May 17, 1906] 



NA TURE 



>i 



fiQeological Society, April 25.— Dr. J. E. Marr, F.R.S., 

 vici'-prcsident, in the chair. — Trilobites from Bolivia, col- 

 lected by Dr. J. W. Evans in 1901-1902 : P. Lake. Several 

 horizons are represented by these fossils. Descriptions are 

 given of the new species and other forms mentioned. It is 

 worthy of remark that, while the earlier forms show affini- 

 ties with the contemporaneous European fauna, the 

 Devonian species are much more closely allied to those of 

 South Africa and North America. — Graptolites from Bolivia, 

 collected by Dr. J. W. Evans in igoi-1902 : Dr. Ethel 

 M. R. Wood. In black pyritic shales from three localities 

 several specimens of Didymograptus were collected : one 

 referable to bifutiis, one of the type of afjinis, and one of 

 the Nicholsoiii type. Phyllograptus, Glossograptus, Crypto- 

 graptus, and Diplograptus were also obtained. A pale, 

 silky-grey shale shows also rare graptolites, belonging to a 

 species comparable with Climacograptus confcrtus. These 

 forms indicate that both the black and the pale shales 

 belong to horizons in the Upper Arenig rocks (Lower 

 Llanvirn of Hicks). — The Phosphatic Chalks of Winter- 

 bourne and Boxford (Berkshire) : H. J. Osborne White and 

 Llewellyn Treacher. Data collected in the district dealt 

 with in this paper suffice to show that the more or less 

 Phosphatic Chalks above the Uintacrinus-band lie in a 

 trough or basin, the formation of which antedates the de- 

 position of the Reading beds. When the area of observ- 

 ation is extended, it is found that the Uintacrinus-Chalk 

 of that tract itself lies in a structural depression. The 

 Phosphatic Chalks of Winterbourne and Taplow evidently 

 mark places on the sea-floor particularly liable to the 

 impingement of strong currents, and may mark places above 

 which the water commonly had a gyratory motion. In any 

 c:ise, their zonal range argues a marked degree of stability 

 in the current-system of the body of water in which they 

 were laid down. 



Physical Society, April 27. — Dr. C. Chree, F.R.S., vice- 

 president, in the chair. — Some simple questions on the 

 images of microscopes and telescopes: W. B. Croft. It 

 may have been noticed that when a microscope is focused 

 visually, an image is formed on the focusing-glass of a 

 camera, into which the microscopic eye-piece is inserted 

 after removing the camera-lens. This image remains more 

 or less in focus at variable positions of the camera-screen. 

 .\Uhough it is not always perhaps true, yet it is surprising 

 how often the pencil emerging from a microscope eye-piece 

 behaves like a single concentrated line of light. Several 

 photographs of microscopic details were exhibited to 

 intimate how often the author had found, when projecting 

 from an optical eye-piece, that no change can be detected 

 in the definition of the image as the screen of the camera 

 is moved. If a camera-lucida is placed on the eye-piece, 

 the image of a stage-micrometer can be thrown on a scale 

 at 10 inches distance or at 40 inches distance. The parallel 

 rays emerging from the eye-piece give the image of a point 

 along a direction, at no definite position. The image can 

 be imagined at 40 inches distance as easily as lo inches. 

 Mr. Croft also showed some photographs taken from sec- 

 tions of the human eye; he indicated that a divergent pencil 

 from a small aperture or from a convex reflecting surface 

 of large curvature will give the Purkinje figures as bright 

 r.-idiating lines, whereas the usual method of sending light 

 through the side of the sclerotic gives them as shadows. 

 .Several different specimens were shown of magnetic oxide 

 of iron and magnetic sulphides of iron. The power of 

 nickel and cobalt to receive permanent magnetism was 

 illustrated with a compass-needle of nickel. — The lateral 

 vibration of bars subjected to forces in the direction of their 

 axes : J. Morrow. Three cases of unloaded bars are dealt 

 with, namely, those under the following end-con- 

 ditions: — " supported-supported," " clamped-clamped," and 

 " clamped-supported." Expressions are obtained from which 

 the frequencies may be calculated, and the results are 

 stated in a form such that the determination of stresses, 

 terminal couples, &c., may be easilv made. The case of 

 greatest interest is that of a stretched bar clamped at each 

 end. .'\pproximate solutions of this problem have been 

 arrived at by both Seebeck and Donkin. These are on the 

 assumption that the vibration is but slightly affected by 

 the rigidity of the material. An assumption of a very 

 different character, and one generally fulfilled in structural 



NO. 1907, VOL. 74] 



work, is made in this paper, namely, that the longitudinal 

 force is not very great. Solutions are thus found for the 

 period of the fundamental or any harmonic. 



