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In most dynamical investigations relating to conservative forces, in particular those conllected with the principle of least action and the developments of Hamilton and Jacobi, it is essential to group the possible tnotions according to the value of the constant h representing the total energy. The trajectories corresponding to any given value of h are said to form a natural f family. In the case of a partiele moving in a three-dimensional conservative field, a natural family consists of oo4 curves. Examples are the oO4 straight lines of space, corresponding to zero force; and the oo4 vertical parabolas whose directrices are located in a fixed horizontal plane, corresponding to gravity assumed constant. In a paper published in the preceding volume of these T r a n s actions, t the general geometric character of natural families was expressed in terms of osculating circles; and a aertain reciprocity, analogous to that of Scheffers for plane isogonal trajectories, was established. In the present paper, which may be read independelltly, we start from the remarkable theorem due to THOMSON and TAIT which states that the x2 curves of a natural family which tneet any surface orthogonally are necessarily orthogonal to ool surfaces, that is, form a normal congruence. Our main object is to show that thts propexty betongs eseclusively to nat?ral farnilies. The new result may be regarded as a converse of Thomson and Tait's theorem and stated as follows: If a qucldruply-ininite system of curves in Space ts s?beh that x2 curves of the system meet an arbttrary surface orthogonally and always form a normal congruence (that ts, admit ool orthogonal sqxrfaces), then the system is of the ncltural type. Natural families present themselves not only in the study of dynamical tra

The ideal elements of projective geometly are usually introduced by means of the parallel and congruence axioms. The idea of defining the ideal elements without the assumption of the parallel axiom is due to KLEIN.+ It was developed by PASCH; :L by SCHUR; § by BONOLA; || and by VEBLEN.1| I11 all of these developments a three-dimensional geometry is assumed. The problem of defining the ideal elements in a plane geometry satisfying only order relations is closely connected with the probleln of finding the necessary alld sufficient condition that a plane may be a part of a three-space in which the axioms of order are satisfied. This condition is stated by HILBERT** to be the validity of the Desargues theorem in the p]ane. The Desargues theorenl may be proved in a three-dimensional space satisfying only order relations, but can not be proved in the corresponding plane geollletry, without additional assumptions, e. g., of the parallel and congruence axioms. In this paper plane axioms of order will be assumed in the form given them by VEBLEN, the undefined elements being taken as the point, and a relation among points, called order. The first eight axioms are identical, except for notation, with his. Another axiom, one of closure, is then introdueed, limiting the set of lvoints considered to a plane. Two more axiollls are then introduced, foruls of the Desargues theorem, and of its converse, in terms of the set of points satisfying only order relations.

If in the ordinary theory of rational numbers we consider a composite illteger m as modulus, alld if from among the classes of congruent integers with lespect to that modulus we select those which are prime to the lnodtllus, they form a well-knowll multiplicative group, which has been called by WEBER (Alyebra, vol. 2, 2d edition, p. 60), the most ialportant example of a finite abelian group. In the more general theory of numbers in an algebraic field we nsay in a corresponding manner take as modulus a composite ideal, which illeludes as a special case a composite principal ideal, that is, an integer in the field, and if we regard a11 those integers of the field which are congruent to one another with respect to the modulus as forming a class, and if we select those classes whose integers are prime to the modulus, they also will form a finite abeliall groupt under multiplication. The investigation of the nature of this group is the object of the present paper. I shall confine my attention, however, to a quadratic number-field, and shall determine the structure of the group of classes of congruent quadratic integers with respect to any composite ideal modulus whatever. Several distinct cases arise depending on the nature of the prime ideal factors of the modulus; for every case I shall find a complete system of independent generators of the group. Exactly as in the simpler theory of rational numbers it will appear that the solution of the problem depends essentially on the case in whicls the modulus is a prime-power ideal, that is, a power of a prime ideal. The most important {:ase, however, is probably that in which the modulus is a rational principal ideal or in otller words a rational integer; therefore a separate discussion will be given of this case. Allother interesting case is that in which the group is

