Hyperelliptic Integrals Research Articles

A problem in the reduction of hyperelliptic integrals

The reduction of hyperelliptic initegrals to elliptic has been treated in numerous memoirs. In the more general case of the reduction of abelialn (non-hyperelliptic) integrals, the properties of the table of normal periods have been investigated by WEIERSTRASS t and M. POINCARP, while the case }v = 3 has been treated in detail by Mme. KOWALEWSKY t by WEIERSTRASS' methods. Professor KL6NIGSBERGER ? has drawn attention to the resemnblance between JACOBI'S transformation theory of elliptic integrals and the reduction theory of hyperelliptic integrals, which is such that the latter appears as a generalization of the former. A fundamental memoir by M. PICARD 1I contains new results and suggests new problems. It is shown that when, to the relation y2 = R6 (x), R6 (x), denoting a polynomial of the sixth degree, there belongs a reducible integral, there exists also a second one, that these two form a system of normal integrals and that the table of periods corresponding to them is

On the System of a Binary Cubic and Quadratic and the Reduction of Hyperelliptic Integrals of Genus Two to Elliptic Integrals by a Transformation of the Fourth Order

On the System of a Binary Cubic and Quadratic and the Reduction of Hyperelliptic Integrals of Genus Two to Elliptic Integrals by a Transformation of the Fourth Order

Addition Theorems for Hyperelliptic Integrals

Proceedings of the London Mathematical SocietyVolume s1-34, Issue 1 p. 172-185 Articles Addition Theorems for Hyperelliptic Integrals A. L. Dixon, A. L. DixonSearch for more papers by this author A. L. Dixon, A. L. DixonSearch for more papers by this author First published: March 1901 https://doi.org/10.1112/plms/s1-34.1.172AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Volumes1-34, Issue1March 1901Pages 172-185 RelatedInformation

On the system of a binary cubic and quadratic and the reduction of hyperelliptic integrals of genus two to elliptic integrals by a transformation of the fourth order

1. If a hyperelliptic integral of genus 2 is reducible to an elliptic integral by a transformation of order k then there always exists a second integral t with the same irrationality which is reducible by a transformation of the same order. In the algebraic treatment of the problem the difficulty lies in the determination of the second integral after the first has been constructed. For k = 4 the problem has been treated by Professor BOLZA : and in the sixth section of his dissertation the formulhe for the two reducible integrals are established algebraically. Professor BOLZA used a particular system of variables x1 x2. These are chosen so that if

On the Reduction of Hyperelliptic Integrals (p = 3) to Elliptic Integrals by Transformations of the Second and Third Degrees

On the Reduction of Hyperelliptic Integrals (p = 3) to Elliptic Integrals by Transformations of the Second and Third Degrees

Scientific Serials

Bulletin of the American Mathematical Society, December 1897.—In accordance with the amended by-laws of the Society (May 29, 1897), by which it was arranged that four New York meetings annually should be held instead of eight, as previously, viz. on the last Saturdays of October, February and April, and an annual meeting in the last week of December, a meeting was held on October 30. The object of the change is to secure greater prominence and interest for each meeting, and to afford the members of the Society a better opportunity for scientific and social intercourse. Each meeting now extends through two sessions, held in the morning and afternoon. Forty-one persons (thirty-seven members) were present, and this after the recent successful meeting at Toronto. Nine papers were read, of which abstracts are given here. Some of the papers are printed in the present number, and others will be published in journals whose titles are given.—Note on hyper-elliptic integrals, by Prof. A. S. Chessin, is one of the papers. If Xr is a polynomial in x of degree r; and Pm(x), Qn(x)1 … polynomials in x of degrees m,n, . . ., we know that the integration of

