Sophus Lie Research Articles

The metrical aspect of the line-sphere transformation

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.,

Societies and Academies

PARIS. Academy of Sciences, September 13.—M. Bouchard in the ochair.—H. Deslandres communicated a telegram from P. Lowell, stating that the presence of free oxygen has been proved in the atmosphere of Mars. The oxygen band B is clearly stronger in the Mars spectrum than in that of the moon.—The movements of the upper solar atmosphere above and round the faculae. The cellular vortices of the sun: H. Deslandres. Details of the work done with the new spectrograph at the Observatory of Meudon. A diagram is given showing the radial movements of the upper K3 layer of the solar atmosphere above and round a facula.—The study of sea temperatures: A. Bouquet de la Grye. A knowledge of the temperatures of the sea over a wide area is an important factor in weather forecasts.—The trypanolytic power of the blood of some coldblooded vertebrates with respect to Trypanosoma evansi: A. Laveran and A. Pettit. The blood of some of the cold-blooded vertebrates contains active trypanolytic substances, and there seems to be a relation between the presence of these substances and the toxicity of the serum. Closely related vertebrates showed differences in the trypanolytic power, the case of Rana esculenta and R. temporaria being especially remarkable in this respect.— The problem of Sophus Lie: N. Saltykow.—Practical formulas for the calculation of aerial helices: M. Drzewiecki.—The magnetic rdle of oxygen in organic compounds: P. Pascal. The constants given in this paper enable the value for the magnetic susceptibility of an organic compound containing oxygen to be used as a guide to its structure.—The estimation of phosphorus in combustible substances by the calorimetric bomb: P. Lemoult. If certain precautions are taken, details of which are given, the determination of phosphorus in organic compounds by combustion in the calorimetric bomb possesses advantages over the methods in general use both in rapidity and accuracy.—The law of the fading of mnemonic traces as a function of the time in Limnaea stagnalis: Henri Pieron.—The natural means of defence of certain cold-blooded vertebrates against the trypano-some of surra, Trypanosoma evansi: A. Massagrlia. Phagocytosis appears to play no part in the destruction of the trypanosomes.

The Development of Geometrical Methods

Among those who have contributed to the development of Infinitesimal Geometry, Sophus Lie is distinguished by several important discoveries which place him in the first rank. He was not one of those who showed from childhood a very marked aptitude, and when he was leaving the University of Christiania in 1865, he was still hesitating between Philology and Mathematics. It was Plücker’s works which made him fully conscious for the first time of his vocation. He published, in 1869, his first paper on the interpretation of imaginaries in Geometry, and by 1970 he was in possession of the ideas which guided the whole of his career.

On Sophus Lie's Representation of Imaginaries in Plane Geometry

THE first published paper of Sophus Lie appeared in the Transactions of the Academy of Christiania in February, 1869, and was concerned with the subject of this article.* So great a r6le does this memoir play in the scientific career of the author that a reference to it in the Leipziger Berichte, 1897, p. 726, elicits from him the observation that it formed the starting point of all his mathematical investigations.t In fact, the point of view there adopted leads in the most natural way to a point-line transformation of space which specialized gives the celebrated line-sphere transformation, and generalized leads to a new duality defined by two aequationes directrices in two sets of point coordinates, upon which is based the monumental memoir in the fifth volume of the Mathematische Annalen (1872): Ueber Complexe, insbesondere Linienund Kugel-Complexe, etc. I propose to give a presentation of some of the results in the paper referred to above, and I am led to do this not only for the reason that the original lacks clearness fnd continuity, but also from the fact that Lie's work has been apparently overlooked by writers on this subject. ? Moreover, I think the method to be expounded possesses merits not shared by such other representations as I have seen. The essential difference consists in this, that Lie assumes the straight line as geometric element in the plane, while other representations are based upon point geometry. Thus the latter give a real line in space as image of a real or imaginary point in the plane, 11 while Lie's method also leads