In the solution of the problerll of the strain of a homogeneous elastic sphere, as given by LAME and bv IVELVIN, the bodily forces are supposed expressible as known functions of the coordinates of position. Wllen self-gravitation is among the forces considered, however, the bodily forces depend in part upon the state of strain, and it is only in the case of assumed incompressibility that the usual method of solution is applicable. The present papel has for its main object the solution of the problem of the strain of an isotropic elastic sphere, initially homogeneous;, due to disturbing forces of a certaill type, takillg into account the changes in the gravitational forces whieh result from the strain. The contents of the paper fall under the following four heads: I. Strain of a gravitatillg, compressible elastic sphere under the action of tidal or centrifugal forces, with numerical consplltations for tlle case in which the surface is free from stress. II. Strain of a gravitating, compressible elastic sphere covered by a shallow ocean, under the action of tidal or centrifugal forces. III. Effect of compressibility upon estimates of the rigidity of the earth. IV. Strain of a gravitating, compressible elastic sphere under the action of small disturbing forces derivable from a potential which is any spherical harlllonic of degree not less than 2.

The recogllition of the abstract identity of geonletry and analysis, which results from the llotion of coordinates on the one hand and from the classical work of VON STAUDT t on the algebra of throws on the other, and which recent work oll the foundations of geometry has ftllly established, has brought with it a broader conception of the colltellt of geometry. It has meant llot only the introduction of imaginary elements alld the resulting conception of a complex space (of any number of dimensions), but it has also led to the consideration of geometries with respect to any number-system (finite or infinite), i. e., of spaces the elemellts of which may be determined by sets of numbers (coordinates) belonging to a givell number-system. t An important result of the recognition of the identity referred to is the etnphasis it places on the possibility of using geometric or synthetic methods in the solution of analytic problems. It seems likely that the fact that such methods have received comparatively little attention hitherto has resulted in a loss of power. The present paper, it is hoped, will telld to substantiate this assertion. We are llere concerned with certaill fundamental problems in the projective geotnetry on a comptex plane, i. e., a plane the points of which are determined by sets of llomogeneous coordinates (xl, X2, X3), where the xi are any ordinary complex numbers. not all zero. Though the illvestigation is essentially geometric alld the results are susceptible of immediate application to problems of importance in geoInetry, these results are of even greater interest in the theory of functions of two complex variables. To make this clear we shall glance briefly at the corresponding problems on a complex line, tlle results of which are well-known ill the theory of functions

Our purpose in this paper is first to discuss nonlinear operators and nonlinear projections in Banach spaces which are related to the resolvents of $m$-accretive operators and maximal monotone operators. Some of these operators in Banach spaces are new. Next, we discuss some properties for such nonlinear operators and nonlinear projections in Banach spaces. Further, using these properties, we prove strong convergence theorems by hybrid methods for nonlinear operators with equilibrium problems in Banach spaces.

This paper deals with the system of curves each of which possesses the following property: A particle, starting from rest and moving along the curve, under the action of a given force, will reach a certain point, the center of tautochronous motion, in the same time from whatever point on the curve it starts. Friction is neglected and it is assumed that the actiing force depends only on the position of the particle. The first part of the paper deals with plane systems. It is assumed that within the region considered the rectangular components of the force, p (x, y), 4 (x, y), are uniform functions which possess first and second derivatives and do not vanish identically. In 1673 Huyghens found that when the acting force is gravity the tautochrones are cycloids with horizontal bases and concavity upwards. Since that time many extensions of the problem have been made and the tautochrones due to more complicated laws of force and resistance have been found. This paper, however, does not deal with the problem of finding the tautochrones due to a general positional force, but investigates the form of the differential equation and the derivable geometric properties of the system. This differential equation, which is of the third order and represents the total triply infinite system of tautochrones in the plane, is found in Article 1 from the well-known physical property of tautochrones, namely, that the tangential component of the acting force along the curve is proportional to the arcual distance of the moving particle from the center of tautochronous motion. The factor of proportionality must be negative for actual tautochrones, but, as it is eliminated in finding the differential equation of the system, the system includes virtual as well as actual tautochrones.

The study of systems'of lines in space of four dimensions is so closely related to the study of systems of circles and other geometric configurations that it has seenmed worth while to develop this geometry and to interpret the results in the geometry of the circle. Klein* has suggested the study of lines in space of four dimensions as a possible approach to the study of systems of circles. The only extensive study made of systems of lines in S4 is that of Castelnuovo;t but this is concerned only with linear systems of lines, and as linear systems of circles have already been studied by Koenigs I and Cosserat, ? no further reference will be made to linear systems of lines. The properties of lines treated in this paper involve the first derivatives.

