the generating function r' for all values of .'. Furthermore, the numerators and denominators sepa- rately converge, the former to the limit <"''' "' l , the latter to t~ r w ~ l . This smooth result is not, however, a typical one, not even for entire functions. ft is duv at least in part to the fact that e x is 144 THE BOSTON COLLOQUIUM. an entire function without zeros. This will be apparent after an examination has been made of the vertical and horizontal lines of Parte* s table, which we now proceed to consider. It is obvious that the first /> + q + 1 terms of the given series (4) determine an equal number of terms of the series for its re- ciprocal. If, therefore, in the table each approximant is replaced by its reciprocal and the rows and columns arc then interchanged, we shall obtain the table for the reciprocal series. The problems presented by the horizontal and vertical lines of the table are con- sequently of essentially the same character, and our attention may be confined henceforth to the horizontal lines alone. By the interchange just described the zeros and poles of (4) become the poles and zeros respectively of the reciprocal function. In the case of the exponential function the reciprocal series has the same character as the initial series, each defining an entire function without zeros, and the simultaneous convergence of rows and columns for all values of a; was therefore to be expected ; but in general this does not hold. In investigating the convergence of the horizontal lines the first case to be considered is naturally that of a function having a number of poles and no other singularities within a prescribed distance of the origin. It is just this case that 3Iontessus [33, "] has exam- ined very recently. Some of you may recall that four year.- ago in the Cambridge colloquium Professor Ow/ocxl* took HtuhnaanVs thesis t as the basis of one of his lectures. This notable thesis is devoted chiefly to series defining functions with polar singularities. }[i,ntissiis builds upon this thesis and applies it to a table possess- ing a normal character. Although his proof is subject to this limitation, hi- conclusion is nevertheless valid when the table is not normal, as I shall -how in some subsequent paper. The first horizontal row of the table scarcely need- considera- tion, for it consist- of the polynomials obtained by taking suc- cessively 1, 'J, .'!, terms of the series. Consequently the con- tinued fraction obtained from the first row, /;-///. n/tlo An,.r. Slnth. Sr., vol. "), pp. 7-1-7S. i Ju ,r. ,' Moth., -.T. I. vol. S I |-'.i-J |. DIVERGENT SERIES AND CONTINUED FRACTIONS. 145 a o a i x a o a > x a x a 3 x 1 a v x + a a 2 x + a y a s x + a 2 is identical with the series, and its region of convergence is a circle. Let 7t, be the radius of this circle and q { the number of poles of (4) which lie upon its circumference. Suppose also that the next group of poles, q 2 in number, lie upon a circle of radius R 2 , hav- ing its center in the origin ; that q s poles lie upon the next circle (jR 3 ) ; and so on indefinitely or until a circle is reached which con- tains a non-polar singularity. Hadamard (/. c, 18) has proved that the denominators D of the approximants of the (q Y + l)th row, of the (q l -f- q 2 + l)th row, and so on, approach a limiting form as we advance in the row, and that the limiting polynomials give the positions of the first q v q l + q 2 , poles respectively. Thus if, for example, and lim 5 q x' n . Using this result of Hadamard, Montes- sus shows that in a normal table the approximants of the (g, + l)th row converge at every point within the circle (A* 2 ) excepting, of course, at the q l poles but not without this circle ; that the approximants of the ( 1+1+1+1 +"' * Tin- coefficients in the continued fraction of Stieltjcs (discussed later in the lecture i can lie easily so determined as to give a ease of this sort, the region of convergence of (7) being the entire plane with the exception of the negative half of the real axis. \\Y suppose, with Pade* that the absolute term of 1> is taken equal to 1 . t It is perhaps worth noting that the coefficients in the tirst type of continued fractions can not he selected arbitrarily it" it is to be connected with such a table as I'ade constructs. In the cither two tvpes the coefficients are entirely arbitrary. divergent series and continued fractions. 147 a with increasing a approaches a limit, as for instance in the con- tinued fraction of Gauss where liin a ti = \. The significance of the existence of such a limit I first pointed out for a comprehen- sive class of cases in 1901 [32, ], and since then I have shown by simpler methods [32, c] that the result is perfectly general. Let lim a = I:. Then the continued fraction converges, save at isolated points, over the entire plane of x with the exception of the whole or a part of a cut drawn from x = 1 j Ah to x = oo in a direction which is a continuation of the vector from x = to x = 1 1 Ah. Within the plane thus cut the limit of the continued fraction is holomorphic except at the isolated points which (if they exist) are poles. When there is no limit for a n but only an upper limit U for its modulus, the continued fraction (see [32, 6]) is mero- morphic or holomorphic at least within a circle of radius 1/4 U having its center in the origin.* A special case is that in which lim a n = 0. The limit of the continued fraction is then a function which is holomorphic or meromorphic over the entire plane. A comparison of this last result with that of Montessus shows that a much greater region of convergence has now been obtained. This is doubtless, in general, a reason for preferring the second and third types of continued fractions to the first. As another illustration of the second type of continued fraction I shall choose the celebrated continued fraction of Stieltjes [20, a]. In this each coefficient a is positive. By putting x = l/zm (2), the continued fraction, after dropping a factor z, can be thrown into the form 11111 which is the form preferred by Stieltjes. To every such con- tinued fraction there corresponds a series *A demonstration of this property within the circle (1/4T) has been pre- viously given in a dissertation hy Worpitzhj [18 bis], which lias come to my notice for the first time during the examination of the proof-sheets of these lec- tures. This dissertation hears the date I860 and appears to he the earliest pub- lished memoir treating of the convergence of algebraic continued fractions. 