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THERE appeared in your pages some three years ago (vol. xv. p. 515) an article of mine “On the Simplest Continuous Manifoldness of Two Dimensions and of Finite Extent.” In a succeeding number a correspondent (Mr. Monro, of Barnet) propounded a query which may be shortly stated as follows:— “How does it happen that the perpendicular on a geodesic from a point moving along another geodesic changes sign without passing through either the value zero (0) or the value infinity (∝)?” The problem here suggested is a peculiarly knotty one. In the case of the Euclidian plane the perpendicular of course changes sign by passing through the value ∝, while in the case of a spherical surface it is equally obvious that the perpendicular passes through zero, since the two geodesies intersect twice. But what are we to say of the strange hybrid surface which formed the subject-matter of my paper? Your correspondent appeared to insinuate that the problem was insoluble, and that the definition of the surface must therefore involve a logical contradiction. For a while I was greatly puzzled by this unforeseen difficulty, but after a little thought came to the conclusion that the perpendicular changes sign by passing through the value l/2√−1, where l is positive and represents the absolute length of a complete geodesic. In other words, I conceived that the sign of the perpendicular changed from + to − by a continuous variation of the real numbers a and b in the complex number a + b√- 1. I conceived a to diminish continuously till, passing through o, it became − -a, while b at the same time increased with simple harmonic motion from o to a maximum, and then decreased from a maximum to o.

IN a paper published in the Messenger of Mathematics (May, 1878), Prof. Sylvester has given a rule for abbreviating the calculation of (w:i,j) (w -1:i,j) ; where, to fix the ideas, let (x:i,j) be regarded as the number of modes of coinposino x with j of the numbers 0, 1, 2, . . . The abbreviation consists in rejectino from the partitionis of w-all partitions whose highest number is not repeated and rejecting from the partitions of w -1. all partitions which do not contain i; the number of partitions thus rejected being shown to be the same in the two cases. This becomes even more obvious if we convert the above (i, j) partitions into (j, i) partitions: that is, replace each of the above partitions by a corresponding one consisting, of i of the numbers 0, 1., . ..j. This, as is well known, can be done by decomposing each number into a column of l's and then recomposing by rows. Now, if we do this, it is plain that those partitions whose highest number was not repeated become partitions containing,r 1; and that those partitions which did not contain i become partitions having less than the full number of parts, or, in other words, partitions containing 0. So that Prof. Sylvester's abbreviation is equivalent to rejecting from the partitions of w those partitions which contain 1, and from the partitions of w 1 those which contain 0. And it is plain that the number of partitions of w which contain 1 is equal to the number of partitions of w 1 which contain 0; for the two sets of partitions are interchanged by the interchange of 0 and 1. Obviously, instead of rejecting the partitions of w which contain 1 andi those of w -1 which contain 0, we may reject the partitions of w which contain m (where qn is any one of the numbers 1, 2, . i (or j)) and those of w -1 which contain m1; the reason being the same as above. 187

IN order to avoid a clumsy circumlocution, I have ventured to use the word quantification to denote in general that class of operations expressed in the several special cases by the terms, rectification, quadrature and cubature. Some of the quaternion formulae for quantification were given in a paper entitled "Investigations in Quaternions," communicated to the American Academy of Arts and Sciences, 9th January, 1878. They are here reproduced in a more general form together withi the formnulae for barycentres.

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If we apply to a set of quaternions the definitioni of a group as given in the Theory of Substitutions, then a quaternion group of the mth order means a set of m quaternions (scalar unity always included) whose products and powers are also quaternions of the same set or group. The object of the present paper is to determine all the possible finite quaternion groups. These groups, or rather their analogues in the ordinary Theory of Functions, have usually been interpreted geometrically as the linear transformations of the plane of complex variables in itself-automorph1dc traisformnations. The forimulae for these linear transform-nations were first given by Professor G0ordan. in his paper "Ueber endliche Gruppen linearer Traiisforinationen einer Veriinderlichen." * The quaternion formulae, although they have their exact correspondents in Gordan's algebraic formulae, have, when interpreted geometrically, a more general character, in that they represent certain automorphic linear transtbrmations of a three-dimensional infinite homoloidal space, or what is the same thing, of a three-fold extended sphere. The determination of these formulae might be made to depend upon Gordan's solution, but the following is a shorter and simpler solution than that of Gordan, and includes it as a special case.

