Bi 7 ' THE PRINCETON COLLOQUIUM u J LECTURES ON MATHEMATICS DELIVERED SEPTEMBER 15 TO 17, 1909, BEFORE MEMBERS OP THE AMERICAN MATHEMATICAL SOCIETY IN CONNECTION WITH THE SUMMER MEETING HELD AT PRINCETON UNIVERSITY, PRINCETON, N. J. BY GILBERT AMES BLISS AND EDWARD KASNER NEW YORK PUBLISHED BY THE AMERICAN MATHEMATICAL SOCIETY 501 WEST 116TH STREET 1913 PRESS OF THE NEW ERA PRINTING COMPANY LANCASTER. PA PREFACE. Soon after its expansion in 1894 into a national organization, the American Mathematical Society inaugurated the series of Colloquia which have been held in connection with its summer meetings since 1896, at intervals of two or three years. These Colloquia consist of courses of lectures delivered by specialists on selected chapters of their fields of work. Their purpose is to enable the members of the Society to keep in touch with the most recent advances of mathematical science and to stimulate a wide interest in its development. The list of Colloquia thus far held is as follows : I. THE BUFFALO COLLOQUIUM, 1896. (a) Professor MAXIME BOCHER, of Harvard University : " Linear Differential Equations, and Their Applications." This Colloquium has not been published, but several papers appeared at about the time of the Colloquium, in which the author dealt with topics treated in the lectures.* (6) Professor JAMES PIERPONT, of Yale University: "Galois's Theory of Equations." Published in the Annals of Mathematics, series 2, volumes 1 and 2 (1900). II. THE CAMBRIDGE COLLOQUIUM, 1898. (a) Professor WILLIAM F. OSGOOD, of Harvard University: "Selected Topics in the Theory of Functions." Published in the Bulletin of the American Mathematical Society, volume 5 (1898), pages 59-87. *Two of these papers were: "Regular points of linear differential equa- tions of the second order," Harvard University, 1896; " Notes on some points in the theory of linear differential equations," Annals of Mathematics, vol. 12 (1898). i 11 PREFACE. (6) Professor ARTHUR G. WEBSTER, of Clark University: "The Partial Differential Equations of Wave Propagation." III. THE ITHACA COLLOQUIUM, 1901. (a) Professor OSKAR BOLZA, of the University of Chicago : " The Simplest Type of Problems in the Calculus of Variations." Published in amplified form under the title: Lectures on the Calculus of Variations, Chicago, 1904. (6) Professor ERNEST W. BROWN, of Haverford College: "Mod- ern Methods of Treating Dynamical Problems, and in Particular the Problem of Three Bodies." IV. THE BOSTON COLLOQUIUM, 1903. (a) Professor HENRY S. WHITE, of Northwestern University: "Linear Systems of Curves on Algebraic Surfaces." (6) Professor FREDERICK S. WOODS, of the Massachusetts Insti- tute of Technology: "Forms of Non-Euclidean Space." (c) Professor EDWARD B. VAN VLECK, of Wesleyan University: "Selected Topics in the Theory of Divergent Series and Continued Fractions." This Colloquium was published for the Society in the volume: The Boston Colloquium Lectures on Mathematics, New York, Macmillan, 1905. V. THE NEW HAVEN COLLOQUIUM, 1906. (a) Professor ELIAKIM H. MOORE, of the University of Chicago : "On the Theory of Bilinear Functional Operations." (6) Professor ERNEST J. WILCZYNSKI, of the University of Cali- fornia: "Projective Differential Geometry." (c) Professor MAX MASON, of Yale University : " Selected Topics in the Theory of Boundary Value Problems of Differential Equations." PREFACE. Ill Published by Yale University in the volume: The New Haven Mathematical Colloquium, New Haven, Yale University Press, 1910. VI. THE PRINCETON COLLOQUIUM, 1909. (a) Professor GILBERT A. BLISS, of the University of Chicago : "Fundamental Existence Theorems." (6) Professor EDWARD KASNER, of Columbia University: "Dif- ferential-Geometric Aspects of Dynamics." This Colloquium is published here in full. The Colloquia of the Society are to an extent comparable with the reports regularly presented to Section A of the British Associa- tion for the Advancement of Science and to the Deutsche Mathe- matiker-Vereinigung, and in so far play a role complementary to those of the Bulletin and Transactions. The Society will doubt- less adopt the custom of publishing the lectures of each Colloquium in a corresponding volume. The Seventh Colloquium will be held in connection with the twentieth summer meeting of the Society at Madison, Wisconsin during the week September 8-13, 1913. Courses of lectures will be given by Professor LEONARD E. DICKSON, of the University of Chicago, and Professor WILLIAM F. OSGOOD, of Harvard University. Thus for the first time an interval of four years has elapsed between successive Colloquia. As a suitable reflection and desirable stimulation of the mathematical activity of this country, it would seem desirable that the Colloquia should be held oftener. To avoid collision with the meetings of the Inter- national Congress of Mathematicians, the Colloquia might per- haps be arranged for every odd numbered year. E. H. MOORE. FUNDAMENTAL EXISTENCE THEOREMS BY GILBERT AMES BLISS CONTENTS Pages INTRODUCTION 1 CHAPTER I ORDINARY POINTS OF IMPLICIT FUNCTIONS 1. The fundamental theorem 7 2. Equations in which the functions are analytic 12 3. Goursat's method of approximation 16 4. Bolza's extension of the fundamental theorem 19 5. The unique sheet of solutions associated with an initial solution 21 6. Auxiliary theorems and definitions 28 7. A criterion that a sheet of solutions be single- valued . . 33 8. Transformations of n variables and a modification of a theorem of Schoenflies 37 CHAPTER II SINGULAR POINTS OF IMPLICIT FUNCTIONS Introduction 43 9. The preparation theorem of Weierstrass 49 10. The zeros of ^b 772, , r]n) is given, then in a neighborhood of these values there is one and but one continuous arc y a = y a (x) (a = 1,2, ,) satisfying the differential equations dy a -fa = 9*( x > Vi> 2/2, , y) (a = 1, 2, , n) and passing through the initial values 77 when x = . 2 1 2 THE PRINCETON COLLOQUIUM. The formulation and first satisfactory proofs of these theorems, at least for the case where only two variables x, y are involved, seem to be ascribed with unanimity to Cauchy. For the implicit functions his proof rested upon the assumption that the function / should be expressible by means of a power series, and the solution he sought was also so expressible, a restriction which was later removed with remarkable insight by Dini. For a differential equation, on the other hand, Cauchy assumed only the continuity of the function g and its first derivative for y, and his method of proof, with the well-known alteration due to Lipschitz, retains to-day recognized advantages over those of later writers. In the following pages (1, 16) the two theorems stated above are proved with such alterations in the usual methods as seemed desirable or advantageous in the present connection. The proof given for the fundamental theorem of implicit functions is applicable when the independent variables x are replaced by a variable p which has a range of much more general type than a set of points in an m-dimensional z-space.* It is not necessary always to know an initial solution in order that others may be found. In the treatment of Kepler's equation, for example, which defines the eccentric anomaly of a planet moving in an elliptical orbit in terms of the observed mean anomaly, one starts with an approximate solution only and determines an exact solution by means of a convergent succession of approximations. This procedure is closely allied to a method of approximation due to Goursat (3), suggested apparently by Picard's treatment of the existence theorem for differential equations. One of the principal purposes of the paragraphs which follow, however, is to free the existence theorems as far as possible from * The notion of a general range has been elucidated by Moore, The New Haven Mathematical Colloquium, page 4, the special cases which he partic- ularly considers being enumerated on page 13. An application of the method of 1 of these lectures when the range of p is a set of continuous curves, has been made by Fischer, "A generalization of Volterra's derivative of a function of a line," Dissertation, Chicago (1912). FUNDAMENTAL EXISTENCE THEOREMS. 3 the often inconvenient restriction which is implied by the words " in a neighborhood of," or which is so aptly expressed in German by the phrase " im Kleinen." It is evident from very simple examples that the totality of solutions (x; y) associated con- tinuously with a given initial solution of a system of equations / = of the form described above, can not in general have the property that the variables y are everywhere single-valued functions of the variables x, and the result of attempting, perhaps unconsciously, to preserve the single-valued character of the solutions has been the restriction of the region to which the existence theorems apply. In order to avoid this difficulty and to characterize to some extent the totality of solutions associated continuously with a given initial one in a region specified in advance, the writer has introduced (5) the notion of a particular kind of point set called a sheet of points. In a suitably chosen neighborhood of a point (a; 6) of the sheet there corresponds to every set of values x sufficiently near to the values a exactly one point (x; y) of the sheet, and the single- valued functions. y so determined are continuous and have continuous first de- rivatives. This condition does not at all imply that there are no other points of the sheet outside the specified neighborhood of the point (a; b) and having a projection x near to a. With the help of the notion of a sheet of points it can be concluded that with any initial solution (a; 6) of the equations / = there is associated a unique sheet S of solutions whose only boundary points are so-called exceptional points where the functions / either actually fail, or else are not assumed, to have the continuity and other properties which are demanded in the proof of the well-known theorem for the existence of solutions in a neighbor- hood of an initial one. It is important oftentimes to know whether or not a sheet of solutions is actually single-valued throughout its entire extent, and a criterion sufficient to ensure this property has also been derived ( 7). On the basis of these results some important theorems con- cerning the transformation of plane regions into regions of 4 THE PRINCETON COLLOQUIUM. another plane by means of equations of the form *i = t\(y\, 2/2), 3-2 = ^2(2/1, 2/2), as in the theory of conformal transformation, have been deduced (8). If the functions ^ have suitable continuity properties and a non-vanishing functional determinant in the interior of a simply closed regular curve B in the z/-plane, and if B is trans- formed into a simply closed regular curve A of the .r-plane, then the equations define a one-to-one correspondence between the interiors of A and B, and the inverse functions so defined have continuity properties similar to those of \f/i and ^ 2 - This is but a sample of the theorems which may be stated. Others are also given ( 8) which apply to the transformation of regions not necessarily finite, and to systems containing more than two equations. The theory of the singularities of implicit functions is of con- siderable difficulty and has been but incompletely developed. For a transformation of the form above in which the functions ^i, ^ 2 are analytic, the singular point to be studied, at which the functional determinant D = d($\, 4 / z)/d(yi, 2/2) vanishes, as well as its image in the .r-plane, may both without loss of gener- ality be supposed at the origin. The most general case under these circumstances is that for which the determinant D does not vanish identically and the equations \f/i = 0, ^ 2 = have no real solutions in common near the origin except the values y l = y 2 = themselves. It is found that the branches of the curve D = bound off with a suitably chosen circle about the origin a number of triangular regions. Each of these regions is transformed in a one-to-one way into a sort of Riemann surface on the z-plane which winds about the origin and is bounded by the image of the boundary of the triangular region (see 11, Fig. 6). If the signs of D in two adjacent triangular regions are opposite, then their images overlap along the common boundary; otherwise they adjoin without overlapping. At any point of one of the Riemann surfaces the inverse functions defined FUNDAMENTAL EXISTENCE THEOREMS. 5 by the transformation are continuous and in the interior of the surface they have everywhere continuous derivatives. These results are obtained by means of applications of the theorem described above for the transformation of the interior of a simply closed curve B; and the same method of procedure would un- doubtedly be of service when the curves \f/i = 0, ^ 2 = have real branches through the origin in common, which must occur whenever they have common 1 points in every neighborhood of the values y\ = yi = 0. The case where the determinant D vanishes identically is also considered ( 12). For the singularities of implicit functions defined by a sys- tem of equations / = there is a generalization of the prepara- tion theorem of Weierstrass ( 9) suggested to the writer by some remarks in the introduction of Poincare's Thesis, and by a study of the elimination theory of Kronecker for algebraic equations. The theorem is presented here (13) for two equa- tions and two variables y\, y-i in the form originally given at the time of the Princeton Colloquium, but the method of proof is similar to that of a later paper* and applies with suitable modi- fications to a system containing more equations and independent variables. These results can not by any means be said to afford a complete characterization of the singularities of implicit functions, but it is hoped that they may be useful in paving the way for researches of a more comprehensive character. The writer published some years ago a paper f concerning the extensibility of the solutions of a system of differential equations, of the form specified above, from boundary to boundary of a finite closed region R in which the functions g a are supposed to have suit- able continuity properties. In the last chapter of these lectures the character of the region has been generalized so that no restrictions as to its finiteness or closure are made, and it is shown that the approximations of Cauchy converge to a solution over an interval * See the footnote to page 73. t " The solutions of differential equations of the first order as functions of their initial values," Annals of Mathematics, 2d series, vol. 6 (1904), page 49. 6 THE PRINCETON COLLOQUIUM. in the interior of which the limiting curve is continuous and interior to R, while at the ends of the interval the only limit points of the curve are at infinity or else are on the boundary of the region. The solutions so defined are continuous and differenti- able with respect to their initial values, a property which once proved is of great service in many of the applications of the existence theorems. One situation in which these results have an important bearing is related to% partial differential equation of the first order F(x, y, z, dz/dx, dz/dy) = 0. When this equation is analytic, any analytic curve C, which is not a so-called integral curve, defines uniquely an analytic surface containing the curve and satisfying the differential equation. The uniqueness in this case is a consequence, in the first place, of the fact that an analytic surface is completely determined when an initial series defining its values in a limited region is given, and, in the second place, of the theorem that at a given point and normal of the initial curve C satisfying the differential equa- tion there is but one series defining an integral surface including the points of C and having the given initial normal. It is not self evident in what sense a solution of a non-analytic equation is uniquely determined by an initial curve, as may be seen by very simple examples. An initial curve which is not an integral curve will in general have associated with it, however, a strip of nor- mals which satisfy the partial differential equation, and whose elements as initial values determine a one-parameter family of characteristic strips simply covering a region R xy of the xy-p\ane about the projection of the initial curve C. There is one and but one integral surface of the differential equation with a continu- ously turning tangent plane and continuous curvature, which is defined at every point of the region R xy and contains the initial curve C and its strip of normals ( 19). CHAPTER I ORDINARY POINTS OF IMPLICIT FUNCTIONS 1. THE FUNDAMENTAL THEOREM The fundamental theorem of the implicit function theory states the existence of a set of functions which satisfy a system of equations of the form (1) f a (xi, :r 2 , -, .r m ; j/i, z/2, , 2/n) = (a = 1, 2, , ri) in a neighborhood of a given initial solution (a; 6). Dini's method,* for the case in which the functions /are only assumed to be continuous and to have continuous first derivatives, is to show the existence of a solution of a single equation, and then to extend his result by mathematical induction to a system of the form given above, a plan which has been followed, with only slight alterations and improvements in form, by most writers on the theory of functions of a real variable. In a more recent paperf Goursat has applied a method of successive ap- proximations which enabled him to do away with the assumption of the existence of the derivatives of the functions / with respect to the independent variables x. One can hardly be dissatisfied with either of these methods of attack. It is true that when the theorem is stated as precisely as in the following paragraphs, the determination of the neighbor- hoods at the stage when the induction must be made is rather inelegant, but the difficulties encountered are not serious. The introduction of successive approximations is an interesting step, * Lezioni di Analisi infinitesimale, vol. 1, chap. 13. For historical remarks, see Osgood, Encyclopadie der mathematischen Wissenschaften, II, B 1, 44 and footnote 30. If Bulletin de la Societe mathematique de France, vol. 31 (1903), page 185. 7 8 THE PRINCETON COLLOQUIUM. though it does not simplify the situation and indeed does not add generality with regard to the assumptions on the functions /. The method of Dini can in fact, by only a slight modification, be made to apply to cases where the functions do not have derivatives with respect to the variables x. The proof which is given in the following paragraphs seems to have advantages in the matter of simplicity over either of the others. It applies equally well, without induction, to one or a system of equations, and requires only the initial assumptions which Goursat mentions in his paper. Where it is possible without sacrificing clearness, the row letters /, x, y, p, a, b' will be used to denote the systems / = (/l,/2, ' ' -,/n), X = (Xi, *2, ' ' ', Xm), y = (y\, 2/2, , yn), a = (ai, a 2 , , dm), b = (bi, 62, , bn), p = (ai, a 2 , , a m ; bi, b z , , &). In this notation the equations (1) have the form /(*; y) = o, the interpretation being that every element of / is a function of xi, x 2 , -, x m ; j/i, j/ 2 , , y n , and every /,- is to be set equal to zero. The notations p t , a t} b t represent respectively the neigh- borhoods x a < e, y b <*; x a < e; \y b\ < e of the points p, a, b. With these notations in mind the fundamental theorem which is to be proved may be stated as follows: Hypotheses : 1) the functions f(x; y) are continuous, and have first partial derivatives with respect to the variables y which are also continuous, in a neighborhood of the point (a; 6) ichich will be denoted by p; 2) /(a; 6) = 0; 3) the functional determinant D = d(fi, / 2 , , fn)/d(yi, yz, , J/n) is different from zero at p. FUNDAMENTAL EXISTENCE THEOREMS. 9 Conclusions : 1) a neighborhood p t can be found in which there corresponds to a given value x at most one solution (x; y} of the equations /(*; y) = 0; 2) for any neighborhood p t with the property just described a constant 5 ^ e can be found such that every x in a s has associated with it a point (x; y) which satisfies the equations f(x; y) = 0; 3) the functions y(x\, # 2 , , x m } so found are continuous in the region a s . For the neighborhood p f let one be chosen in which the continuity properties of the functions / are preserved. If (x; y) and (x; y') are two points in p t , it follows, by applying Taylor's formula to the differences f(x ; y'} f(x; y), that A(*; y') - /i(*; y) = (y/ -*)++ (y-' - y), /n(s; y') -/n(z;y) = ^ (2/1' - yO + ---- h ^7 (y n ' - y B ), where the arguments of the derivatives dfjdy$ have the form z; V + Qa.(y' ~ y}, and < a < 1. The determinant of these derivatives is different from zero when (x; y'} = (x; y} = (a;b), and hence must remain different from zero if p is restricted so that in it the functional determinant D remains different from zero. It is then impossible that (x; y} and (x; y') should both be solutions of the equations f(x; y) = 0, if y is distinct from y'. In the corresponding region b f the function m > when 77 ranges over the closed set of points 77 forming the boundary of b f , on account of the continuity of m 10 THE PRINCETON COLLOQUIUM. remains true for all values x in a suitably chosen domain a & . Hence for a fixed x in a s the minimum of , x m } converge. The coefficients in the functions Y will be greater numerically than the corresponding coefficients of the series y(xi, x 2 , ", Xm), and hence the series y will also converge. To show this suppose that p is a positive constant smaller than the radii of convergence of the functions h(x; y). Then 14 THE PRINCETON COLLOQUIUM. the series h(p; p) are convergent, and each term is numerically smaller than a constant M chosen greater than the sum of the absolute values of the terms in any one of the series h(p; p). The coefficient of any term in h(x; y) is less than M/p" where v is the degree of the term. The series = are similar to the series h(x; y] in the matter of missing terms, and dominate them in the manner described above, since the coefficient of any term of degree v is M/p" or greater. The unique series satisfying equations (6) will evidently be convergent if a convergent series u in x can be determined satisfying _ _ Xm \ ( 1 _ nU\ for then every series y can be put equal to that series u. The latter equation is however a quadratic in u and has the solution 4Jfn(p+3fn) = 1 ' I ^ vanishing with x. This will certainly be representable by a convergent series in x provided that since then the second term under the radical is numerically less than unity. FUNDAMENTAL EXISTENCE THEOREMS. 15 The two theorems which have just been proved enable one to make the following statement concerning the solutions whose existence was proved in 1 : // the functions f(x; y) are analytic in the region p f , then the solutions (4) of the equations f(x; y] = are analytic at every point of the region a s . It is only necessary to transform the origin of coordinates to the particular point (x; y) of the solution which it is desired to investigate. Furthermore when the domain in which the equations / = are to be studied is the domain of complex numbers, a theorem analogous to that of 1 may be stated. // in the domain of complex numbers the functions f(x; y} are analytic at a point p(a; 6) at which f, i^ n r/ t\ I "(/l>/2> '''ijn) f(a; 6) = 0, D(a; b) = -^ x=n * 0, LtffJ/l, 2/2, ' ' ', y n ) J y=b then there exists a neighborhood p e in which any x corresponds to at most one solution (x;y), either real or complex, of the equations f(x; y) = 0. For any such choice of p f a neighborhood a s (5 ^ e) can be found such that every point x in a s has associated with it a solution (x', y) of the equations f = in p f , and the values y for these solutions are defined by a set of functions (7) y a = y a (xi, xt, -, x m ) (a = 1, 2, , ri) which are expressible as series in the differences x a convergent in the region a&. The existence of the neighborhood p f is provable by the ar- gument used in 1, since for any two points (x; y} and (x; y') in the common domain of convergence of the functions /, equa- tions of the form (a= 1,2, ,) hold, where the coefficient A^ is a convergent series in the dif- 16 THE PRINCETON COLLOQUIUM. ferences x a, y b, y f b with constant term equal to a a/3 . The existence of the coefficients A can be established by con- sidering two analogous terms in f(x; y) and f(x; y'). The difference of such a pair of terms will always be linearly expressible in terms of the differences (y.' - to - (y. - to = y.' - y. (o - 1, 2, -, n). Furthermore for (a-, ?/, y') = (a, 6, 6) the derivative of the first member with respect to y ft ' reduces to a a/3 , while that of the second is the constant term in A aB . Hence for these values of the variables the determinant A aft reduces to D(a, 6) 4= 0. By transforming the origin of coordinates to the point (a, 6) and applying the first two theorems of this section, it follows that there exists a set of convergent series (7) satisfying the equations / = identically; and for a sufficiently small region a a the points (x; y} which they define will all lie in the neighborhood p f . 3. GOURSAT'S METHOD OF APPROXIMATION The method of approximation which is to be presented in the following paragraphs is of interest primarily because it affords a direct method of finding the values of implicit functions, and justifies computations sometimes used in the applications of the theory. In order to exhibit this method suppose again that the functions / have the properties described in the principal theorem of 1, and consider the following set of equations suggested by Taylor's formula: fi(x ; y) + an(2/i' yi) + 012(2/2' - 2/2) + + a ln (y n ' y n ) = 0, (8) fn(x ; y} + a n i(yi y\) + a n z(yz 2/2) + + a nn (y n ' y n ) = 0, in which the coefficient a a& is the value of dfjdy^ at the point p. When solved for the variables y', these equations take the form (9) y a ' = (a, b, a, b) b < e(l 6), then the sequence y (k) defined converges uniformly as before in a neighborhood a s of the point a and determines a solution As an example consider the equation (12) y - e sin y = x (0 < e < 1), which in the theory of elliptic orbits determines the value of the eccentric anomaly y in terms of the mean anomaly x. The func- tion