148 THE BOSTON COLLOQUIUM. (9) -. ,,2 C 2 -? + - for which c o C l (1 2 C -l A n c l c 2 C 3 C ft (10) B = C l C 2 C 3 * C a C c o, c ,. n+1 n + 2 The correspondence is also a reciprocal one. To every series which fulfills these conditions there corresponds a continued frac- tion of the above type with positive coefficients. From the con- ditions (TO) it follows that c . > and that c lc ,>c ,/c ,, V / I 71/ 71 1 "^ ri1/ n 2 If, therefore, the increasing ratio o n /o n _ l has a finite limit, the series is convergent. On the other hand, if it increases without limit, the series is divergent. In investigating the convergence of the continued fraction the especial skill of Stieltjes was shown. From the relation connect- ing three consecutive denominators (numerators) of the conver- gents it was shown easily that either set of alternate denominators (numerators) made a Sturm's series, whence it follows that all the roots of the denominators (numerators) lie upon the negative half of the real axis of z. This leads naturally to the conjecture that the region of convergence will be the entire plane of z with the exception of the whole or a part of the negative half axis, and that the functional limit will have no zeros exterior to this half of the axis. First the convergence is examined when z is real and positive. The criterion of Seirfef, cited previously in this lecture, then applies. If, namely, SrV is divergent, the continued fraction will converge along the positive axis, while if 2^ is con- DIVERGENT SERIES AND CONTINUED FRACTIONS. 149 vergent, the two sets of alternate convergent* have limits which are distinct. The conclusion is next extended by Stieltjes to the half of the complex plane for which the real part of z is positive. This brings him to the difficult part of his problem, the exten- sion of the result to the other half-plane but with exclusion of the real axis. Here, particularly, Stieltjes [20, a, 30] shows his ingenuity. He overcomes the difficulty by establishing first a preliminary theorem which is of vital importance for sequences of polynomials or rational fractions. The theorem is as follows. Let f 1 {z),f 2 { z )> be a sequence of functions which are holomor- phic within a given region T, and suppose that ^' l=l f n (z) is uni- formly convergent in some part T" of the interior of T. Then if j\(z) +/,(?) + -f f n (z) has an upper limit independent of n in any arbitrary region T' which includes T" but is contained in the interior of 7", the series ^fjz) will converge uniformly in T' and therefore has as its limit a function which is holomorphic over the whole interior of 7 1 *. In the application of this theorem Stieltjes decomposes each convergent X n {z)j D n (z) into partial fractions, M x M, M r -hi eh , + z + , + " ' + z ir>o. a.^o. Ear = From this it follows that N n /D n has an upper limit independent of /; in any closed region of the plane which does not contain a point of the negative half-axis. If now in either the sequence of the odd convergents or of the even convergents we denote the nth term of the sequence by N n fD n and place /iW+/ a W + ---+/ B () = ^ the series 2* =1 / n (2) converges uniformly in any portion of the plane * For a further extension of this line of work, see Osgood, Annats of Math., ser. 2, vol. 'i (1001), p. 25. 150 THE BOSTON COLLOQUIUM. for which the real part of z is positive. All the conditions of the lemma of Stieltjes are now fulfilled, and the region of convergence may be extended over the entire plane with the exception of the negative half-axis. On account of the uniform character of the convergence the limit of either sequence is holomorphic at every point exterior to the negative half-axis. When 2a,', is divergent, the two limits coincide and the continued fraction itself is convergent. On the other hand, if S/ is convergent, the two limits are distinct. Stieltjes shows also that in the latter case the numerators and the denominators of either sequence converge to holomorphic functions p(z), (j(z) of genre 0, and the two pairs of functions are connected bv the equation 7(*W*) - #0/> = 1, which corresponds to the familiar relation D 2a X 2n _ l -D 2a _ l N 2n = 1. A more direct method [31] of demonstrating the convergence results of Stieltjes is by an extension * of the criterion previously cited for the convergence of continued fractions in which the partial fractions l/(a ( -f i/3 n ) have an a n of constant sign and a (3 n of alternating sign. The introduction of the lemma of Stieltjes is consequently unnecessary, hut I wish nevertheless to emphasize its fundamental importance. Other notable results which it will be impossible to reproduce here are also contained in his splendid memoir. -' If namely, S, I " n -\- i^n I is divergent and the condition concerning the signs either of the or of the 3 is fulfilled, the continued fraction will converge pro- vided |,i|/| 1 n \ has a lower or an upper limit respectively. Put now z n- 2 in is') so that it becomes '( ' ' ' ) "' V "i"' : ""' : "3"' ; / When V,r' is divergent, this falls under the extended criterion if we put a'w a \ i '1 , except when z is negative. < >n the other hand, when 5"' ' s con- vergent, the criterion applies without extension directly to (H / ), In cither case the uniform character of the convergence follows with the addition of a few lines. DIVERGENT SERIES AND CONTINUED FRACTIONS. 151 It is interesting to bring this work of Stieltjes into connection with the table of Pu(-v) and ty(x), when expanded formally, give rise to the same divergent series, so also will <(.r ) + c f(x) ~ 1+ c " ' in which < denotes an arbitrary constant. Special properties, however, attach themselves to the two functions picked out by the continued fraction of Stieltjes, upon which we can not linger here. This result of Stieltjes seems to me to be especially significant, since it indicates a division of divergent series into at least two classes, the one class containing the series for which there is prop- erly a single functional equivalent and the other comprising the *Loc. cit., p. 428. 