May 24, 1894] 



NATURE 



i possession of a real invariant sub-group. This remark obviously 

 refers to translations, and in fact it appears to have been pre- 

 viously noticed that in the elliptic and hyperbolic geometries, 

 the transformations that correspond to translations do not form 

 a group. In the present communication a namber of repre- 

 -entations of elliptic and hyperbolic geometry are described and 

 illustrated with the object of making this kinemalicai distinction 

 between the Euclidean and the other geometries intuitively 

 obvious. — Permutations on a regular polygon, by Major P. A. 

 MacMahon, F.R.S. — The stability ol a tube, by Prof. Green- 

 hill (IJr. J. Larmor, F. R.S., pro lem. in the chair). The 

 difficulties of constructing a theory for the stability of a tube, 

 subject to external pressure and end thru-t, have been discussed 

 by Mr. A. B. lUsset in the Phil. Ma:;. September 1S92. 

 Similar investigations have been undertaken by Mr. Love and 

 Mr. Bryan in the Proceeiiings of the London Math. Society. 

 The analytical difficulties due to the difference of pressure on 

 the two sides of the plate, have not yet been overcome, so that 

 the investigation of the present paper must be taken as pro- 



jvisional, as it proceeds on the old theory, as laid down in 



iThomson and Tail's " Natural Philosophy." The chief 



jobject is to determine the number of segments or waves into 

 which the cross section of the tube will tend to break, as the 

 supporting influence of the ends is made to operate at sections 



■which are brought closer and closer together ; the influence of 

 the end thrust is also taken into account. A differential equa- 



|tion is obtained for;;', the infinitesimal normal displacement of 



ithe tube, of the form 



ld_^ 



U.V 



■V 2 



d-w 

 Jx-dy' 



+ 2 



d-w 

 d'd^ 



a/> 



dx' 



j^dhv_ 

 dy* 



./.<- V dy- 



S) 



;)"■ 



(A) 



'where v is measured parallel to the axis of the tube, and j cir- 

 jcumferentially ; a denotes the radius of the tube, b its thickness, 

 A the flexural rigidity, a- Poisson's ratio, X the longitudinal 

 ,thrust in the tube per unit length of cross section, and Z the 

 iexternal applied pressure ; the inch and pound are taken as 

 lanits of length, so that the theoretical results may be compared 

 limmediately with experimental values ; to do this it is assumed 

 larovisionally that we may put A = ,',, Mi^!(l - a-), where M 

 lenotes Young's modules of elasticity. If the tube breaks 

 :ircumferentially into n waves, we put 



d-cv _ __ n'-w d*'ui _ «';(/ _ 

 dy- a- ' dy* a* ' 



ind equation (A) becomes 



a-dx- a* 



dx'' 



I ^ Xa2\ d"-iv , „ 

 \ A / rt-</.i- 



A a* 



. (B) 



For cylindrical collapse, when the supporting influence of the 



nds is left out of account, — ^is zero, 

 dx' 



Za3 

 A 



I, «Z = 



and therefore 

 M 





But if the ends of the tube are supported or strengthened, the 

 ollapsing pressure is obviously increased, so that 



Za' 

 A 



(«= - I) 



. . (C) 



n practice X is proportional to Z, when it is not zero ; and to 

 elermine the number « of segments into which the tube 



i positive. If the supporting influence is due to a series of 

 quidiitant strengthening rings as is a caisson, / inches apart, 

 'reserving accurately the circular form at the corresponding 

 eclion, while permitting slight changes of direction in the 

 angitudinal seams, we put 



collapses, we may put Zo'/A = y, and (ira//,- = x, and draw 

 the hyperbolas represented by (C) for values of k = i, 2, 3, ... ; 

 and the points of crossing of these hyperbolas will rep-esent the 

 separating states when an integral change in « is about to lake 

 place. The case of « ^ i would only occur when the tube was 

 used as a long cylindrical column, on the point of buckling 

 sideways, without crippling ; we now find that the formula 



assigns a critical thrust which is only ~{/>/a)- of that given by 



3 

 the usual theory, due to Euler. — -Researches in the calculus of 

 variations, Part v., the discrimination of maxima and minima 

 values of integrals with arbitrary values of the limiting varia- 

 tion? ; Part vi,, the theory of discontinuous or compounded 

 solutions, by Mr. E. P. Culverwell. 