y remains less than 6 when 1 - e _ e For any given x = a, a value y = b can be determined, by graph- ical methods for example, so that i , T x , | b - e sin b - a (a, b, a, b) b \ = 1 e cos 6 The process described above therefore converges in a suitably chosen neighborhood of x = a, and a solution of equation (12) can be found when an approximate solution only has been de- termined in advance. 4. BOLZA'S EXTENSION OF THE FUNDAMENTAL THEOREM* The neighborhood P, of a set of points P in the space (x; y) is the totality of points (x; y} which satisfy inequalities of the form x a | < e, \y b\ < e, * Vorlesungen iiber Variationsrechnung, page 160: also Mathematische Annalen, vol. 63 (1906), page 247. The theorem was proved independently by Mason and Bliss, " Fields of extremals in space," Transactions of the American Mathematical Society, vol. 11 (1910), page 326. 20 THE PRINCETON COLLOQUIUM. where (a; 6) is some point of P. The sets of points (a) and (b) which belong to points (a; 6) of P are the projections of P in the .r- and ?/-spaces, and will be denoted by A and B, respectively. The fundamental theorem of 1 remains true if in its statement the single point p is replaced by a set of points P which is finite and closed, and ichich furthermore has the property that no two distinct points (a; b), (a'\ b'} of P haw the same projection a' = a. According to the conclusions of the theorem there exists then a neighborhood P e in which no two solutions of the equations f(x; ?/) = have the same projection x, and a neighborhood A s in which every x surely belongs to a solution (x; y) in P e . The single-valued functions y(xi, a- 2 , , Xm) so defined in A s are continuous, and if the func- tions f(x; y] have continuous derivatives of the n-th order in a neighborhood of P, so have the functions y(xi, .r 2 , , x m } in A$. To prove the theorem suppose first that a sequence of positive constants & (k = I, 2, ) approaching zero has been selected arbitrarily. If the first part of the theorem were not true, then in any neighborhood P fk there would be two distinct solutions (.r; y}k and (x; y'}k of the equations f(x; y) = 0, which would satisfy, respectively, inequalities of the form x a < k , y ft < A-; (13) x-a'\c which would belong to no solution in P g . To each (x)k there would correspond a point (a)k in A satisfying the inequalities x a < with the values (x)k, and the points (a)k would have a point of condensation a in A, which would also be a point of condensation for the sequence (x)k, since A is finite and closed when P is so. But by the original theorem of 1, again, it is known that a neighborhood a s of a can be chosen in which every point x has associated with it a solution (x; y) in p e , where p(a;b) is the point of P having the projection a. Consequently the existence of the sequence (x)k is contradicted. If now the region P e is so restricted that the functional de- terminant D(x; y) remains different from zero throughout it, then the original theorem of 1 can be applied to show that the functions y(xi, x 2 , , x n ) are continuous at any point of the region A & and possess as many continuous derivatives as are pos- sessed by the functions f(x; y). 5. THE UNIQUE SHEET OF SOLUTIONS ASSOCIATED WITH AN INITIAL SOLUTION The points of the space (x; y) may be divided into two classes, ordinary points and exceptional points, with respect to the func- tions /. An ordinary point is one at which the first and third hypotheses of the theorem of 1 are postulated, that is, one near which the functions/ and their first derivatives with respect to y are continuous and the functional determinant D = d(/i,/ 2 , , 22 THE PRINCETON COLLOQUIUM. fn)/d(y\, yz, ' , y n ) is different from zero. An exceptional point is one at which some of these conditions are not fulfilled or are not presupposed. A sheet of points in the (ra -f- w)-dimensional space (x; y} may be defined as a point set S with the property that for any point p(a;b) belonging to the set a neighborhood p f can always be found such that no two points of S in p f have the same pro- jection x. In other words, the variables y are single-valued functions y(x\, ar 2 , , x m ) in the neighborhood of the point p, for points of the sheet. If for any neighborhood 6, of the kind just described, a region a s (5^ ) can be found in which every point x belongs to a point of S in p t , then p is said to be an interior point of the sheet S. A boundary point is a limit point of points of the sheet, which is not itself an interior point and may not even belong to S. A sheet is said to be connected if every pair (x';y f ), (x"\y") of its interior points can be joined by a continuous curve *=*(*), y = y(t) (t'^t^t"), consisting entirely of interior points of the sheet. In the following pages it is always to be understood that the sheets considered are continuous and have continuous first derivatives, or in other words at any interior point of one of them the functions ^(a-i, z 2 , -, x m } mentioned above have these properties. A sheet will be said to become infinite near a point x' if x' is the limit of the projections of a sequence of points (x; y) of the sheet for which one at least of the variables y approaches infinity. With the preceding agreements as to nomenclature in mind, it is possible to prove the following theorem: // a point p(a; 6) is an ordinary point for the functions f and satisfies the equations f = 0, then there passes through p one and only one connected sheet of solutions of these equations, with the properties : 1) all points of the sheet are ordinary points of the functions f; 2) all points are interior points', FUNDAMENTAL EXISTENCE THEOREMS. 23 3) the only boundary points of the sheet are exceptional points for the system f. The set of points [*i, x 2 , , x m ; y\(xi, x 2) -, x m ), , y n (xi, z 2 , - , x m }} defined over the region a & by the principal theorem of 1, is a sheet *Si of solutions of the equations/ = which satisfies all the requirements of the theorem just stated except possibly the last. Its points are all interior points since the region a s is defined by inequalities only. If any boundary point p'(a f ; b') of S\ is an ordinary point of the functions / it must satisfy the equations / = 0, since the /'s are continuous and p' is a limit point of points on Si. Consequently the theorem of 1 can be applied in the neighborhood of p', and the sheet S' so determined near p' forms with Si a new set S 2 . This process may be repeated any number of times, and the totality of points which can be attained by a finite number of such extensions, constitutes the sheet S required in the theorem. The set of points S so determined constitutes a sheet, since any point q of it is an ordinary point and a solution of the equa- tions / = 0, and according to the theorem of 1 the solutions of these equations in the neighborhood of q have the property which is characteristic of a sheet. From the manner of its construction the sheet is evidently connected and consists entirely of interior points. If any boundary point q of S were an ordinary point of the functions /, the sheet could be extended to include q as an interior point by the process described in the preceding paragraph. There could not be a second sheet S containing a point TT not in S and having the properties stated in the theorem. For there would in that case be a continuous curve x = x(t), y = y(t) (h^t^ # 2 ) in S joining p with TT and consisting entirely of ordinary points. In a neighborhood of t = ti all of the points defined on the curve would also be points of S, since the solutions of the equations 24 THE PRINCETON COLLOQUIUM. / = near the initial point p of the curve are all in S. The values of t defining points on the curve and in S would therefore have an upper bound T ^ i* such that T would define on the curve a boundary point of S. But this is impossible since all of the points of the curve are ordinary points. If the functions / are known to be continuous and to have con- tinuous derivatives in a region R, then it follows readily from what precedes that through any ordinary solution of the equations/ = interior to R there passes one and only one sheet of solutions having the property that the only boundary points of the sheet are boundary points of R, or interior points of R at which the functional determinant vanishes. If R is finite and closed and consists entirely of ordinary points for the functions/, then there can not be more than a finite number of points of the sheet on any ordinate x. Otherwise the points common to the ordinate and the sheet would have a point of condensation p, also in R. Since p is an ordinary point there can be at most one solution of the equations in a properly chosen neighborhood p e . It is interesting to determine a criterion which shall characterize a sheet which is at most single-valued on any ordinate. Such a criterion is derived in 7 in connection with a theorem due originally to Schoenflies, and afterwards proved by Osgood. The proof of it involves the auxiliary notions described in 6 and the following corollaries to the preceding theorem: // the initial point of a continuous arc (C*) Xi = Xi(t) (i = 1, 2, , m\ t' . For if so, there would be an interval t% < t ^ t 2 in which the curves would be distinct while at t = t 3 they coincide. This is, however, impossible since in a neighborhood of the point corresponding to t 3 there can be but one solution of the equations / = corresponding to a given set of values x. Suppose that a continuum X of points (x\, x 0, and the constants e* could not therefore decrease indefi- nitely in size. A similar argument shows the existence of the constant 6. FUNDAMENTAL EXISTENCE THEOREMS. 27 Suppose now that the interval u' ^ u ^ u" is divided by values Uk (k = 1, 2, , v) into sub-intervals so small that the points of any arc ak-i has the equations Since the functions defining C x are continuous, and therefore uniformly continuous, in t' ^ t ^ t", it is possible to take the points of division t', t%, t 3 , , t v , t" so close together that the differences x (fc) , for any point x on the arc W( fc+1 > of C x , are uniformly less than an arbitrarily assigned positive constant 5; and the preceding theorem shows that the curve C xy and the Continuation curve along the polygon both lead from p' to p". If the sides <*><*+ and ,-/dwjfc|| (i = 1, 2, , m; k = 1, 2) do not all vanish simultaneously at any point of U. A simply connected region in two dimensions is defined above, and a connected region A" in a space of points (x it x 2 , , x m ) has a definition quite similar to that for two dimensions. In order to specify conveniently the properties of a region X which is simply connected, the term elementary curve will also be used. By an elementary curve in X is meant a simply closed continuous curve which either lies in a normal subspace of two dimensions entirely in the interior of A', or else is such that in every neighbor- hood of it there is a simply closed continuous curve having this property. It is thus seen that while an elementary curve may not itself be imbedded in one of the two-dimensional normal sub- spaces interior to X, it can nevertheless be approximated as closely as may be desired by one which does. The word neighbor- hood is here used in the sense described in connection with the fourth theorem of 5 (see page 26) . If a region X is connected, then any simply closed continuous curve in its interior may be developed into two such curves by an auxiliary arc joining two of its points, and the process of development may be continued on the two arcs so formed. // a region X is such that any simply closed continuous curve in its interior is an elementary curve, or may be developed into a number of elementary curves by means of auxiliary arcs, as just described, then X is said to be simply connected* * For a discussion of the connectivity of higher spaces, see Picard and Simart, Th6orie des Fonctions alg^briques de deux Variables independantes, Chapitre II, in particular 11 ff. If every simply closed continuous curve interior to R lies in a normal subspace of two dimensions interior to R, one sees intu- itively that a second neighboring subspace of the same kind can be passed through the curve. The closed two-dimensional subspace so formed is FUNDAMENTAL EXISTENCE THEOREMS. 33 7. A CRITERION THAT A SHEET OF SOLUTIONS BE SINGLE- VALUED Consider in the first place a set of equations (16) f a (xi, x 2 ; yi, y 2 , - -, y n ) = (a = 1, 2, , ri) in which there are but two independent variables x. If a connected sheet S of solutions of equations (16) consists only of ordinary points of the functions f, and furthermore has a simply connected projection X in the XiX^-plane such that no interior point of X is either a point where S becomes infinite or the pro- jection of a boundary point of S, then the sheet S is single-valued over the interior of X. Suppose, in contradiction to the theorem, that over any interior point of X there were two points, p' and p", of the sheet. Since S is connected there would be a continuous curve (C xy ) xi = xi(t), x 2 = x 2 (f), y a = y a (t) (? t i"\ a= 1, 2, .--,71) consisting entirely of interior points of the sheet and joining p' with p" in the space (x; y). The projection (C.) an = antf), x, = x 2 (t) (f t t") of this curve would necessarily be a closed curve in the and by the second theorem of 5 the arc C xy is the only one associated with C x in the sheet S and having the initial point p'. The curve C x may be simply closed and regular; but if it is not, there will nevertheless be a curve in the region X having these properties, and for which the continuation curve analogous to C xy is not closed. For, in the first place, from 5 it is seen that the curve C x may be supposed to be a polygon no two ad- jacent sides of which have more than an end point in common, provided that it is desired only to secure a continuous curve in separated into two parts by the curve, and hence the number which Picard and Simart designate by p\ is equal to unity for a simply connected region of the kind denned in the text above. 4 34 THE PRINCETON COLLOQUIUM. the sheet passing from p' to p". Let the corners of this polygon in the a>plane be denoted by 1, 2, , , where is a symbol for a point (x\, x 2 ). The side ,,1 touches i 2 at its end point 1, and it can be argued therefore that there will be some first side AA+I which touches some one of the preceding sides elsewhere than at its initial point A . Let the side so touched by A A +I be +! where K + 1 is necessarily less than \, and let the first point of AA+I which lies on ,*+! be . If the portion of the curve C xy which corresponds to the polygon (17) , .+i, . +2 , -, A, is not closed, then the polygon (17) itself is a simply closed curve in X of the kind desired above, that is, one along which there exists a continuation curve in the xy-space whose end points are different. If the portion of C xy which corresponds to (17) is closed, then that part of C xy which belongs to the polygon (18) 1, 2, , , , A+I, **, , 1 is also continuous and leads from p' to p". Since K + 1 < X the side (C +i )t +2 at least is missing in (18), and the number of sides is at least one less than that of the original polygon. By an alteration of the kind suggested in the proof of the last theorem of 5, which also reduces the number of sides, it can be brought about, if not already true, that the polygon (18) still has no two adjacent sides with more than an end point in common. By continuing this process one must come at some stage to a simply closed regular curve in the ar-plane with a corresponding continuation curve in the xy-space which is not closed. In order not to complicate the notation too much it may be supposed that the curve C x itself is such a curve. Every point of C x is an interior point of the region X since the corresponding point of C xy is an interior point of the sheet S. The interior of C x is therefore also composed entirely of interior points of X, since X is simply connected. If the interior of C x is subdivided into FUNDAMENTAL EXISTENCE THEOREMS. 35 two parts by a segment of a straight line, as described in the pre- ceding section, the dividing segment will also have a continu- ation curve on the sheet S throughout its entire length, by the second theorem of 5. For its initial point on the curve C x corresponds to an interior point of the sheet S and, by the hy- pothesis of the theorem which is to be proved, none of its points can be a point where S becomes infinite or can correspond to a boundary point of S. Hence one of the simply closed curves formed by the curve C x and the dividing segment is a curve retaining the property that it has a continuation curve on the sheet S which is not closed. Suppose that C x r is this curve. By continuing the process a sequence of curves { C x (k) } , with diameters approaching zero, can be found, each lying in the interior of C x and having an unclosed continuation curve C XV (K * onS. If a point p (k) is selected arbitrarily on the curve C xy (k) , the sequence {p (k) } (k= 1, 2, , oo ) will have a finite point of conden- sation 7r(a; j8) in the xy-space which is an interior point of the sheet S. For the projections x (k) of the points p (k) all lie in the in- terior of C x and hence must have a point of condensation a. Fur- thermore the points of the sequence p (k) whose projections are in the neighborhood of a can not become infinite or approach a boundary point of the sheet, since a. is interior to X. They must therefore have at least one limit point ir which is an interior point of the sheet, and with which there are associated two neighborhoods Tr e and a s by the principal theorem of 1. Some of the points p ( lie in ir f , and have corresponding curves C x (k) in a s . For such points the continuation curves C xy (k) also lie in ir e and can not be unclosed, since to any point x in a s there corresponds in TT at most one solution of the equations / = 0. The original assumption that S is multiple-valued in the interior of X is therefore contradicted. The theorem remains true for any system of equations of the form (19) /.(an, x 2 , -,*; yi, y 2 , , #) = (a = 1, 2, -, n). 36 THE PRINCETON COLLOQUIUM. In this case the curves C xv and C x have equations (CW) *i = *t(0, 2/a = 2/a(0 (i= 1,2, , m; a = 1, 2, , n; t' ^ t ^ O, (C x ) .T, = *,-(*), and the question asked in the proof of the theorem just stated is whether or not the latter curve may be closed while the former has distinct end points. It is a part of the hypothesis of the theorem that the region X is simply connected according to the definition of the preceding section; and, according to the arguments made in the paragraphs above, the curve C x may be supposed a simply closed polygon. In any neighborhood of C x there will be, according to 6, on account of the simple connectivity, an elementary curve C x lying in a normal subspace of two dimensions (20) Xi = gi(ui, M 2 ) (i = 1, 2, -, ra) entirely interior to X. If the continuation curve C xy is not closed, and if C x is taken sufficiently near to C x , then the corresponding continuation curve C xy will also not be closed. The normal subspace (20) is defined over a simply connected domain U of points (MI, M 2 ), and has no multiple points. To every point of C x there corresponds therefore a single pair of values (C u ) u\ = Mi(0, M 2 = and the functions so defined are continuous, by the principal theorem of 1, since at every point some pair of the equations (20) has a functional determinant for MI, M 2 which is different from zero. The curve corresponding to the curve C xy in the space (M; y) may be denoted by (C uv ) ui = Mi(0, M 2 = w 2 (0, 2/a = 2/a(0, (a = 1, 2, -, n), and its initial point, corresponding to p', by p u ' (M'; y'). Every point of C uv is an ordinary solution of the equations (21)

