82 



NATURE 



[May 28, 1891 



Such a comparison I have endeavoured to make, or rather to 

 indicate the basis on which it may be made, so far as systems of 

 geometrical algebra are concerned. As a contribution to analysis 

 in general, I suppose that there is no question that Grassmann's 

 system is of indefinitely greater extension, having no limitation 

 to any particular number of dimensions. 



J. WiLLARD GiBBS. 



The Flying to Pieces of a Whirling Ring. 

 In Nature of May 14 (p. 31) I notice a letter by Mr. C. A. 

 Carus-Wilson on the rotation of a hollow steel flask, composed 

 apparently of a spherical shell mounted on an axis constituting a 

 diameter. Mr. Carus-Wilson speaks of this body as being 

 under a " tension " of " 31 '5 tons per square inch " at a certain 

 speed of rotation. He does not, however, specify what is the 

 tension to which he refers, nor where it is found, neither does 

 he give the density and elastic constants of the material nor 

 indicate the method by which he arrived at his result. 



So far as I know, the only problem of the kind which has 

 yet been solved is that of an isotropic spherical shell ^ rotating 

 about an imaginary axis through its centre at speeds at which 

 the strains follow Hooke's law. This differs from the case Mr. 

 Carus-Wilson speaks of, inasmuch as the existence of a real 

 material axis must introduce conditions somewhat different from 

 those assumed by the mathematical theory, and further the 

 results obtained by this theory cannot legitimately be applied to 

 speeds exceeding that where bulging becomes sensible, if indeed 

 so far. 



This solution is probably, however, the nearest to the 

 practical problem at present attainable. 



According to it the strains and stresses vary throughout the 

 shell with the distance from the centre, and the angular distance 

 from the axis of rotation. They also depend on the density and 

 on the elastic properties of the material. There are also at 

 every point three principal stresses, whereof one it is true 

 vanishes over the surfaces. Thus such a statement as Mr. 

 Carus- Wilson's requires further explanation. 



According to the two theories most commonly entertained, the 

 quantity which determines the limiting safe speed is the maxi- 

 mum value of either the greatest strain or the maximum stress- 

 difference, — i.e. the algebraical difference between the greatest 

 and least principal stresses at a point. Over the surfaces of 

 the shell the absolutely greatest values of both these quantities 

 are found, for shells of all degrees of thickness, in the equatorial 

 plane — or plane through the centre perpendicular to the axis 

 of rotation. 



Denoting the angular velocity by w, the radii of the outer and 

 inner surfaces respectively by a and a', the density by p. Young's 

 niodulus by E, the greatest strain by s, the maximum stress- 

 difference by S, and the stress at right angles to the meridian 

 plane by *, the three last quantities being measured in the 

 equator, the following are some of the results I found for 

 materials in which Poisson's ratio is 1/4 : — 



Inner Outer Inner Outer 

 surface, surface, surface, surface. 



<P/OJ-prt- 



Inner Outer 

 surface, surface. 



a'/ a = 0*9 



— ^^-^ negligible 



0-950 0-833 



■064 o'866 o'9i2 o'866 



Apparently in the case mentioned by Mr. Carus-Wilson, 

 «'/«= 15/16 = 0-9375, Supposing the material to have Poisson's 

 ratio = 1/4, which seems to accord fairly with experiments on 

 steel, the approximate values of j, S, and *, for this case could 

 be obtained by interpolation from those I give above. The dif- 

 ferences between the values of corresponding strains and stresses 

 at the two surfaces are less, of course, for a'/a = 15/16 than for 

 a' la = 0-9, but still are far from negligible. Mr. Carus- Wilson's 

 numerical result rather suggests that the tension he refers to is 

 the stress *, measured as above in the equator, and that he 

 employed the formula * = w-pa^. This formula (see Cambridge 

 Philosophical Transactions, vol. xiv. p. 300), is correct for 

 the value of * in the equator in an infinitely thin shell, but it 

 does not strictly apply to any shell whose thickness is comparable 

 with its radius. In the paper in the Cambridge Transactions 

 first referred to, there are given tables of the numerical measures 

 of the strains and stresses over the surfaces for a series of values 



' Cambridge Philosophical Society's Transactions, vol. xiv. pp. 467-483/ 

 NO. II 26, VOL. 44] 



of a'ja for materials in which Poisson's ratio is 1/4. 'ihese give 

 by interpolation fairly accurate values for all values of a'/a. 