The line-sphere transformation of Sophus Lie, first described ill llis relnarkable article, ;sUeber Cortlplexe, insbesolldele Linien und Kugelcomplexe, rnit Anwendung auf die Theorie partieller l)ifferentialgleichungen " t dezives its interest frozn two causes. Fil stly, lines and spheres are the simplest follrdimensional geometrical mallifolds with which we have to deal; it is illteresting to establish connections between them. SecoIldly, intersecting lines correspond to tallgent spheles. The consequence usually dlawn from this is that the linesphere transformation falls under the genel al type of coiltact transf{rlllation, and has the beautiful property of carrying asymptotic lines over into lines of ourvature. These facts are of capital importallce when the subject is apploached from the point of view of Pfaff equations. But we may adopt a different pOillt of view, alld then the question naturally arises: " Can not the correspolldence of intersecting lines with tangent spheres be made to appear as a special case of sotne more general relation conllecting the distallcess of lines with the allgles of intersection of spheres ? " It is the object of the present paper to answer this question affirmative]y, to develop as far as pQssible the relations between the metrical properties of line and sphere systx3ms, alld to bring the two into the closest possible coslnection. :0: Before proceedillg to our task, two words of caution are necessary. Filst, as the angles between spheres appear most naturally through their trigonometric fuIlctions, the corresponding line-functions are found ulost elegantly in non-euclidean space, where distances usually appear in trigonometric form. Secondly, the group of contact transformations that carries lines into lilles and keeps distances invariant depends upon six pararneters, the group that leaves the angles of spheres intact is a ten parameter group, isomorpllic with the quinary orthogonal oIle. We must, therefore, introduce some seemingly arbitrary restrictions, i. e.

In this paper we give a sufficient colldition that the characteristic numbers of an unsymmetric kernel exist and be real, and prove the expansibility of arbitrary functions in terms of the corresponding characteristic functions. This sufficient condition is stated in terms of a functional transformationt T(f) defined by eertain general properties (§1), and for the special case T(f) _f we obtain the known theory of the orthogonal integral equations. The method employed is that of infinitely many variables and is based to some extent on an earlier paper.t However, the present paper, with the exeeption of a few footnotes, can be read independently of (I) if Theorems 20 and 8 which are here stated in §1 are accepted; and if the suggestion in Remark 1 of §1 is carried out, the only reference to (I) that is necessary is Theorem 20.

The theory of lirlear diflerence equations with ratiollal coefficients was in a very backward state until POINCARG t in 1882 developed the notion of asymptotic representation, and its application to this brallch of mathematics. Further important progress ill this direction has been made only very recently, notably by GALBRUN,? who by means of the Laplace transformation has investigated the properties of a set of fundamental solutions in the entire plane of the complex variable t. The ainl of the presellt paper is first to study the nature of these solutions to which I have been led by mealls of direct ulethods, and secondly to show that there exists a purely Riemannian theory of linear differellce equations. In particular certain rational functions of e27r8-lX are shown to play a part like that of the monodromic group constants of an ordinary linear differential equation. To the best of my knowledge, the importance of the functional standpoint in the field of difference equations was emphasized first by VAN VLECK in an inspiring series of lectures given at the University of Wisconsin in the spring of 1909, in which he conjectured the existence of sets of solutions analytic on either the left or the right side of the complex plane. On account of the extreme simplicity of the matrix notation, I have found it convenient to deal with a linear difference system of n equations of the first order

To the theory of multiply transitive groups Bochert has contributed formulae for the upper limit of the degree of a group when the number of letters displaced by a substitution in it is known, f But »for simply transitive primitive groups a corresponding theorem of such elegance does not exist. J However, in the present paper is offered a theorem with some claims to simplicity. It may be stated thus : If a simply transitive primitive group contains a substitution of prime order p and of degree pq ( q less than 2p 43 ), its degree is not greater than the larger of the two numbers qp 4q2 — ?, %q2 —p2Also when the given substitution of prime order has more than 2p 42 and less than p2 cycles or when p is 2 or 3, our results are capable of fairly concise expression ; but when q exceeds p2—1 and p is greater than 3 the corresponding upper limit is unfortunately a more complicated function of p and q. The subject matter of this study is not restricted to the simply transitive primitive groups, but it is proposed to find an upper limit for the degree of some transitive subgroup of a primitive group known to contain a substitution of order p on qp letters. It seems unnecessary to state that a simply transitive primitive group has no transitive subgroup of a lower degree.