-,n). FUNDAMENTAL EXISTENCE THEOREMS. 37 With a continuous curve C joining (u\, Uz) to an arbitrarily chosen point (u\, u^) of U there is always associated a continu- ation curve of solutions of the equations (21), having the initial point p u f and defined throughout the whole of C, since any such curve defines a curve in the a>space interior to X along the whole of which there is a corresponding continuation curve for the equations (19) in the sheet S. Hence there is a unique sheet S u of solutions of the equations (21) whose projection in the UiU Z - space is U; and no interior point of U is a point where the sheet becomes infinite or corresponds to a boundary point of the sheet, since the same is true of 5 with respect to X, The preceding argument can therefore be applied to show that the sheet S u is single-valued over the region U, and the existence of the curve C uy with the distinct end points p u ' and p u " is contradicted. Hence C xy can not have distinct end points p' and p", and the theorem last stated is proved. 8. TRANSFORMATIONS OF n VARIABLES AND A MODIFICATION OF A THEOREM OF SCHOENFLIES It is interesting to deduce by means of the preceding theorems some conclusions concerning a system of equations of the form (22) f a (x;y) = x a -^ a (y b y 2 , ,*/) = (a = 1, 2, ., n). The functions \J/ are once for all assumed to be single-valued, continuous, and to have continuous first derivatives in a con- tinuum Y in which the functional determinant D = dtyi, fa, -, y a = y a (0, defined by equations (22), is a curve interior to the sheet S and joining (x'; y') to (x"\ y"}, so that S is evidently connected. Any boundary point (a;j3) of space joining x' with x" and containing no points of the set A. All of the points of X are points of X. For any set of values x in X can be joined to x' by a continuous curve C x lying entirely in X and containing therefore no points of A. By the second theorem of 5 the corresponding continuation curve C xy must extend along the entire arc C x , since otherwise the values of y for points on C xy would approach infinity or else have a limit point on the boundary B of Y, and some point of C x would in that case necessarily be a point of A. It follows that x, like x', is the image of some point y in Y. From the initial theorem of the last section, for the case when there are more than two variables, it follows that If A is distinct from X, and X is simply connected in the sense of 6, then the sheet S is single-valued. In other words the continuum Y is tranformed in a one-to-one way into a continuum X by means of the equations (22), and the functions (23) y a = y a (xi, 3-2, , &) (a = 1,2, -, ri) so defined over X are single-valued, continuous, and have continuous first derivatives. The character of the functions (23) near any point of X follows at once from the theorem of 1. Let it be supposed that the set of points A divides the x-space into exactly tico continua X, H such that every point of A is a bound- 40 THE PRINCETON COLLOQUIUM. ary point for each of them, and suppose furthermore that there is a particular point in H which does not correspond by means of the equations (22) to any point of Y. Then the image X of Y is distinct from A and coincides with X. If X is simply connected the other conclusions of the last theorem follow at once. In the first place it can be shown that if any point ' of H corresponds to a point of Y then every other point " of E would also have this property. For ' and " can be joined by a continuous curve x a = *.(/) (a=l, 2, .-., n; t' ^t ^") entirely interior to H. The corresponding continuation curve of solutions of equations (22) must be defined along the whole of the interval t' 5^ t ^ t", since otherwise as < approached any upper bound T of the values t which could be reached by con- tinuation, the corresponding points y of the curve would have to approach infinity or else have a point of condensation on the boundary of Y. But this is impossible, since for a sequence of points x corresponding to a sequence of points in 7 approaching infinity or a boundary point of Y, the only limiting points possible are at infinity or else in the set A. It follows at once, on account of the hypothesis of the theorem, that no point of H can correspond to a point of Y, and neither can any point of A, since in any neighborhood of such a point of A there are points of H which in that case would also correspond to values y in Y. The image of the region Y in the z-space is a single continuum whose only boundary points are points of A. According to the preceding argument it cannot be E and it must therefore be X. A modification of a theorem of Schoenflies can be deduced readily from the results which precede. The theorem has to do with a pair of equations of the form (24) xi - fa(yi, y 2 ), *2 FUNDAMENTAL EXISTENCE THEOREMS. 41 in which the functions \{/ are single-valued, continuous, and have continuous derivatives on a simply closed regular curve B of the y-plane and in the interior 7 of B. The functional determinant D = dtyi, ^z)/d(yi, 2/ 2 ) is supposed to be different from zero in Y. If the curve A in the x-plane formed by transforming the simply closed regular curve B in the y-plane, by means of the equations (24), is distinct from the image X of the interior Y of B, then X is a simply connected continuum whose only boundary points are points of A, and the correspondence defined between X and Y is one-to-one. The single valued functions (25) 2/1 = y\(xi, xz), 2/2 = yz(xi, x 2 ), so determined in the region X, are continuous and have continuous first derivatives.* From the preceding theorems of this section it follows that the complete image X of Y is a single finite continuum whose only boundary points are points of A. It remains to show that X is simply connected and that the correspondence between X and Y is one-to-one. If any simply closed regular curve C x is drawn in X, its interior must consist entirely of points of X. Otherwise there would necessarily be a boundary point of X, a point of the curve A, interior to C x , and there would also be points of A exterior to C x since X is finite. Hence there would necessarily be a point of the continuous curve A on C x itself, which contradicts the as- sumption that A and X are distinct. It follows at once from the first paragraphs of 7 and the simple connectivity of X just proved, that only one point y in Y corresponds to a given x in X, and by the theorem of 1 it may be seen that the functions * Schoenflies assumed only the continuity of the functions \l/i, ^2, adding, however, that the correspondence denned between the regions X and Y of the two planes is to be one-to-one. In the theorem here proved \}>\ and ^-2 are subjected to further continuity restrictions, but the correspondence is proved to be unique. See Schoenflies, " Ueber einen Satz der Analysis Situs," Gottinger Nachrichten (1899), page 282. The theorem was later proved by Osgood and Bernstein in the same journal (1900), pages 94 and 98, respectively. 42 THE PRINCETON COLLOQUIUM. (25) have the continuity properties described in the theorem in the neighborhood of any particular point x. Another theorem, slightly different in form, may be stated as follows: If the images of the points of the simply closed regular curve B in the y-plane all lie on a simply closed regular curve A in the x-plane, then the equations (24) define a one-to-one correspondence between the interior X of A and the interior Y of B, and the functions (25) so defined have the same continuity properties as before. In this case it can first be shown that the image x' of any point y' in Y must be distinct from A, and the rest of the proof is the same as before. For, if x' were a point of A, every point of a properly chosen neighborhood of x' would also be the image of a point of Y, since at (x'; y') the functional determinant of equations (24) does not vanish. It would follow then, by con- tinuation, that every point exterior to the curve A would also be the image of a point of Y, which is impossible since thefunctions ^ are finite. The continuum X is therefore identical with the interior of A, by the preceding theorems, and the correspondence between X and Y is one-to-one. An example applying some of the theorems of 5, 8 is given at the end of 14. CHAPTER. II SINGULAR POINTS OF IMPLICIT FUNCTIONS The theorems which have been developed in the preceding pages of these lectures have to do with the behavior of implicit functions at ordinary points, or in regions which have no singular points in their interiors. For singular points where the functional determinant vanishes the theory is much more complicated, and no methods which can be comprehensively applied have so far been developed. There are, however, many special cases in widely different fields which have been studied with success, and it may not be out of place to glance at a few of them before proceeding to the further theorems with which these pages are primarily concerned. Perhaps the most complete single theory which has been developed is that which has to do with the singularities of an algebraic function y of x determined by an equation of the form (1) P(x, y) = 0, where P is an irreducible polynomial in the two variables x and y. Suppose for convenience that the singular point to be considered is at the origin, and that the polynomial P(0, y} has a lowest term of degree n in y. Then it is known that for each value of z in a sufficiently small neighborhood of x 0, there exist exactly n solutions y of equation (1) in the neighborhood of y = 0, and the values of these solutions are given by k cycles of the form (2) y=a^ + a/z^+... (j =1,2, ,&), where the numbers n, p are positive integers satisfying the relations [ij < m' < ju," < , p! + p 2 + + p k = n. 43 44 THE PRINCETON COLLOQUIUM. The series is one member of the cycle; the others are found by replacing x vp ' by u v x llp > (v= 1, 2, , p, 1), where a; is a primitive py-th root of unity. The number PJ has no factor in common with the exponents /i,-, ju/, . Otherwise the expansion would be in terms of a root of x of lower order than PJ. Thus there are in all n series in fractional powers of x which define the roots of the algebraic equation in the neighborhood of the origin. The coefficients of the series may be computed by means of the well-known Newton polygon,* or by methods due to Ham- burgerf and Brill. J If the substitution x = t K is made in the series (2), the points (x, y) which it defines may be expressed in the parametric representation x = f', y = r> [otj + a/P/-"' + >} (j = 1, 2, , k). All the solutions of the equation (1) in the neighborhood of the origin evidently belong to a finite number of such branches. With the help of the preparation theorem of Weierstrass, which is to be studied in the following pages, results similar to those just given may be proved for the solutions of an equation F(x, y) = in the vicinity of any point where F is analytic. The singularities of a surface F(x, y, z) = at a point where the function F is analytic have also been ex- tensively studied. The points of the surface in the neighbor- hood of a singular point are determined by means of a finite number of expansions of the form x = P(u, i>), y = Q(u, ), where P and Q are analytic in the parameters u and . * See Appell and Goursat, Th6orie des Fonctions algebriques, pp. 184 ff. t Weierstrass, Werke, vol. 4, Kapitel 1. J Munchener Berichte, vol. 21 (1891), p. 207. See Black, " The parametric representation of the neighborhood of a singular point of an analytic surface," Proceedings of the American Academy of Arts and Sciences, vol. 37 (1902), p. 281. FUNDAMENTAL EXISTENCE THEOREMS. 45 In the calculus of variations the construction of " fields of extremals " in the plane requires the study of the real solutions of a system of equations of the form (3) * = (Q, a), y = iKO, a) will then be the equations of C. If the curvatures of the two arcs at their point of contact are always different, then the extremal arcs E simply cover a portion of the plane N on one side of C and adjacent to it. In other words, the equations (3) define a one-to-one correspondence between the points of a region ad- joining the axis t = in the to-plane, shown in the accompanying figure, and a certain neighborhood N on one side of the arc C. FIG. 2. In the interior of the region N the functions t(x, y), a(x, y) have continuity properties similar to those of

' * > %m with coefficients which are series in y. It is desired to determine b so that the identity (7) holds, and so that the expressions pk have coefficients which contain y only to the degree n- 1. By substituting the expressions (8) in the identity (7) and equating terms of the same degree in the z's, it follows that bo(a yf Q }y n = a y n , bi(a yf )y n = 6 /i Pi, bz(a yfo)y n = 60/2 + &i/i Pz, ' ) bk(a yfo)y n = b fk + bif k -i + + &t-2/2 + b k -ifi Pk, These equations are to be identities in x and y. The first one determines 60 uniquely with constant term unity, and further- more so that each coefficient is a quotient, in fact a polynomial with positive integral coefficients in a finite number of the coef- ficients of /, divided by a power of a . In the second equation pi must be chosen equal to the terms of 6 /i which contain y to the degree n 1 or less, after which 61 is uniquely determined. Similarly in the fcth equation pk must first be chosen to cancel the terms on the right of degree n 1 or less in y, and then bk is unique. It only remains to show that the series 6 and a& are convergent in any numerical case for which / converges. There is no loss of generality in assuming that the series / converges in the domain \Xi\ ^1, \y ^1 (i = 1, 2, , m), since this can always be effected by a substitution of the form Xi = piXi, y = try' (i = 1, 2, -, TO). Suppose then that K is a number greater than the absolute value of any term in the series /(I, 1, , 1, 1), that is, greater 52 THE PRINCETON COLLOQUIUM. than the absolute value of any coefficient in /. If A is the absolute value of ao, the series n _ R y n+l _ K where dominates / in the sense that every coefficient except the first has a numerical value equal to or greater than K; and the series B satisfying the relation BF = A y n + A,y n ~ l + + A n analogous to (7) has coefficients numerically greater than the absolute values of those of b. Hence if B converges the same will be true of 6. But it is easy to show that the series B converges. It will certainly do so if convergent series Ak, C, D can be found satisfy- ing the relation Aoy n (l-y)-Ky n+l -KX=(A Q y n +A l y*- l + +A n )(Cy + D), because then B would have the value Cy+D' On comparing the coefficients of the two highest terms in y in the next to last equation, and for convenience denoting by a the constant value ~AT> it is found that C = aA , D = 1 -aAi. By comparing the other powers of y and substituting these values, FUNDAMENTAL EXISTENCE THEOREMS. 53 we have AI + a AI + A n -i + aA Q A n = aAiA n -i, A n = aA^An - KX. But these equations have linear terms in AI, A%, , A n with functional determinant different from zero, and hence have solutions, by the theorems of 2, which are convergent series in Xi, x 2 , , x m and have no constant terms. It is evident, in any numerical case for which / is convergent, that a neighborhood of the origin may be chosen in which the series b is everywhere different from zero. In such a neighbor- hood all of the values (xi, x 2 , , x m , y] which make/ vanish are roots of the equation p = 0, and vice versa. If/Gri, 0, , 0, y) has its terms of lowest degree homogeneous and of degree n, then the polynomial p(x if 0, , 0, y] has the same initial terms, since the first coefficient of the factor series 6 is unity. 10. THE ZEROS OF (0, ) has a first term of the form i m with coefficient unity. According to the results of the preceding section, all of the roots of $(u, v) in a neighborhood of the origin will be roots of a certain polynomial (10) 54 THE PRINCETON COLLOQUIUM. where the coefficients a* are series in u having no constant terms. The polynomial P may be equal to the product of two poly- nomials of similar form, boV k -f- 61^* l -{- -}- bk, C T m ~ k + Citf n ~ k ~ l + + where the coefficients b and c are convergent series in u. In that case the product boCo must be identically unity, and by dividing the first polynomial by 6 and multiplying the second by the same series, the two factors will have the form * + fciV-i + - - - + b k ', ---- h c m . k ', The coefficients b' and c' are now series in u without constant terms. Otherwise the product P would have a term of lower degree than tf", with a coefficient series whose constant term would be different from zero. It is readily seen from this that the polynomial P is either irreducible in the sense that it can not be decomposed into a product of polynomials of the same sort, or else it is the product of a number of irreducible polynomials of lower degree. Suppose that Q(u, r) is a polynomial of the form (10) which is irreducible in the sense just described. Then its discriminant with respect to r is a series in u which does not vanish identically, since otherwise Q and Q v would necessarily have a common factor of the form (10), and Q would not be irreducible. There is a neighborhood < u ^ u\ in which the discriminant is every- where different from zero, and for any value u satisfying these inequalities the values of v making Q = are all distinct. According to the results which have been stated above in the introduction to this chapter of the lectures, the values of v which make Q vanish for different values of u will be defined by m series of the form (11) r = aw (t/p + aV /p + ; FUNDAMENTAL EXISTENCE THEOREMS. 55 and these series must all be distinct, since for sufficiently small values u 4= 0, as has been seen, the roots of Q are all distinct.* It is evident then that all the roots of (u, v) in the neighbor- hood of the origin, including those which correspond to the fac- tor u k in equation (9), are given by a finite number of elements of the form u = at p , v= fa" + &'r'+ , where a and b do not vanish simultaneously, and p, p, \i! , are positive integers having no common factor. The product of factors of the form (12) {v - cm u/p - a'u' lp - }, corresponding to the elements of a cycle, is a polynomial Q\(u, v) of the form (10). For the product Qi is a series in u llp and v which is unchanged when u llp is replaced by is meant the number of times its factor (12) is repeated in the product u k P. The order is evidently equal to the multiplicity in u k P of the irreducible factor to which the element belongs. If

satisfies the equation (u, ) = 0, are the points of a finite number of distinct elements of the form (18) u = at', r = 6r + 6Y* 1 + (0 ^ * ^ <0 FUNDAMENTAL EXISTENCE THEOREMS. 59 whose coefficients are real and such that a and b are not both zero. It may be of interest to note in passing that if an element of

) = k' lie. By means of these results it can now be shown that any two distinct points of the region OP\Pz are transformed into two distinct points of the xy-plane. For if (u', ') and (u", v") defined the same point (x f , y'} they would both give r = ^lx 2 -\-y 2 the same value k', and hence must lie on the same curve PiP 2 . But in that case the values of w corresponding to the two points would necessarily be different, as has been seen above, and hence (x r , y') and (x", y") could not be the same. From the final theorem of 8 it follows at once that the theorem last stated above is true, provided that the circle of radius p\ is altered so that the arc of it which lies between the branches OEi and OE% lies also within the region 0PiP 2 . The region into which S is transformed must lie entirely in one quadrant of the xy-plane, since the values of co which correspond to points of S are all in one quadrant. In the interior of the image of S the inverse functions (20) are analytic, since at interior points of S the determinant D is different from zero. Some conclusions with regard to the distribution of the elements of on the branch OE\ would be everywhere ir/2, or else everywhere 7r/2, and the same is true for OE%. But this is impossible since along the arc P\Pi the value of w varies monotonically through an interval less than 7T/2. A similar remark holds for the elements of \l/. Hence it follows easily that Between any elements of D the elements of

; if the signs of D are the same, the regions 2i and 2 2 adjoin along 07r 2 without over- lapping. The adjoining figure illustrates the case when D has four real elements and the signs of D are opposite in any two adjoining regions S. Further illustrations of the theorem are given in 14. FIG. 6. It has not been proved above that the functions (24) are continuous on a boundary OTT of one of the regions 2. Suppose that TT is a point of such a boundary, and let (25) l> 7T 2 , be any sequence of points of 2 with limit TT. The corresponding points (26) pi, pi, p 3 , " of S have condensation points in S, one of which may be denoted by p. There is then a sub-sequence Pi PS FUNDAMENTAL EXISTENCE THEOREMS. 67 among the points (26) whose limit is p; and on account of the continuity of the functions (23), the corresponding points (27) 7T/, 7T 2 ', 7T 3 ', of the sequence (25) must have as limit point the image of p in S. But the limit of (27) is necessarily TT, and TT is therefore the image of p. It follows at once that the sequence (26) has a unique limit point p which is the image of IT, and from this property the continuity of the functions (24) in the ordinary sense can be readily deduced. The functions , \f/, and D can be expanded in the form

) = 68 THE PRINCETON COLLOQUIUM. identically in u and v. It is possible to show that such a relation exists also near a singular point at which the four derivatives above all vanish. If a relation can be found after a substitution of the form u = mil + |3i'i, v = yui + 8vi, for which a8 /3y does not vanish, then it will surely be satisfied when MI and v\ are replaced by the original variables u, v. Suppose then that the analytic functions

