February 9, 1883.] 



SCIENCE. 



15 



the specific descriptions, it appears to us that 

 Mr. Herriclj trusts too much to such charac- 

 ters as the number and arrangement of tlie 

 joints of the antennae, which change with the 

 growth of the individual. Even sexual ma- 

 turity in these animals does not determine 

 the limit of structural change. 



Besides the microscopic forms, two species 

 of craj'-flsh are recorded, — Cambarus virilis 

 Hagen and C. signifer sp. nov. Attention is 

 again drawn to the curious fact that size does 



not govern the transition from the ' second 

 form ' or sexuall.y immature ( ?) male to the 

 ' first form ' or perfected state ; the second 

 form often exceeding the first in its dimensions. 

 Zoologists 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. Tj-pes of the ' new ' 

 species, C. signifer, kindly communicated by 

 Mr. Herrick, prove to be C. immunis Hagen. 

 Eleven plates accompanj' this memoir. 



WEEKLY SUMMARY OF THE PROGRESS OF SCIENCE. 



MATHEMATICS. 



Quadrature of the circle. — In vol. xx. of the 

 Mathematisclie aniialeii, Lindemann gave a proof of 

 the fact that tt cannot be a root of an equation of auy 

 degree with rational co-efficients. This is a most re- 

 markable paper, as it thus contains the first direct, 

 absolute proof that has ever been given of the im- 

 possibility of the quadratvtre of the circle. M. Linde- 

 mann's investigation is based tipon, and presupposes 

 a knowledge of, Hermite's earlier paper, in which he 

 showed that e, 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 ac- 

 count, the author of the article at present referred to, 

 M. Kouche, 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 sim- 

 plify the processes in both cases. M. Rouche 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 simpler proof will yet 

 be given of the fact that tt 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- 

 miihl, considers the case of geodesies upon triaxial 

 surfaces of the second order. He derives first Weier- 

 trass' formulas for a general geodesic, and obtains 

 forms for the entering constants in terms of the dou- 

 ble i/ieia-functions, rendering them easy of compu- 

 tation. 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, con- 

 tains a derivation of the equations of the envelopes 

 of geodesies, 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 hyperboloid. Numerous references 

 are given to previous investigations. — {Math, annalen, 

 XX., 1882.) T. 0. [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 ijerfect form) for the quartic curve of the differen- 

 tial expression dlT, (in Clehsch and Gordan's nota- 

 tion) or d Hia (in Prof. Cayley's notation) in the 



r/3 



integral of the third kind 



ind / dn. 



in the final normal 



f/5 rv 



form for which I d 11, = j d V^a 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. [3 



PHYSICS. 



Aconstics. 

 Instrument for measuring the intensity of 

 aerial vibrations. — The instrument is based on an 

 experiment described by the author (Lord Eayleigh) 

 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 aerial currents. A brass tube 

 is closed at one end wtth 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 throTigh the side of the tube by a glass window. 

 A lens is so placed as to throw an image of the slit 

 upon a scale. The opposite end of the tube, prolonged 

 to a distance equal to that between the slit and mirror, 

 is closed by a diaphragm of tissue-paper. A sliding 

 tube extends for some 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 



