Annals of Mathematics (July and October, 1901).—Concerning Du Bois Reymond's two relative integrability theorems. The two theorems considered by E. H. Moore are, (1) a continuous function of (properly) integrable functions is integrable; (2) an integrable function of an integrable function is integrable, (1) was announced in 1880 and a proof published two years later (Math. Ann., vols. xvi. and xx.). In connection with this proof (2) was announced. Dr. Moore in this note shows, by means of a simple example, that (2) is not true. Reference is made to a proof of (1) by Dini with an extension which is not applicable to the general case, but Dr. Moore extends Du Bois Reymond's general proof (1882).—P. Saurel, on a theorem of kinematics, gives an elementary demonstration of the well-known theorem that every displacement of a rigid body is equivalent to a rotation followed by a translation parallel to the axis of rotation.—The collineations of space which transform a non—degenerate quadric surface into itself, by Ruth G. Wood, discusses the ∞6 collineations of space which transform the surface.—J. Westland contributes a note on multiply perfect numbers, with a view to determine all numbers of multiplicity 3 of the form m = p1a1p2a2p2 where p1, p2, p2 are three distinct primes and p1 < p2 < p3.—The isoperimetrical problem on any surface, by J. K. Whittemore, gives a generalisation of the problem known to Pappus (see W. Thomson, “Popular Lectures and Addresses,” vol. ii. p. 78). He solves Pappus's problem by the calculus of variations, and then: solves, by an apparently novel method, the problem “Find a curve, v = φ(u)”, joining the two given points (u0, v0) and (u1, v1) having a given length L, and such that the area of the portion of the surface between the two curves, v = f(u) and v = φ(u), shall be a maximum.”—On a surface of the sixth order which is touched by the axes of all screws reciprocal to three given screws, by E. W. Hyde, has for its main object the determination and discussion of the envelope of a certain conicoid, which is touched by the axes of all screws of a certain system, so enabling one to grasp the nature of the system. The surface possesses other features of interest. The paper is illustrated with diagrams.—D. Sintsof, in a note sur revaluation d'une integrate definie, discusses a previous note by M. Pell (evaluation of a definite integral, Annals (2), tome 1, No. 3).—The October number opens with a lengthy article (18 pp.) on the convergence of the continued fraction of Gauss and other continued fractions, by E. B. Van Vleck. Numerous references are given.:—M. B. Porter supplies a short note on the differentiation of an infinite series term by term.—A note on geodesic circles, by J. K. Whittemore, discusses these circles in Bianchi's sense, viz. their definition is the locus of a point on a surface at a constant geodesic distance from a fixed point of the surface (“Vorlesungen über Differentialgeometrie,” p. 160). Darboux (“Théorie Generate des Surfaces,” vol. iii. p. 151) calls such a circle a curve of constant geodesic curvature. Mr. Whittemore gives three theorems—the first is, if, on a surface, there exists a family of concentric geodesic circles, such that the geodesic curvature of each curve of the family is constant, then the total curvature of the surface is constant along each curve of the family, and the surface is applicable to a surface of revolution, so that the geodesic circles fall on the circles of latitude of this surface.—Prof. Osgood gives a note on the functions defined by infinite series whose terms are analytic (functions of a complex variable, with corresponding theorems for definite integrals. References to other memoirs abound.—Mr. C. L. Bouton gives an account of a game which he entitles “Nim” (a game with a complete mathematical theory). It is a game played at a number of American colleges and fairs and has been called “Fan-tan,” though it does not correspond with the Chinese game of that name. He gives a description of the game (too curt, we think), and also discusses the theory of it.—Dr. G. A. Miller discusses the groups generated by two operators of order three whose product is also of order three, a short note, as is also the concluding one, on the invariants of a quadrangle under the largest subgroup, having a fixed point, of the general projective group in the plane, by W. A. Granville.