\, 02, ' , a m , and are therefore convergent series in v, x. The product (39) ffot, , *) = *(, *) *=i is a convergent series in u^, v, x, also symmetric in the variables Wfc, and hence expressible as convergent series in v, x. The function h(v, 0) does not vanish identically, on account of the hypothesis that f(u, v, 0, 0, , 0) and g(u, v, 0, 0, , 0) have no common factor. If it did vanish identically, then for every sufficiently small value of v one at least of the expressions g(uk, v, 0) would vanish. But in 10 it was seen that when f(u, v, 0) and g(u, n, 0) have no factor in common, there is always an interval < v ^ v\ in which there is no value v belonging to a pair (u, 0) making both of these functions vanish. The preparation theorem of Weierstrass can therefore be applied also to the function h(v, x), and the polynomial so found is the one desired in the theorem. For, in the first place, a constant can be chosen so small that every root (u, v, x) of / and g in the region (36) must be one of the sets (u k , v, x), and must make * A proof that the values of u and v belonging to the roots of a system of equations of the form (34) are roots of polynomials similar to (35) was given by Poincar6 in the introduction to his Thesis, " Sur les proprie'te's des fonctions d^finies par les Equations aux differences partielles," Paris (1879). FUNDAMENTAL EXISTENCE THEOREMS. 73 the product (39), and hence p, vanish. In the second place, a constant 5 ^ e can be taken so small that every root v of p as well as the corresponding sets (uk, v, x) lie in the domain (36). One at least of these sets must evidently satisfy g = as well as / = 0. The restrictions on 5 and e have been stated somewhat roughly, but the reader will readily convince himself that these quantities may be selected so that the convergence of the different series and their equivalence with the corresponding polynomials are properly adjusted. Finally, the polynomial p is linearly expressible in the form described in the theorem, in terms of / and g. To prove this, suppose that the above process has been applied to the functions / a and g /3. A polynomial P(v, x, a, /3) with coefficients which are series in x, a, /3 is then found, which may be written in the form P(v, x, a, 0) = P(v, x, 0, 0) + Ca + Z>/3, where C and D are convergent series in the arguments of P. The series P(u, x, f, g) vanishes identically in u, v, x since P = must be satisfied by every set of variables (u, v, x, a, /3) in a neighborhood of the origin which make / a and g vanish, certainly then by the set (u, v, x,f, g). Hence P(t>, x, 0, 0) = - Cf - Dg is an identity in u, v, x, when a and |8 are replaced in C and D by the series/, g. But P(v, x, 0, 0) is precisely the polynomial p(v, x) found above, since for a = = the steps in the con- struction of P(v, x, 0, 0) are identical with those used in finding p. If the series f(u, v, 0, 0, , 0) and g(u, v, 0, 0, , 0) begin with homogeneous polynomials having no common factor of degrees m and n, respectively, then the degree of the polynomial pis v= mn.* * In a paper of recent date the writer has developed a generalization of this theorem and the results which follow, for a system of equations of the form fi(xi, x 2 , -,x m ; ?/i, 7/2, , y n ) = (i = 1, 2, , n). See Transactions of the American Mathematical Society, vol. 13 (1912), p. 133. 74 THE PRINCETON COLLOQUIUM. Let the lowest terms of f(u, v, 0,0, , 0) and g(u, r, 0, 0, , 0) be denoted by

m , has a term involving u alone with coefficient different from zero, since

i, Uz, , u m . The expression for such a symmetric function is isobaric and has the weight a in the coefficients of the poly- nomial (35). When x = the terms of lowest degree in U will be at least of degree a in v, since each coefficient a* of (35) begins with the coefficient of u m ~ k in the polynomial (p m (u, v). The terms of lowest degree in v alone in the product (39) will there- fore be those of the product and they have the value v mn R/ao, in which a is the coefficient of v m in m (u, v) and R is the resultant of

*\_1 F (u f f T) in powers of z "; and the value of G(u\, z, , T) is a series p n r> ri -FulJz f 2 lJu p ~ ( 2 D "T * ' * r u whose first term is different from zero, since for the values (u', f , , r) we have " r\ = \*n ( ?J I' S {*// !'' tv * ' *Ju vj; >Qu\u j sJ fl'vi.w j ^ > ?/ as is readily seen from equations (40). Hence the quotient H(z, , T)/(Z f) is different from zero, and neither H(z, , t) nor its polynomial P(z; ^, can have more than one factor 2 - tu' - v'. 14. APPLICATIONS OF THE PRECEDING THEORY The real transformation . x = (u, n) = aiou + ooifl + a zo ir + y = \l/(u, v) = biou + b iv + 6 20 w 2 + FUNDAMENTAL EXISTENCE THEOREMS. 79 has a singular point at the origin when (51) = 0. |010 001 If one of the elements of the determinant is different from zero, it may be assumed without loss of generality to be a; then after two transformations V = V, , , b w x - x, y = -- x+ y io the equations (50) take the form x = u -j- azou 2 -f a\\uv + aoa* 2 + , (52) y = bzoU 2 + bnuv + 6 2 2 + For convenience the primes have been dropped, and the notation for coefficients of terms of higher degree than the first is the same as that in the original equation. It may further be supposed that the polynomials i = u, \f/ 2 = bzov? + bnuv + &020 2 have no common factor, in other words that 6 2 ={= 0. The origin is then a singular point for the transformation (50) of a very general type, since aside from the assumption (51) only inequalities on the coefficients of the series have been exacted. The functional determinant has the expansion D(u, v) = b n u + 2&020 + -i and hence has a single branch 611 v = -, u + , &02 along which D vanishes and on opposite sides of which D has different signs. The image A of this curve in the zy-plane has 80 THE PRINCETON COLLOQUIUM. an ordinary point at the origin, as shown by its equations x= u+ -, y = - -bn 2 'OJ The region S in the figure has in it one real element of

dv du' Such a transformation might be called a monogenic transforma- tion. It follows at once that

m (u, v) and \f/ m (u, v) have each m real linear factors in u, v, and that no factor of

axis being drawn triply between x = and x= 32/3 since this segment is described three times by the point (54) with varying u. To the auxiliary arc o 2/n) 03 = 1, 2, -, ri) when the symbols y and/ in equations (1) are interpreted as row letters in the way apparently first introduced for differential equations by Peano.f An interesting deduction from the theorems for a system of equations is the proof of the existence of a solution of a partial differential equation dz dz which is not necessarily analytic in its five arguments, by means of the well-known theory of characteristic curves, as described in 19. * Vorlesungen iiber Variationsrechnung, page 179. t " Integration par series des equations differentielles line'aires," Mathe- matische Annalen, vol. 32 (1888), p. 450. 88 THE PRINCETON COLLOQUIUM. 15. THE CONVERGENCE INEQUALITY There is an inequality which is of frequent service in the existence proof of the following sections and which can be readily deduced from a simple preliminary theorem. If u is a single-valued function of i with a well-defined forward derivative u' at each point of the interval ^ t ^ t\, and if \u'\ < k\u + /, k and I being two positive constants, then u also satisfies the inequality M ^ \u \e kt + ^ (e kt - 1), where u is the initial value of u at t = 0. Consider the function v = \u \e kt + ~ (e kt - 1) satisfying the differential equation v' = kv + I and having u \ as its initial value. The value of u is never greater than that of v, since otherwise the difference u v would vanish and have a positive or vanishing forward derivative at some point. At a point where u and v are equal, however, M'| < k\u + I = kv + / = v', which is a contradiction. A similar argument shows that u is always less than v. If u is a single-valued function of x with well-defined forward and backward derivatives at each point of an interval x ^ x ^ x\, and such that u k\u then, for any and x in the interval, u also satisfies the inequality (3) u\ ^ |tt($)|e*"- + -I (*'-*' ~ 1). K FUNDAMENTAL EXISTENCE THEOREMS. 89 This may be proved from the preceding paragraphs by putting t = x % for values of x greater than , and t = x -f- for values less than . 16. THE CAUCHY POLYGONS AND THEIR CONVERGENCE OVER A LIMITED INTERVAL It is proposed to consider a differential equation (1) for which the function f(x, y) is continuous in the interior of a certain region R of the xy-p\ane, and such that the quotient f(x,yT)-f(x,y) y'-y is finite when (x, y) and (x, y'} lie in any closed region whose points are all interior to R. A so-called Cauchy polygon for the equation (1) through a point (, 77) interior to R is defined by means of equations of the form y\ = y = y n -i+f(x n -i, y n -\)(x z n _i). The division points < xi < x 2 < may be taken for convenience at equal distances 5 from each other. Any value x > will lie on one of the intervals x n -ix n , and the polygon will either be well-defined for all such values, or else there will be a constant ft such that for every x in the in- terval ^ x < ft the points of the polygon are interior to R, while for x = ft the corresponding point (x, y) will be a point of the boundary of R. The polygon defined by the equations above may be denoted by PI(X), and the analogous one when the division points are distant 5/2 n-1 from each other by P n (x). A common interval ^ x ^ a for two functions P(x), Q(x) with respect to any region R may be defined as one over which 90 THE PRINCETON COLLOQUIUM. both are interior to R, and one such that on any ordinate of the interval all the points between (x, P(a*)) and (x, Q(x)) are also interior points of R. Consider now a closed region RI interior to R and containing the point (, 77), and let m and k be two constants greater respec- tively than the absolute values of f(x, y) and the quotient (4) in the region R\. If I > is given in advance, the partitions for any tico polygons P(x}, Q(x) through (, 77) can be taken so small that (5) \P(x)-Q(x)\ jfr*-* 1 -- 1) for all values of x in any common interval of P(x) and Q(x) with respect to RI. For at the point (x, y), where y = P(x), the equa- tion P' = f(x, P) + {/(*_!, 2/ n _0 - f(x, P)} = f(x, P) + p is satisfied by the forward and backward derivatives of the polygon P. On account of the continuity of f(x, y) there exists for any / a constant /* such that x x imply \yy'< whenever the points (x, y) and (x, y') are in RI. If the subdivi- sions for P(z) are taken less than /z and nfm in length, it follows that on the polygon P(x) x y n -\ m x x n - and hence the absolute value of p is less than 1/2. Similarly Q(x) satisfies an equation where a < 1/2, provided that its intervals are less in length than ju and n/m. The difference P Q has forward and back- FUNDAMENTAL EXISTENCE THEOREMS. 91 ward derivatives which satisfy the relations \P'-Q'\ |/(*,P)-/(*, arbitrarily, and / so small that Then i_ 1 < . |P n ,(*) - PGr)| < {*' - 1} < 6, 92 THE PRINCETON COLLOQUIUM. provided that the intervals 6/2 n ' -1 and 5/2"" 1 are each less than the constant /x corresponding to /. Hence the sequence P n (x) converges uniformly to a continuous function y(x] on the interval The equations Pn(x) = 77 + f Pn'Wdx = 77 + f {/(*, P n ) + Pn }dx J( J t hold for every n, and the sequences {f(x, P n ) } and { p n } approach uniformly the limits f(x, y(x}) and zero, respectively. Hence y(x) = 77 + J /(a:, from which it follows by differentiation that y(x) is a solution of the differential equation. It is easy to show by means of the convergence inequality that there is only one continuous solution y = y(x) of the dif- ferential equation (1) in the region R and passing through (, 77). For suppose there were another, Y(x), distinct from y(x) at a value x' > . There would then be a value 1 < x' at which 2/(i) = Y(l-i), and such that the two solutions would be distinct throughout the interval 1 < x ^ x'. In a neighborhood of the point of intersection (1, 771) interior to R a relation H/(*, Y) - f(x, y)\ < k\Y - y\ dx would be satisfied, and hence, from the convergence inequality (3), Y - y\ ^ 0. This contradicts the hypothesis that y(x) and Y(x) are distinct throughout the interval 1 < x ^ x'. FUNDAMENTAL EXISTENCE THEOREMS. 93 17. THE EXISTENCE OF A SOLUTION EXTENDING TO THE BOUNDARY OF THE REGION R It has been proved in the preceding section that, on a certain interval ^ x ^ + a\, the polygonal curves y = P n (x) con- verge uniformly to a continuous solution y = y(x) of the differ- ential equation (1) lying entirely within the region R. The in- terval for which the proof has been given may not be the largest one on which the sequence of polygons has this property. There will, however, be a number ft ^ + a\, possibly infinity, with the property that on any interval ^ x ^ fti, where ft\ < ft, the sequence of polygons converges uniformly to a continuous solution interior to R. A continuous curve y=y(x) is thus defined which has a derivative and satisfies the differential equation for all values of x in the interval ^ x < ft. As x approaches ft the points (x, y(x}} of the solution can have no limit point (ft, 7) interior to the region R. If they did, there would be for any given c a value x' < ft such that \y(x'} - y , and an integer N such that, whenever n ^ N, the inequality would hold for all values of x in the interval ^ x ^ x'. At the value x' in particular |P n (*') - 7 ^ |PGO - y(x')\ + \y(x') - 7 < e; so that for n ^. N the points (#', P n (x'}} would all lie in the e-neighborhood of the point (ft, 7). About the point (ft, 7) as center a rectangle x - ft\ ^ A, \y-y\B could be described entirely within the region R, and in the portion 94 THE PRINCETON COLLOQUIUM. RI of R which lay within the rectangle or within the region SZxZx', 2/(.r) - e ^ y ^ y(x) + e the absolute values of f(x, y) and the quotient (4) would be less than two constants m and k, respectively. It can be shown without great difficulty that every polygon P n (x) for n ^ N would be defined and lie within the region R for an interval extending beyond ft at least a distance A\, where A\ is the smaller of the numbers A and (B e me)/w. A proof similar to that of 16 would then show that the polygons P n (x) converge uniformly to a continuous solution of equation (1) interior to R\ over an interval x ft -{ Ail an d consequently ft could not be the upper bound described above. As x approaches ft, therefore, the only limiting points of the solution y = y(x) are at infinity or else are boundary points of the region R. If R is further a closed region, that is, one con- taining all of its limit points, then there is but one limit point for the curve y = y(x) as x approaches ft. For suppose (ft, 7) to be a finite point in any neighborhood of which there are points on the curve. About (ft, 7) a rectangle (7) \x-ft\^A, \y - 7 ^ B can be chosen arbitrarily, and the points of R lying in it form a finite closed set in which \f(x, y)\ remains always less than a constant M. On the interval ft A\ < x < ft, where A\ is the smaller of the numbers A and B/M, all the points of the curve y = y(x) satisfy the inequality (8) \y - 7! M(ft - x}. For if (x f , y') is any point of the curve in the rectangle (7) and also in an e-neighborhood of the point (ft, 7), then the inequality \y - y\ ^ \y' - 2/1 + \y' - T| < M(x' - x) + e FUNDAMENTAL EXISTENCE THEOREMS. 95 must be satisfied by any preceding point P (x, y) of the curve y = y(x) for which the arc PP f is interior to the rectangle. It follows that the solution must lie interior to the rectangle and satisfy the last inequality, at least on an interval x' A,?)| the initial values of the two solutions being taken at x = % + A. If A and AT; are sufficiently small the expression on the right is less than 5 for all values of x belonging to the region R s , and hence S must be defined and interior to R& for all such values. Otherwise, for some interior value of x, it would attain one of the values (f>(x, , 77) =*= 8, which is seen to be impossible on account of the choice just made of A and AT;. Consider now the difference By a step similar to (12), and the inequality (11), it is seen to be less than {|ATT| + whenever A and AT; have been so chosen that S lies entirely in the region R s . Hence the continuity of + Jo is a continuous function of x, A, AT;, the values , rj being con- 8 98 THE PRINCETON COLLOQUIUM. sidered as constant for the moment. Hence A^ = ceS****. When A = or AT; = 0, the constant c has respectively the values (t, , 77 + AT;) i/ A c = & It may be remarked in conclusion that the theorems which have been proved in 16-18 are true for systems of equations as well as for a single one. 19. AN EXISTENCE THEOREM FOR A PARTIAL DIFFERENTIAL EQUATION OF THE FIRST ORDER WHICH is NOT NECESSARILY ANALYTIC Proofs have been given by Cauchy, Kowalewski, Darboux, and others for -the theorem that in general there exists one and but one analytic surface 2 = z(x, y} which passes through an arbitrarily selected analytic curve C in the ary-space and, with the derivatives _ dz dz P = dx' q= dy> satisfies a differential equation of the form F(x, y, z, p, 9) = 0, FUNDAMENTAL EXISTENCE THEOREMS. 99 where F is an analytic function of its five arguments. These proofs, however, say nothing about the solutions which may exist through a curve C whose defining functions are not ex- pressible by means of power series; and they are not applicable when F itself has not this property. An existence proof is to be given below which is based upon much less restrictive as- sumptions on the functions F and the curve C. It involves the well-known theory of characteristic strips, which are solutions of a set of ordinary differential equations. If a one-parameter family of characteristic strips intersecting a given curve C is properly selected, it will generate a surface S which is a solution of the differential equation. The existence of the family and the differentiability of the surface depend, however, upon the existence and differentiability of the equations of the character- istic strips with respect to the initial values of the variables which they involve, that is, upon theorems similar to those which have been developed in the preceding sections. Suppose that the function F is continuous and has continuous first and second derivatives in a certain region R of points (x, y, z, p, 0. The differential equations satisfied by the charac- teristic strips have the form dx dy dz du = F " + F z2u + *>u + /> = are satisfied identically, so that the direction p : q : 1 is always normal to the curve defined by the first three equations. Evi- dently if F vanishes at a single point of the strip, it will also vanish at every other point. The solutions (14) along which F vanishes are called characteristic strips, and any one of the strips (14) will surely be of this type if the initial condition 7, r, T, K) = o is satisfied. Consider now a continuous and differentiable strip of elements (17) x = (r), y = rj(v), z = f(r), p = 7r(r), 9 = /c(r) which lies in the interior of the region R and satisfies the con- ditions v i > ._ A TT^t! ~r kTjt, 5 r ' (18) where the arguments in the derivatives of F are the same as those in the second equation. The first two of these conditions imply that the direction TT : K : 1 is normal to the curve (19) x = 00, y = i?(t), 2 = f (), FUNDAMENTAL EXISTENCE THEOREMS. 101 and that the curve and its strip of normals satisfy the differential equation. The third prevents the strip from being a so-called integral strip of the differential equation, through which there does not in general pass a unique integral surface without singularities. To make the situation simpler it will be supposed that the projection of the strip (17) in the xy-p\ane does not intersect itself. When the functions (17) are substituted in the equations (14), a new system x = X(it, v), y = Y(u, v), z = Z(u, v), p = P(u, v), q = Q(u, v) with the initial conditions , 77(0) = F(0, ), f(tO = Z(0, t>), T() = P(0, v), K(V) = is determined. There is a region (/?) A ^U^B, ! ^ V Vz, where A is a negative and B a positive constant, in which the functions (20) are continuous, have continuous first derivatives, and satisfy the relation (22) 4= 0. y y * u -f v For if M is the maximum of the absolute values of the functions on the right in the equations (13), for a closed c-neighborhood of the points of the strip (17) in the interior of R, then the solutions (14) are defined at least over an interval u ^ c/3/, and the absolute values of A and B can be taken at least as great as this constant without disturbing the continuity properties desired for the functions (20) in the region R uv . The condition (22) is satisfied for the values u = 0, t'i ^ v ^ ^ because of the first two of equations (13) and the third of the relations (18); and the 102 THE PRINCETON COLLOQUIUM. region R vv can therefore be chosen so that the determinant is different from zero everywhere in it. By an argument similar to that used in proving the theorem of 4 it can be shown that A and B can be restricted still further, if necessary, so that no two distinct points (u r , v'), (u", v") in the region R uv define the same point (.r, y) by means of equations (20). The boundary of the region R uv is trans- formed then by the first two of equations (20) into a simply closed regular curve in the zy-plane which bounds a portion R xy of the a*t/-plane. The equations establish furthermore a one-to-one correspondence between the points of R uv and those of R xy , and the functions (23) n = u(x, y), v = v(x, y) so defined are continuous and have continuous first derivatives in R xy . The others of the equations (20) define then three functions (24) 2 = z(x, y), p = p(x, y), q = q(x, y) which are also continuous and have continuous first derivatives in R xy , and which with the values (23) for u and v satisfy the equations (20) identically in x, y. The functions (20) satisfy the relations (25) PX V + QY V - Z v = 0, F(X, 7, Z, P, Q) = 0, identically in u, v. The first and third of these follow at once from the equations (15), (16), the second of the equations (18), and (21). The expression QY V - Z v has the initial values (26) 0(0, v) = TT&, + w, - r, = o, FUNDAMENTAL EXISTENCE THEOREMS. 103 which vanish on account of the first of equations (18). Further- more 12 U = P U X V + Q U Y V + PX UV + QY UV , and from the first of equations (25), = P V X U + Q V Y U + PX UV QY VV . By subtracting the last expression from that for 12 U and using the equations (13) which the functions (20) satisfy, it follows that _3F " dv' \u *uX v -f~ *fu* v r v X u *tv* u in which the arguments of the derivatives of F are the functions (20). Hence with the help of the third of equations (25) and the initial values (26), - fl u = - 12F,, 12 = 12(0, v)e * * * u '= 0. The single- valued function z(x, y) defined above over the region R xy has the derivatives Z x = Z u Y u Z v Y v X u = p(x, Z y = ** u A I u X v Y v = q(x, y), found by substituting the functions (23), (24) in the equations (20), differentiating the resulting identities, and applying the first two of the relations (25) . It satisfies the differential equation F = on account of the third of the equations (25). Further- more x, y, z(x, y), p(x, y), q(x, y) reduce to , 77, ", TT, K at any point of the strip (17), since at such a point w(, 77) = and the relations (21) are satisfied. It has been proved therefore that there is a single-valued 104 THE PRINCETON COLLOQUIUM. function (27) z = z(x, y), defined over a region R xy of the a-y-plane, which is continuous and has continuous first and second derivatives, contains the initial strip (17), and satisfies the differential equation F = 0. There is no other surface (28) z = zi(x, y} defined over the region R xy and having these properties. If there were such a one, it would have to contain all of the points of the strips defined by equations (20). To prove this, suppose that (x f , y f , z' y p', "no, To) of the curve there corresponds a direction TT O : K O : 1 satisfying the relations (18), and such that ( > f]o, fo> TTQ, KO) ls interior to R, then there will be a strip of elements of the form (17) along the curve containing these initial values for v = r . For the first two equations (18) have the solution (v , TT O , K O ) 106 THE PRINCETON COLLOQUIUM. when their first members are regarded as functions of v, IT, K, and on account of the third relation (18) their functional de- terminant for TT, K does not vanish at these values. According to the fundamental theorem of 1 there is therefore a pair of functions TT(V), K(V) defined over an interval v\ ^ n ^ v? con- taining v and satisfying, with (), rj(v), f(u), the relations (18). The results of the preceding paragraphs may be summarized as follows: Suppose that (C) x = $(), y = 17(0), 2 = f(t) is a continuous and differentiate curve, at some point (o, *?o, To) = ((*>o), *?(0o), r(*o)) of which there is a normal TO : K O : 1 satisfying the equation r) , fo, TTO, KO) = 0. Suppose furthermore that 4=0, and that the initial element (o> "no, To* TTO> KO) lies in a region R of points (x, y, z, p, q) in which F is continuous and has continuous first and second derivatives. Then there is a strip of the form (S) x = (), y = 77(0), z = (), p = TT(V), q = K(V) vi v ^ containing (% , y , $" , TO, K O ) /or t? = r , an x points, the total number of trajectories, for all initial conditions, is oo 3 . The differential equation of the third order representing this system of trajectories, found by eliminating the time from (1), is . (2) y, - y' _ J+2 ~~ n !+!/'* d+y' 2 ) 3/2 u 2 = ^ 2 1+2/' 2 The functions 0, \j/ depend only on the position of the particle; the auxiliary intrinsic functions N, T, 3i, , 9Ji, 9^2, ^E], ^2, denned above, depend also upon the direction of motion; finally, N,, T,, 9J,, s depend upon the curvature of the path. Cf. Bull. Amer. Math. Soc., vol. 15 (1909), p. 475. t Cf. Trans. Amer. Math. Soc., vol. 7 (1906), pp. 401-424. The result contained in property IV of 3 gives this simple, but apparently overlooked, dynamical theorem: If a particle starts from rest, the initial curvature of the path described is one third of the curvature of the line of force through the initial position. 10 THE PRINCETON COLLOQUIUM. I. If for each of the oo 1 trajectories passing through a given point in a given direction we construct the osculating parabola, at the given point, the locus of the foci of these parabolas is a circle passing through that point. II. The circle that corresponds, according to property I, to a lineal element, is so situated that the element bisects the angle between the tangent to the circle and a certain direction fixed for the given point (the direction of the force acting at the given point). III. In each direction at a given point there is one trajectory which has four-point contact with its circle of curvature: the locus of the centers of the QO l hyperosculating circles constructed at the given point is a conic passing through that point in the fixed direction described in property II. IV. With any point there is associated a certain conic passing through it as described in property III. The normal to the conic at cuts the conic again at a distance equal to three times the radius of curvature of the line of force passing through 0. (The lines of force are defined geometrically by the fact that the tangent at any point has the direction associated with that point in accordance with property II.) V. When the point is moved, the associated conic referred to above changes in the following manner. Take any two fixed perpendicular directions for the x direction and the y direction; through draw lines in these directions meeting the conic again at A and B respectively. Also construct the normal at meeting the conic again at A 7 . At A draw a line in the y direction meeting this normal in some point A', and at B draw a line in the x direction meeting the normal in some point B'. The variation property referred to takes the form _ 3cu 2 where AA' and BB' denote distances between points, and where co denotes the slope of the lines of force relative to the chosen ASPECTS OF DYNAMICS. 11 x direction. This is true for any pair of orthogonal directions, FIG. 1. and therefore really expresses an intrinsic property of the system of curves. 4. The most general system of oo 3 curves in the plane is represented by an arbitrary differential equation of the third order TO y'"=f(x,y,y',y"}. It thus involves one arbitrary function of four arguments. A system of dynamical trajectories, on the other hand, is represented by an equation of the particular form x + coco,, = 0, (/V) K + Xco + M)* - X x = 0. Thus the general system having properties I-IV involves three arbitrary functions of x, y; while that having all five properties involves two such functions. By integrating these relations, we may express the four functions in terms of two arbitrary functions V X= ,