152 THE BOSTON COLLOQUIUM. series which correspond to sets of functions. It is, of course, just possible that this distinction may be due to the nature of the algorithm employed in deriving the functional equivalent of the series, but it is far more probable that the difference is intrinsic and independent of the particular algorithm. If this view be cor- rect, the method of Borel which gives a single functional equivalent, is limited in its application to series of the first class. An extension of the work of Stieltjes has been sought in two dis- tinct directions by modification of the conditions imposed upon his series. Borel [4.'>] so modifies them as to make the series (when divergent) fulfill the requirement imposed in lecture 2 and permit of manipulation precisely as a convergent series. In the last number of the Transaction* * [45] I began a study of series which are subject to only one of the two restrictions expressed in the inequalities (10), but was obliged to bring the work to a hurried close to prepare these lectures. In the main, the corresponding continued fractions have the same properties as the continued fraction of Stieltjes, but a con- siderable difference is shown in regard to convergence. Though the roots of the numerators and denominators of the convergents are still real, they are no longer confined to the negative half of the real axis, and may be infinitely thick along the entire extent of the axis. In certain cases the continued fraction con- verges in the interior of the positive and negative half planes, defining in each an analytic function which has the real axis as a natural boundary. The continued fraction therefore effects the continuation of an analytic function across such a boundary, and gives a natural instance of such a continuation t natural in dis- tinction from artificial examples set up with the express object of showing the possibility of a unique, non-analytic extension. Parfe [1 7, "] has suggested the foundation of a theory of diver- * J uly, lW).'i. i Karlier instances of a natural continuation are also to be found, as, for example, that afforded by across the axis ol reals. DIVERGENT SERIES AND CONTINUED TRACTIONS. 153 gent series upon the continued fractions of his table. The diffi- culties of carrying out the suggestion are undoubtedly very great and have been pointed out by BoreL* Xot only must the con- vergence of the principal lines of approxiraants and the agreement of their limits be investigated, but the combination of two or more divergent series must also be considered. It is not enough to point out, as does Pade, that the approximants of given order for any two series, whether divergent or convergent, determine uniquely the approximants of the same or lower order for the sum- and product-series. For practical application of the theory it must be proved also that the function defined by the table corresponding to the new series is, under suitable limitations, the sum or product of the functions denned by the given divergent series. But great as are the difficulties of such an investigation, even for restricted classes of series, the reward will probably be correspondingly great. So far as it has been yet investigated, the diagonal type of con- tinued fractions seems to have accomplished nearly everthing that can fairly be asked of a sequence of rational fractions. Xot only does it afford a convenient and natural algorithm for computing the successive fractions, but in every known instance the region of convergence is practically the maximum for a series of one valued functions. The continued fraction of Halphen [21 , a\ , so frequently cited as an instance of a continued fraction which diverges though the corresponding series converges, might appear at first sight to be an exception. But this divergence occurs only at special points. In fact, the continued fraction not only converges at the center of the circle of convergence for the series, but, as Halphen himself says, continues the function over the entire plane with the exception of certain portions of a line or curve. If then, continued fractions offer such advantages for known series and classes of functions, is it too much to expect that in the future they will throw a powerful searchlight upon the continuation of analytic functions and the theory of divergent series? Lea Series divergentes, p. 00. 154 THE BOSTON COLLOQUIUM. LECTURE 6. The Generalization of the, Continued Fraction. In the last lecture the algebraic continued fraction was presented under the form of a series of approximants for a given function. An immediate generalization of this conception can be obtained either by increasing the number of points at which an approxima- tion is sought or by requiring a simultaneous approximation to several functions. The latter generalization results at once from an attempt to increase the dimensions of the algorithm or, in other words, the number of terms in the linear relation of recurrence between the successive convergents or approximants. As this generalization is without doubt the more important, I shall make it the chief subject of this lecture. But a few words, at least, should be devoted to the former extension, which is worthy of a more careful and systematic study than it has received. Denote again by N (x)/D,(x) a rational fraction with arbitrary coefficients. These can, in general, be so determined that its ex- pansion at .' = shall agree for n. successive terms with a given series Cq + c v r + c/- + its expansion at x = a for n successive terms with b + h l (x-a l ) + bp-a i y+ , at .' = a., for n., successive terms with k Q + /-,(( - a 2 ) + k.lr - a.,) 2 + , and so on, the total number of conditions thus imposed being equal to [> -f- , <>]. 