August 2, 1906] 



NA rURE 



117 



tainly it looks as if the influence of electricity in radio- 

 active change, and its importance generally in its relation 

 to matter, could be overestimated. 



Frederick Soddy. 

 The L'niversity, Glasgow, July 29. 



Stress in Magnetised Iron. 



TiiK important question whether there is any mechanical 

 stress in an iron rod or ring when magnetised, and, if so, 

 whether the stress is compressive or tensile, was discussed 

 in Nature ten years ago (vol. liii., pp. 269, 316, 365, 462, 

 533), but has not yet, so far as I know, received any 

 generally accepted answer. That a magnetised rod must 

 necessarily be in the same condition as if under a 

 mechanically applied compressive stress tending to shorten 

 the iron, was, I believe, first suggested by myself (Phil. 

 ■J'rans., vol. cl.xxix., p. 216, 18S8). Those who support this 

 view generally speak of the stress as " Maxwell's stress," 

 and assume its value to be B'/Stt. The stress in question 

 seems, however, to be quite unconnected with the " stress 

 in the medium " proposed by Maxwell, and its value is not 

 in general exactly B-/Sir, but (B=-H-)/8:r. I have lately 

 had occasion to consider the problem again, and perhaps 

 I may be allowed to re-state my argument in a slightly 

 .Utered form, and illustrate it by means of an imaginary 

 model. 



If a uniformly magnetised rod is divided transversely, 

 and the cut faces are brought close together, the magnetic 

 force inside the narrow gap will be B = H-i-45rI. The 

 force acting on the magnetism of one of the faces, and 

 urging this face towards the other, will be less than 

 B by 27rl, the part of the total force due to the first face 

 itself ; hence the force per unit of area with which the 

 faces would press against each other if in contact is 

 P = (B-2irI)I = 27rF-|-HI = (B=-H=)/8ir. (In the case of 

 an endless permanent magnet, H=o, and P = B''/87r.) The 

 width of the gap may be diminished until it is no greater 

 than the distance between two neighbouring molecules, 

 when it will cease to be distinguishable ; but, assuming 

 the molecular theory of magnetism to be true, the above 

 statement will still hold good for the intermolecular gap. 

 The same pressure P will be exerted across any imaginary 

 section of a magnetised rod, the stress being sustained by 

 the intermolecular springs, whatever their physical nature 

 may be, to which the elasticity of the metal is due. The 

 whole of the rod, therefore, will be subject to a com- 

 pressive longitudinal stress P, the resulting contraction, 

 expressed as a fraction of the original length, being P/M, 

 where M is Young's modulus for the metal. 



Let a magnetic molecule of iron be represented by a 

 rigid steel sphere, uniformly magnetised and covered with 

 a closely fitting shell of india-rubber, to play the part of 

 the " intermolecular springs." Imagine a straight row of 

 these spheres in contact with one another, and kept in 

 place by a force analogous to cohesion, which, while bind- 

 ing the spheres together, leaves them free to turn on their 

 centres. This arrangement would, for present purposes, 

 serve as a model of a filament of iron one molecule in 

 diameter. If the magnetic axes of the spheres pointed 

 indifferently in all directions, the attractions would be 

 balanced by the repulsions, and the length of the filament 

 would be the same as if the spheres were unmagnetised. 

 If, however, the magnetic axis of every sphere pointed in 

 the same direction aWng the filament, as would be the case 

 when the filament was magnetised, the india-rubber be- 

 tween all the pairs of unlike poles would be compressed 

 and the filament would be shortened. Let F be the com- 

 pressive stress across the rubber between a single pair of 

 poles, and .! the amount, expressed as a fraction of a centi- 

 metre, by which the rubber is contracted ; then, if there are 

 rt spheres, the total contraction will be ns (n being 

 assumed so great that it is sensibly equal to n + i), which 

 is the same as would be caused by an equal compressive 

 stress F applied at the two ends of the unmagnetised 

 filament. The whole filament when magnetised mav there- 

 fore be regarded as under compressive stress due to the 

 magnetic forces, and since Young's modulus M = F/'tis, 

 where I is the length of the unmagnetised filament, the 

 contraction expressed as a fraction of the length is, as 



NO. I918, VOL. 74] 



originally stated, F/M, the value of F in an actual piece 

 of iron being 2irI--|-IU. 



Sometimes there may presumably also be a longitudinal 

 tension, as in the case of an iron rod placed along the 

 lines of force in a uniform field, when the tension would 

 be HI. In a ring electromagnet this would not exist. 



As to what effect would be produced in magnetised iron 

 by Maxwell's distribution of stress in the ether, I cannot 

 venture an opinion. But if there is a tension, it can hardly 

 have the familiar value B°/87r, which is possible only 

 when B is equal to II, and there is no magnetisation 

 (" Electricity and Magnetism," § 643). My point is that 

 an important component of the stress in magnetised iron 

 is a compression which can be calculated and allowed for. 

 The question whether or not this view is tenable is of the 

 highest interest in connection with the possible correlation 

 of magnetic phenomena, and urgently needs an answer. 

 Shelford Bidwell. 



The Mixed Transformation of Lagrange's Equations. 



I SHOULD fancy from the review by " G. H. B." in 

 Nature of July 19 (p. 265) that the papers of Prof. Levi 

 Civita relate largely to the mixed transformation of 

 Lagrange's equations, the complete theory (Proc. Camb. 

 Phil. Soc, vol. vi., p. 117; "Hydrodynamics," vol. i., 

 p. 171) of which was first given by myself so far back 

 as 1887. But what I wish to point out is this, that this 

 theory depends no more on any so-called theory of 

 " ignored " coordinates (or kinosthenic coordinates as 

 Prof. J. J. Thomson [Phil. Trans., 1885, part ii.] calls 

 them) than it does on the existence of the hypothetical 

 personage known as the Man in the Moon. 



The theory is merely the result of a piece of elimin- 

 ation, and is as follows : — Let the coordinates of a 

 dynamical system bo divided into two groups B and x > 

 let and k be the momenta of types and x ; and let 

 T be the Lagrangean expression for the kinetic energy. 

 Then it can be shown that 



T = 3:-l-9{ (I) 



^.=« (2) 



M- '3. 



where !J^i is a homogeneous quadratic function of the 

 velocities 9', 3J is a similar function of the momenta k, and 

 is a linear function of the k's. 



By means of (2) all the velocities and accelerations of 

 typfi Jf can be eliminated from Lagrange's equations, and 

 the result is expressed by means of the modified Lagrangean 

 function 



L = S-F2(0tf)-SK-V (5> 



and 



. m ^/■d&\ ,,, 



^=a^-n«aj <" 



Equations (5) and (6) constitute the mixed transform- 

 ation of Lagrange's equations, and include the equations 

 of Hamilton as well as those of Lagrange. 



When the coordinates x at'e kinosthenic coordinates, 

 that is to say, coordinates which enter into expression for 

 the energy of the system only through their differential 

 coefficients with respect to the time, all the k 's are con- 

 stants, and (5) is sufficient to determine the motion. 



In § '73 of niy " Hydrodynamics," the words " the latter 

 of which does not enter into the expression for the energy 

 of the system " should be omitted. 



A. B. Basset. 



Two Modifications of the Quartz Wedge. 

 Some little time ago I wished to make a quartz wedge 

 for producing interference colours with the polarising 

 microscope. The usual wedge supplied by optical instru- 

 ment makers seldom gives colours lower than " clearer 

 gray " of Newton's colour-scale according to Quincke, 

 while the lower colours are often particularly valuable in 

 petrological work. The quartz wedge is described in the 



