the specific descriptions, it appears to us that Mr. Herrick trusts too much to such characters as the number and arrangement of the joints of the antennae, which change with the growth of the individual. Even sexual maturity in these animals does not determine the limit of structural change.
Besides the microscopic forms, two species of cray-fish are recorded,—Cambarus virilis Hagen and C. signifer sp. nov. Attention is again drawn to the curious fact that size doee not govern the transition from the 'second form' or sexually immature (?) male to the 'first form' or perfected state; the second form often exceeding the first in its dimensions. Zoölogists whose lot it is to live in a cray-fish country cannot be too strongly urged to study the habits and physiology of these so-called dimorphic males. Types of the 'new' species, C. signifer, kindly communicated by Mr. Herrick, prove to be C. immunis Hagen. Eleven plates accompany this memoir.
WEEKLY SUMMARY OF THE PROGRESS OF SCIENCE.
MATHEMATICS.
Quadrature of the circle.—In vol. xx. of the Mathematische annalen, Lindemann gave a proof of the fact that cannot be a root of an equation of any degree with rational co-efficients. This is a most remarkable paper, as it thus contains the first direct, absolute proof that has ever been given of the impossibility of the quadrature of the circle. M. Lindemann’s investigation is based upon, and presupposes a knowledge of, Hermite’s earlier paper, in which he showed that , the Napierian base, cannot be the root of an equation with rational co-efficients. The fact that Lindemann has started from Hermite’s results makes his paper rather hard reading; and on this account, the author of the article at present referred to, M. Rouché, has thought it worth while to give an account of the work done by Hermite, and more recently by Lindemann, and at the same time to simplify the processes in both cases. M. Rouché has really done very little in the way of simplification, but by bringing together the proofs he has produced an interesting and valuable paper. He professes the belief that the last word has not yet been said on the subject, but that another and siinpler proof will yet be given of the fact that + cannot be a root of any equation of any degree with rational co-efficients. Lindemann has certainly done a splendid piece of work in thus absolutely proving the impossibility of “squaring the circle;’ and it is only to be regretted that his work will not carry conviction to the minds of those mistaken individuals, the ‘circle-squarers.’ But it is hardly to be supposed that they will be convinced of the futility of their task, any more than the perpetual-motion inventors were convinced by the discovery and enunciation of the principles of the conservation of energy.—(Nouv. annales, Jan., 1883.) t. c. [1
Geodesic lines. — The author, Herr A. v. Braun-mühl, considers the case of geodesics upon triaxial surfaces of the second order. He derives first Weiertrass’ formulas for a general geodesic, and obtains forms for the entering constants in terms of the double theta-functions, rendering them easy of computation. Examples are given of the computation of geodesic lines in the general and in several special cases. The latter, and newer part of the paper, contains a derivation of the equations of the envelopes of geodesics, and a discussion of the same. The en-velope is determined by aid of the hyperelliptic func-tions, and special applications are made to the ellipsoid and two sheeted hyperbuloid. Numerous references are given to previous investigations. — (Math. annalen, rx., 1882.) 1. Cc. 2
Abelian and theta functions. — Prof. Cayley in this memoir has reproduced with additional de- velopments the course of lectures which he deliv- ered in the Johns Hopkins University, in the win- ter and spring of 1882. The memoir has a special interest as being the first of any consequence upon this subject in the English language, and, indeed, one of the most important in any language. The chief addition to the theory consists in the determination made for the cubic curve, and also (but not as yet in a perfect form) for the quartic curve of the differential expression (in Clebsch and Gordan’s notation) or (in Prof. Cayley’s notation) in the integral of the third kind in the final normal form for which the limits and parametric points interchangeable. The notation and demonstrations of Clebsch and Gordan are much simplified, and the theory is illustrated by examples, in regard to the cubic, the nodal quartic, and the general quartic respectively. The first three chap- ters only of the memoir have yet appeared. — (Amer. journ. math., v., 1883.) t. c.
PHYSICS. Acoustics.
Instrument for measuring the intensity of aérial vibrations. — ‘The instrument is based on an experiment described by the author (Lord Rayleigh) in the Proceedings of the Cambridge philosophical society for November, 1880; from which it appeared that a light disk, capable of moving about a vertical diameter, tends to set itself at right angles to the direction of alternating aérial currents. A brass tube is closed at one end with a glass plate, behind which is a slit through which pass rays of light from a lamp. A light mirror with attached magnets, such as are used for reflecting galvanometers, is suspended by a fine silk fibre so that the light from the slit is incident upon it at an angle of 45°, and, after reflection, passes out throngh the side of the tube by a glass window. A lens is so placed as to throw an image of the slit upon ascale. The opposite end of the tube, prolonged to a distance equal to that between the slit and mirror, is cloved by a diaphragm of tissue-paper. A sliding tube extends for sume distance beyond this. If the instrument is exposed to sounds whose half-wave- length is equal to the distance from the slit to the tissue-paper diaphragm, nodes are formed at each