156 THE BOSTON COLLOQUIUM. I 1 1 X n l 1 z - x ~ (z - J - (a; a { ) ~ z - a t z - a, z - x 1 x a, ( 1 x a 1 + M + - 2 . z-a x z ai \z a 2 z a., z x the series 1 --' -i (-i)(-a 3 ) (*--i)(*-a 2 X-3) is immediately derived, provided that the o. are so distributed as to fulfill proper conditions for the convergence of the series. If now we take successively 1, 2, ?>, terms of the expansion, we obtain the series of polynomials, 1 ,, 1 x-a t and it is evident that ^(V) for the n -f 1 values .> = j, 2 , , a n+ agrees in value with 1 (.: .r). By applying to (1) the well- known formula of Elder [1, a] * for converting any infinite series into a continuous fraction it follows immediately that these poly- nomials are the successive convergents of the continued fraction 1 x a, cc a., a, z a.. _1 - ^ . . ' , ;' a, 1 -1 + v _ l ~l + The generalization of formula (1) can be made at once in the familiar manner by the use of Ccitchy's integral. We get thus , { , 1 rf(z)fk ( : ,_,) r f{z)dz J(J) = 2/tt J z - x =J M + -li-rr J (z - ,) ( 2 - 2 ) + " ' ' which by placing <(') = (''-.)('- "2) (.'- "J may be written *Cf. h'nri/Unpii.li,- ,ln- M>h. Wi,,. } I A :;, [.. i:;i, formula (104). DIVERGENT SERIES AND CONTINUED FRACTIONS. 157 + (-,)(- ' " sna ^ be the values of some analytic function at a set of points a. having a limit point P. We turn now to the generalization of the algorithm of the con- tinued fraction. The first investigation on this subject is found in 158 THE BOSTON COLLOQUIUM. a paper of Jaeobi* published posthumously in 1868. The devel- opments of Jaeobi were, however, of a purely numerical nature. On this side they have been perfected recently by Fr. Meyer [83] . The first example of a functional extension was given by Hermite in his famous memoir [84] upon the transcendence of e, and the theory has been developed since independently of each other by Pineherle and Pade. To explain the nature of the generalization it will be desirable first to refer to the mode in which a continued fraction is com- monly generated. Two numbers or functions, f and/,, are given, from which a sequence of other numbers or functions is obtained by placing A=\A~U (">) A = \A fv A = \A fv in which the X. are determined in accordance with some stated law. For the quotient f jf v we obtain successively /i, 1 1 f ^\ f ^1 1 _ ' ' ' ' A A A and it therefore gives rise to the continued fraction 1 1 By means of the equations (3) each f can be expressed linearly in terms of the initial quantities/^, /',. Thus ( r >) ./',:. = ^ >.../. + ^o. ,,./; in which .1,, , ,, .1, ., are polynomials in the elements A... It is easy to see that these polynomials both satisfy the same difference ' " A 1 1 l: i m m 1 1 h I'hi'oiic dci' ki'itt'tihrtifhiilmliclu'ii Algoritliinen, in wolchen jede Z:i!il :iu- (1 ici v< u'luT^flicnclcn g(.'l)il(k't winl."' Jnurii. fiir Mutii., vol. 69 (l.sf.s), p. _!'.. DIVERGENT SERIES AND CONTINUED FRACTIONS. 150 equation as/!, and for their initial values we have Consequently A ltl and ^1 n are the numerator and denominator of the (n 1 )th convergent of (4). When the generating relations have the form /o = Kfi + ^.4 /, = v; + ^.4 ? the resultant continued fraction is ( } ' \ + \ + ' ' ' ' A distinction then appears between the system of functions (A u +] , A 0> H+1 ) and the system which consists of the numerator and denominator of the ith convergent. Though the quotient of the two functions of either system is the nth convergent, the former pair of functions satisfy the same relation of recurrence as the/*, namely, while the corresponding relation for the other system is The latter equation is called by Pincherle [77, a] the inverse of the former. In the continued fraction (4) we took fx. = 1 so that the two relations were coincident. The immediate generalization of these considerations is obtained by taking m + 1 initial quantities f {) ,f x , ;f m in place of two. With a very slight change of notation we may write 160 THE BOSTON COLLOQUIUM. (6) /, + x 2 / 2 + n 3 f 3 + . . + , m+1 / m+1 =/ m+2 , /- + \- m+1 / t _ m+ l + f M n- m+ 2fn- m +2 + * * ' + "/ =/+! Then by expressing /' ( in terms of the m -f 1 given quantities we have (7) / = ^o, /o + A, /i + + < -*)/!, ? (8) ^0.,H , K i.i - ^,,-, ^0,,,.. A v * m .1 = (-r r The relation which is the inverse of (6) has the form 0') .'/ m + \ff , i + A*. r / jH ,- 2 + ' + ".'/ = .7,,-r To obtain a system of independent integrals of this equation, let DIVERGENT SERIES AND CONTINUED FRACTIONS. 1(51 1\, denote the minor of A 0n in (8), Z 5 , the minor of ^1, n after the first eolumn has been moved over the remaining columns so as to become the last, ]\ n the minor of A., n after the first two columns have been moved over the remaining columns so as to become the last two, and so on. It can be demonstrated easily that the desired system is obtained by placing g { , = P t n (1 = 0, 1, , in), and these new polynomials rather than the Aj are the true analogues of the numerator and denominator of an ordinary continued fraction. The connection between the two systems of polynomials is, however, both an intimate and a re- ciprocal one, for not only is (9) the inverse of (6) but the converse is also true. On this account the two systems can be employed simultaneously with advantage in working with the generalized continued fraction. For all except the very lowest values of n the new polynomials can be found from the equations * (9') 1% + MV, + /"A_ 2 + + vj\ n - m = 1\ . m _x. In place of these relations it will be often found convenient to employ such a process as is indicated in the following equations for m = 2 [83, a, p. 180] .f 1 A 1 _ A, 2 _ 72, 2 A, 3 _ 7i, i ' ., > j) 1x, i <" ,, r l, 1 L 1,2 TZ, 1 J 1, 3 . 73, 72,2 + ".,'' + ., DIVERGENT SERIES AND CONTINUED FRACTIONS. 163 This norm will not be altered in any way by dividing (10) through by S , It is therefore determined uniquely by the ratios of /S , S v $. and conversely the ratios by the norm. Without loss of generality we may set S = 1. Place also , Qn (11) 1\ = B C n c A \A n B, n n . R n = C n+ ,-4+, 1 n+ 1 i A rn Ji If then n -f 3 in (11), is replaced successively by n and n + 1, and the two equations are solved for S l and S 2 , we obtain rv ~h C . . O C JO , y yjii ' t1 n n rl or (12) S } - ?