5>4 



NATURE 



[October 20, 19 10 



spinning', and the other wheel carries a square tin plate 

 on whiih is fixed a card coloured in squares. The machine 

 is so mounted that it can turn freely about its centre of 

 gravity. To throw it out of truth without disturbing the 

 centre of gravity two small bolts are inserted in the rims, 

 one in each, at opposite points. The machine is set 

 spinning about the a.xis of the wheels. When the 

 instantaneous a.xis cuts the card at a point well within a 

 square, a patch of the colour of that square is seen dis- 

 tinctly, and the rest of the card appears confused. As the 

 a.xis moves, a series of distinct patches are seen at short 

 intervals. 



On Friday, September 2, the section divided. In the 

 mathematical department Major P. A. MacMahon opened 

 the proceedings by reading a paper on functions derived 

 from complete and incomplete lattices in two dimensions, 

 and the derivation therefrom of functions which enumerate 

 the two-dimensional partition of numbers. The investi- 

 gation was suggested by the solution of a ballot problem 

 of finding the chance that at every stage of the voting 

 the candidates are in their final order. 



Dr. BaUer in his paper on a certain permutation group 

 said that he had been led to inquire into its properties 

 by becoming interested in a game plaved bv some children. 

 The game consisted in writing down a series of letters 

 and then rearranging Ihem by writing the last first, the 

 first second, the last but one third, the second fourth, 

 and so on. When the rearrangement was completed it 

 was performed again, and so on repeatedly. Finally, i> 

 was found that, after a certain number of rearrange- 

 ments, the origin.'U order of the letters was obtained. 



For instance, the set of seven letters 

 a b i d e f ,.■• 

 gives 



•- fi f h e c d 

 d ,;■ V a e f h 

 h d f ,^ e V a 

 a h ( d e / g 



Dr. Baker showed that when there are n letters the 

 number of rearrangements required to reobtain the 

 original order is the least number r, such that one of 

 the two numbers a*"— i, 2'+i, is divisible by 2n-t-i. 



Dr. Baker also read a paper on the trisection of elliptic 

 functions, in which the problem was discussed in connec- 

 tion with the theory of the quartic equation. 



Lieut. -Colonel Allan Cunningham read two notes of 

 .great interest on the theory of numbers, one on the 

 factorisation of (2"-f-i), and the second on the question 

 whether {2''— 2) is divisible by p- ip a prime). Upon 

 these Dr. Baker made the following remarks : — 



We are often told that the problems of the physicist are 

 set to him by nature itself, but the problems of the 

 mathematician are invented by himself, and therefore 

 w-orthy of less attention. Those to whom this seems a 

 sound criticism will probably admit that the puzzling 

 problems of integral numbers are put to us from without. 

 We sliould therefore regard the theory of numbers with 

 especial concern, quite apart from its own extraordinary 

 interest. In Germany at the present time great progress 

 is being made in the subject ; it touches our reputation as 

 English-speaking mathematicians to encourage, so far as 

 we can, a similar interest in the theory of numbers here. 



Prof. A. W. Conway read a paper on the convergence 

 of a certain series used in electron theory. The series 

 was one obtained by means of Lagrange's expansion. 



Dr. J. W. Nicholson read a paper on some problems 

 of initial motion of electrified spheres, in which he re- 

 ferred to the work of G. W. W'alker and Prof. Conway. 

 Starting with an electron having a small Newtonian 

 mass, it was shown that difficulties are met with when 

 this mass is reduced to zero ; it appears impossible to 

 ascribe an initial .-icceleration to a conducting sphere with- 

 out introducing imperfection in the conductivity, although 

 the electrical distribution on the sphere tends to become 

 uniform very rapidly. These results have a bearing on 

 a possible conception of the electron. These difficulties 

 are absent from the corresponding problem for an insu- 

 lating sphere. 



Dr. Duncan M. \". Somerville pointed out the need of 

 a_ non-F.uclidean bibliography. It is now thirtv years ago 

 since Halsted published the first bibliography' of non- 

 NO. 2138, VOL. 84] 



Euclidean geometry, and one still finds it referred to as 

 a standard work. In the discussion Prof. Love suggested 

 that a report on the subject would be more valuable. 