) The particle may be started from any position, in any direction, with any speed: its motion is then determined by the field of * This conic is not the same as the conic arising in property III. 10 18 THE PRINCETON COLLOQUIUM. force, and it describes a definite trajectory. The totality of trajectories constitutes a definite system of oo 5 curves. (We exclude the case where the force vanishes at every point, the trajectories then being merely the oo 4 straight lines.) What are the properties of such quintuply infinite systems of curves? Obviously an arbitrary system of space curves cannot be obtained as the totality of trajectories connected with any field of force. In fact the most general system of oo 5 curves (assuming that oo 1 curves pass through any point of space in any direction) would be represented by a pair of differential equations, one of the third order and one of the second order, of the general form (2) y'" = f(x, y, z, y', z', y"}, z" = g (x, y, z, y', z', y"), thus involving two arbitrary functions of six arguments; while the dynamical type involves merely three arbitrary functions of three arguments. The differential equations representing the dynamical type, obtained by eliminating the time from the equations of motion, may be written in the form 1 2 trajectories will be obtained. If these form a normal congruence (that is admit a set of orthogonal surfaces), the same will necessarily be true for any other speed r . The trajectories starting out from any -point with * That is, the cubic intersects the plane at infinity in three mutually orthogonal directions. All the quadrics passing through the curve are then of the equilateral type. ASPECTS OF DYNAMICS. 33 a given speed form a normal congruence when, and only when, the field is conservative. The necessity of this condition is included in the Thomson- Tait theorem discussed in the next chapter. Its sufficiency, of course, requires a separate discussion which is connected with the theory of velocity systems. 3. In order to make the preceding test purely geometric, it is necessary to have a geometric method of assembling those tra- jectories which, starting from the same point, correspond to the same initial speed. Such a method is readily found from the fact that the square of the speed varies directly as the radius of curvature and directly as the normal component of the force. The oo 2 trajectories corresponding to a given speed have circles of curvature intersecting each other at the same point on the line of the force vector; that is, the centers of curvature lie in a plane perpendicular to the direction of the force acting at the given point. In the conservative case, the oo 2 trajectories so selected form a normal congruence. 4. Among the oo 2 trajectories considered there are, for any field, three w r hich admit hyperosculating circles of curvature. The three initial directions thus determined will be mutually orthogonal when and only w r hen the field is conservative. Only test 1 is directly connected with the set of properties I-IV of page 19. The other three are suggested by the discussion of velocity systems (cf. 32). 11 CHAPTER II NATURAL FAMILIES: THE GEOMETRY OF CONSERVATIVE FIELDS OF FORCE 28. ORIGIN AND APPLICATION OF THE NATURAL TYPE 28. We now consider the properties of the trajectories gener- ated by conservative fields of force. The total system of tra- jectories will have the general properties previously considered for an arbitrary field of force, together with the additional proper- ties stated in 26, 27, peculiar to the conservative case. An entirely new feature presents itself, due to the fact that the differential equations of motion admit an integral of the first order, namely, the energy equation. During any motion of the particle in the given field, the sum of the kinetic and potential energies is constant; thus each motion corresponds to a definite value of the constant h, representing the total energy. The motions may therefore be grouped according to the values of h. Those corresponding to a given value form what may be termed, following Painleve, a natural family. Thus, in space of two dimensions, the complete system of trajectories for a given conservative field of force consists of oo 3 curves grouped into oo 1 natural families, each composed of oo 2 curves. For example, in the case of ordinary gravity the tra- jectories are the 3 vertical parabolas (in a given vertical plane), and the natural families are formed by grouping together those parabolas which have the same (horizontal) line as directrix. In space of three dimensions, the complete system contains oo 5 trajectories grouped into oo 1 natural families, each containing oo 4 curves. Examples are the oo 4 parabolas with vertical axes whose directrices are situated in a fixed horizontal plane; and the oo 4 circles orthogonal to a fixed sphere. The simplest ex- ample, corresponding to the case of zero force, is the oo 4 straight lines of space. 34 ASPECTS OF DYNAMICS. 35 This grouping of the trajectories according to the values of the total energy constant, that is, into natural families, is funda- mental in most dynamical investigations relating to conservative forces, in particular, those connected with the principle of least action and the developments of Hamilton and Jacobi. From this point of view, dynamical problems relating to the same field of force, but having distinct values of h, are considered as essentially distinct problems. Quoting Darboux: "This re- striction is in accordance with the spirit of modern mechanics which attaches less importance to force than to energy, and which permits us to regard as distinct two problems in which the force function or work function is the same, but the total energy is different." It therefore seems of interest to work out the purely geometric properties of natural families. According to the principle of least action, such a family is made up of the extremals defined by the variation problem J ^W + h ds = minimum, that is, the curves which cause the first variation of the integral to vanish. This follows from the fact that the speed v, in the action integral jvds, is determined by the energy equation #= 2(W + h). Abstractly, a natural family of curves may be defined as one which can be regarded as the totality of extremals connected with a variation problem of the form J Fds = minimum, where F is any point function, that is, any function of x, y, z in the three-dimensional case. Such families arise not only in the discussion of trajectories, but also, for example, in the discussion of brachistochrones, catenaries, optical rays, geodesies, and contact transformations. 36 THE PRINCETON COLLOQUIUM. The brachistochrone problem for a conservative field with any work function W leads to the integral /*-; ds + h Thus the complete system of brachistochrones is made up of oo 1 natural families, one for each value of h. When a homogeneous, flexible, inextensible string is suspended in the conservative field, the forms of equilibrium, which are termed catenaries in the general sense of the word, are obtained by rendering the integral f(W+K)ds a minimum. Hence here also we have oo 1 natural families, one for each value of h.* Consider an isotropic medium in which the index of refraction v varies arbitrarily from point to point. The paths of light in such a medium, according to Fermat's principle of least time, are determined by minimizing the integral J vds and hence form a single natural family. This is the most concrete way of defining a natural family. The connection with the theory of geodesies is obvious. Thus in the two-dimensional case the geodesies of the surface whose squared element of length (first fundamental form) is X(:r, y}(dx i + dy~) are found by minimizing the integral f VX ds, and hence the representing curves in the x, y plane constitute a natural family. Hence if any surface is represented conformally on a plane, the geodesies are pictured by a natural family of curves in that plane. The extension to more variables is evident : * The complete systems of oo 5 brachistochrones and oo 5 catenaries have geometric properties distinct from each other and from those of the s trajectories: no quintuply infinite system of curves can be at the same time the system of trajectories for some field and the system of brachistochrones or catenaries in either the same or a different field. The distinctive properties for an arbitrary field are given in 107, p. 94. Cf. 103. ASPECTS OF DYNAMICS. 37 any natural family in any space may be obtained by conformal representation from the geodesies of some other space.* As a last application we consider the transformations which Sophus Lie has termed the infinitesimal contact transformations of mechanics. In the plane case, such a transformation is defined by a characteristic function of the special form 12(ar, */)(! + y' )* and is characterized by the fact that the lineal elements at each point are converted into the elements of a circle about that point as center. The path curves of every contact transformation of this category form a natural family. 29-31. CHARACTERISTIC PROPERTIES A AND B 29. Osculating Circles Property A. We now consider the general geometric properties of a natural family in ordinary space, that is, the totality of oo 4 extremals connected with an integral of the form (1) fF(z, y, z) Vi + y * + 2 ' 2 dx. The differential equations of the family are then the corresponding Euler-Langrange equations 2/'' =(, -(x, y, z), t(r, y, z), x(x, y, z) ; ASPECTS OF DYNAMICS. 39 then the requisite condition is ~v = X-g> X + *Y + X Z = 1, where X, Y, Z denote the coordinates of the center relative to axes with the common point as origin. Using the general formulas for the center of curvature, and combining with (5), we find THEOREM 2 : The differential equations of any system of curves possessing property A are of the form y"= (^- (7) 2"= (X- where <, \J/, x are arbitrary functions of x, y, 2. The converse is valid also. The equations (2) are seen to be included in this form, hence the result certainly holds for our natural systems, as stated in theorem 1. 30. Hyper osculations Property B. The circles of curvature at a given point, for any system of the form (7), constitute a bundle. We now inquire whether any of these circles correspond to four-point, instead of three-point, contact. If a twisted curve is to have an hyperosculating circle of cur- vature at a given point, two conditions must be satisfied, namely, (8) 1 y' z' y" z" y'" z'" dr = 0, 40 THE PRINCETON COLLOQUIUM. The first of these states that the osculating plane has four-point contact with the curve; the second, in which r denotes the radius of curvature, is the condition for the existence of an osculating helix, i. e., one with four-point contact. When both conditions hold the helix is simply the circle of curvature, which then has hypercontact. Applying these conditions to the curves defined by (7), we find, from (8), and, from (9), (11) (1 + yf + 2 ' 2 )2' -( + y't + z'x) ,' _|_ 7 yU' _|_ R'Y' 1 = o, where the indicated summations extend over <, \(/, x and where ', for example, denotes 2 + y'$ v + z' z . Since we wish to discuss the oo 2 curves through a given point, we may simplify our equations considerably by taking the axis of abscissas in the special direction (4). Then, at the selected point, \f/ and x vanish, and the above equations reduce to (10') y'x' - z'f = 0, (11') (/ + z' 2 )(4>' - 2 ) - W + z'x'} = 0. Neglecting the trivial solutions for which y' 2 + z' 2 vanishes, we may reduce this pair of simultaneous equations to the form y % This set of equations for the determination of y', z' is of a familiar type, namely, that arising in the determination of the fixed points of a collineation, and is easily shown to admit three solu- tions.* Hence * Of course in special cases some of these may coincide, or the number of solutions may become infinite. The theorem stated is true " in general " in so far as it omits these cases which are definitely assignable. ASPECTS OF DYNAMICS. 41 THEOREM 3 : The curves defined by equations of the form (7) are such that through each point there pass three with hyperosculating circles at that point. Since the form (7) is characterized by property A, it follows that the existence of three hyperosculating circles in each bundle is a consequence of property A. We state two further properties, found by considering the conditions (10') and (11') separately. The tangents to those curves of a system (7) which pass through a given point and there have an hyperosculating plane form a quadric cone. This cone passes through the special direction (4). The tangents to those curves which have an osculating helix at the given point form a cubic cone. This cone passes through the special direction (4) and through the minimal directions in the plane normal to that direction. These properties hold for natural families since they hold for all systems with property A. By comparing (7) with (2), we see that the functions 0, \f/, x i n the case of a natural family are (13) = L x , \f/ = L v , x = L 2 ; and hence are connected by the relations (14) ^ - x v = 0, x*-* = 0, 4> y -tx = 0. We now inquire what is the effect of these relations on the directions of the hyperosculating circles. Introducing, for symmetry, (15) X : Y : Z = 1 : y' : z', we may write our equations (12) in the homogeneous form (16) - x x X 2 + 2 Z 2 + y YZ- x v XY+ (0*-0 2 - Xz)XZ= 0, In virtue of (14), each of the quadric cones (16) is seen* to be * The condition for such a cone is that the sum of the coefficients of X 2 , Y 2 , and Z 2 shall vanish. 42 THE PRINCETON COLLOQUIUM. of the rectangular type. Hence the three generators common to the cones must be mutually orthogonal. This gives THEOREM 4 : In the case of any natural family the three hyper- osculating circles which exist in any bundle are mutually orthogonal. We refer to this property as property B. 31. The relations (14) are seen to be necessary as well as sufficient for the orthogonality in question. Hence property B is the equivalent of (14), and serves to single out the natural families from the more general class defined by equations of form (7). The latter form was characterized by property A; hence we have our FUNDAMENTAL THEOREM: A system of oo 4 curves, one for each direction at each point of space, will constitute a natural family when, and only when, it possesses properties A and B: that is, the osculating circles at any given point must form a bundle, and the three hyperosculating circles contained in such a bundle must be mutually orthogonal. 32. GENERAL VELOCITY SYSTEMS 32. The most general system with property A is represented by differential equations of the form y /2 + A and thus involves three arbitrary functions. Only in the case where these functions are the partial derivatives of the same function is the system a natural one. We now point out a dynamical problem that leads to the general type (7): this justifies the term velocity system which we hereafter employ to denote any system of this type. Consider a particle (of unit mass) moving in any field of force, the components of the force being 6, \{/, x- The equations of motion are then x = (x, y, z), y = \l/(x, y, z), z = x(x, y, z}. If the initial position and the initial velocity are given the motion ASPECTS OF DYNAMICS. 43 is determined. If only the initial position and direction of motion are given, the osculating plane will be determined but the radius of curvature r will depend for its value on the initial speed v. Hence, in addition to the usual formula there must be a formula expressing v 2 in terms of x, y, z, y', z', r. This is furnished by the familiar equation v 2 = rN, where N denotes the (principal) normal component of the force, so that A72 ^2_L /2_l 2 (4> + tfty + *'xf = *-+* 2 +x 2 - ' The result may be written in the two (equivalent) forms 2 _ Qfr-yfrXl + y'S+Z*) _ ( X - 2 (l + y' 2 + 2 ' 2 ) " y" z" In the actual trajectory v varies from point to point. If now we replace v 2 in this result by some constant, say l/c, the resulting equations may be written The curves satisfying these differential equations they are not in general trajectories we define as velocity curves. For any field a curve is a velocity curve corresponding to the speed v , provided a particle starting from any lineal element of the curve with that speed describes a trajectory osculating the curve. In a given field of force there are oo 5 trajectories and oo 5 velocity curves.* If c is given we have oo 4 velocity curves. In particular * The properties of a complete system of oo 5 velocity curves are analogous to, but distinct from, those of a complete system of trajectories. Cf. p. 94. 44 THE PRINCETON COLLOQUIUM. if c (and hence v) is taken to be unity, our equations become precisely (7). Any system of oo 4 curves possessing property A, that is, any system (7), may be regarded as the totality of velocity curves cor- responding to unit velocity in some (uniquely defined] field of force. Only when the field is conservative do the velocity systems for each value of v (or c) become natural systems. The trajectories also are in this case made up of oo 1 natural families, one for each value of the energy constant h; but the two sets of natural families are distinct. The determination of a velocity system in one conservative field is equivalent to the determination of a tra- jectory system in another conservative field, and vice versa. We find in fact the following explicit result: // two conservative fields with work functions W\ and JF 2 satisfy the relation* "iWi W z = ae^l - h, then the oo 4 velocity curves for the speed VQ in the first field coincide with the oo 4 trajectories for the constant of energy h in the second field.^ 33. RECIPROCAL SYSTEMS 33. With any velocity system S y= there is connected a definite point transformation T: for in virtue of property A to any point p corresponds a definite point P, the osculating circles constructed at the first point all passing through the second point. The transformation T is explicitly (T) * We note that if W\ is left unaltered and r c varied, TFj takes quite distinct forms. The oo 1 velocity systems in a given field do not constitute the com- plete system of oc 6 trajectories in any field whatever. t It is seen that the two fields have the same equipotential surfaces and therefore the same lines of force. (Central fields therefore correspond to central fields.) ASPECTS OF DYNAMICS. 45 It is thus entirely general. To an arbitrary transformation* corresponds a definite velocity system. In particular, to the inverse transformation T~ l there corresponds a certain system S', which we define as reciprocal to S. Hence to a general^ velocity system S, that is, any system possessing property A, there corresponds a definite reciprocal velocity system S'. The osculating circles of those curves of system S which pass through any point p are at the corresponding point P the osculating circles of the curves of the system S' passing through P. Consider the bundle of circles determined by two corresponding points p and P. We know that three of these circles have hypercontact with ^-curves at p, and three have hypercontact with 2 rays (in general curved) starting out normally from anys urface admit oo 1 orthogonal surfaces. These present themselves as surfaces of equal time. We shall describe them as a set of wave fronts or wave surfaces. 37. The geometric part of the theorem of Thomson and Tail may be stated as follows : In any natural family of oo 4 space curves, the oo 2 curves which meet any surface orthogonally always form a normal congruence. Is this geometric property, which we shall refer to as the Thomson-Tait property, characteristic? This is in fact the case. We shall prove, namely, the following CONVERSE THEOREM. // a quadruply infinite system of curves in space is such that oo 2 of the curves meet an arbitrarily given surface orthogonally* and always form a normal congruence (that is, admit an infinitude of orthogonal surf aces) , then the system is of the natural type, that is, it may be identified with the extremal system belonging to an integral of the form jF(x, y, z)ds. 38. The result is simple but the proof is rather long. We give the essential steps. Consider an arbitrary quadruply infinite system of curves in * This means the same as requiring that one curve of the system passes through each point of space in each direction. ASPECTS OF DYNAMICS. 51 space, assuming that one passes through each point in each direction. Such a system may be defined by a pair of differential equations of the second order (1) y" = F(x, y, z, y', 2'), z" = G(x, y, z, y', z'), where F and G are uniform functions which we assume to be analytic in the five arguments. Denoting the initial values of x, y, z, y', z', which may be taken at random, by x, y, z, p, q respectively, and j employing X, Y, Z as current coordinates, we may write the solutions of (1) in the form 7 = y + p(X - x} (2) Z= z+q(X-x) + Here F and G are expressed as functions of x, y, z, p, q', and M and N, found by differentiating (1), are given by M = F X + P F y + qF, + FF P + GF q , N = G x + pG y + qG z + FG P + GG q . The terms of higher order will not be needed in our discussion. Equations (2) involve five arbitrary parameters but of course represent only oo 4 curves. Consider now an arbitrary surface 2 (4) z=/(*,y). At each point of this surface and normal to it a definite curve of the given family (1) may be constructed. A certain congruence will thus bt determined. We wish to express the condition that this shall be of the normal type, that is, that the oo 2 curves shall admit a family of orthogonal surfaces. The direction normal to the surface S at any point is given by 1 -P -q = fx :/ : - 1, so that (5) p = P(x, y) } q = Q(x, y), where (50 P =///*, Q--I//X. 52 THE PRINCETON COLLOQUIUM. These functions are connected by the relation (5") PQ X - q? x - Q v m 0. The equations of the 2 curves corresponding to the given initial conditions may now be written X = z+t, (6) Y = y + Pt + IF? + where t takes the place of A' x in (2), and where the bars indicate that the substitution (4), (5) has been carried out, so that, for example, (7) F(x,y] =F(x,y,f,P,Q). The coefficients of the powers of t in (6) are thus functions of the two parameters x, y. The general condition for a normal congruence given in para- metric form is* (8) (Y'XY) - (Z'ZX) + Y'(Z'YZ) - Z'(Y'YZ} = 0, where the parentheses denote jacobians taken with respect to t, x, y, and Y', Z' denote the derivatives of Y, Z respectively with respect to t. Expanding (8) in powers of t in the form (9) a + fli< + W- + , we find that vanishes in consequence of (5")- This is as it should be, since our oo 2 curves are orthogonal to S by con- struction. The terms containing the first power of t give * We may also use the convenient form due to Beltrami. Cf. Bianchi- Lukat, Differentialgeometrie, p. 340. ASPECTS OF DYNAMICS. 