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In the year 1849 the great paper of Stokes "On the Dynamical Theory of Diffraction " was read before the Cambridge Philosophical Society, and this has remained until the present day the standard upon this important subject. The method of Stokes was based upon the old elastic solid theory of light, and gave the following conclusions: First. That when the incident light was plane polarized, the diffracted light from a small orifice was also plane polarized in such a manner that the displacement was in the same plane as that of the medium at the orifice. So that if a sphere was drawn with the orifice as a center and meridians drawn on the sphere with the axis in the direction of the vibration at the orifice, then these meridians represented the direction of displacement in the diffracted light. Second. The intensity of the polarized light was represented as follows: Let 3 represent the angle between the incidenlt ray prolonged and the diffracted ray, and let q be the angle between the diffracted ray and the direction of the displacement at the orifice. Then the intensity of the diffracted light around a very small orifice will be proportional to

The notation adopted in this paper for the triangle ABC is: G the centroid. I the centre of the inscribed circle. I 1 , I 2 , I 3 the centres of the escribed circles within angles A, B, C. O the centre of the circumscribed circle. D, E, F; D 1 , E 1 , F l ; D 2 , E 2 , F 2 ; D 3 , E 3 , F 3 the points of contact of the inscribed and escribed circles with BC, CA, AB. The Ds lie all on BC, the Es on CA, and the Fs on AB. H, K, L the mid points of BC, CA, AB. X, Y, Z the feet of the perpendiculars from A, B, C, on BC, CA, AB. A′, B′, C′ the vertices, opposite to A, B, C, of the triangle formed by drawing through A, B, C parallels to BC, CA, AB.

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In my former paper on Fourier's double-integral I remarked that Poisson's form of the integral gave the same incorrect; result as Fourier's form in an example by which I tested it, and seemed subject to the same limitations.

The communication on this subject, as originally made to the society, consisted of a series of theorems, giving (1) expressions for the radii of a great many sets of circles, (2) identities connecting several sets of these radii, and (3) miscellaneous identities closely related thereto. As, however, the paper culminated in a general theorem which may be looked upon as fundamental, and the proof of which makes evident the mode of arriving at the said expressions for radii, and as the relations connecting sets of radii are easily found when attention has been directed to their existence, I have thought it best to print little more than the fundamental theorem and a few auxiliary notes.

1. In the first part of this paper we have considered merely the contours of curves, that is, contour points, and the method of obtaining the various physical diagrams. In this part we shall consider chiefly the contours of surfaces; that is, contour lines. If any curve be cut by planes parallel to that of ( x, y ) and if the various points of intersection be projected on any one of these planes, say z = 0, the contour points so obtained will evidently lie on a definite line, and the line will be more accurately indicated in proportion as the number of intersecting planes is greater and their mutual distance is less. It will be given without any break in continuity by projecting every point of the curve upon the plane z = 0. But such a line may be regarded as the intersection, by the plane z = 0, (see fig. 48) of a cylindrical surface whose generating lines are parallel to the z -axis and are drawn from the given curve to meet that plane. We have here then the intersection of a given surface by a surface over which z is constant. But this satisfies our definition of a contour line. This case of a cylindrical surface supplies the simplest system of contour lines by giving z different values. The contours are all superposed in the diagram, but are not in general conterminous. The only case in which they would be conterminous is that in which the same values of the x and y co-ordinates of a point on the curve correspond to different values of the z -co-ordinate.

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The following paper contains little that can be regarded as new mathematical information. It aims only at showing, or rather at emphasising, the correspondence which exists between two geometrical theories which are related to each other in the same way as the arithmetical theories of multiplication and division. Such value, therefore, as it possesses is primarily pedagogical.