 For other values of Poisson's ratio, recourse must be had to the 

 general formulae given in the paper, unless c, ~ I - a'ja, is very 

 small, when the greatest values of s and S are given approxi- 

 mately by Ej/w-pa^ = I _ jg(i _ ^)^ S/wV" = » + ^/(l + '»). 

 where r/ is Poisson's ratio (see Camb. Trans. , vol. xiv. p. 304). 

 May 16. C. Chree. 



A Comet observed from Sunrise to Noon. 



A SHORT time ago I got the loan of an old number of Harper's 

 Monthly (March 1889), good reading matter being very accept- 

 able, however old, in this outlandish place, in which I read an 

 article, on the origin of celestial species, by J. Norman Lockyer, 

 F. R. S., Cor. Inst. France, that set me thinking of what I 

 observed of the great comet of 1882, when it made its tremen- 

 dous plunge round the sun, on September 18. At that 

 time I was master of a small vessel, trading in the Society 

 Islands; and on the day mentioned — in latitude 16° 25' S., 

 longitude 151° 57' W. of Greenwich, a position about midway 

 between the two islands Bolabola and Maupiti (the Maurua of 

 Cook)— I saw, with the naked eye, the comet travel about 90° of 

 the circle of the sun's disk, between sunrise and noon ; but what 

 made it most remarkable to us was that it should be possible 

 for us, in a perfectly clear sky, to be able to watch it all, from 

 sunrise to noon, with very little more distress to the eye than if 

 in a clear night looking at a full moon. 



Now, Sir, may it not be that this is partly a proof of the 

 theory set forth by Norman Lockyer in the article above men- 

 tioned, viz. that comets are swarms of meteorites in collision, 

 travelling through space, and that the outer invisible part of 

 the swarm that formed this comet's nucleus had partially eclipsed 

 the sun, like a veil over it ? I am not aware if it was noticed 

 by any competent astronomer or not, but the chances are that 

 none had the splendid opportunity that we had to see the 

 phenomena ; so. Sir, knowing that men of science are always 

 glad to get facts from observers in all parts of the world is my 

 excuse for writing this to you, not knowing Mr. Lockyer's 

 address. Thinking this, although late, may probably be of some 

 interest to the scientific world, I leave you to do what you may 

 think proper with it. Wm. Ellacott. 



Raiatea, January 30. 



Graphic Daily Record of the Magnetic Declination or' 

 Variation of the Compass at Washington. 



I BEG to call your attention to the enclosed reprint from the 

 May Pilot Chart of curves of magnetic declination as recorded 

 at the United States Naval Observatory at Washington. This 

 reprint admits of reproduction more readily than the curves as 

 shown on the Pilot Chart, being in black and white, and only 

 reduced to two-fifths of true size (the reduction on the Pilot 

 Chart itself being one-quarter). It will be interesting to this 

 Office to elicit expressions of opinion relative to the advantages 

 of the prompt publication of these curves. The experiment is to 

 be tried for three months, but it is not likely to be continued 

 longer unless certain decided advantages develop. It may be of 

 sufficient interest to Nature to republish these curves, and thus 

 assist us in giving them wide publicity. 



Richardson Clooer, 



Washington, D.C., May 6. Hydrographer. 



[We are unable to print the curves, but we may note that 

 they are issued with the following explanation : — " 1 hese curves 

 indicate graphically the true direction in which the magnetic 

 needle at the Naval Observatory pointed during each instant 

 from noon, March 29, to noon, April 30. The base-line shows 

 a slight break at the end of each two hours, 75th meridian time, 

 and the amount of westerly variation at any time is a^ plus the 

 number of minutes represented by the height of the curve above 

 the base line at that time, measured by the scale at the right or 

 left margin of the diagram. The slight breaks in the curve 

 itself occur when the chronograph sheets are changed. Although 

 the daily change of variation at any one place, even in magnetic 

 storms such as those that have occurred during the past month, 

 is too small to be of any importance in practical navigation, yet 

 it is thought that the prompt publication of these curves cannot 

 fail to interest masters of vessels, as well as scientific men. The 

 mean daily curve, which can be drawn by taking the average of 

 many such curves, shows that there is a regular, though slight. 