Physical Society, May 11. — Walter Baily, Vice-President, 

 in the chair. — .\ mathematical communication on electro- 

 magnetic induction in plane, cylindrical, and spherical current 

 sheels and its represenlalion by moving trails of images, by 

 G. H. Bryan (part I, general equations), was read by Dr. C. 

 V. Burton, who also explained some of the parts in greater 

 detail, .\fter mentioning that the magnetic field due to induced 

 currents in thin conducting sheets placed near moving magnetic 

 poles could be represented by moving iraiK of images of those 

 poles, the author goes on to say that in the paper, the surface- 

 conditions which hold at the surfaces of the sheets are deduced 

 directly from the fundamental laws of electromagnetic induction, 

 (i) The total current across any enclosed portion of a surface 

 which always contains the same particles is equal to i 47r of the 

 line-integral of the magnetic force round the curve bounding 

 the surface ; and (2) the rate of decrease of the surface integral 

 of magnetic induction across any enclosed surface which always 

 contains the same panicles to equal to th"! lins-integral of electro- 

 motive force round the curve bounding the surface. By working 

 wiih the scalar magnetic potential instead of vector-potential, 

 the investigation is sim|ilified. In addition to the above laws, 

 the author makes the usual assumptions that di-placement cur- 

 rents in the dielectric aie so small as to be negligible, and that 

 the induced currents are distributed uniformly through the thick- 

 ness of the sheet. On thes^ suppositioni the surface c in- 

 diiions satisfied by the potentials at the two sides of plane, 

 cylindrical, or spherical sheets are determined, and with an addi- 

 tional limitation as to the thickness of the sheet fulfilling certain 

 conditions, extended to current sheets of other forms. In the 

 latter part of the paper a synthetic determination of the images 

 in a plane sheet is given and expressed in the form of a definite 

 integral. In reading the paper to the meeting Dr. Burton 

 pointed out several misprints in the proof. — Prof. Minchin 

 showed that equation (i) of the paper (n2-^i= 4 ir <() -f con- 

 stant, where H., and n, are the m.agnetic potentials at the two 

 sides of the sheet, and (p the current function), could be de- 

 duced by purely mathematical reasoning instead of being based 

 on the laws of electromagnetic induction. Moreover, it was 

 true for any function whatever and did not depend on <l> being 



the current function. Equation (2) 



NO. 1282, VOL. 50] 



V </--' dz ) 



— J- ) followed 



immediately from the fact that the magnetic force was continuous. 

 The latter part of the paper might be simplified by integrating 

 the linear partial differential equation (15) 



\dzdt dz' dzdt J 



in the ordinary way, for the form was one for which the auxi- 

 liary equations are well known. Dr. Burton, in reply, said he ' 

 thought Mr. Bryan's reason for developing the equations from 

 the laws of electromagnetic induction was to give his work a 

 physical rather than a mathematical basis. — .\ paper on di- 

 electrics was read by Mr. Hollo Appleyard. In tes'ing the 

 insulation resistance of celluloid, by having a sheet pressed 

 between two metal plates, the author noticed that the resistance, 

 which was very high, decreasui as the time the testing battery 

 was left on increased. The "electrification" (using the 

 word to indicate the rate of diminution of galvano- 

 meter deflection) was therefore negiilive. The resistance 

 also diminished greatly with increas; of battery power, 

 and a considerable amount of hysteresis was observed, the 

 resistance at any given voltage, after a minute's electrification, 

 depending on the previous history of the specimen. On making 

 contact with the surfaces of the celluloid by mercury instead of 

 by solid metal, the abnormal results disappeared, little or no 

 resistance- hysteresis or "electrification" being present, and 