/ = 4" ( x = C, +l S -C S .,), v / II \ n. n+l n n n+\/> and (13) # 2 ~ / = (A*, = *, # +1 - JB^flf). An examination of P n , Q n , R n , \ n , fx, )t will show that their degrees in x are 7il, n '2, h 3, / 1 , /. (n = 2r), 1, a 2, to 3, /!, / 1 (to=2/-+1). Hence the expansions of ^) ; 7 J u and RjP n in descending powers of #, agree with $j and *S' 2 to terms of degree 3/' 1 and 3/' 2 in- clusive if a ='li', and of the 3/'th degree if n = 2r-f-l. The generalized continued fraction therefore affords a solution of the problem : to find two rational fractions with a common denom- inator which shall give as close an approximation to the given functions #, and S 2 as is consistent with the degrees prescribed for their numerators and denominators. When three series in ascending powers of x, % = %>+%*+%&+..- (/= 1,2,3), 164 THE BOSTON COLLOQUIUM. are chosen as the initial functions, a more comprehensive algorithm can be introduced. Pade [79, a] takes three polynomials A^ } (x) } Aft(x), Ap,(x ) with undetermined coefficients, the degrees of which are indicated by their subscripts, and requires that their coefficients shall be so determined that the expansion of A^S. + AVS, + A F" n IF n have limits for n = oo, and these limits are different from 0. (2) There is also one particular integral called by Pincherle the integrate dlstinto the ratio of which to every other integral of (14) has the limit zero. Pincherle's interest is evidently concentrated upon this prin- cipal integral. It seems to me, however, more natural to call the algorithm convergent when the ratios QJP ti and R n /P n (cf. Equations 12 and 13) converge to finite limits for n= oo. Under ordinary circumstances these limits will doubtless coincide with the ratios of the generating functions, fjf^ and/!,//^. In the case of an ordinary continued fraction the two definitions coalesce. For suppose that the nth convergent N n jD n of (4') has the limit L. Then N n LI) is such an integral of the differ- ence equation, that its ratio to any other integral, /;,iV -f k 2 D n , has the limit 0. Conversely, if the principal integral N n LP n exists, there must be a limit I for the continued fraction. Possibly the case in 166 THE BOSTON COLLOQUIUM. which the principal integral is D n might be called an excep- tion, since the continued fraction is then convergent by Pincherlds definition, but lim N n /D n = oo. A study of the conditions of convergence, so far as I am aware, has at present been made in only two special cases. Fr. Meyer [83, a, 7] has made a partial investigation when the coefficients X n , , v n in equations (6) are negative constants. Pincherle [82] has examined the case in which the coefficients of the recurrent relation /, +('V' + <)/,+ > + ^,./U 2 =/,- ; -3 have limiting values and finds that the generalized continued frac- tion is convergent for sufficiently large values of a*. Let the limits of the coefficients be denoted by a, a', and b respectively. To demonstrate the convergence he avails himself of the notable the- orem of Poincare, already cited in Lecture 4. If, namely, no two roots of the equation (15) - bz~ - (ax + ')/- 1 = are of equal modulus, f\f n _ x will have a limit for n = oo, and this limit will be one of the roots of the auxiliary equation (15), usually the root of greatest modulus. From this it follows di- rectly that AJA n _ v B n /B H _ V C n /C n _ i as quotients of integrals of the difference equation last given, also P n jP n _ v Q n j Q n _ v l\JR n _ x as integrals of the inverse equation, have each a definite limit. The existence of limits for Q,J P n and of PIP n is then established for sufficiently great values of x, and the analytic character of these limits is finally argued. Let them be denoted by I\x) and IV). Then A" n = A n + BJJ(x) + CV(.r) is the principal in- tegral of the difference equation, and has the following distinctive property : Its expansion in powers of 1 /.r begins with the highest possible power consistent with the degrees of A n , />' , C h , and coincides with./j for each successive value of n. DIVERGENT SERIES AND CONTINUED FRACTIONS. KIT Bibliography of Memoirs relating to Algebraic ( '<).\ti.\tki> Fractions. In the following bibliography only works in Latin, Italian, French, Gorman, and English are included. In )V6lffi?if/s Mathe- matischer Biichersehatz (heading Kettenbruche) several dissertations, etc., are mentioned which may possibly relate to algebraic con- tinned fractions but which are not accessible to the writer. They are therefore not included here. The writer would be glad to have his attention called to any noteworthy omissions in the bibliography. In many cases it has been extremely difficult to draw the line between inclusion and exclusion, especially under divisions vi IX. Any classification of the material which may be adopted will be open to objections, but even an imperfect classification will prob- ably add greatly to the usefulness of the bibliography. Since much of the work relating to algebraic continued fractions appears elsewhere under other headings, it is believed that such a bibliog- raphy as is here given may be of service. For a brief resume of the theory of algebraic continued frac- tions the reader is referred to Osgood's section of the Encyklopadie tier Math. Wissenschaft, iibi, 38-30. I. On the Derivation of Continued Fractions from Power Series. General Theory. A. Early Works. 1. Euler. (a) Introductio in analysin infinitorum, vol. 1, chap. 18, 1748. (b) De transformatione serierum in fractiones continuas. Opus- cula analytica, vol. 2, pp. 138-177, 1785. -2. Lambert, (a) Verwandlung der Briiche. Beytrage zum Gebrauche der Mathematik und deren Anwendung, vol. 2 V p. 54 ft'., p. 161, 1770. (b) Memoire sur qnelques propriety remarqnables des quantity transcendentes circulaires et logarithmiques. Histoire de l'Acad. roy. des sciences et belles-lettres a Berlin, 1768. 16. 640-643, 1889. The test for convergence given here is included in a more general criterion given later by Bringsheim, No. 29. 