I propose in this paper to investigate the geometrical nature of a certain species of differential equation, which includes, as a particular case, Clairaut's well worn equation

In the first section of Kneser's book on Integral Equations and their Applications to Mathematical Physics, he applies that theory to the solution of some of the problems which arise in the Linear Flow of Heat. The object of this paper is to illustrate Kneser's use of Integral Equations in the Mathematical Theory of the Conduction of Heat by the discussion of one of the classical problems of Linear Flow which he leaves untouched.

The main points of this paper are :—(i) The construction of a linear substitution from its poles or linear invariants and its multipliers (§§3, 7);(ii) a formula for r repetitions of a substitution (§ 4);(iii) a specification of the types of linear substitutions of order r, with examples of the simplest of those types(§§9ff.);(iv) The geometrical illustration of the case of three variables (§§15 ff.)

The last ten years have witnessed many changes in the methods employed in the teaching of arithmetic, which to a very considerable degree are the direct product of the activity and exertions of this Association. Two signal benefits may be mentioned ; firstly, uniformity of method in elementary processes has been secured ; and, secondly, the educational value of practical applications to mensuration, physics, etc., has received due recognition. The object of this paper is to raise for discussion a question of quite a different character.

1. Def. The reciprocal (1/N) of any number (N) will be termed a Binal, Tertial, Quintal, ... Decimal, ... r-mal fraction, when expressed in the scale whose radix is r = 2, 3, 4, ... 10, ... r respectively. In this paper the developments will be given for the most part in a general form in the r-scale; the examples will be in the scales of r = 3 and 5. The notation is similar to that of ordinary decimals. Thus (see Tables at end)

n/a

The time has long passed when the teacher of elementary geometry, however elementary, can efficiently exercise his individuality in devising improved methods of exposition without taking some cognisance, however small, of the study of non-Euclidean geometry which has grown up out of the failure to prove what may be described as “Euclid’s Parallel Postulate.”A paper has recently been published in the Proceedings of the Edinburgh Mathematical Society by Prof. Carslaw, of Sydney, N.S.W., containing an elementary interpretation of the Bolyai-Lobatschewsky geometry, which is so simple and so easily intelligible that I strongly recommend every teacher of mathematics or science to read it. It does away altogether with the necessity of forming conceptions of space which appear contradictory to experience, and shows the possibility of constructing a non-Euclidean geometry in ordinary Euclidean space.

In most treatments of the theory of olle-parameter families of plane curves attention is concentrated altogether upon the discussion of tlle envelope. More recently, some authors, especially LILIENTHAL t and SCHEFFERS, t have considered some other problems associated with such families of curves, but all of these are of a metric nature. In the plesent paper, the point of view is that of projective diffelential geometry and the analytic treatment is based upon the invariant theory of a certain completely integrable system of partial diSerential equations of the second order. Incidentally a simple new geometrical interpretation is follnd for the well-known Laplaciarl traxlsformation of a linear partial diSerential equatioll of the second order, which enables one to make use of the results of that theory for the purpose of studying nets of plane curves. The osculating conics of the curves of the net are also determined and their properties illvestigated for the first time.

In 1858, D. Bierens de Haan published as "Vierde Deel" of the Verhandelingen der Koninklijke Akademie van Wetenschappen (Amsterdam) his "Tables d'Int6grales Definies." In this volume there are 7300 formulas, and these are accompanied by complete bibliographical references. De Haan then undertook the revision of these formulas, and the consideration of the underlying theory of Definite Integrals. Accordingly, in 1862, his "Expos6 de la Theorie des Proprietes, des Formules de Transfornmation, et des Methodes d'Evaluation des Integrales D6finies" was issued as Volume VIII in the same Verhandelingen. In this is included all his critical work of the intervening four years. "Nouvelles Tables d'Integrales Definies par D. Bierens de Haan" were published in 1867 (Leide). These tables contain 8339 formulas, of which 4200 come from the original tables, 2620 from the " Expose " and the remainder from notes published at various times. In this paper it is proposed to examine from the modern rigorous standpoint certain evaluations in the "Nouvelles Tables," and the theorems of the "Expose" on which they depend. In nearly every case the formulas here considered will involve the principal value integral.