53 From (6') we find F x = F x + FJ X + F P P X + F q Q x , with corresponding results for G x and G v . Substituting these values, and observing from (5) and (5') that /* = - l/, g\ = X - q, where 0, \f/, x are functions of x, y, z only. The general solution of the last three equations of the set (11) is therefore (12) F=(^-^)(l+p 2 +g 2 ), G=( x -)(l+p 2 +7 2 ). We have still to satisfy the first equation of (11), which now reduces to (13) *. - x, + p(x* - 0.) + q(4>y - U = 0. The functions , \f/, x must therefore satisfy the equations (13') ,k - Xv = o, x* - * = o, />, - ^ = o, and hence are expressible as the derivatives of a single function in the form (13") = L x , t=L y , x = L z . The solutions of the set (11) are therefore F= (L y -pL x )(l + p* + q*), G= (L 2 - P L z )(l + p 2 + 9 2 ), involving an arbitrary function L of x, y, z. The resulting system (1) is thus recognized to be a natural family. This gives our fundamental converse theorem. 39. In the above discussion use has been made, not of the complete condition for a normal congruence, but only of con- dition (10') derived from the terms of the first order in t. We may therefore state a stronger converse result as follows: ASPECTS OF DYNAMICS. 55 The only systems of oo 4 curves which have the property that the curves orthogonal to any surface are always orthogonal to some infinitesimally adjacent surface are those of the natural type. If a congruence of curves meets two neighboring surfaces orthogonally it need not meet oo 1 surfaces orthogonally, and therefore it approximates to, but need not coincide with, a normal congruence. The above theorem shows however that if the weak requirement of approximate normal character be imposed on all the congruences obtained from the given quadruply infinite system, they will all be exactly normal. 40. We may further strengthen our theorem by demanding the orthogonality property for some instead of all surfaces. Our fundamental equations (11) resulted from the fact that x, y, z, f, P, Q, P x , P y , Q x might receive arbitrary numerical values. It will therefore be sufficient to take a manifold of surfaces sufficiently large to leave these quantities, or the equivalent quantities X, y, Zy Z x , Zy, Z xx > Zj;y, Zyy, unrestricted. Since these quantities define a differential surface element of the second order, we may state the result as follows: The converse theorem remains valid if, instead of considering all base surfaces, we employ a manifold of surfaces sufficiently large to include all the oo 8 possible differential elements of the second order. 41. The Thomson-Tait theorem holds of course even when the base S shrinks to a curve or a point: there will still be a normal congruence orthogonal to the curve or point (in the latter case orthogonality means simply passage through the point). We state a number of results obtained in this connection. If for an arbitrary curve as base the corresponding oo 2 orthog- onal curves of a given quadruply infinite system always form a normal congruence, the given system is necessarily natural. If we require each of the congruences here considered to be of approximately normal character, a more general type of system 56 THE PRINCETON COLLOQUIUM. is obtained, namely the velocity type of 32. The velocity type is thus characterized by the fact that those curves of the system which meet an arbitrary curve orthogonally are orthogonal to some infinitesimally adjacent (of course tubular) surface. We may even restrict ourselves to the case where the base is a curve of the given system, or the case where it is any straight line. 42. Suppose next that the base is an arbitrary point. Are natural families the only families of co 4 curves such that the oo 2 curves passing through any point form a normal congruence? A discussion shows that this is not the case. There exist families not of the natural type, for example, that defined by the dif- ferential equations y" = y'\ *" = o, with the restricted property stated. To find all such systems would be a rather difficult, but certainly an interesting, under- taking. The result would of course include the natural type as a special case. 43. It will not however be the velocity type. It may be shown in fact that the only velocity systems for which the curves passing through an arbitrary point constitute always a normal congruence are those of the natural type. Recalling the fact that the velocity type is characterized by property A, we may give a new characterization of the natural type as follows: Natural families are the only quadruply infinite systems of curves in space such that the oo 2 curves through an arbitrary point admit an infinitude of orthogonal surfaces, and such that the osculating circles constructed at the common point form a bundle. 44. It may also be shown that if for every point and every straight line as base the corresponding congruence is normal, the system will be natural. To have a velocity system it is sufficient to demand that the congruence corresponding to an arbitrary straight line shall be approximately normal. To have a natural system it is sufficient to demand approximate normality for the congruences corresponding to arbitrary straight lines and plane?. ASPECTS OF DYNAMICS. 57 45-53. WAVE PROPAGATION IN AN ISOTROPIC MEDIUM : PROPERTIES OF WAVE SETS 45. The optical interpretation of a natural family and the Thomson-Tait property suggest certain sets of surfaces which we shall now study. Consider a given medium defined by its index of refraction v(x, y, z) given as a function of position. The rays (in general curved lines) are the oo 4 extremals of (1) J v ( x > y> 2 )^* = minimum; they form the natural family, whose differential equations are distinct media. The question then arises, just what will uniquely determine a medium. A natural family is uniquely determined if we are given one set of wave fronts and a single extra trajectory. This means a tra- jectory not belonging to the congruence defined as the orthogonal trajectories of the wave set. 49. The extra curve however cannot be taken at random; it must be related in a certain way to the wave set. If the wave set is f(x, y, z) = constant, then the condition on the curve is that it satisfy the Monge equation of second order (3) 2Az" fl 4- v' 2 4- z'VA z'A ") / z'f ""** V X I * J Ai / i _A j /w _i - ./ Z & J X where (3') As /,*+/,*+/,*. 60 THE PRINCETON COLLOQUIUM. Here/, and hence A, are given, and y and z are unknown functions of x. The interpretation is obvious from property A. In order that an extra curve shall be consistent with a given wave set (that is, in order that both shall belong to a single medium) it is necessary and sufficient that the curve shall cross the surfaces (of course obliquely) in such a way that at any point of intersection the circle of curvature of the extra curve shall intersect the circle of curvature of the curve orthogonal to the surfaces. When the curve satisfies this restriction, it defines with the given wave set a unique natural family. 50. If we are merely given one wave set, the number of possible media is oo" (since v involves arbitrary functions). Each of these has oo 4 rays (forming a natural family). The totality of media give rise to a totality of oo 00 rays, namely the solutions of the Monge equation of second order (3). This equation is of the type Ay" + Bz" + C = (where the coefficients are functions of x, y, z, y', z'}, which the author has shown to be characterized by the Meusnier property:* Those curves which pass through a given point in a given direction have circles of curvature (constructed at the common point) generating a sphere. 51. The inverse problem connected with natural families, namely, given the oo 4 trajectories to construct the generating field of force, is solved immediately in connection with property A. The force acting at any point p acts in the line joining that point to the corresponding point P, and its intensity is proportional to the reciprocal of the distance between the two points, f This construction may be carried out if we know a sufficient number of trajectories, without knowing the whole system. 52. The greatest number of rays which two distinct media *Kasner, Butt. Amer. Math. Soc., vol. 14 (1908), pp. 461-465. The result includes the extension of Meusnier's theorem made by Lie, and is in fact the largest generalization possible. t The determination of the potential function W(x, y, 2) or, what is equiv- alent, the index of refraction v(x, y, 2), requires a quadrature. ASPECTS OF DYNAMICS. 61 can have in common is oo 2 (one through each point of space). If two media have that many in common, it is easily shown that the resulting congruence is necessarily normal. Any normal congruence can be obtained in this way, for, as stated above, it belongs, not only to two, but to oo* distinct media. 53. We mention only one special problem: the determination of those media in which disturbances are propagated by Lame families of surfaces ; that is, every wave set is to be of the Lame type (thus forming part of a triply orthogonal family of surfaces). The index of refraction is found to vary inversely as the power of the point with respect to a fixed sphere; the rays then are the oo 4 circles orthogonal to that sphere. Since the radius of the sphere may be zero, real, or imaginary, these media yield well known interpretations of parabolic, hyperbolic, and elliptic geometries. (See Transactions of the American Mathematical Society, volume 12 (1911), pages 70-74.) 54-61. A SECOND CONVERSE PROBLEM CONNECTED WITH THE THOMSON-TAIT THEOREM 54. Consider the general conservative field, defined by its work function W(x, y, z). With any motion of the particle there is associated a definite value of the constant of total energy 1 V 2 _ w = h. If h is not assigned the complete system of trajectories is made up of oo 5 curves. Consider now an arbitrary surface, which we term the base surface, (2) z = /Or, y). From each of its points we may draw normal to the surface oo 1 trajectories since the initial value of the speed v is arbitrary. We thus have in all oo 3 trajectories normal to 2. In order to have a congruence we must assign the value of v at each point of 2, that is, we must give a law of distribution of the initial speed. The question arises: What form of law will make the corresponding 62 THE PRINCETON COLLOQUIUM. congruence a normal congruence? Of course for any law the congruence will be orthogonal to the base surface, but usually it admits no other orthogonal surfaces. The Thomson-Tait theorem (in its complete dynamical form) gives one such law: it states that if the initial speed is selected so as to make h have the same value at all the points of 2, the congru- ence will be normal. It thus gives a plan for constructing oo 1 normal congruences for a given base, one for each value of h. We shall refer to any one of these as "constructed according to the Thomson-Tait law." Is this the only answer to our question? If oo 2 trajectories are drawn orthogonal to 2 and if they form a normal congruence, does it follow that the distribution of values of the initial speed is precisely such that the sum of the kinetic and potential energies has the same value at all points of 2? The requisite discussion is not simple. We shall merely state the results we have obtained. 55. The answer to our question is " in general " in the affirma- tive. The first converse theorem, discussed in 37, is true without exception. The present is true with exceptions which may be definitely limited. For a " general " base surface 2 in a given conservative field of force, the only congruences, formed by oo 2 trajectories orthogonal to 2 (one draicn at each point), which admit an infinitude of orthogonal surfaces, are those constructed according to the Thomson-Tait law (so that the total energy has a constant value). 56. To make this precise we must of course limit the class of exceptional surfaces connected with a given field. These appear in the analytical discussion as the solutions of a certain partial differential equation of the second order* _ W x + qW z W x + qW. * The expanded result is of the form PIT + P z s + P 3 t + P 4 =Q, w here r, s, t denote the derivatives of second order of z = f(x, y). ASPECTS OF DYNAMICS. 63 where W is the given work function, and w = pW x + qW, - W, Vi + P Z + ? 2 This differential equation defines a class of surfaces which is seen to depend only on the equipotential surfaces W(x, y, z) = constant. The result may be put into geometric form and stated as follows : The only surfaces 2 which may be exceptional in the theorem of 55 (that is, which may give rise to normal congruences not included in the Thomson-Tait law} are those with this property: along each of the equipotential lines* of the surface the component of the acting force normal to the surface is constant. 57. Observe that it is not stated that the surfaces described, which exist in any field, actually give rise to additional normal congruences. To understand the situation more precisely, it is necessary to observe that in the analytic discussion the condition for a normal congruence is developed in the form i Q! + * 2 ft 2 + = 0, where t is the parameter which varies along the curve, starting with the value zero on the surface 2, and the coefficients 12 are functions of the two parameters defining the initial points on 2. By assumption the congruence is orthogonal to 2, so the term Q independent of t, will not appear. For a normal congruence all the coefficients ft must vanish. If only a certain number vanish the congruence may be described as approximately normal (the approximation being of degree n if fti = ft 2 = ft n = 0) : the curves are then orthogonal not only to 2 but also to one or more (infinitesimally) adjacent surfaces. 58. If now we impose on the congruence of trajectories normal to 2 the condition fti = 0, we find that this may be fulfilled for * The equipotential lines of any surface are the lines cut out by the equi- potential surfaces W = const. 64 THE PRINCETON COLLOQUIUM. any surface: the restriction is merely on the law of initial speed and means that the total energy must be the same, not necessarily over the entire surface, but along each equipotential line of the surface.* 59. If we further impose the condition Q 2 = 0, then for a " general surface " the law of speed must be the Thomson-Tait law, but for an " exceptional surface " the law is the more general one just stated. 60. The discussion of the higher conditions 12 3 = 0, etc., we have not completed. It is therefore not known precisely in which cases normal congruences (in the exact sense) may arise. For central and parallel fields it may be shown that the exceptional surfacesf actually give rise to normal congruences (in addition to those included in the Thomson-Tait theory): for such fields the vanishing of the higher coefficients follows from the vanishing of the first two. 61. The principal results of the converse problem may be formulated as follows: // oo 2 trajectories (of a conservative field), meeting a surface 2 orthogonally, are also orthogonal to an infinitesimally adjacent surface, then the total energy along each equipotential line of S is constant. If oo 2 trajectories, selected from the complete system of oo 5 , form a normal congruence, then in general they will all belong to the same natural family (that is, the total energy will be the same for all the curves); except possibly when the oo 1 orthogonal surfaces* are ex- ceptional in the sense defined in 56 (the additional congruences then and only then are normal to at least the second degree of approximation). Normal congruences not of the Thomson-Tait type (that is, not * If, in particular, the surface is one of the equipotential surfaces, the dis- tribution of speed is thus entirely arbitrary. t In the case of ordinary constant gravity the exceptional surfaces are those termed moulure surfaces by Monge: they are generated by rolling the plane of any plane curve about a vertical cylinder of arbitrary cross section. J If one of these surfaces is exceptional, all will be. ASPECTS OF DYNAMICS. 65 selected from within a natural family) actually arise for central and parallel fields. 62-67. GEOMETRIC FORMULATION OF SOME CURIOUS OPTICAL PROPERTIES 62. In Thomson and Tait's Natural Philosophy* the character- istic function of Hamilton is applied to the motion of a particle in a conservative field of force, and certain results are obtained which we shall try to restate as purely geometric properties of a natural family of trajectories. To what extent these properties are characteristic is not settled. We quote the principal passages referred to. " Let two stations, and 0', be chosen. Let a shot be fired with a stated velocity, V, from 0, in such a direction as to pass through 0'. There may clearly be more than one nat- ural path by which this may be done; but, generally speaking, when one such path is chosen, no other, not considerably diverging from it, can be found; and any infinitely small deviation in the line of fire from 0, will cause the bullet to pass infinitely near to, but not through, 0'. Now let a circle, with infinitely small radius r, be described round as center, in a plane perpendicular to the line of fire from this point, and let all with infinitely nearly the same velocity, but fulfilling the condition that the sum of the potential and kinetic energies is the same as that of the shot from bullets be fired from all points of this circle, all directed infinitely nearly parallel to the line of fire from 0, but each precisely so as to pass through 0'. Let a target be held at an infinitely small distance, a', beyond 0', in a plane perpendicular to the line of the shot reaching it from 0. The bullets fired from the circum- ference of the circle round 0, will, after passing through 0' ', strike this target in the circumference of an exceedingly small ellipse, each with a velocity (corresponding of course to its position, under the law of energy) differing infinitely little from V, the common velocity with which they pass through 0'. Let now a circle, equal to the former, be described round 0', *Part I (Cambridge, 1903), pp. 355-359. 13 66 THE PRINCETON COLLOQUIUM. in the plane perpendicular to the central path through 0', and let bullets be fired from points in its circumference, each with the proper velocity, and in such a direction infinitely nearly parallel to the central path as to make it pass through 0. These bullets, if a target is held to receive them perpendicularly at a distance a = a'V/V, beyond 0, will strike it along the circum- ference of an ellipse equal to the former and placed in a " cor- responding " position; and the points struck by the individual bullets will correspond; according to the following law of " cor- respondence ": Let P and P' be points of the first and second circles, and Q and Q' the points of the first and second targets which bullets from them strike; then if P' be in a plane containing the central path through 0' and the position which Q would take if its ellipse were made circular by a pure strain; Q and Q' are similarly situated on the two ellipses." 63. The second passage is as follows : " The most obvious optical application of this remarkable result is, that in the use of any optical apparatus whatever, if the eye and the object be inter- changed without altering the position of the instrument, the mag- nifying power is unaltered." ..." Let the points and 0' be the optic centers of the eyes of two persons looking at one another through any set of lenses, prisms, or transparent media arranged in any way between them. If their pupils are of equal size in reality, they will be seen as similar ellipses of equal apparent dimensions by the two observers. Here the imagined particles of light, projected from the circumference of the pupil of either eye, are substituted for the projectiles from the circumference of either circle, and the retina of the other eye takes the place of the target receiving them, in the general kinetic statement."* * This fact and many other applications are included in the following general proposition. " The rate of increase of any one component momentum, corresponding to any one of the coordinates, per unit of increase of any other coordinate, is equal to the rate of increase of the component momentum cor- responding to the latter per unit increase or dimension of the former coordinate, according as the two coordinates chosen belong to one configuration of the system, or one of them belongs to the initial configuration and the other to the final." ASPECTS OF DYNAMICS. 67 64. The statement in the first passage is not purely geometric; for it involves not only the curves described, but also the speeds V and V at the points and 0'. We therefore try to formulate the part of the theorem which is really geometric. We have a natural family made up of < 4 curves in space, one for each initial lineal element (point and direction) of space. Select any one of these curves c and any two points and 0' upon it. Construct the planes p and p' normal to this curve at and 0'. For each direction through 0, a curve of our family is deter- mined; this strikes the plane p' at a definite point. We thus have a certain correspondence between the bundle of directions through and the points of p'. For directions infinitesimally close to the direction of c at 0, and for points close to 0', this correspondence is linear; and by a proper selection of cartesian axes at and 0', we may write the correspondence in the canon- ical form where (x r , y'} denote the coordinates of the point in the plane p', and the corresponding direction at has direction cosines proportional to ( : 77 : 1). In an entirely analogous way, by considering the curves of the natural family which go through 0', and the points of inter- section with the plane p, we obtain a second linear correspond- ence which may be reduced to the form ' = ^2 = iK*i y)' If an arbitrary point transformation, unaccompanied by any change in the time, is applied, the new differential equations will usually involve not only x and y, but also the velocity com- ponents dxfdt, dyfdt. In fact the only exception is where the point transformation is merely affine: .1-1 = ax + by + c, yi = a'x + Vy + c'. Appell showed that if a general collineation (f)} ax+ by + c a'x + b'y + c' Xl ~ a"x + b"y + c'" yi ~ a"x + V'y + c" is accompanied by a change of the time of the form at (2') dt, = k(a"x+ V'y + c") 2 ' 73 74 THE PRINCETON COLLOQUIUM. the new differential equations will be of the original form d?Xi ? (3) - 2 and therefore define motion in some new positional field of force. The relation between the new field and the original field is explicitly as follows *i = V(a"x + b"y + c"r-{C'W 1 ft = P(a".r + b"y + c") 2 { - C(x} - y, ^), the first two defining the initial point, and the last two giving the components of the vector. The coordinates of the new vector are (x\, y\, 4 bound vectors of the first plane, represented by the independent coordinates (x, y, 3 trajectories of the new field; it is to be noticed however that, during any correspond- ing motions, positions which correspond according to the col- lineation will usually not correspond to the same instant of time; in fact from (2') dt h J k 2 (a"x + b"y 75. If X, Y denote the velocity components at the position x, y and if the corresponding velocity in the second plane is Xi, YI, acting at the position x\, y\, then we find, from the ca- nonical form (5), , Y l = ky^xY - yX). Thus we have a different vector transformation which may be termed the phase* transformation (in distinction from the force transformation of 74) : it gives the relation between the corre- sponding phases in the two planes. If we speak of points and vectors which correspond in the two planes according to the given collineation as projectively related, then the result may be stated in this form: The new phase vector does not coincide with the projection of the given phase vector: it has the same initial point, but the ratio of the actual length to the length of the projected vector is k 2 times the product of the distances from the ends of the original vector to the vanishing line of the collineation. 