 (The general committee has since appointed a small com- 

 mittee to consider this question and draw up a report if 

 considered advisable.) 



Mr. H. Bateman read a paper on the present state of 

 the theory of integral equations, in which he sketched the 

 history of the subject and indicated some of the physical 

 applications. Prof. Conway and Prof. Webster remarked 

 that in the problems they had tried it was very difficult 

 to get a simple solution by means of integral equations. 

 In answer to this. Dr. Hobson pointed out that in the 

 problems referred to the theory of integral equations fails 

 to give a simple solution because a simple form of the 

 solution does not exist ; but by studying the theory we 

 can hope to obtain some idea of the form and behaviour 

 of the solution, although the analytical expression for it 

 is not suitable for calculation. This important, exhaustive 

 report of Mr. Bateman 's has been ordered to be printed 

 in cxtenso. 



Mr. Bateman also read a paper on the foci of a circle 

 in space and some geometrical theorems connected there- 

 with. Special attention was paid to twisted polygons 

 formed of isotropic lines. 



Prof. J. C. F'ields read a paper on the theory o- ideals. 

 .Starting from Hensel's power-series, he defined adjoint- 

 ness relative to a prime p in a. manner analogous to that 

 in which he has defined the property in connection with 

 the algebraic functions. If € is the solution of an algebraic 

 equation, it was shown that we can construct a rational 

 function R(t) possessing any assigned set of adjoint orders 

 of coincidence corresponding to a prime p. It is deduced 

 that we can construct a general function R(e) which 

 represents only integral algebraic numbers, and possesses 

 a single coincidence with the branches of an assigned 

 one of the cycles corresponding to any prime, while 

 it is not conditioned with regard to any other 

 specific prime or any other cycle corresponding to the 

 prime in question. The aggregate of numbers so repre- 

 sented is a prime ideal. 



The report of the committee on the further tabulation 

 of Bessel functions was taken as read. This committee 

 is proceeding to calculate the functions I (.v) and K„(;r'i. 

 Its scope has been extended so as to empower it to proceed 

 to the calculation of any necessary functions. 



Meanwhile, a joint meeting of the phvsics department 

 and Section B (Chemistry) was being held. The proceed- 

 ings of this meeting will be in part reported by the 

 chemical section. Two papers only will be dealt with 

 here. A. paper was re.ad by Mr. J. A. Crowther on the 

 number of electrons in the atom. From the mean scatter- 

 ing of $ particles in passing through a substance, it is 

 deduced that the number in question is three times the 

 atomic weight, provided that the positive electricity in 

 the atom has a volume comparable with the atom itself. 

 The substances considered are carbon, aluminium, copper, 

 silver, and platinum. The numbers obtained, if the posi- 

 tive electricitv be assumed to be divided into small particles 

 comparable in size with the negative, are not propor- 

 tional to the atomic weight — a result which would be in 

 conflict with experiments on the scattering of Rontgen 

 rays — and it is thence concluded that this alternative hypo- 

 thesis is not correct. 



The second paper was by Dr. R. D. Kleeman, on the 

 attractive constant of a molecule of a compound and its 

 chemical oroperties. Making use of previous deductions 

 from surface tension and latent heat data. Dr. Kleeman 

 shows that the various chemical compounds can be divided 

 into groups, and it is found th.it this grouping corresponds 

 with that obtained from purely chemical considerations ; 

 for example, the amines fall into three groups. The 

 property speciallv studied is the ratio T,,/2\/(.A), where 

 T,. is the critical temperature and the denominator is the 

 sum of the square roots of the atomic weights of the 

 components of the compound. 



The proceedings on Monday, September ?, began with 

 a demonstration by Dr. H. J. S. Sand of vacuum-tight 

 seals between iron and gla.ss. An iron wire is sealed into 

 a glass tube. While the glass is hot a small piece of 

 heated steel tube surrounding the wire is pushed a few 

 millimetres into the glass. .After cooling, the tube is 



