November 20, 1902] 



NA TURE 



the council, Mr. Campbell, Lieut. -Colonel Cunningham, Dr. 

 GlaUher, Prof. Greenhill, Mr. Macdonald, Major MacMahon, 

 Mr. Western, Mr. Whittaker, Mr. A. Young. — Prof. Lamb 

 having taken the chair, Dr. Hobson delivered an Address on the 

 infinite and the infinitesimal in mathematical analysis. He 

 sketched briefly the history of the attempts that had been made 

 at various times to deal with questions of the infinite, and dwelt 

 especially upon the critical work of the latter half of the nine- 

 teenth century, pointing out that pertinent criticism of funda- 

 mentals almost invariably gives rise to new construction. He 

 explained how the system of analysis, connected with the title 

 " arithmetisation," had turned a difficulty, to which all previous 

 systems were liable, in that they were unable to give a proof 

 of the existence of the limit. He described the character 

 of the numerical continuum, and contrasted its properties wiih 

 those of other aggregates, which possess unlimited divisibility. 

 He proceeded to recount the objections that had been raised 

 to the introduction of infinite numbers, as opposed to variables 

 which become indefinitely great ; and he concluded with an out- 

 line of the theory of transfinite numbers. — The following papers 

 were communicated : — Prof. D. Hilbert, Ueber den Satz von 

 der Gleichheit der Basiswinkel im gleichschenkligen Dreieck. 

 The paper forms part of a critical discussion of geometrical 

 axioms. The possibility of setting up various systems of axioms, 

 so that the axioms of a system shall be mutually consistent and 

 mutually independent, has been proved ; and it becomes im- 

 portant to ascertain the relations of the more fundamental 

 geometrical propositions to the possible systems of axioms. — 

 Prof. Burnside, On linear homogeneous groups. The charac- 

 teristic determinants of any simply transitive, and of any transi- 

 tive, linear homogeneous group are discussed, and general forms 

 of the determinants are given ; the results are applied to simplify 

 the proofs of known propositions concerning the continuous 

 group that is defined by any given group of finite order. — Prof. 

 Lamb, On wave-propagation in two dimensions. The divergence, 

 in two dimensions, of waves from a source, of a more or less 

 transient character, is worked out in detail and illustrated 

 graphically. The disturbance begins suddenly at a place when 

 the wave reaches it ; but it does not cease suddenly after a time 

 equal to that during which the source is in action. The 

 existence in two-dimensional wave motion of a sort of " tail " 

 to a wave, which does not occur in the case of waves in one 

 dimension or in three dimensions, is further elucidated by 

 various comparisons between the characters of the three cases. — 

 Prof. A. C. Dixon, (I) Summation of a certain series; (2) 

 Expansions by means of Lame's functions. The first of these 

 papers is a development of previous work by Morley on the 

 hypergeorcetrfc functions that arise from the consideration of the 

 sum of the cubes of binomial coefficients. The second paper 

 contains a discussion of the use of Lame's functions to determine 

 a potential from its singularities and boundary values for the 

 following regions : — (a) the interior of an ellipsoid, (d) the 

 exterior of an ellipsoid, (c) the space between two confocal 

 ellipsoids, \d) two distinct regions, bounded by confocal ellip- 

 soids, wholly or partly coextensive and connected together 

 through the area of the focal ellipse. — Mr. W. H. Young, (i) 

 On sets of intervals, (2) Note on unclosed sets of points defined 

 as the limit of a sequence of closed sets of points. The first of 

 these papers aims at developing the theory of sets of intervals 

 on the straight line in a systematic manner ; it is pointed out 

 that, although the discussion of such sets forms a natural intro- 

 duction to some parts of the theory of aggregates, only a few 

 isolated theorems about such sets have been formulated hitherto. 

 The object of the second paper is to obtain the necessary and 

 sufficient condition that the content of the set obtained by 

 closing an unclosed set, which is the limit of a sequence of 

 closed sets, may be the limit of the contents of the closed sets 

 of the sequence. — Prof. Hill, The continuation of certain funda- 

 mental power series. The object of the paper is to illustrate 

 the theory of continuation in simple cases in which the work 

 need not be artificial. The continuations, along arbitrary 

 circuits, of the binomial series, the logarithmic series, the 

 series for arc tan x, are developed in detail. The methods of 

 the paper depend upon theorems proved by Abel in his 

 classical memoir on the binomial series. — Prof. L. Crawford, 

 A geodesic on a spheroid and an associated ellipse. The length 

 of the arc of a geodesic drawn from a given point on a spheroid 

 in a given direction is found as the length of an arc of an ellipse, 

 and the difference of longitude of any point on the geodesic 

 and the given point is expressed as an elliptic function of an 

 angle connected with the corresponding points on the same 



NO. I725, VOL. 67] 



ellipse ; an expression is found for the change in longitude on 

 return along the geodesic to the same latitude.— Prof. A. W. 