does not vanish in the point ( z ) = ( a ), and for the case in which J vanishes identically in the neighborhood of this point. This paper deals with the intermediate case in which J vanishes in the point (x) = ( a), but does not vanish identically in the n variables x. Concerning this case, Professor OSGOOD has pointed out that, as examples show, the solution of the system of equations (1) for x1, * * *, xn may lead to multiple-valued functions which are not analytic in the point (y) = (b), and has made the suggestion, "Dies durfte wohl stets der Fall sein." This turns out in fact to be the case whenever an inverse exists throughout the neighborhood of the point (x) = (a) . Some phases of the problem here suggested have been studied by LONGLEY,1: DEDERICK,§ and URNER|| (see also the note to Theorem 4). For the most of their work, fi is a real function of real variables. For the case treated here fi

The occurrence of rational curves in pairs is a well-known fact: thus, given a rational curve pp, of order n, in a space of p dimensions, there is uniquely determined, to within a collineation, a curve pn_p-l n of order n, in a space of n-p-1 dimensions, by requiring all hyperplane sections of either curve to be apolar to the hyperplane sections of the other.t We call two curves associated in this way conXugate curves. If the rational curve pp is regarded as the projection of the norm-curve pn in a space Sn of n dimensions, from an Sn_p_ln the interpretation of this fact is immediate. An Sn_l in Stl meets Pn in n points which may be regarded as given by a binary form of order n: dually, a point of Sn determines on pn a set of n points, which may be given by a second binary form. The condition of apolarity of the two forms is precisely the condition of incidence of point and Sn--l * All Sn lns having n-point contact with ptl meet Sn_p_l in the hyperplanes of a curve rn_p_l of class n. The curves obtainable by projection from S7t_ and section by Sn_l are conjugate curves. In this paper we shall deal with the case n = 5, p-2, the rational plane quintic. If our curve r5 is given parametrically by

The origin of the following paper is to be found in the application of the iterative principle to the classification of collineations. To this I have been unexpectedly led by my preceding paper on the Ewtension of a Theorem of Poincare. Other topics closely connected with my subject which will be treated here are as follows: 1. The determination of a normal form for one-parameter collineation groups in n homogeneous variables, also for homogeneous linear groups in n variables. The form consists namely of components of the structure (3) below, where m is the parameter of the group. The total number of equations of all the components must, of course, be equal to n. 2. The determination of a real normal form for real groups of the character above described. It consists of components of form (3), in which p is to be taken positive, and of components of the form (14). 3. The enumeration and classification of these groups for any value of n; -in particular, for n = 3, 4. 4. The varieties and form of the real path curves (the so-called U'-curves) for real one-parameter collineation groups in space of three dimensions. The quadratic complex of tangents is also briefly considered. The prevalent mode of classifying collineations has been developed by SEGRE and others and is based upon the normal form into which the finite equations of collineation may be transformed in accordance with the theory of elementary divisors. Geometrically, this form is dependent upon the invariantive configuration of points, lines, planes, etc. LIE, on the other hand, uses the infinitesimal transformation for classification.t The results vorrespond to those obtained by the preceding method. An obvious inconvenience or difficulty arises when a collineation is given, as is most commonly

Each covariant locus of a rational plane curve contributes its share toward the covariant and invariant tlleorST of rational curves of higher order by reason of the fact that osculantst of rational curves play the same role in regard to rational curves as polars do in regard to binary forms. Especially is this true of covariant points and covariant lines of the rational curve of order n (whicll we shall call Rn) because corresponding to each covariant line or point of all Rn is a series of covariant rational curves of any Rk, where D > n; the parametric equations of these covariant rational curves are easy to obtain. In tlle first section of this paper a new kind of covariant loclls is defined for Rn and illustrated by important loci associated witll tlle R4. Section II is devoted to combinant curves of the R5. The theory of combinant curves of the R5 in the plane corresponds to the invariant theory of the R6 in space; tllis correspondence is pointed out and illustrated. Finally a sufficient number of combinants and combillative sl-iCS of two line sections of tlle plane R) have been foulld to form the algebraic basis of an analytic treatment of combinant curves of the Ro in the plane, which is shown to be identical with the illvariant theory of the Ro in space.