Scientific Serials

Bulletin of the American Mathematical Society, vol. ii. No. 1, October 1895.—The number opens with accounts of the proceedings at the second summer meeting of the American Mathematical Society, held at Springfield, Mass., on August 27 and 28, and of the proceedings at the meeting of Section A of the American Association, held at the same place, from August 29 to September 4. The papers at each meeting are given in abstract, and two of them are printed in full, viz. on the differential equations of certain systems of conies, by R. A. Roberts, and asymptotic lines on a circular ring, by Prof. Maschke. The results in the former paper are principally deduced by means of elliptic integrals and the first class of hyper-elliptic integrals, and from these are derived theorems concerning doubly infinite porisms of curvilinear polygons. The latter paper contains an application of elliptic functions to curves drawn on the surface of a circular ring.—Prof. F. Morley communicates a short note on a generalisation of Weierstrass's equation with three terms.—The notes contain various items of interest, and the list of publications is unusually full.

On the Form of Hyperelliptic Integrals of the First Order, which are expressible as the Sum of two Elliptic Integrals

On the Form of Hyperelliptic Integrals of the First Order, which are expressible as the Sum of two Elliptic Integrals

Note in connexion with the hyperelliptic integrals of the first order.

Note in connexion with the hyperelliptic integrals of the first order.

Proceedings of the London Mathematical Society

THE papers in this volume, as usual, are mostly purely analytical in their character. Prof. Cayley's contributions are very short: the binomial equation xp - I = 0; quinquisection; on the flexure and equilibrium of a skew surface; on the geodesic curvature of a curve on a surface, and on the Gaussian theory of surfaces. Sir J. Cockle continues his remarks on binomial biordinals. Mr. Glaisher's papers are also few and short, viz. on some definite integrals expressible in terms of the first complete definite integral, and of gamma-functions; note on certain symbolic operators and their application to the solution of certain partial differential equations. Messrs. Crofton and J. J. Walker have some points of contact, the former writing on operative symbols in the differential calculus, the latter continuing his theorems in the calculus of operations. Mr. Walker also contributes a quaternion proof of a problem discussed by Mr. S. Roberts, viz. certain tetrahedra specially related to four spheres meeting in a point. Mr. Roberts also gives a historical note on Dr. Graves's theorem on confocal conies.” Mr. W. R. W. Roberts has a paper on the periods of the first class of hyper-elliptic integrals, and a note on the coordinates of a tangent line to the curve of intersection of two quadrics. Mr. T. Craig has a note on Abel's theorem. Papers bearing on geometry are contributed by Prof. Genese, on a system of co-ordinates; by Mr. H. Hart, on the general equation of the second degree in tetrahedral co-ordinates; by Mr. H. M. Jeffery, on bicircular quartics, with a triple and a double focus, and three single foci, all of them colinear; and on spherical quartics, with a quadruple cyclic arc and a triple focus; by Prof. Mannheim, sur les surfaces parallèles; by Mr. R. A. Roberts, on the tangents drawn from a point to a nodal cubic; and note on a system of cartesian ovals, passing through four points on a circle. Signor Brioschi writes sur une propriété du paramètre de la transformée canonique des formes cubiques temaires; and Mr. Carpmael renews an old discussion in his some solutions of Kirkman's 15-school-girl problem. The subject of kinematics on a sphere is ably treated by Mr. E. B. Elliott. Mr. Routh contributes some applications of conjugate functions, and Mr. W. D. Niven writes on the electrical capacity of a conductor bounded by two spherical surfaces cutting at any angle. The presidential address is by Mr. C. W. Merrifield, and is entitled “Considerations respecting the Translation of Series of Observations into Continuous Formulæ.” We have sketched out a bill of fare appealing to many diverse tastes, and we can assure our readers that the dishes are all of admirable quality.

On the Periods of the First Class of Hyper-Elliptic Integrals

On the Periods of the First Class of Hyper-Elliptic Integrals

New Demonstration of the Reduction of Hyperelliptic Integrals to the Normal Form.

New Demonstration of the Reduction of Hyperelliptic Integrals to the Normal Form.