23. Pincherle. Sur les fractions continues algebriques. Ann. de l'Ec. Nor., ser. 3, vol. 6, pp. 145-152, 1889. An incomplete result is here obtained. See No. 32c for the complete t heorem. 24. Pade. Sur la convergence des fractions continues simples. Comp. Rend., vol. 112, p. 988, 1891. Also found in 45-47 of No. 16a. DIVERGENT SERIES AND CONTINUED FRACTIONS. 171 25. Banning. Ueber Kugel- und Cylinderfunktionen und deren Ket- tenbruchentwickelung. Dissertation, Bonn, 1894, pp. 1-33. 26. Stieltjes. (a) Rechercbes sur les fractions continues. Annales de Toulouse, vol. 8, J. pp. 1-122, and vol. it, A, pp. 1-47. 1894-95. Published also in vol. 32 of the Memoires presentes a l'Acad. des sciences de 1'Institut National de France. A rich memoir, developing particularly the connection between an important class of continued fractions and the cor- responding integrals. (') Sur un developpement en fraction continue. Comp. Rend., vol. 99, p. 508, 1884. (a") Same subject. Ibid., vol. 108 (1889), p. 1297. (a f// ) Sur une application des fractions continues. Ibid., vol. 118 (1894), p. 1315. (a ,v ) Rechercbes sur les fractions continues. Ibid., vol. 118 (1894), p. 1401. Markoff. (b) Note sur les fractions continues. Bull, de l'Acad. imp. des sciences de St. Petersbourg, ser. 5, vol. 2, pp. 9-13, 1895. This gives a discussion of the relation of his work to that of Stieltjes. 27. H. von Koch, (a) Sur un theoreme de Stieltjes et sur les fonctions definies par des fractions continues. Bull, de la Soc. Math, de France, vol. 23, pp. 33-40, 1895. (a r ) Sur la convergence des determinants d'ordre infini et des fractions continues. Comp. Rend., vol. 120, p. 144. 1S95. 28. Markoff. Deux demonstrations de la convergence de certaines frac- tions continues. Acta Math., vol. 19, pp. 93-104, 1895. Contained also in his Differenzenrechnung (deutsche Ueber- setzung), chap. 7, 21-22. This discusses the convergence of the usual continued frac- tion for %h f_Wy z y when/(i/) > between the limits of integration. 29. Pringsheim. Ueber die Convergenz unendlicber Kcttenbruche. Sitzungsberichte der math.-pbys. Classe der k. bayer'schen Akad. der Wissenschaften, vol. 28, pp. 295-324, 1898. The most comprehensive criteria for convergence yet obtained are found in 29, 31, and 32ft. 172 THE BOSTON" COLLOQUIUM. 30. Bortolotti. Sulla eonvergenza delle frazioni continue algebriche. Atti della R. Accad. dei Lincei, ser. 5, vol. 8 pp. 28-33, 1899. 31. Van Vleck. On the convergence of continued fractions with com- plex elements. Trans. Amer. Math. Soc, vol. 2, pp. 215-233, 1901. 32. Van Vleck. (a) On the convergence of the continued fraction of Gauss and other continued fractions. Annals of Math., ser. 2, vol. 3, pp. 1-18, 1901. (b) On the convergence and character of the continued fraction a x z a.,z a 3 z "f +' 1'+ 1 -}- Trans. Amer. Math. Soc, vol. 2, pp. 476-183, 1901. (c) On the convergence of algebraic continued fractions whose coefficients have limiting values. Ibid., vol. 5, pp. 253-262, 1904. 33. Montessus. (a) Sur les fractions continues algebriques. Bull, de la Soc. Math, de France, vol. 30. pp. 28-36, 1902. The content of this memoir was discussed in lecture 5. (6) Same title. Comp. Rend., vol. 134 (1902), p. 1489. See also 37a', 41. III. Ox Various Continued Fractions of Special Form. A. The Continued Fraction of Gauss. 34. Gauss. Disquisitiones generales circa seriem infinitam , a -.3 a(a + 1)3(3+ 1) ., .r x -*- 1> l-2j(vfl) Deutsche Uebersetzung von Simon, or YYerke, vol. 3, pp. 134- 138, 1812. 34, bis. Vorsselman de Herr. Specimen inaugurate de fractionibus con- tinuis. Dissertation. Utrecht, 1833. Numerous references are given here to the early literature upon continued fractions. 34, ter. Heine. Aus/.ug eines Schreibens iiber Kettenbriiche von Herrn K. Heine an den Herausgeber. Jour, fur Math., vol. 53, pp. 2S4 2S5, 1S57. See also 40c. p. 231. 35. Euler. (a) Commentatio in fractionem continuam in qua illustris Lagrange, pot estates binoiniales expressit. Memoires de l'Acad. imp. (\v> sciences de St. Pctersbourg, vol. 6, pp. 3-11, 1818. DIVERGENT SERIES AND CONTINUED FRACTIONS. 173 Pade. (b) Sur la generalisation des developpements en fractions continues, donnes par Gauss et par Eider, de la function (1 4- x)"\ Comp. Rend., vol. 129, p. 753, 1899. (c) Sur la generalisation des developpements en fractions contin- ues, donnes par Lagrange de la fonction (1 4- .r)"'. Ibid., vol. 129, p. 875, 1899. (d) Sur l'expression generale de la fraction rationuelle approchee de (1 + a;)". Ibid., vol. 132, p. 754, 1901. See also Nos. 11, 32a, 65. B. The Continued Fractions for c r . 36. Winckler. Ueber angeiraherte Bestimmungen. Wiener Berichte, Math.-naturw, Classe, vol. 72, pp. 646-652, 1875. 37. Pade. (a) Memoire sur les developpements en fractions continues de la fonction exponentielle, pouvant servir d' introduction a la theorie des fractions continues algebriques. Ann. de l'Ec. Nor., Ser. 3, vol. 16, pp. 395-426, 1899. (a') Sur la convergence des reduites de la fonction exponentielle. Comp. Rend., vol. 127, p. 444, 1898. See also Nos. 16a". 106, and pages 243-5 of 40c. C. The Continued Fraction of Bessel. 38. Gunther. Bemerkungen iiber Cylinder-Functionen. Archiv der Math, und Phys., vol. 56, pp. 292-297, 1874. 39. Graf, (a) Relations entre la fonction Besselienne de 1" espece et une fraction continue. Anuali di Mat., ser. 2, vol. 23, pp. 45-65, 1895. Giving references to earlier works where the continued frac- tion of Bessel is found. Crelier. (b) Sur quelques proprietes des functions Bessel iennes, dives de la theorie des fractions continues. Annali di Mat., vol. 24, pp. 131-163, 1896. See also Nos. 25, 32a. D. The Continued Fraction of Heine. 40. Heine, (a) Ueber die Reihe x ( - 1) {q _ 1) {q y _ J)"' {q _ 1) (,,-< __ 1} {q y _ 1} {qy r 1 _ 1} *-- 174 THE BOSTON COLLOQUIUM. Jour, fur Math., vol. 32, pp. 210-212, 1846. (b) Untersuchuug iiber die (selbe) Reihe. Ibid., vol. 34, pp. 285- 328, 1847. (c) L T eber die Zahler und Nenner der Naherungswerthe von Ket- tenbruche. Ibid., vol. 57, pp. 231-247, 1860. Christoffel (d) Zur Abhandlung " Ueber Zahler und Nenner" (u. s. w.) des vorigen Bandes. Ibid., vol. 58, pp. 90-91, 1861. 