The chief purpose of this paper, carried out in its second part, is the determination of numerical limits for certain constants, hitherto undetermined, which figure in the principal theorems of the first two parts (Abschnitte) of the author's thesis,t concerning the degree of approximation to a given continuous function f (x) that can be attained uniformly in an interval by a polynomial of the nth degree in x, or for all values of x by a trigonometric sum of the nth order. By a " trigonometric sum of the nth order at most " is meant an expression of the form

The greater part of this paper consists of generalisations of well-known theorems regarding perpendiculars to the sides of a triangle, or other base-lines, the perpendiculars being replaced by isoclinals.

In investigating the properties of the cubic curve C, we are led to consider its Hessian C′, its Cayleyan Γ and the Hessian of its Cayleyan Γ′. In the present paper we propose to deal with the intersections of the systems C + λC and Γ + μΓ′.

I propose in these papers to demonstrate, or rather to indicate the nature of the demonstration of, the proposition that—All Geometry, whether projective or metrical, may logically be regarded as merely a particular concrete application of the general theory of Order, as defined by Boundaries. It is true that Geometry, as ordinarily understood, is something more than this; it is concerned with the actual subjective conceptions we entertain of Space, and so far provides matter of discussion to the Psychologist; and it is concerned with the actual objective measurements we make upon physical bodies in Space, and so far concerns the Physicist; while in both these respects it extends beyond the region of pure Logic into those of Epistemology and Philosophy generally. But so far as the pure Mathematician is concerned, every geometrical theorem is embraced in the Theory about to be discussed.

(In reference to A. C. Dixon’s note on lm + 2m + 3m +...+nm, on pp. 283-4 of this volume, the following further notes may be of interest. They are taken, with some alterations, from a paper communicated to the London Mathematical Society in February, 1910.)

It is the purpose of this paper to demonstrate three theorems and two corollaries, whoseformal statement will be preceded by a brief explanation. The treatment is suggested bv recent papers of LEsEsGuE,t in which corresponding problems are discussed for the case of trigonometric series. A part of the work of LEBESGUE is reproduced at length in my thesis,: and where the developments of the present paper are closely parallel to those in the trigonometric case I shall occasionally refer the reader to one of those other presentations.

Suppose X X is a metric space, and Y Y is a complete metric space. In this paper a sufficient condition is given to insure that a sequence of continuous functions from X X into Y Y converge to an embedding from X X into Y Y .

This paper considers the delayed age replacement policy, in which the lifetimes of all units are assumed to be uncertain variables, and the lifetime of the first unit has an uncertainty distribution which is different from the others. A delayed age replacement model which is concerned with finding the optimal replacement time to minimize the expected cost is developed. In the policy, the optimal replacement time is irrelevant to the uncertain distribution of lifetime of the first unit over the infinite time span.

The motivation for this paper lies in the following remarkable property of certain probability distributions. The distribution law of the r. v. (random variable) X is exactly the same as that of 1/ X, and in the case of a r. v. with p. d. f. (probability density function) f(x; a, b) where a, b are parameters, the p. d. f. of 1/X is f(x; b, a). In the latter case the p. d. f. of the reciprocal is obtained from the p. d. f. of X by merely switching the parameters. The existence of random variables with this property is perhaps familiar to statisticians, as is evidenced by the use of the classical 'F' distribution. The Cauchy law is yet another example which illustrates this property. It seems, therefore, reasonable to characterize this class of random variables by means of this rather interesting property.

It is generally recognized that in order to show that an actual solution of the physical problem is given by the formal series arising in many problems of mathematical physics, it is necessary to apply the theory of convergence factors to these series. Results in this direction of the greatest generalitr have been obtained by a method originated by FEJER,t and applied by him fo problems involving the ordinary Fourier's series. In a previous paper t the writer has applied FEJER'S method to problems that involve developments in Bessel's functions. The object of the present paper is to make such an application to problems that involve the development of a function of two variables in a double Fourierns series. Such a discussion naturally involves the consideration of certain general facts in the theory of the summability of double series, a theory that has as yet been scarcely touched on. We have not made, however, any attempt to found a very broad theory of this sort. Such an attempt would have led us too far from the central aim of the paper, and in any case it seems preferable that the theory of summability of double series should be developed at first from the point of view of its applications, and that the study of such a theory from the abstract point of view can be made to better advantage when the facts of widest use in the application of the theory have been brought to light. The consideration of the summability of double series naturally suggests the consideration of the summability of multiple series of any order, and the study of the summability of the double Fourier's series suggests the study of the summability of the triple Fourier's series and the problems connected with it. Generalizations of the results of the present paper to such cases may be obtained by methods similar to those employed in the subsequent discussion.

