May 26, 1898] 



NATURE 



93 



he proposes to do in the present paper 5 in a subsequent paper 

 he hopes to discuss the cases in whicli certain functions are 

 discontinuous either within or at an extremity of the intervals 

 within which they are considered. The paper was read at the 

 December meeting, and within its limits appears to be a 

 thorough discussion of the matter. — Another paper read at the 

 December meeting is by C. L. Bouton, on some examples of 

 dift'erential invariants. It is founded on Lie's methods. The 

 invariants are those occurring in projective transformations, 

 and the treatment for the plane is given in full ; the method for 

 the corresponding solid problem is sketched in, and the results 

 given. In the author's opinion all the invariants are new. — 

 T'apers read at the February meeting are on an extension of 

 Sylow's theorem, by Dr. G. A. Miller. — Note on the tetra- 

 hedroid, by Dr. J. L. Hutchinson. The writer points out the 

 connection between a certain quartic surface, discussed by him 

 in the Annals of Mathematics (vol. ii. p. 158), and the above- 

 named surface. — Note on integrating factors, by P. Sarel. — 

 Early history of Galois' theory of equations, by Prof. J. Pier- 

 pont. This is a very interesting bibliographical paper, which 

 treats of (i) Galois' relations to Lagrange, and (2) how Galois' 

 algebraic theories became public. Galois' estimate of his dis- 

 coveries is thus stated : " j'ai fait d«s recherches qui arreteront 

 bien des savants dans les leurs." — Reviews follow of Love's 

 theoretical mechanics, of Schell's tortuous curves, and of Page's 

 differential equations. — There are a few slight notes, and the 

 useful list of mathematical publications. 



Wiedemann'' s Annalen der Physik nnd Chemie, No. 3. — 

 Conductivities of electrolytes, by F. Kohlrausch, L. Holborn, 

 and H. Diesselhorst. The authors point out that the modern 

 advances in the measurement of temperatures and resistances 

 have made it necessary to redetermine the conductivities of elec- 

 trolytes in terms of the units now adopted. As standard 

 electrolytes they take solutions of sulphuric acid of density 

 I "223, magnesium sulphate of density I'igo, and sodium chlo- 

 ride saturated at 18°. The resistance of i cc. as a cube is 

 07398, o 04922, and 021605 in the three cases, which represent 

 the maximum conductivities of those salts at the temperature 

 mentioned. — The foundations of the electric unit of resistance, 

 by W. Jaeger and K. Kahle. The authors describe the methods 

 adopted in the Physikalisch-Technische Reichsanstalt for puri- 

 fying the mercury and calibrating the tubes of standard resist- 

 ances. The tubes must be filled in a vacuum. The resistances 

 show a secular diminution of about 0'oooo3 ohms in five years. — 

 Absorption and emission of steam and carbonic acid in the infra- 

 red spectrum, by II. Rubens and E. Aschkinass. The infra-red 

 rays separated out by five successive reflections at fluospar sur- 

 faces are absorbed by carbonic acid and water vapour in thick 

 layers. Their wave-length is about 24 fx. Their absorption by 

 the atmosphere accounts for their absence in the solar spectrum. 

 — On the transparency of some liquids for rays of great wave- 

 length, by the same authors. Water shows considerable absorp- 

 tion, but benzol is more transparent even than silver chloride. — 

 On light nodes in a kathode ray bundle under the influence of a 

 magnetic field, by E. Wiedemann and A. Wehnelt. When the 

 lines of force are parallel to the axis of the tube, the kathode rays 

 are twisted into a bundle having successive nodes. The pheno- 

 menon is completely in accordance with the projected- particle 

 theory of kathode rays. — Visibility of Rontgen rays, by E. Dorn. 

 Proves that the light effects seen are not due to an accommoda- 

 tion-strain or to electrical discharges in the neighbourhood of the 

 observers head. 



SOCIETIES AND ACADEMIES. 



London. 



Royal Society, May 5. — "Observations on the Action of 

 Anaesthetics on V^egetable and Animal Protoplasm." By J. B. 

 Farmer, M.A., and A. D. Waller, M.D., F.R.S. Received 

 March 9. 



The object of the investigation was to observe simultaneously 

 and comparatively the effects of certain ana;sthetics (carbon 

 dioxide, ether, and chloroform) upon vegetable and upon animal 

 protoplasm. 