41. Thomae. Beitrage zur Theorie der durch die Heine'sche Reihe darstellbaren Funktionen. Jour, fur Math., vol. 70, 1869. See pp. 278-281 where the convergence of Heine's continued frac- tion is proved. See also 32a. 42. (On Fisenstein's continued fractions). Heine, (a) Verwandlung von Reihen in Kettenbriiche. Jour, fur Math., vol. 32, pp. 205-209, 1846. See also vol. 34, p. 296. Muir. (6) On Eisenstein's continued fractions. Trans. Roy. Soc. of Edinburgh, vol. 28, part 1, pp. 135-143, 1877. Muir plainly was not aware of the preceding memoir by Heine. E. The Continued Fraction of Stieltjes. (See No. 26.) 43. Borel. Les series de Stieltjes, Chap. 5 of his Memoire sur les series divergentes. Ann. del'Ec. Nor., ser. 3, vol. 16, pp. 107- 128 ; and also chap. 2 of his treatise, Les Series divergentes, pp. 55-86, 1901. 44. Pade. Sur la fraction continue de Stieltjes. Com]). Rend., vol. 132, p. 911, 1901. 45. Van Vleck. On an extension of the 1894 memoir of Stieltjes. Trans. Amer. Math. Soc. vol. 4, pp. 297-332. 1903. See also Nos. 27, 102. F. The Continued Fraction for 1 -- nix r "'('" " " )'*'"' !" "*('" f" ") (" l ' 2/ij.r 1 - and its special cases. 46. Euler. () De seriebus divergentibus. Novi eommentarii Acad. scientiarum imperialis LVtropolitaine, vol. 5. pp. 205-237, 1754- 5 : in particular pp. 225 and 2,'52-237. (//) De transformatione seriei divergentis 1 ui.r ' iii(ui n).r- m()ii /()('" \- 2/i).r' DIVERGENT SERIES AND CONTINUED FRACTIONS. 17 in fractionem continuam. Nova acta Acad, scientiarum im- perialis Petropolitanae, vol. 2, pp. 36-45, 178-4. Gergonne. (c) Recherches sur les fractions continues. Gergonne'a Annales de Math., vol. 9, pp. 261-270, 1818. 47. Laplace. (a) Traite de mecanique celeste. Oeuvres, vol. 4, pp. 254-257, 1805. e dx evolvere licet. Jour, fur Math., vol. 12, pp. 346-347, 1834, or AVerke, vol. 6. p. 76. See also p. 79 of No. 20, and the first note under lecture 2. O. Periodic Continued Fraction*, and Continued Fractions Connected with the Theory of Elliptic functions. 48. Abel, (a) Sur 1' integration de la formule differeutielle pdxlV R, R et p etant des fonctions entieres. Jour, fur Math., vol. 1, pp. 185-221, 1826, or Oeuvres, vol 1, p. 104 If. Dobnia. (b) Sur le developpeinent de \R en fraction continue. Nouvelles Ann. de Math., ser. 3, vol. 10, pp. 134-140, 1891. 49. Jacobi. (a) Note sur une uouvelle application de l'analyse des fonctions elliptiques a falgebre. Jour, fur Math., vol. 7, pp. 41-43, 1831, or Werke, vol. 1, p. 327. Borchardt. (h) Application des transcendantes abeliennes a la theorie des fractions continues. Ibid., vol. 48, pp. 69-104, 1854. 50. Tchebychef. Sur 1' integration des differentielles qui contiennent une racine carree d'un polynome du troisieme on du quatrieme degre. Memoires de l'Acad. imp. des sciences de St. Peters- bourg, ser. 6, vol. 8, pp. 203-232, 1857. 51. Frobenius und Stickelberger. Ueber die Addition und Multiplication der elliptischen Functionen. Jour, fur Math., vol. 88, pp. 146- 184. 1880. 52. Halphen. Sur les integrales pseudo-el liptiques. Comp. Rend., vol. 106 (1888), pp. 1263-1270. 53. Bortolotti. Sulle frazioni continue algebriche periodiche. Rendi- conti del Circolo Mat. di Palermo, vol. 9. pp. 136-149, 1895. See also Nos. 21, 26(a), 40. II . Miscellaneous. 54. Euler. () Speculatioues super formula integrali x"d.r \ a 1 2bx ex'' /. 170 THE BOSTON COLLOQUIUM. ubi simul egregire observationes circa fractiones continuas occur- rent. Acta Acad, seientiarum imperialis Petropolitanae, 1784, pars posterior, pp. 62-84, 1782. (/>) Suraraatio fractionis continuce cujus indices progressionem arithmeticam constituunt. Opuscula Analytica, vol. 2, pp. 217- 239, 1785. 55. Spitzer. (a) Darstellung des unendlichen Kettenbruchs i_ X _1 _ X X ~ r x + 1 -t- x + "2 + x +~ '3 + in geschlossener Form, nebst anderen Bemerkungen. Archiv der Math, und Pbys., vol. 30, pp. 81-82, 1858. (b) Darstellung des unendlichen Kettenbruchs 1 1 1 2x + 1 4- 2x + 34- 2x 4-54- 2x + 7 + in geschlossener Form. Ibid., vol. 30, pp. 331-334, 1858. (c) Note iiber eine Kettenbriiehe. Ibid., vol. 33, pp. 418-420, 1859. ((7) Darstellung des unendlichen Kettenbruches Wx) = n(2x+ 1U ? " M in geschlossener Form. Ibid., vol. 33, pp. 474-475, 1859. 56. Laurent, (a) Note sur les fractions continues. Nouvelles Ann. de Math., ser. 2, vol. 5, pp. 540-552, 1866. This treats the continued fraction xxx 1 + 1 Mr"" E. Meyer. (b) Ueber eine Eigenschaft des Kettenbruches x . Archiv der Math, und Phys., ser. 3, vol. 5, x x p. 287, 1903. Meyer's results will be found on p. 548 of Laurent's memoir and differs only in that x lias been replaced by \/x'-. 57. Schlomilch. () Ueber den Kettenbruch fiir tan z. Zeitschrift fur Math, und Phys., vol. 16, pp. 259-260, 1871. Glaisher. (/>) A continued fraction for tan nr. Messenger of Math., ser. 2, vol. 3, p. 137. 1874. (c) Note on continued fractions for tan nx. Ibid., ser. 2, vol. 4, pp. 65-5.K, 1875. DIVERGENT SERIES AND CONTINUED FRACTIONS. 177 58. Schlomilch. I'eber die Kettenbruchentwickelung fur unvollstan- dige Gamma-function. Zeitsehrift fiir Math, und Rhys., vol. 16, pp. 261-262, 1871. This gives the development of ( f-~ ] e~ f dt. Jo 59. Schendel. Uebereine Kettenbruchentwickelung. Jour, fiir Math., vol. 80, pp. 95-96. 1,875. 60. Lerch. Note sur les expressions qui, dans diverses parties du plan, representent des fonctions distinctes. Bull, des sciences Math, ser. 2, vol. 10, pp. 45-49, 1886. 61. Stieltjes. (a) Sur quelques integrales definies et leur developpement en fractions continues. Quar. Jour, of pure and applied Math., vol. 24, pp. 370-382, 1890. (b) Note sur quelques fractions continues. Ibid., vol. 25, pp. 198- 200, 1891. 62. Hermite. Sur les polynomes de Legendre. Jour, fiir Math., vol. 107, pp. 80-83, 1891. This connects D\ y> P (n) (x) with a continued fraction. IV. On the Connection of Continued Fractions with Differen- tial Equations and Integrals. A. Riccati's Differential Equation. 63. Euler. (a) De fractionibus continuis observationes. Commentarii academia: scientiarum imperialis Petropolitame, vol. 11. see pp. 79-81, 1739. (b) Analysis facilis sequationem Riccatianam per fractionem con- tinuam resolvendi. Memoires de 1' Acad, imperiale des sciences de St. Petersbourg, vol. 6, pp. 12-29, 1813. 