In a recent paper Dr. ROSENBLATT gives two interesting examples of algebraic surfaces which are invariant under an infinite discontinuous group of birational transformations and at the same time are not envelopes of quadric surfaces.t In an earlier paper I mentioned 1: that all surfaces belonging to such groups which have thus far been noticed were examples of surfaces defining an ordinaxy elliptic ( 2, 2 ) correspondence. Practically all the memoirs bearing on the problem are mentioned in the articles just named. The surfaces discussed by DR. ROSENBLATT have a pencil of elliptic curves, and the transformations are expressed by a linear transformation of the parameters u, v of elliptic functionsn in terms of which the coordinates of a point on either surface can be rationally expressed. The treatment is transcendental, the transformation is defined only for points on the surface and no explanation of the geometric meaning of the transformation is given. It is a curious fact that these transformations are birational for all space, that they have a simple geometric interpretation in terms of the (2 X 2 ) correspondence, and that surfaces of any order or of as high a geometric genus as may be desired can be constructed which are invariant under this group, or a larger group under which that discussed by Dr. Rosenblatt is a subgroup of infinite index. Consider the surface whose equation is of the form

One fundamental characteristic of that discipline which is called by the vague title of " higher geometry " consists in taking as element, not necessarily a point in two or three dimensional space, but any object which can be determined by a finite number of parameters. The student of analytic geometry begins in Cartesian fashion with a point and its coordinates. A little later line and plane coordinates are introduced, and the student feels that the content of his subject has been multiplied by two. Further extensions soon follow, the Plucker line geometry, the coordinates of a circle or sphere, the linear line-complex as space element, the projective geometry of n dimensions. The process is capable of indefinite extellsion. In line geometry a very fruitful idea was introduced by Study,t that of taking as element, not a single line, but a pair of lines called a "cross." One line of the pair is finite, the other, in general, at infinity, and each plane through one is perpendicular to the other. The cross takes a more symmetrical shape in non-euclidean space where under the previous definition the two lines are merely mutually polar with regard to the absolute quadric.: It is the object of the present paper to extend this concept to circles. A circle cross will be composed of a pair of circles so related that every sphere through one ts orthogonal to the other. In the first section we shall give a number of preliminary theorems, some well known, concerning the angles of circles. In the second we take systems of circles orthogonal to one sphere. The analogr to the corresponding problem in non-euclidean line geometry is here so close as to be almost identity, and we shall give our results with all possible brevity. In section three we shall discuss circle crosses in general position in space. The simple analytical treatment of the cross as element does not seem possible in this case, but a number of interesting results are obtainable. In the last section we discuss a new transformation which carries circles into circles, but not spheres into spheres. Theorems already familiar are marked with an accent to the number, as Theorem 2', etc. Others not so marked are believed to be new.

Very important progress has recently been made in the analytic theory of homogeneous linear diSerence equations. GALBRUN t has used the Laplace transformation to derive important existence theorems, and has investigated the nature of certain principal solutions for large values of the variable. At about the same time NORLUND t applied the theory of factorial series to linear difference equations, showed the existence of the same solutions, and gave their asymptotic form. In addition to the case of polynomial coefficients he has considered the case where the coefficients can be expressed in factorial series. In a more recent paper he has investigated an equation with still more general coefficients.§ BY means of a method of successive approximation, and a suitable extension of a contour integral due to GUICHARD, CARMICEAEL 1t has independently shown the existence of solutions, and has found the bounds of increase and decrease of the solutions in a direction parallel to the real axis. The latest important contribution to the general theory is by BIRKHOFF*1T He employs a matrix notation, and shows in a direct manner the existence of certain intermediate solutions and of the principal solutions. The asymptotic form of these solutions is determined throughout the complex plane. A modification of the integral used by CARMICHAEL plays an important r81e.