The following note may be considered as an addendum to the paper by me on pp. 42–47 of this volume of the Proceedings . In that paper it is shown how to inscribe in a triangle ABC, a triangle DEF, such that the perpendiculars to the sides of ABC, drawn through the points D, E, F, shall be concurrent in a point P. This is done by constructing on each of the sides of ABO a triangle similar to DEF; then O the point of concurrence of the three lines joining the vertices of ABC to the vertices of these triangles is the point “inverse” to P. The question, then, naturally arises, What must be the shape of the triangle DEF in order that the point P may be one of the Brocard points, and, as a consequence, O the other one? and the answer is easily seen to be that DEF must be similar to ABO. Hence the following construction:—

[At the first meeting of this Session a paper was read on the value of cos 2π/17, which evidently may be made to depend on the solution of x17 – 1 = 0.* The present paper is the outcome of a suggestion then made, that a sketch of Gauss's treatment of the general equation might prove interesting. To give completeness to the subject the necessary theorems on congruences have been prefixed. The convenient notation introduced by Gauss is here adopted; thus, when the difference between a and b is divisible by p, instead of writing a = Mp + b, we may write a ≡ b (mod p), the value of M seldom being of importance. It is evident that if a ≡ b, then na ≡ nb, and an ≡ bn, n being any positive integer, and the same modulus p being understood throughout. Also a/n ≡ b/n provided n be prime to p. Other properties (similar to those of equations) are easily seen, but only the above are needed here.

[The substance of this paper will appear in the second volume of Professor Chrystal's Algebra.]

§1. Since the former paper on this subject was read, Prof. Cantor has published the second volume of his history of Mathematics. This has necessitated various additions to the paper, which can perhaps be best given as an appendix.On page 413 Prof. Cantor says that the construction of Dürer's pentagon is found in a book called Geometria deutsch, which was lately discovered in the town library at Nürnberg, and gives 1487 as the upper limit to its date. The construction is said to be “mitunverrücktem Zirckel,” the same expression that Schwenter applies to Dürer's solution.

§ 1. It is proposed to discuss in this paper partial differential equations involving two independent variables x and y, and a dependent variable z. The method of reduction which is explained can be applied to certain equations involving more than two independent variables, but such application is subject to too many restrictions to be of much general utility.

In this paper the laws of addition and subtraction of vectors were considered, and examples of their extreme usefulness in geometrical applications were given.

1. My only justification for presenting this paper to the Society lies in the fact that, so far as I am aware, the uniform convergence of the Fourier Series is nowhere alluded to, and far less discussed, in any English textbook; while the precautions that are necessary in differentiating the series are hardly ever mentioned even in treatises which give a very thorough treatment of its convergence. I have confined myself almost exclusively to what may be called ordinary functions, as a complete discussion of what has been done in recent years for functions that lie outside the category of “ordinary” would make the paper much too long. For information as to the original authorities, I would refer to the paper which I communicated to the Society last session On the History of the Fourier Series. It is sufficient to say here that the proof I now give is simply an adaptation of that of Heine (Kugelfunctionen, Bd. I. 57–64, Bd. II. 346–353) and of that of Neumann (Uber die nach Kreis … Functionen fortsch. Entwickelungen, 26–52).

The notation employed in the following pages is that recommended in a paper of mine on “The Triangle and its Six Scribed Circles”* printed in the first volume of the Proceedings of the Edinburgh Mathematical Society. It may be convenient to repeat all that is necessary for the present purpose.

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In this paper the following expansion will be obtained: in which n and r are positive integers. The Series in the square brackets are Hypergeometric Series with a finite number of terms.