64. Lagrange. Sur l'usage des fractions continues dans le calcul inte- gral. Nouveaux Mem. de l'Acad. roy. des sciences et belles- lettres de Berlin, 1776, pp. 236-264, or Oeuvres, vol. 4, p. 301 ff. One of the few important early works. See 546 ; also No. 66a for work on differential equations of the 1st order. B. Miscellaneous Differential Equations of the Second Order. In a numerous class of continued fractions the denominators of the convergents satisfy allied (Heun, gleichgrujipiije") differ- ential equations of the second order. Early instances are found in works of Gauss (No. 114), Jacobi (No. 65) and Heine (No. 72). The theory, from two different aspects, is furthest developed in 66a and 76. 178 THE BOSTON COLLOQUIUM. 65. Jacobi. Untersuchung iiber die Differentialgleichung der hyper- geometrischen Reihe. Nachlass. Jour, fur Math., vol. 56, 1859 ; see in particular 8, pp. 160-161, or Werke, vol. 6. p. 184. 66. Laguerre. (a) Sur la reduction en fractions continues d'une frac- tion qui satisfait a une equation differentielle lineaire du pre- mier ordredont les coefficients sont rationnels. Jour, de Math., ser. 4, vol. 1. pp. 135-165, 1885. This is a comprehensive memoir which incorporates substan- tially all the following memoirs : (b) Sur la reduction en fractions continues d'une classe assez et endue de fonctions. Comp. Rend., vol. 87 (1878), p. 923, or Oeuvres, vol. 1. p. 322. (c) Same title as (a). Bull, de la Soc. Math, de France, vol. 8 (1880), pp. 21-27. or Oeuvres, vol. 1. p. 438. (d) Sur la reduction en fraction continue d'une fraction qui satis- fait a une equation lineaire du premier ordre a. coefficients ration- nels. Comp. Rend., vol. 98 (1884), pp. 209-212 or Oeuvres. vol. 1, p. 445. 67. Laguerre. (a) Sur 1' approximation des fonctions d'une variable au moyen de fractions rationnelles. Bull, de la Soc. Math, de France, vol. 5 (1877), pp. 78-92 or Oeuvres, vol. 1. p. 277. (b) Sur le developpement en fraction continue de ." (7)-/.^- Ibid., vol. 5 (1877). pp. 95-99 or Oeuvres. vol. 1, p. 291. (c) Sur la function I I . Ibid., vol. 8 (187!)), pp. 36-52, or Oeuvres. vol. 1, p. 345. (d) Sur la reduction en fractions continues de c F[J > ) F(.r) desig- nant iin polyndme entier. Jour, de Math., ser. 3. vol. 6 (1880), pp. 99 110, or Oeuvres, vol. 1, p. 325. (d') Same subject. Com]). Rend., vol. 87(1878), p. 820, or Oeuvres, vol. 1. p. 318. 68. Humbert. Sur la reduction en fractions continues d'une classe de fonctions. Bull, de la Sue. Math, de France, vol. 8. pp. 182- 1S7. 1871) 1SS0. 69. Hermite et Fuchs. Sur un developpement en fraction continue. Acta Math., vol. I. pp. 89-92, 18S4. See also No. 20. 34 In- 71 76. DIVERGENT SERIES AND CONTINUED FRACTIONS. 179 C. Differential Equations of Order Higher than the Second. 70. Pincherle. Sur la generation u the roots of functions connected by a linear recurrent relation of the second order. Annals of Math., ser. 2. vol. 3, pp. 55 70. 1902. Ser dim Nos. 20. 2) Sur une formule d'analyse. Bull. Phys. Math, de l'Acad. des sciences de St. Petersbourg, vol. 13, pp. 210-211, 1854, or Oeuv- res, vol. 1, pp. 701-702. (c) Sur une nouvelle serie. Ibid., vol. 17. j>p. 257-2G1, 1858, or Oeuvres. vol. 1, pp. 381-384. (d) Sur 1' interpolation par la methode des moindres carres. Mem. de l'Acad. des sciences de St. Petersbourg, ser. 7, vol. 1, pp. 1-24, 1859, or Oeuvres. vol. 1, pp. 473-498. (e) Sur le developpement des fonctions a une seule variable. Bull, de l'Acad. imp. des sciences de St. Petersbourg, ser. 7, vol. 1, pp. J 94-199, 1860, or Oeuvres. vol. 1, pp. 501-508. TX. Miscellaneous. 122. Tchebychef. () Sur les fractions continues algebriques. Jour, de Math., ser. 2, vol. 10. pp. 353-358. 1865, or Oeuvres, vol. 1, pp. 611-614. (b) Sur le developpement des fonctions en series a l'aide des frac- tions continues, 1866. Oeuvres, vol. 1, pp. 617-636. (c) Sur les expressions approchees. lineares par rapport a deux polynomes. Bull, des sciences Math, et Astron., ser. 2. vol. 1, pp. 289, 382 ; 1877. Hermite. (J) Sur une extension donnee a la theorie des fractions continues par 31. Tchebychef. Jour, fur Math., vol. 88, pp. 12-13. 1880. 123. Tchebychef. (a) Sur les valours limites des integrales. Jour, de Math., ser. 2. vol. 19, pp. 157-160, 1874. (h) Sur la representation des valeurs limites des integrales par des residus integraux (1885). Acta. Math. vol. 9. pp. 35-56, 1887. Markoff. (<) Demonstration de certaines inogalitos de M. Tcheby- chef. Math. Ann., vol. 24, pp. 172 178, 1881. (<1 ) Nouvelles applications des fractions continues. Math. Ann., vol. 17, pp. 579-597, 1896. 124. Laguerre. (a) Sur le developpement de (r z) m suivant les puis- sances de (;;'-' 1). Comp. Rend., vol. 86 (1878), p. 956, or < (euvres, vol. 1 . p. "> 1 5. DIVERGENT SERIES AND CONTINUED FRACTIONS. 187 (b) Sur le developpement d'une fonction suivant les puissances d'une polynome. Jour, fur Math., vol. 88 (1880) ; in particular, p. 37, or Oeuvres, vol. 1, p. 298. (c) Same subject. Comp. Rend., vol. 86, (1878) p. 383, or Oeuv- res, vol. 1, p. 295. (d) Sur quelques theoremes de M. Hermite. Extrait d'une lettre addressee a M. Borchardt. Jour, fur Math., vol. 89 (1880), pp. 340-342, or Oeuvres, vol. 1, p. 360. 125. Sylvester. Preuve que it ne peut pas etre racine d'une equation algebrique a coefficients entiers. Comp. Rend., vol. Ill, pp. 866-871, 1890. A fundamental error in the proof has been pointed out by Markoff. See p. 386 of vol. 30 of the Fortschritte der Math. 126. Gegenbauer. Ueber die Naherungsnenner reguliirer Kettenbriiche. Monatshefte fur Math, und Phys., vol. 6, pp. 209-219, 1895. 127. Bortolotti. Sulla rappresentazione approssimata di funzioni alge- briche per mezzo di funzioni razionale. Atti della R. Accad. dei Lincei, ser. 5, vol. l i; pp. 57-64, 1899. Addendum to I A. 128. Euler. De fractionibus continuis dissertatio. Comment. Petrop., vol. 9, ]). 129 fi'., 1737. THE LIBRARY UNIVERSITY OF CALIFORNIA Santa Barbara THIS BOOK IS DUE ON THE LAST DATE STAMPED BELOW. IHJ/n,n^.^."i,^. GlC ^ AL UBRARV FACfLITV A A 000 088 434 r