Scientific Serials

Transactions of the American Mathematical Society, vol. i. No. 3.—Wave propagation over non-uniform electrical conductors, by M. I. Pupin, is a paper read before the society in December last. The main object of it is the solution of a problem which, looked at from a purely mathematical point of view, can be stated as follows:—Find the integral of the partial differential equation Ld2y/dt2+Rdy/dt = 1δ2y/Cδs2 and determine it so as to satisfy k + 2 boundary conditions, where k + 1 is the number of coils. The principal difficulty is to determine the proper mathematical formulation of these sundry conditions so as to obtain a system of equations which can be readily solved. The paper is illustrated by diagrams which put the problems discussed in a clear light.—“Ueber systeme von differentialgleichungen dessen vierfach periodische functionen genüge leisten,” by M. Krause, was presented at the Chicago (April) meeting of the present year. References are given to Hermite (“Sur quelques applications de la théorie des fonctions elliptiques,” 1885), and to a paper by Picard (Comptes rendus, Band 89), and to previous work by the author.—E. B. van Vleck follows with a paper on linear criteria for the determination of the radius of convergence of a power series. Its object is to establish criteria for the convergence of a power series when the (n+1)th coefficient An is connected with the preceding coefficients by a linear relation which tends to take, a limiting form as n increases indefinitely. The criteria include Cauchy's ratio-test as a special case, and may be looked upon as an extension of the test, and are applicable in cases in which the simple ratio-test fails. The paper closes with two theorems which are an extension for the case of two variables, criteria for the convergence of power series in such a case are stated to be very rare.—On the existence of the Green's function for the most general simply connected plane region, by W. F. Osgood.—A short but suggestive note “D” lines on quadrics, by A. Pell. These lines, so named by Cosserat, were originally considered by Darboux. They are the lines drawn upon a surface in such a way that the osculating sphere at every point is tangent to the surface at that point. In addition to the above, the lines have been studied by Enneper and Ribaucour (for surfaces in general). In the present paper the, author applies, the theory of elliptic functions to the integration of Darboux's differential equation, and obtains an idea of the appearance of the lines and also some of their properties.—Starting from an article, by Prof. F. Morley, in the previous number of the Transactions, F. H. Loud gives sundry metric theorems concerning n lines in a plane. By giving a different interpretation to formulæ got by Prof. Morley, Mr. Loud obtains a new series of theorems and other results of some interest.—An application of group theory to hydrodynamics, by E. J. Wilczynski. It was observed by Sophus Lie that the stationary motion of a fluid can serve as a perfect picture of a one-parameter group in three variables. Apparently this fact has not been utilised for the purposes of hydrodynamics. This paper does this. Amongst other advantages, the treatment, from the new standpoint, leads to special cases of exceptional interest and importance, which otherwise appear to be difficult and unpromising.—Dr. L E. Dickson, following up work recently published in the Proceedings of the London Mathematical Society (vol. xxxi. pp. 30, 351), contributes an article on the determination of an abstract simple group of order 27.36.5.7, holohedrically isomorphic with a certain orthogonal group and with a certain hyperabelian group (contributed to the Chicago [April] meeting of the society).

An application of group theory to hydrodynamics

It has been observed by SOPHUS LIE that the stationary motion of a fluid can serve as a perfect picture of a one-parameter group in three variables. So far as I know, neither he nor any of his followers utilized this fact for the purposes of hydrodynamics. It is the purpose of the present paper to do this. One of the advantages gained for hydrodynamics by this standpoint lies in the general conception. But another advantage is, as is always the case when a class of problems is investigated from a new standpoint, that from the group-theoretical point of view, certain special cases are of exceptionial interest, simplicity, and im portance, cases which otherwise would appear difficult and unpromising.

On surfaces enveloped by spheres belonging to a linear spherical complex

Surfaces enveloped by spheres cutting orthogonally a fixed sphere were first studied by MOUTARDt in the paper: Sur la transformation par rayons vecteurs reciproques, Nouvelles Annal es de Mathematiques, ser. 2, vol. 3, 1864. In the discussion there given, the transformation of space named in the title is all important. The volume of DARBOUX, Sur une class remarquable de courbes et de surfaces, 1873, treats the surfaces and curves of MOUTARD more at length, with especial reference to the case of order 4, the well-known cyclides and bicircular quartics. These surfaces belong to the general class to be studied in this paper, viz.: Surfaces enveloped by spheres belongilng to a linear spherikcal complex. The configuration of o3 spheres, the Kugelcomplex, owes its origin to SOPHUs LIE.* In the surfaces of MOUTARD the complex involved consists of all spheres intersecting a fixed sphere orthogonally. Other special cases are: 10. Complex of all spheres of constant radius. The surface is now either a parallel or tubular surface. 20. Complex of all spheres cutting a fixed plane under constant angle. This case has been treated in a paper by the author, Ont a transformation of Laguerre, Annals of Mathematics, ser. 2, vol. 1, July, 1900. As in the investigations of MOUTARD and DARBOUX the familiar transformation, inversion in a sphere, serves as the main instrumenit, so in the following pages the properties of a more general contact-transformation under which spheres remain spheres, suffice for the derivation of the principal theorems.

Sophus Lie's Transformation Groups: A Series of Elementary, Expository Articles

Sophus Lie's Transformation Groups: A Series of Elementary, Expository Articles

Sophus Lie's Transformation Groups: A Series of Elementary, Expository Articles

Sophus Lie's Transformation Groups: A Series of Elementary, Expository Articles

Sophus Lie's Transformation Groups: A Series of Elementary Articles

Sophus Lie's Transformation Groups: A Series of Elementary Articles

Sophus Lie's Transformation Groups: A Series of Elementary, Expository Articles

Sophus Lie's Transformation Groups: A Series of Elementary, Expository Articles

Sophus Lie's Transformation Groups: A Series of Elementary, Expository Articles

Sophus Lie's Transformation Groups: A Series of Elementary, Expository Articles

Scientific Serials

Bulletin of the New York Mathematical Society, Vol. ii. No. 4 (New York, 1893).—The contents of this number are an abstract of a paper (read before the Society, June 4, 1892) by Prof. W. Woolsey Johnson, entitled “On Peters's Formula for Probable Error” (pp. 57-61). A clear abstract of Engel and Sophus Lie's Theorie der Transformationsgruppen, by C. H. Chapman (pp. 61-71), and a similar account of U. Dini's work on the theory offunctions of a real variable, by J. Harkness (pp. 71-76). Notes and new publications complete the number.