The graphical representation of a group given by Cayley* leads to a diagram consisting of several lines of different colors, a so-called color-group, which affords a very clear insight into the structure of the group. Cayley himself applied his method only to groups of comparatively low orders, and it seems that the inethod has never been used for more complicated cases.t The purpose of the present paper is to show how readily Cayley's method can be applied to the construction and investigation of numerous groups of higher orders. In particular, the color diagrams for the rotation groups of the regular bodies can be arranged in such a way that they lend themselves miuch easier, at least in some respects, to a study of the groups concerned, than even the models of the regular bodies. The most prominent feature of these diagrams, to which their high degree of perspicuity is due, consists in the fact that their color lines do not intersect each other, so that the diagrams, when described on the sphere, constitute convex polyhedrons. I determine, in the first part of the paper, all the color-groups thus defined and show that, apart from two other cases, they are identical with the rotation groups of the regular bodies. In the second part I study in detail the connection between the rotation groups and the corresponding diagrams. The third part of the paper contains some extensions of the

The following paper was written last summer, and was submitted to Dr Mackay with a view to eliciting his opinion particularly on the curious passage referred to in § 3, and on the remarks contained in § 8. I was not aware of the intention of Mr T. L. Heath to follow up his excellent edition of Apollonius by an edition of Archimedes on similar lines, and when I saw the announcement of his Archimedes in the month of October, I at once concluded that the notes I had made would have been anticipated by him. Since reading his masterly work, however, I am disposed to think there is still sufficient interest in the notes I have written to justify me in laying them before the Society; I therefore submit them in their original form, although I should have omitted certain details had I been acquainted with Mr Heath's work before writing the paper.

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It is to be expected that the systematic introduction of irrational covariants will enrich geometry with curves and surfaces, not previously observed or discussed, allied to any given fundamental system by projective relations. Well known irrationalities also must be expected often to reappear. In 1888 Dr. HILBERT t remarked the existence of two nets of conics covariantly related to a general curve of third order in a plane. Two additional systems, nets of curves of the second class, can easily be defined by anl equation closely analogous to Hilbert's. These systems of conics and the four cubics whose polars they are prove to be not entirely unknowvn hitherto, for their dually equivalent loci and envelopes form the basis of the one-to-one correspondences upon the point-cubic and the line-cutbic respectively. That only two of every three such correspondences are found here is due to the domain of rationality that is assumed. By employing the irrationality that occurs in HESSE'S canonical fornm of the cubic I ani able to identify Hilbert's two systems of irrational covariant conics and to exhibit their relation to the other two systems just mentioned. As a consequence it is found possible to give explicitly covariant equations of definition for the two cubics which have the same Hessian and for those which have the same Cayleyan as a given fundamental cubic. These results are here derived by the aid of a canonical form of the cubic containing Hesse's irrationality. The desirable invariantive proofs of these results are given by Professor GORDAN in a paper presented by him to the Americani Mathematical Society for publication in its Transactions.4

Introduction. In 1772 LAGRANGE published a celebrated meinoiron the Problem of Three Bodies, which contained all the solutions in which the ratios of the mutual distances of the bodies are constants. He found two distinct configurations. In the one, the three bodies always lie in a straight line; in the other, they are always at the vertices of an equilateral triangle. Their distribution upon the line depends upon their masses, being explicitly defined by the real positive root of a certain quintic equation. The equilateral triangular configuration is possible for all distributions of the masses. In both cases the three bodies move in the same plane, in conic sections with respect to each other or with respect to their common center of gravity, and in such a manner that the law of areas is true for each body considered separately. No other periodic solutions of the motion of three or more bodies were discovered for more than a century, although many splendid papers appeared on the Problem of Three Bodies. A new impetus was given to the subject by the celebrated memoir of Dr. HILL on the Lunar Theory,t in which he discussed a new species of per iodic solutions. He made the restrictions that one body should be infinitesimal, and that the finite bodies should describe circles around their center of gravity. For the purposes of the applications to the Lunar Theory he neglected the ratio of the distances of the iiioon and the sun, thus introducing errors in the numerical results which in certain cases became important. Professor DARWIN has more recently taken up the subjectt by a method not essentially different from that employed by HILL, and discussed a great many orbits in detail, not neglecting the solar parallax. He has considered only cases in which the infinitesimal body moves in the plane of the finite bodies with direct motion. To these must be added the masterful researches ? of POINCARP, who has