 Conway, The propagation of light in a uniaxal crystal. New 

 forms of integrals of the equations of propagation are obtained. 

 The results are applied to the discussion of the direction of 

 vibration and the flow • of energy ; it appears that the ray 

 direction is not the direction of the energy flux in waves 

 diverging from a source within the crystal. Applications of the 

 integrals are also made to discuss the passage of parallel and 

 of divergent beams of light through a thin crystalline plate. — 

 Mr. E. T. Whittaker, On a new connection of Bessel functions 

 with Legendre functions. A symbolic relation, which connects 

 the functions in the case where the order of the Bessel functions 

 is half an uneven integer, is transformed into an expression for 

 the Bessel functions of unrestricted order as definite integrals 

 involving Legendre functions of unrestricted order. 



Chemical Society, November 6. — Prof. McLeod, F.R.S., 

 in the chair. — The following papers were read : — The specific 

 heats of gases, by Mr. H. Crompton. An extension of the 

 application of Le Chatelier's formula for the specific heats of 

 elementary gases to the vapours of complex substances. — The 

 action of nitric acid on bromophenolic compounds, by Mr. W. 

 Robertson. An investigation of the effect produced by dis- 

 placement of the hydroxyl group by methoxy- or acetoxy-groups 

 in inhibiting the replacement of bromine by nitroxyl. — 3:5- 

 dichloro-fl-xylene and 3:5-dichloro-o-phthalic acid, by Drs. 

 Crossley and Le Sueur. — The combination of carbon monoxide 

 with chlorine under the influence of light, by Drs. Dyson and 

 Harden. These gases when dried, mixed in equal quantities 

 and exposed to light, undergo first a period of photochemical 

 induction and finally reach a stage of equilibrium with the 

 carbonyl chloride formed. — The constituents of commercial 

 chrysarobin, by Dr. Jowett and Mr. Potter. — The constituents 

 of oil of rue, by Dr. Power and Mr. Lees. The following new 

 constituents have been obtained : — methyl «heptylcarbinol, 

 methyl K-nonylcarbinol, methyl salicylate, cineol, limonene and 

 pinene. — Methyl 8-methylhexyl ketone, by Mr. H. Lees. — Di- 

 indigotin, by Dr. Moir. This substance was obtained by the 

 application of Baeyer's process for the synthesis of indigotin 

 from 0-amidocinnamic acid to the diphenyl analogue, benzidine 

 dicarboxylic acid. — The localisation of phosphates in the sugar- 

 cane, by Mr. Sprankling. — On the non-existence of the gaseous 

 sulphide of carbon described by Deninger, by Messrs. Russell 

 and Smith. — Isometric anhydrous sulphates of the form 

 M"S0 4 ,R'S0 4 , by Mr. F. R. Mallet.— The catalytic racemisation 

 of amygdalin, by Dr. J. W. Walker. The optically active glu- 

 coside is converted by the hydroxyl ions of aqueous alkaline 

 solvents into racemic amygdalinic acid. — On asymmetric opti- 

 cally active selenium compounds, and on the hexavalency of 

 selenium and sulphur, by Prof. Pope and Mr. Neville. Methyl- 

 phenylselenetine has been obtained in dextro- and laivo-modifi- 

 cations by fractional crystallisation of the o'-bromcamphorsul- 

 phonate. — The transformation of acetylchloroaminobenzenes into 

 the isomeric chloroacetanilides, by Drs. Chattaway and Orton. 



Paris. 

 Academy of Sciences, November 10. — M. Albert Gaudry 

 in the chair. — On uniform transcendentals defined by the equa- 

 tion y" = 6y- + x, by M. Paul Painleve. — On quasi- waves, by 

 M. P. Duhem. From the theoretical examination of the 

 velocity of propagation of sound waves in air, it is shown that 

 if the coefficient of conductivity has a finite value, however small, 

 the waves will be propagated in accordance with the formula of 

 Newton, and it is only in the case where the conductivity co- 

 efficient is rigorously zero that the waves will travel in accordance 

 with the formula of Laplace. But although the conductivity of air 

 is small, it is not zero, and this leads to a serious discrepancy 

 between theory and experiment. The author shows that the 

 existence of viscosity in air, although small, renders impossible 

 the propagation of waves properly so called, and examines the 

 conditions of transmission of the quasi-waves which are possible, 

 and succeeds in showing that an explanation of the discrepancy 

 becomes possible. — Further observations and experiments re- 

 lating to the determination of the velocity of the X-rays, by 

 M. R. Blondlot. According to the theory put forward by 

 Wiechert and Sir G. G. Stokes, the X-rays consist, not of con- 

 tinuous ether vibrations, but of extremely short, isolated pulsa- 

 tions, and this hypothesis has been shown to give a complete 

 explanation of the absence of refraction and reflection and of 

 the diffraction phenomena shown by the rays. All the ex- 

 perimental results obtained by the author are also in accord 