76. Having studied the Appell transformation and its geo- metric interpretation in terms of force vectors and phase vectors, we now ask whether other more general transformations can play a like role. Appell proved the following converse theorem: * The phase of a particle at any instant, in the sense of Gibbs, is its position together with its velocity: it is defined by the four numbers (x, y, x, y). ASPECTS OF DYNAMICS. 77 The only transformations of the form xi = *(*, y), yi = V(x, y), dt^ = n(x, y)dt which convert every set of differential equations d 2 x d?y (i) ^ = v(x, y), M = *(*,y), into one of the same form are those defined by (2), (2'). 77. By eliminating the time from (1), giving the differential equation of the trajectories in the form (page 7) (7) (t - y' l trajectories, will form a pencil (property I). As the initial direction (that is y'} varies about 0, the vertex of the pencil just described will move along a straight line] passing through (property II). The other properties, especially the fifth, are much more complicated. 81. In conclusion we point out another way in which the projective group enters in dynamics. If an arbitrary point transformation zi = 3>(x, y), yi = V(x, y) is applied to the differential equations x = xx x 2 + 23> xv xy + 3> yy f will usually define a motion due to a positional force together with a force depending on the velocity x, y. If this latter force is to be absent the transformation will be affine, as already re- marked ( 74). If, instead, we demand that the latter force shall act in the direction of the velocity (and thus be in the nature of a resistance), we find that the transformation may be any collineation. More generally, projective transformations are the only point * In the original metric statements these are of course the circular points at infinity. t The force direction will be determined protectively as the harmonic of this line with respect to the lines joining O to A and B- ASPECTS OF DYNAMICS. 81 transformations which leave invariant the type x = 1 natural families which are not usually related, that is, do not form the complete system of trajectories belonging to a conservative field. The only case in which the new families are related arises when W = \, for then the new systems are due to the work function W, = 1/X. We then reach the conclusion that in any conformal representation (excluding the trivial homothetic case*) there is a unique conservative force whose complete system of oo 3 dynamical trajectories is con- verted into the complete system of some (usually distinct) conserva- tive force. The work function of the force in question is defined by the squared ratio of magnification, 85. Similar statements may be made for brachistochrones. Every system of oo 2 brachistochrones due to any work function and a given value of h of course becomes such a system, for any natural family may be regarded as a family of brachistochrones. But there is only one complete system of 2 such curves, defined by the property that the curvature at each point and direction equals the curvature of a free particle starting out from that point and direction with the speed v . The differential equation of this velocity system is This is recognized as a natural family; it corresponds to the geo- desies of the surface whose first fundamental form is By varying VQ we obtain the oc 1 velocity systems belonging to the given field ; they are pictured by the geodesies of oo : surfaces. Consider now a conformal representation of the xy-p\ane upon itself. This converts dx- + dy- into where H(x, y), by known theory, is a harmonic function. We thus obtain oo * new natural families corresponding to the geo- desies of the oo x surfaces 86 THE PRINCETON COLLOQUIUM. These l natural families cannot usually be regarded as related velocity systems for some new field: the requisite condition is that W shall be the same as // except for a constant factor. Hence for a given conformal transformation of the plane (which is not merely a similitude), there is a unique complete velocity system belonging to a conservative field of force which is converted into a complete system. The unique work function is W = H = log X, where X denotes the squared ratio of magnifaction in the given conformal representation. The fields obtained are Laplacian, that is, satisfy the condition W xx + W n = 0. As an example, the transformation Zi = log z converts the oo 3 velocity curves of the field W = log r (in which the force varies inversely as the distance from the origin) into the 3 velocity curves of the field W\ = Xi (force vertical and constant) . 90. It was shown above that conformal transformations are the only point transformations which convert every natural family into a natural family. Natural families are characterized by properties A and B of 31. It is of interest to notice that property A by itself is conformally invariant. The most general system having this property (that osculating circles constructed at any point have another point in common) is what we have termed a velocity system. We now prove that The only point transformations which convert every velocity system into a velocity system are the conformal transformations. Consider, say the three-dimensional case, where the general velocity system is y' 2 + A *" - fc - *V)U + / + A The only curves which are common to all such svstems must ASPECTS OF DYNAMICS. 87 satisfy 1 + / + z' 2 = 0, y" = 0, z" = 0, and are therefore the minimal straight lines of space. Since the only transformations converting minimal lines into minimal lines are conformal, we have the result stated. That conformal transformations actually leave the velocity type invariant is easily verified analytically*. The result is obvious synthetically (in the case of more than two dimensions) since the conformal group converts circles into circles and bundles of circles into bundles. Hence if the original system possesses property A, the same will be true of the transformed system. 91. It may be shown that, for any given non-conformal trans- formation, there exists one and only one velocity system which is converted into a velocity system. 92-94. CONTACT TRANSFORMATIONS 92. With each natural family, or, what is the same, with each isotropic medium, there is associated a definite infinitesimal contact transformation. This connection, which appears im- plicitly in Hamilton's fundamental memoir of 1835, was worked out in detail by S. Lie.f If the index of refraction is v(x, y, z), the associated contact transformation has the characteristic function (1) v(x, y, z) Vl + p 2 + 9 2 , where x, y, z, p, q are considered as the coordinates of a surface element. If the one-parameter group generated is applied to an arbitrary surface the resulting oo 1 surfaces form a wave set. The trajectories or rays appear as the path curves of this group. Lie show r ed that the category of transformations which thus * Cf. American Journal of Mathematics, vol. 27 (1906), p. 213, for the two- dimensional case. t " Die infinitesimalen Beriihrungstransformationen der Mechanik," Leip- ziger Berichte (1889), pp. 145-153. A very elegant discussion, with new results, is given by Vessiot, Bull. Soc. math, de France, vol. 34 (1906), pp. 230-269. 88 THE PRINCETON COLLOQUIUM. appears, with a characteristic function of type (1), and which he termed " the infinitesimal contact transformations of mechanics," is distinguished geometrically by the fact that the so-called* transversality relation reduces to orthogonality. 93. The following simple and easily proved theorem appears to be new. The alternant (or Klammerausdruck of Lie) of the contact trans- formations associated with any two media is always a point trans- formation. 94. Here we are dealing with two natural families in the same three-dimensional space. In connection with the most general problem of dynamics (page 70), spaces of any dimensionality must be considered, with arbitrary variable curvature. The space depends on the quadratic form defining the kinetic energy: this determines the quadratic expression appearing under the radical in the generalization of (1). The potential! determines the factor v which may be any point function. The general theorem is then as follows: The alternant of the contact transformations associated with two dynamical problems (or natural families} will be a point transforma- tion when, and only when, the two expressions for the kinetic energy are either the same or differ by a factor (which may be any point function) ; the two potential energie^ remain entirely arbitrary. In particular, if any two natural families are constructed in the same space (which space is entirely arbitrary), the alternant will be a point transformation. For a detailed discussion of the two-dimensional case, in- cluding a number of converse results, the reader is referred to the author's paper, cited in the first footnote below. * Lie does not use this term. The author borrows it from the closely connected problem in the calculus of variation. See " The infinitesimal contact transformations of mechanics," Bull. Amer. Math. Soc., vol. 16 (1910), pp. 408-412. t Here considered as including the energy constant h, which is fixed, since we are dealing with a natural family. ASPECTS OF DYNAMICS. 89 95-97. A GROUP OF SPACE-TIME TRANSFORMATIONS 95. In the fundamental transformation of the relativity theory, known as the Lorentz transformation, the position coordinates x, y, z and the time coordinate t are merged: the new position and the new time appear as functions of both the original position and the original time. The Lorentz group is composed of the linear transformations of the four variables x, y, z, t which leave invariant the quadric ar 2 + 2/ 2 + z 2 ~ c 2 * 2 = 0. Its importance is due to the fact that it leaves unaltered the form of the Maxwell equations. We consider in this section an entirely different group of space- time transformations, depending on arbitrary functions instead of arbitrary constants. It arises in connection with ordinary (newtonian) dynamics in the theory of forces depending on the time as well as position. We confine the discussion for the sake of simplicity to the case of two dimensions. What transformations of the three vari- ables x, y, t will convert any set of equations of the form ds*c d u (1) -ftp = (x, y, t), jp = t(x, y, t) into another set of the same form? An arbitrary transformation would produce equations representing* a force depending, not only on x, y, t, but also on the velocity dx/dt, dy/dt. The problem is to find those peculiar transformations which do not introduce the velocity in the final equations. The result is as follows : The only space-time transformations which convert every space- time field of force into a space-time field are those of the form (2) t, = f(t), x, = (ax + by) 4f'(t) + 0(0, The group thus involves three arbitrary functions f(t), g(t], h(t) as well as four arbitrary constants a, b, c, d. 90 THE PRINCETON COLLOQUIUM. 96. Another representation of the same group, which has the advantage of avoiding radicals, is -n = (ax (3) y, = (ex + dt)\(t) + v(t}. When such a transformation is applied to equations (1), the new equations are found to be X^ = (XX - 2X 2 )(aa: + by) -f X 2 (a^ + 6 l curves may be regarded as the lines of force in some field, in fact in an infinite number of different fields. The triply infinite system of trajectories has peculiar properties which have been discussed in Chapter I. Other noteworthy systems of curves are connected with the field, for example, brachistrochrones, catenaries, velocity curves, and tautochrones. 99. Omitting the tautochrones, the other three systems named, together with the trajectories, may all be obtained as special cases of this simple general problem : to find curves along which a con- strained motion is possible such that the pressure is proportional to the normal component of the force. 100. If an arbitrary curve is drawn in the plane field of force, and the particle, of say unit mass, is started along it from one of its points with a given speed, the constrained motion along the given curve is determined. The acceleration along the curve is given by T, the tangential component of the force vector. So the speed at any point is determined by (1) tf = fids. The pressure P (of course normal to the curve, since the curve 91 92 * THE PRINCETON COLLOQUIUM. is considered smooth) is given by the elementary formula (2) P=*-N. If we increase the initial speed, the effect is to increase v 2 by a constant tf; and hence P changes by the addition of a term of the form cfr. 101. If the given curve is a trajectory, the initial speed may be so chosen that the pressure vanishes throughout the motion; that is, trajectories may be defined as curves of no constraint. Of course, if a different initial speed is used, P will be of the form cfr; but, as regards the curves, they are completely characterized by P = 0. 102. If the given curve is a brachistochrone and if the motion along it is brachistochronous, Euler proved (assuming the force to be conservative) that the pressure was double the normal component of the acting force and opposite to it in direction, that is, P = 2N. If the force is not conservative, the real brachistochrones, as defined by a problem of the calculus of varia- tions, form a quadruply infinite system. The curves defined by the property P = 2 A T then form a triply infinite system of what should be called pseudo-brachistochrones. These curves are really brachistochrones only in the conservative case. No ambiguity however will arise by terming the system here con- sidered brachistochrones instead of pseudo-brachistochrones. 103. The general problem suggested is to find curves such that P shall be proportional to N. So P = kN. To a given value of k there correspond =o 3 such curves: the system so obtained will be denoted by Sk- The four special cases of physical interest are as follows: k = gives So, the system of trajectories; k = 2 gives S-%, the system of brachistochrones; k = 1 gives Si, the system of catenaries; k = oo gives S^, the system of velocity curves. ASPECTS OF DYNAMICS. 93 104. The last case requires a justification in terms of limits which is easily carried out analytically. 105. The third case follows from the known fact that when an inextensible flexible homogeneous string is suspended in any field of force, the resulting form of equilibrium, called a catenary in the general sense of the term, has the dynamical property that when a particle, started out with the proper initial velocity, rolls along the curve, the pressure at any point equals the normal component of the force: that is, catenaries are defined by P = N, corresponding to k = 1. 106. Of course a triply infinite system Sk exists for any value of the parameter k. The differential equation of the system, in intrinsic form, is easily obtained by eliminating v from the equations (3) v*/r = (k + 1)N, vv s = T. The result is (4) Nr s = (n+ 1)T - rW, where (4') n We may readily find various properties from this intrinsic equation, but in order to obtain a complete set it is necessary to have recourse to the equivalent equation in cartesian coordinates (* - y W" = {*. + Gh, - v*W - 3 + - -i- 2 - \y ". 1 + r This obviously reduces to the familiar trajectory equation of 1 when n 2, corresponding to k = 0. Brachistochrones cor- respond to n = 2, catenaries to n = 1, velocity curves to n = 0. 107. We now state the characteristic properties of a system of the above type for any value of n, that is, any value of k. 94 THE PRINCETON COLLOQUIUM. Characteristic Properties of the System Sk Property 1. For any given element (x, y, y') the foci of the osculating parabolas of the single infinity of curves determined by the given element lie on a circle passing through the given point. Property 2. At any point the tangent of the angle which the focal circle makes with the given element is to the tangent of the angle which the given element makes with a certain direction fixed at (the direction of the acting force) as 3 is to n + 1, that is, as 3k + 3 is to k + 3. Property 8. Through a given point there pass a single infinity of curves admitting hyperosculating circles of curvature; the centers of these circles lie on a conic passing through the given point in the direction of the force vector. Property 4- The normal at the given point cuts the conic described in property 3, at a distance equal to n + 1, that is (k + 3)/(fc + 1), times the radius of curvature of the line of force passing through 0. Property 5. This is of the same form as property V ( 3) obtained in the discussion of trajectories, the number 3 being replaced by the number n + 1. In the notation of page 11 d 1 j?._l COCOsy COzWy _ dxAA'^~ dyBB'^ (n+ l)co 2 108. The special case where n equals 1, that is, the system 8-3, is exceptional and requires a separate discussion; but as we do not need the results, this case is omitted. 109. While the properties corresponding to different values of k are analogous, they are of course not identical. The first property is common to all the systems. But the second property involves the parameter k. Thus, while for trajectories the con- stant ratio that appears is 1 (bisection), it is 3 for brachisto- chrones, 3/2 for catenaries, and 3 for velocity curves. Not only are the triply infinite systems Sk, corresponding to different values of k, distinct in any given field of force, but also no two ASPECTS OF DYNAMICS. 95 systems arising in two distinct fields can ever coincide. For example, if a certain system of o 3 curves arises as trajectories in one field, it cannot also arise as catenaries in either the same or another field. 110. If we combine all the systems Sk, in a given field ( 2 straight lines of the plane form part of every system Sk. On the other hand, if r approaches zero, then r a approaches a definite limit Remembering that the tangent of the angle of deviation is one third of r t , we may state the result obtained as follows: In any system Sk if we take any lineal element and let r approach zero, the tangent of the corresponding angle of deviation is to the tangent of the angle which the force vector makes with the normal to the given element in the fixed ratio of n -f 1 to 3. The special values of this ratio for the four special systems of physical interest are respectively 1, 1/3, 2/3, 1/3. In the case of tra- jectories, it is noteworthy that the limiting position of the axis ASPECTS OF DYNAMICS. 97 of deviation coincides with the direction of the force acting at the given point. 115-116. CURVES OF CONSTANT PRESSURE 115. We now consider a second simple generalization of the problem P = 0, defining trajectories. We consider, namely, curves corresponding to P = c, where c denotes any constant. The curves obtained may be termed curves of constant pressure : only along such a curve is a constrained motion of a particle possible such that the pressure against the curve remains constant. For a given value of c a system of 3 such curves is obtained, whose intrinsic equation, found by differentiating the relation P = tf/r - N = c, is (c+ N)r a = 3T- 9t. We see that this system for any value of c retains property I of the system of trajectories. Omitting the discussion of the higher properties of these triply infinite systems we consider the quad- ruply infinite system whose differential equation, found by elimi- nating c, may be written in either of the intrinsic forms - 37V)r.. = (2r9i - 7> s 2 + [9V + (9? 2 - 3)r - 3N]r s , - 37> 2 = (3r9t - 4Z>i 2 + [9V + (9t 2 This gives the totality of oo 4 curves of constant pressure defined by a given field. As regards special cases of interest, we note, in addition to c = 0, giving trajectories, the case c = QO which gives r s = 0, defining circles; hence for any field of force the oo 4 curves of constant pressure include the oc 3 circles of the plane, which arise in fact as curves of infinite pressure. 116. The quadruply infinite system which here arises, as w r ell as that obtained in the previous problem P = kN, comes under 15 98 THE PRINCETON COLLOQUIUM. the category represented by a differential equation of the type* f = Ay'" 2 + By'" + C. It therefore enjoys the property, previously stated in the other problem ( 112), that the locus of the centers of the osculating conies corresponding to any element (x, y, y', y") is a conic touching the element (x, y, y'}. Of course, since the forms of A, B, C in the two problems are quite distinct, the systems are distinguished in their higher properties. 117-118. TAUTOCHRONES 117. Tautochrones are not included in either of the previous problems. They are not distinguished by any simple law of pres- sure, f The condition for a tautochrone is that the resulting con- strained motion of a particle along the curve be harmonic, that is, (1) T = k(s - ), where k is a constant (which is negative for actual and positive for virtual tautochrones) and s So denotes the arc reckoned from a fixed point of the curve, the center of the tautochronous motion. From this (2) T u = and hence, by expansion, the general equation of the system of 3 tautochrones in any field is\ (3) tfr.-^f +(, + 5R)r- T, where the notation is that of 2. * This type (noteworthy in that it unifies many distinct mathematical and physical problems) first presented itself in the author's study of "Systems of extremals in the calculus of variations," Bull. Amer. Math. Soc., vol. 13 (1907), p. 290: the extremals of any integral of the second order J f(x, y, y', y")dx form a system of that type. In these lectures other physical problems leading to species included in this type are treated in 110, 135, 137. t It may be shown that during any tautochronous motion P = k(s - s ) 2 /r - N. t " Tautochrones and brachistochrones," Bull. Amer. Math. Soc., vol. 15 (1909), pp. 475-483. ASPECTS OF DYNAMICS. 99 We see that r is a quadratic function of r, and not a linear function as in the case of trajectories and the other systems Sk- For a discussion of the geometric properties of tautochrones, we refer to the dissertation of H. W. Reddick.* 118. There is no field in which the tautochrones coincide with the trajectories, or with any of the systems Sk, in either the same or some other field, except for the case k = 2 corresponding to brachistochrones. The classical work of Huygens and J. Bernoulli showed that for a uniform field the system of tauto- chrones is identical with the system of brachistochrones. The author has shown that the only other field where such duplication occurs is that in which the force is central and varies directly as the distance. The only case of duplication in two distinct fields is as follows: The tautochrones of the field