The highly fruitful conception of adjoint pairs of linear systems, each consisting of a homogeneous linear differential equation of the nth order and of n homogeneous linear boundary conditions, was first introduced in an explicit and general manner by BIRKHOFF.t That we have here a conception of fundamental importance is hardly yet generally appreciated. It is my purpose in the present paper to make an application of this idea to an important question in the theory of linear boundary problems which has so far been treated only in very special cases.+ I will mention in passing that the results may also be obtained by the use of integral equations; though, if we wish to include all cases, not perhaps so simply or obviously as might at first sight appear. In § 1 the conception of adjoint system and a few other fundamental matters concerning linear systems are explained and a few preliminary questions disposed of. In § 2 come the main results of the paper, namely Theorems I and II, which apply to the case of complex as well as to that of real independent variNble. These theorems are then applied in some detail to the equation of the second order, whereby the relation to MASON'S results becomes apparent. In § 3 an extension to systems of equations of any order is briefly indicated; and a somewhat generalized type of differential equation is also considered. Finally, ill § 4, the conception of adjoint system is extended to the case in which the number of boundary conditions is not the same as the order of the differential equation.

provided that a proper determination of the constant c1 be made. In the present paper I wish to consider the solutions of (1) in the vicinity of x = so, but under the restriction that p ( x ) and q ( x ) have the form ( 2)t The method of attack is essentially the same as that which I have employed earlier in the consideration of singular points.t In this special case the results may be given a more striking form. The last two theorems have no analoguesin my earlier paper and are susceptible of wide generalization.

It is proposed in this paper to show now the well-known Laplace's transformation,which is of great help in finding the solution of linear differential equations, gives also interesting results concerning the theory of integral equations. In §2 we shall study its application to certain differential equations, and find a large class of equations which remain unchanged by this transformation. Then, (§3), taking instead of eazt, a more general function of the product zt, we shall find a solution for some homogeneous integral equations ; in § 4 we shall describe a method of solving a very general type of integral equation of the first kind, namely,a further extension to integral equations with the kernel ef(z)f(t) the object of §5. Then, studying an extension of Euler's transformation, we shall (§ 6) consider equations such aswhich will prove to be singular; and finally, in §7, we shall give other examples of singular integral equations.

In a previous paper (q.v.) by Mr W. L. Marr and the present writer, it was shown that, in accordance with Morley's Theorem, the angles A+2pπ, B+2pπ, C+2rπ of the triangles ABC be trisected, the three groups of six lines at the vertices give rise to 27 triangles DEF, the biangular coordinates of D with respect to BC being (B/3+2qπ/3, C/3+2rπ/3) or (βq, γr), and similarly for and F with respect to CA and AB.

THE present paper is a supplement to the mernoir of the same title issued last year*. It is not proposed to give here any account of the work which has been done in the attempt to elicit information on the more difficult subject of the natutre of selection in man, but only to publish the correlations and regressions obtained by using an alternative measure of environment, and by varying the periods in which the effects of a selective death-rate can be detected.

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The list of the primitive groups of the first thirteen classes prepared by JORDAN has long been known to be defective in the case of class 12; in fact, JORDAN states explicitly that the calculations for that case, having been gone through but once, may very well contain errors.* In spite of the considerable advances in substitution theory in recent years, the complete a priori determination of all the groups of this class still involves a good deal of labor. In order therefore not to intrude too far upon the space of this journal, the writer avoids a redetermination of the primitive groups of class 12 on less than 21 letters.1 In a series of memoirs and in his lectures Professor MILLER has gone over this ground without use of the list of the groups of class 12. He has thus checked the results of MiSS MARTIN, of JORDAN, and of Mliss BENNETT on the degrees 18, 19, and 20, respectively, according to whiclh there is no primnitive gr'oup of class 12 on either of these three degrees. The method here followed will be much the same as that used by the author in treating the classes 6, 8, and 10, with the advantage of having at hand the processes employed in the paper entitled "On the Limit of the Degree of Primitive Groups." This last paper, as well as that "On Mlultiply Transitive Groups," and that "On the Order of Primitive Groups," will be used freely without specific reference.t In addition to the 25 primitive groups of class 12 of degree less than 18, there are four others, one of degree 27, two of degree 28, and one of degree 36.