Two gas chambers in series, through which anaesthetic and 

 other vapours can be passed, contain : the first, a leaf of Elodea 

 Canadensis under the microscope ( x 300) ; the second, a sciatic 

 nerve of Nana tentporaria connected with an inductorium and 

 galvanometer (or upon occasion a galvanograph). 



NO. 1 49 1, VOL. 58] 



The actual movements of chlorophyll bodies in a cell of the 

 leaf were observed and measured by one observer, while the 

 other took readings of the galvanometric deflections in response 

 to excitation of the nerve. To establish comparison between 

 the two classes of effects we took as measures : — the number of 

 chlorophyll bodies that crossed a cobweb in the eye-piece durintr 

 each successive minute, and the magnitude of galvanometric 

 deflections at intervals of one minute, before, during, and after 

 the action of the vapour. The number of bodies passing per 

 minute gives measure of the rate of movement in the vegetable 

 protoplasm, while the magnitude of successive galvanometric 

 deflections gives measure of the mobility of the animal proto- 

 plasm. 



The results obtained from a study of Chara, Elodea, and Plas- 

 modium of Badhainia were quite consistent, but owing to the 

 greater ease in making a quantitative determination, Elodea was 

 used for the more exact comparative experiments. 



The action of carbon dioxide was to produce an initial slight 

 acceleration, followed speedily by a complete cessation of move- 

 ment. On disconnecting the CO^ apparatus and aspirating 

 air through the chamber the protoplasm, after the lapse of two 

 or three minutes, began to show signs of recovery. Fitful 

 movements of the granules first occurred, and then they soon 

 resumed their processional motion around the cell ; at first very 

 slowly. The movement rapidly became accelerated and consider- 

 ably exceeded the normal rate. This acceleration was not of 

 long duration, and was followed by a slowing down to the 

 ordinary speed. 



The results of experiments with chloroform and ether were 

 also given. 



May 12. — "On the Connection of Algebraic Functions with 

 Automorphic Functions." By E. T. Whittaker, B.A., Fellow 

 of Trinity College, Cambridge. Communicated by Prof. A. R. 

 Forsyth, Sc.D., F.R.S. 



If « and z are variables connected by an algebraic equation, 

 they are, in general, multiform functions of each other ; the 

 multiformity can be represented by a Riemann surface, to each 

 point of which corresponds a pair of values of u and z. 



Poincare and Klein have proved that a variable t exists, of 

 which u and z are uniform automorphic functions ; the existence- 

 theorem, however, does not connect t analytically with u and z. 

 When the genus [genre, Geschlechl) of the algebraic relation is 

 zero or unity, t can be found by known methods ; the auto- 

 morphic functions required are rational functions, and doubly 

 periodic functions, in the two cases respectively. But no class 

 of automorphic functions with simply connected fundamental 

 polygons has been known hitherto, which is applicable to the 

 uniformisation of algebraic functions whose genus is greater than 

 unity. 



The present memoir discusses a new class of groups of pro- 

 jective substitutions, such that the functions rational on a 

 Riemann surface of any genus can be expressed as uniform 

 automorphic functions of a group of this class. Groups are first 

 considered which can be generated by a number of real sub- 

 stitutions of period two, whose double points are not on the real 

 axis, and whose product in a definite order is the identical 

 substitution. A method is given for dividing the plane into 

 curvilinear polygons corresponding to such a group ; these 

 polygons are simply-connected, and cover completely the half of 

 the plane which is above the real axis. Sub-groups of these 

 groups are found, whose genus is greater than unity, and 

 which are appropriate for the uniformisation of any algebraic 

 curves. 



The sides of the polygons, into which the half-plane is divided, 

 are formed of arcs of circles orthogonal to the real axis. 



The analytical connection between the variables of the alge- 

 braic form and the uniformising variables is given by a differen- 

 tial equation of the third order. A certain number of the 

 constants in this equation have to be determined by the condition 

 that the group of substitutions associated with the equation leaves 

 unchanged a certain circle. When any arbitrary values are given 

 to these constants the solution of the differential equation is 

 termed a quasi-uniformising variable. The properties of quasi- 

 uniformising variables, and their relation to the uniformising 

 variable, are discussed in the last section of the pa|>er. 



Physical Society, May 13.— Mr. Shelford Bidwell, 

 President, in the chair.— A paper by Prof. W. E. Ayrton 

 and Mr. T. Mather, on galvanometers, was read by Prof. 

 Ayrton. It is a sequel io Proc. Physical Soc., vol. x. p. 393, 