3 tautochrones in a given field has asso- ciated* with it a certain time of oscillation, determined by the value of the constant & in (1). To each value of the period, that is, to each value of k, corresponds a certain family of oo 2 tauto- chrones, whose differential equation, in implicit form, is r(k- ) ='N, or, expanded, We pass over the easy geometric interpretation ; and note merely the special family, corresponding to the value k = 0, for which * Amer. Jour, of Math., vol. 33 (1911). t The corresponding problem in space is treated in Reddick's paper and gives a class of fields involving twenty parameters. 100 THE PRINCETON COLLOQUIUM. the period is infinite. This separates the actual from the virtual tautochrones. 119. NON-UNIFORM CATENARIES 119. It is a familiar fact that vertical parabolas appear in elementary dynamics in two distinct discussions; first, as trajec- tories of a cannon ball, and secondly as forms of equilibrium of a chain in which the mass (or load) of any element is propor- tional to the horizontal projection of that element. Here the force is ordinary gravity. The question arises whether any other fields of force give rise to a like duplication. We first consider the following general problem of non-uniform catenaries. If a flexible string or chain, in w r hich the mass of any element of length is proportional to some given function p of x, y, y', is suspended in a positional field, the possible forms of equilibrium are defined by the equation This represents the 3 non-uniform catenaries for a given field ) - A>l where the argument v of / is to be expressed in terms of x, y, y't y" by means of y" Consider now the l trajectories starting from a given element (x t y, y'}. The focal locus, that is, the locus of the foci of the osculating parabolas, varies in shape with the function /, that is, with the law of resistance. W T e know that, if there is no resistance, property I of 3 holds, that is, the focal locus is a circle passing through the given point. Are there any resisting media for which this property is pre- served? A simple discussion shows that there are, the appro- 102 ASPECTS OF DYNAMICS. 103 priate media being those for which R is of the form Av 2 -f- B. For such media, property II will not usually be fulfilled; in fact the only medium preserving the properties I and II is that in which the resistance varies as the square of the speed. If we impose also property III, both A and B must vanish, that is, the resistance vanishes and the force is purely positional. It is of interest to examine the case where the resistance varies as any power v n of the speed. The differential equation of the trajectories is then of the form y'" = ay" + by"* + cy" m , where m = |(4 n). The focal locus is a curve whose inverse with respect to the given point is X= at+biY This becomes a straight line (as in the case of no resistance), when m is 1 or 2, that is, when n is 2 or 0. The curve is a conic when m is 3 or or 3/2, that is, when n has one of the values 2 or 4 or 1. When n = 2 the conic is a parabola with its axis parallel to the given element. When n = 4 it is a hyperbola, asymptotic to the line of the given initial element. When n = 1 it is a parabola touching the initial line (not at the given point). 121. We now state briefly the corresponding results in ordinary space. No matter what the law of resistance is, property I (of the set of four properties for space given in 11) is fulfilled; for the osculating planes necessarily pass through the force vector. The only laws for which property II is preserved are those included in R = Av 2 + B. If property III is also to be preserved, the resistance must vanish. 122. The results may be derived easily from the intrinsic equations (3) v* = rN, vv s = T + R, 104 THE PRINCETON COLLOQUIUM. obtained by taking components of the acting forces along the normal and tangent to the trajectory. The geometric equation, resulting from the elimination^ v, is of the form* (4) Nr a = - r91 + 3r+2fl. This gives the relation between r a (the rate of variation of r with respect to *) and r (the radius of curvature). The resistance R, which is given as a function of v, is here to be expressed in terms of r by means of the first of the relations (3). If prop- erty I, of plane trajectories, is to hold, r s must be a linear integ- ral function of r; this will be the case not only when R vanishes, but also, as stated above, when it is of the form AT? + B. 123-126. PARTICLE ON A SURFACE 123. The motion of a particle on any constraining surface x = ~~ ' dt\dv " dv where T is the kinetic energy 2T = Etf+IFuv+Gi? and U, V are the components of the force given as functions of u, fl.f The explicit equations of motion are of the form u = v = * From this we may obtain the following dynamical result : If a particle starts from rest, the initial radius of curvature of the trajectory is to the radius of curvature of the line of force passing through the initial point as 37 1 + 2R is to T. When R vanishes we have the simple result previously stated. t See for example Whittaker, Analytical Dynamics, p. 390, and Hada- mard, Jour, de Moth. (5), vol. 3, p. 331. ASPECTS OF DYNAMICS. 105 where 3>, ^ define the force and the A's and B's are functions of u y v depending only on the given surface. 124. We observe that here u, v depend not only on the position u, v but also upon the velocity u, v. Hence the motion in the wfl-plane corresponding to the actual motion on the surface is not usually generated by any positional force in that plane. The only exception arises when the A's and the B's vanish identically: this is the case only if the given surface is develop- able, and if its representation on the wt>-plane differs from its development on the plane by at most an affine transformation. Another problem including this as a special case is to deter- mine when the motion in the wi>-plane can be regarded as due to a positional force together with a resistance acting in the direction of the motion. The condition for this is A v? + 2Aiuv -f- A^v 2 u B u 2 + 2B l uv + B z v 2 ) ' Expanding, we find four conditions on the six functions A, B, which turn out to be precisely the conditions that the geodesies of the surface shall be pictured by straight lines, a result which may be proved directly. Hence the only case in which the motion on the surface is pictured in the z<-plane by a motion due to a positional force together with a resistance depending on the velocity components and acting in the direction of the motion, is that in which the surface has constant curvature and the rep- resentation is geodesic. 125. We proceed with the general equations of motion. If we eliminate the time, we obtain the differential equation of the third order defining the o 3 trajectories in the form = {5 / where the coefficients are functions of u, v. We confine our- selves to the observation that the picture curves in the uv- 106 THE PRINCETON COLLOQUIUM. plane come under the type where the coefficients are lineal-element functions: the focal locus is thus not a circle, but a special quartic. Hence if we consider the oo ! trajectories on the surface obtained by starting a particle at a given point in a given direction with different speeds, the picture curves in the wo-plane have osculating parabolas at the common point whose foci lie on a special quartic curve. 126. What is the simplest property of the actual trajectories described on the surface? -What is, in particular, the locus of the osculating spheres of the oo l trajectories considered? To answer this we take our surface not in -parametric form, but in the explicit form z = f(x, y). We may take the given point as origin, the tangent plane as the zy-plane, and the fixed initial direction as that of the axis of x. We find, by differentiating the equation of the surface and making use of y' = 0, z' = 0, that z" = a> z'" = b + cy", where a, b, c are constants, equal respectively to the values of the partial derivatives f xx , f xxx , 4f xy at the origin. Again, from the general equation of the trajectories, we have a relation of the form y'" = a + ft" + yy"\ The center of the osculating sphere of the trajectory is then X = 0, z'" = b + cy" = y"z"' - z"y'" = y"( b + C y") - a(a+py" + yy" z ) ' " = .'//" -.",."' ~ y"z'" - z"y'" y "(b + cy") - a (a ASPECTS OF DYNAMICS. 107 Here y" enters as parameter, varying from curve to curve: eliminating it, we find the locus, lying in the plane X= 0, to be a7 2 + /3F(l-aZ) + 7(l - aZ) z + Z{bY + c(l - aZ)\ = 0. Hence for any positional force on any surface, the l trajectories starting from a given lineal element of the surface have osculating spheres, at the common point, whose centers lie on a (general) conic in the plane normal to the element. This conic passes through the center of curvature of the normal section of the surface determined by the given element. If the element is in one of the principal directions of the surface, the conic touches the normal to the surface. 127-130. THE GENERAL FIELD IN SPACE OF ^-DIMENSIONS 127. Any dynamical system with n degrees of freedom may be represented by a particle in space of n dimensions. For example, an arbitrary rigid body in ordinary space is represented by a particle in six-dimensional space, and the astronomical problem of three bodies in the most general case leads to a representative particle in space of nine dimensions. For conservative forces, or natural families, the general dis- cussion for any dimensionality has already been given ( 69). We shall not attempt a complete discussion for arbitrary posi- tional forces (corresponding to that given in Chapter I for two and three dimensions). The equations of motion for an arbi- trary field are X n = We confine ourselves to the simplest questions. If the initial position and initial direction are kept fixed, and only the initial speed v is varied, what are the properties of the oo 1 trajectories obtained? The simplest geometric result is that r s (the rate of variation of the radius of curvature with respect to the arc length) varies as a linear function of r. The locus of the centers of the osculating spheres is a straight line, just as in the case where n is three. 108 THE PRINCETON COLLOQUIUM. 128. A general curve in n-space has at each point an osculating plane, an osculating 3-flat, and so on up to an osculating (n 1)- flat. It is obvious that our oo 1 trajectories have the same os- culating plane since this is determined by the given initial direction and the direction of the force. It can be shown that the osculating 3-flat is also fixed; the 4-flat varies, generating a pencil; the 5-flat varies, generating a quadratic system; and so on, with more complicated variations. 129. Consider next the connection between the various cur- vatures and the speed. In the plane (n = 2) there is only one curvature 71, and this varies inversely as the square of v. In space (n = 3) the first curvature 71 varies as above, and the second curvature or torsion y z remains fixed. If n = 4, we have three curvatures. The laws for 71 and 72 are as above, while 73 = ci -f c 2 y~ 2 , where c\, c 2 are constants (depending of course on the given initial lineal element). If n = 5, we have 71 = av~~, 72 = b (these forms are valid for any dimensions) and d\ + d z v* + e? 3 4 = V ci + c 2 tT 2 + 73 = *v ci -h c z v -f W -, 7,, = + If n = 6, 73 remains the same, the numerator in 74 is replaced by the square root of a polynomial involving v 8 , and 75 is given by a rational formula. It is easy to write down the general formulas for the n 1 curvatures in n space. All except the first, second, and the last are irrational. These results are to be regarded as general- izations of the elementary fact (included in the formula for centrifugal force v*/r), that the ordinary curvature varies as v~ 2 . 130. By eliminating v from any two of the formulas, we can obtain purely geometric results. For example, in space of four dimensions, 73 = A -f- Byi, where A and B depend only on the ASPECTS OF DYNAMICS. 109 common initial element. But in higher spaces 73= V^ + fl Tl +C7i 2 . This is the form required in particular in the application to the problem of three bodies, since the representative space has nine dimensions. 131-132. INTERACTING PARTICLES IN THE PLANE AND IN SPACE 131. We consider the motion of n + 1 particles, denoted by M, MI, , M n , moving in the plane under the action of any forces depending on the position of the particles. The dif- ferential equations of motion are then of the form x = l * Since the forces in the three-body problem are conservative, we may decompose the motions into natural families, and interpret each family in a flat space of eight dimensions. The circles of curvature at a given point will meet again; eight of them will be hyperosculating, and these will be mutually orthogonal. Cf. 70. 112 THE PRINCETON COLLOQUIUM. trajectories, one for each value of the initial speed. The follow- ing properties are obtained: (1) We find that the focal locus (that is, the locus of the foci of the oo l osculating parabolas) is a quartic curve whose inverse with respect to the given point is a parabola which is tangent to the given direction line (the point of contact is not usually at the given point). (2) As y' varies (x, y, t being held fixed) this point of contact describes a cubic curve whose inverse is a conic passing through the given point in the direction of the force. (3) The initial direction of y' bisects the angle between the direction of the force and the direction of the axis of the parabola described in (1). 134. The total system of trajectories, for all initial conditions, consists of oo 4 curves. Only in the case where the force does not depend upon the time does the system consist of oo 3 tra- jectories. In the properties stated above, the initial time is kept fixed. In a certain sense then the results are not purely geometric: they would not appear in a photograph of the complete system of trajectories. This system will be represented by a cer- tain differential equation of the fourth order; but it is not possible to carry out the requisite eliminations in explicit form, and hence the derivation of purely geometric properties involves essentially new difficulties. A complete characterization is however ob- tained, by projection from space curves, in 136, 140. 135. There is an interesting special case in which the elimina- tion can be carried out: namely, the problem of the motion of a particle of variable mass in a positional field of force. The time then appears only through the mass, so the equations of motion are of the form (3) /(O* = v(x, y}, As the result of the elimination is complicated, we shall here consider only the case where the function f(t), representing the mass, is of one of the special types t 4 , t z , e', (log O 2 - The equa- ASPECTS OF DYNAMICS. 113 tion of the fourth order representing the trajectories is then found to be of the form (4) y = Ay'" 2 + By" 1 + C, where A, B, C involve only x, y, y', y". We see that the fourth derivative is a quadratic function of the third derivative. This category of equations of the fourth order arises in a number of different connections, in particular in the inverse problem of the calculus of variations, as stated in 116. The characteristic geometric property may in the present case be stated as follows: If the particle, whose mass varies according to one of the four laws stated, is projected into a field of force from a fixed initial position in a fixed direction at different times, with the initial speed for each time so adjusted as to cause the initial curvature of the trajectory to have a fixed value, and if for each of the oo l trajectories thus obtained we construct the osculating conic (having five-point contact), the locus of the centers of these conies is a conic passing through the given conic in the given direction. Of course not every system of oo 4 curves having this property can be regarded as a trajectory system corresponding to equations of motion of the form considered. We do not, however, attempt a complete characterization. 136. Space-time Curves. When we integrate the equations of motion, either in the special case where the forces depend only on the position (!') x = 3 trajectories arise, since the system of space- time curves, whose number is still oo 4 , now admits the group of translations along the -axis. If we project the space-time curves orthogonally on the xt- plane and on the yt-p\ane, we obtain in each case a system of oo 4 plane curves. What are the properties of the system of oo 4 space-time curves? The following two properties are characteristic: (1). The osculating planes of the oo 2 space-time curves through a given point go through a fixed line parallel to the ary-plane. (This line is parallel to the direction of the force acting at the projected point in the xy-plane.) (2). If the oo 2 space-time curves through the given point are orthogonally projected on any plane perpendicular to the in/- plane, the oo 2 plane curves obtained are such that those which have the same tangent also have the same curvature. Another complete characterization may be given as follows: (3). If the oo 2 space-time curves through a given point are orthogonally projected on either the .rtf-plane or the 2/2-plane, the oo 2 plane curves obtained have their centers of curvature located on a special cubic of the form 1?=a(x 2 -\-t 2 ) or fl=b(y-+( i ). A corresponding cubic locus will then necessarily arise by pro- jection on any plane perpendicular to the a-y- * It may be remarked that if, in problem (1), the force is multiplied by a constant c (or, what is equivalent, the mass of the particle is multiplied by lie), a distinct system of oo 4 space-time curves will be obtained. The totality of oo 5 space curves, thus related to the oo 1 plane problems x - c(* y> 2 )> ^ = vfa y> 2 )- The curves in space x, y, z are then the space-time curves. Their projections on the i/z-plane are the trajectories (whose explicit properties have not been derived). Their projections on the xy- plane (or on the o-z-plane, or on any plane parallel to the z-axis) are curves whose properties have just been stated ( 137, 138). The differential equation in the xy-p\ane is where the coefficients involve only x, y, and y 1 '. 140. We have not attempted a complete direct characterization of the systems of curves arising in any one of the coordinate planes. Such a characterization has however been given ( 136) for the system of 4 space-time curves. Indirectly this really solves all the problems. A system of curves in the plane can be regarded as trajectories of a force depending on time and position if and only if the curves can be obtained by orthogonal projection from some system of oo 4 curves in space having the properties (1) and (2) of 136. If, furthermore, the space system is invariant under translation perpendicular to the given plane, the plane system, then consisting of only oo 3 curves, belongs to a posi- tional field. 141. For any force depending on time and position x =