5k« 



NA TURE 



[April 3, 1890 



product of the principal radii of curvature of the 

 surface at any point remains constant) that a diminished 

 curvature along the axis will be accompanied by a nearer 

 approach to a circular section, and reciprocally. Since a 

 circular form has the largest area for a given perimeter, 

 internal pressure tends to diminish the eccentricity of the 

 elliptic section, and with it the general curvature of the 

 tube. Thus, if one end be fixed, a pointer connected with 

 the free end may be made to indicate the internal pres- 

 sure." Lord Rayleigh adds, " It appears, however, that 

 the bending of a curved tube of elliptical action cannot 

 be pure {i.e. unaccompanied by stretching), since the 

 parts of the walls which lie furthest from the circular 

 axis are necessarily stretched. The difficulty thus 

 arising may be obviated by replacing the two halves 

 of the ellipse, which lie on either side of the major axis, 

 by two symmetrical curves which meet on the major axis 

 at 2i finite angle ^^ 



In fact some Bourdon gauges, notably those required 

 for low pressures only, and requiring great sensibility 

 but not much strength, are constructed in this manner, 

 and the difficulty of manufacture is thereby considerably 

 reduced. Barometers are constructed in this way, and 

 give good results ; the tube is partially exhausted of 

 air, and closed at both ends ; and now an increase of 

 external atmospheric pressure tends to flatten, and thus 

 curl up the tube. 



In constructing any theory, we are then immediately 

 brought up by the great difficulty at present engaging the 

 attention of our mathematical elasticians, such as Ray- 

 leigh, Basset, Pearson, and Love ; who are not agreed as 

 to how far it is legitimate to theorize on the equilibrium 

 of elastic shells, by treating separately the bending and 

 the stretching as independent of each other, and con- 

 sidering the first — the bending — of the most importance. 

 If we take a piece of thin sheet metal in our hands, we 

 find we can bend it with comparative ease, but any 

 stretching we can produce is quite insensible ; and it is 

 thence argued that bending only is likely to take place, 

 as so easily produced ; and apparently reversing the 

 ordinary mathematical procedure, the large stresses due 

 to any stretching are neglected, as not likely to be in 

 existence. These difficulties confront us in any attempt 

 at a rigorous theory of the instrument, which would give 

 quantitative results, enabling us to graduate the instru- 

 ment from a formula. 



The Rev. E. Hill has given in the Messenger of Mathe- 

 matics, vol. i., 1872, an explanation of the Bourdon 

 metallic barometer, treating the question as one of pure 

 bending, and giving a quantitative formula for the change 

 of .curvature a of the total curvature Q in terms of the 

 change .i" in the semi-minor axis b, viz. alQ = xjb. But 



wall will cause this wall to elongate ; and thus an increase 

 of internal pressure would cause the tube to curl up, the 

 opposite effect to what happens when the bending effect 

 due to the outward bulging of the flat walls is considered 

 the leading phenomenon. 



Even with a circular cross-section the stretching 

 hypothesis would prove that the tube curls up under 

 internal pressure ; but this effect would be so small 

 as to be imperceptible, because of the enormously 

 greater stresses required for stretching than for bending 

 in a thin tube ; and this is found to be practically the 

 case, inasmuch as the circular cross-section of the tube 

 destroys all indications ; and further, that che indications 

 of the tube are reversed in direction when the axes of the 

 elliptical cross-section are interchanged so that the minor 

 axis is perpendicular to the plane of the circular axis of 

 the tube. 



The action of Bourdon's gauge is a differential effect ;. 

 the bending of the surface changes the curvature one 

 way, and the stretching produced by the same pressure 

 the other way ; but the bending effect is so much greater 

 than that of stretching, that the latter may be left out of 

 account. 



In Gunnery we have, in a similar manner, two ant- 

 agonistic causes producing a tendency for an elongated 

 rifled projectile to deviate from a vertical plane of motion. 

 If fired from a gun rifled with a right-handed screw, the 

 vortex set up in the air by the spinning of the projectile 

 causes difterences of pressure, tending to deviate the 

 projectile to the left, and this effect is sometimes very 

 noticeable with golf or tennis balls ; but, in addition, the 

 forces set up by the tendency of the projectile to fly with 

 its axis in the tangent of the trajectory urge the projectile 

 to the right, and these latter forces are found to prepon- 

 derate in practice. 



A mathematician might be tempted to apply to the 

 problem of Bourdon's gauge the formulas on the equi- 

 librium of elastic plates and their change of curvature, 

 anticlastic and synclastic, which are given in Thomson 

 and Tail's " Natural Philosophy" (§§ 711-720), but these 

 formulas apply only to a plate originally plane ; and, 

 besides, the applied pressures of the liquid complicate 

 the analysis of the question to an extent which has not 

 yet been overcome by elasticians. 



The final conclusion would thus appear to be, that any 

 quantitative formula cannot be hoped for yet, for a long 

 time ; but that Lord Rayleigh's reasoning, quoted above, 

 gives a clear and concise descriptive explanation of the 

 action. 



The analogous practical problem of the resistance of 

 flues to collapse still stands in need of a rational theory, 

 when the supporting influence of the ends or of collapse 



the determination of xjb for a given change of pressure rings is taken into account. When this question has 



is as yet an intractable mathematical problem, even for 

 the simplification of supposing the tube a straight elliptic 

 cylinder. 



When we attempt to determine mathematically the 

 pure bending produced in an elliptic cylinder by an 

 increase of internal pressure and consequent tendency of 

 the cross-section to the circular form, we are baffled by 

 the analytical difficulties of determining the change in 

 the length of the axes of the section, subject to the con- 

 dition of keeping the peiimeter unchanged in length, 

 this length being expressed by a complete elliptic integral 

 of the second kind, of which the modulus is the eccen- 

 tricity of the ellipse. This problem was mentioned by 

 Sir W. Thomson at the British Association in 1888 ; but 

 we have not yet seen any development of it published by 

 him. 



Mr. Worthington, on the other hand, treats the ques- 

 tion from the point of view of pure stretching ; and now, 

 with rectangular cross-section of the tube, as he supposes, 

 a thrust in the inner wall due to the internal pressure will 

 cause this wall to contract, while the pull in the outer 



received satisfactory treatment at the hands of theorists, 

 we may hope to pass on to the far more difficult quanti- 

 tative theory of Bourdon's gauge. 



A. G. Greeihhill. 



NOTES. 

 The half-yearly general meeting of the Scottish Meteoro- 

 logical Society was held in the hall of the Royal Scottish 

 Society of Arts, Edinburgh, on Monday afternoon. The follow 

 ing papers were read : — Influenza and weather, with special 

 reference to the recent epidemic, by Sir Arthur Mitchell and 

 Dr. Buchan ; the temperature of the high and low-level Ob- 

 servatories of Ben Nevis, by T. Omond, Superintendent; 

 thunderstorms at the Ben Nevis Observatory, by R. C. 

 Mossmann. In the last Report presented by the Council, refer- 

 ence was made to a proposed systematic observation of the 

 numbers of dust-particles in the atmosphere with the instrument 

 recently invented by Mr. John Aitken, and an opinion was 