Mathematical Society, May 10. — Prof. A. R. Forsyth, 



president, in the chair. — The substitutional theory of classes 

 and relations : Hon. B. Russell. The object of the paper 

 is to explain a solution of the contradictions discovered by 

 Burali-Forti and the author. The solution is sought in 

 the substitutional theory, sketched in a previous paper by 

 the author, according to which statements apparently about 

 a class are significant only when they can be analysed 

 into statements about all or some of the members of the 

 class. The substitutional theory is extended to propositions 

 and relations. — The expansion of polynomials in series of 

 functions : Dr. L. N. G. Filon. The question is that of 

 expanding a function /(.v) in a series of functions of the 

 form <^(k„, ,v), where the numbers «,, k.^, . . . are the roots 

 of a transcendental equation. The method is analogous to 

 Cauchy's method of expansion of functions in Fourier's 

 series, and depends upon the calculus of residues. In 

 Cauchy's method a subsidiary function F((() is introduced 

 through a knowledge of the form of the coefficients of 

 Fourier's series; in the present paper a rule is given for 

 determining this subsidiary function o priori, and the rule 

 is shown to be applicable to many classes of functions 

 <p(k,.i) when the function fix), of which the expansion is 

 desired, is a polynomial. The subsidiary function F(it) 

 being known, the coefficients In the expansion of f{x) can 

 be obtained explicitly. — The motion of a swarm of particles 

 the centre of gravity of which describes an elliptic orbit of 

 small eccentricity round the sun : Dr. E. J. Routh. It 

 Is proved that for a spherical swarm the period equation 

 takes the Lagrangian determinantal form, and the con- 

 ditions of stability can be completely exhibited. The 

 problem of a swarm of unequal thicknesses in different 

 directions Is illustrated by a discussion of the case in which 

 the boundary is ellipsoidal ; and the changes of length of 

 two diameters in the plane of motion, one of which passes 

 through the sun, are investigated in detail. — The theory 

 of integral equations: H. Bateman. The partial Integral 

 equation 



j\(s, x)Ax, t) d.r = j J{s, x) k(x, t) dx 



Is regarded as the characteristic equation of a transform- 

 ation by which the properties of the function h{s,t) are 

 deducible from those of k{s,1). This transformation leaves 

 unaltered the numbers A„ for which the homogeneous 

 equation 



„(j) = 



\„ / k{s, /) <t, (t):il 



possesses a solution different from zero. The numbers A„ 

 are important in the theory of the potential and in con- 

 nection with a certain theory of the origin of spectral 

 lines. — Linear differential equations of rank unity : E. 

 Cunningham. The paper is concerned with an extension 

 of Laplace's method of solution of linear differential equa- 

 tions by means of definite Integrals. The proposed solu- 

 tion takes the form of a double integral involving a sub- 

 sidiary function which satisfies a certain partial differential 

 equation. Particular forms of this subsidiary function are 

 developed, and the appropriate domains of integration 

 determined. 



Paris. 

 Academy of Sciences, April 30. — M. H. Poincare in the 

 chair. — Diphenyl or alkylphenyl camphomethane and 

 methylene : .\. Haller and E. Bauer. — Simple relations 

 between the " statical actions " of muscle with the energy 

 which produces them : .-\. Chauveau. — The doubly Infinite 

 varieties of points of a quadric in space of four dimensions 

 applicable to a plane : C. Guichard. — Contribution to the 

 study of the infra-red spectrum : Milan Stephanik. In 

 the study of the infra-red region of the spectrum three 

 methods have hitherto been used, the thermometric method, 

 the photographic method, and the utilisation of the pheno- 

 mena of phosphorescence. In studying the eclipse of 

 -August 30, 1905, the author noticed that when -a deep red 

 screen was placed In front of the slit of the spectroscope 

 a portion of the infra-red spectrum became visible. This 



