THE ELECTRIC AHD LUMINIFEROUS MEDIUM. 
299 
extraneous tensile force of intensity Q per unit cross-section : let M denote the 
modulus of elastic extension of its material, which will be an even function of 1 . 
'.fhe mechanical work expended on the wire in a slight alteration of its circumstances 
is })er unit volume 
§W = (M-i 8 Q + 8 r;) Q + H SI = + Q d-qlciq) SQ + (H + Q drildl) SI, 
where 17 is its intrinsic magnetic elongation when magnetized to intensity I under 
tension Q, this magnetization practically not altering the extraneous field in the case 
of a wire. To avoid perpetual motions, 8 W must in the absence of hysteresis be a 
perfect differential of the independent variables I and Q: hence = 
(dH/dQ)^ + drjidl. Here I is a function of H and Q, so that to determine dH/dQ 
when I is constant we have (dI/c/H)Q + (dI/dQ)H (dQ/dH)i = 0 , the subscript 
denoting the variable that is constant in the differentiation. Thus on substitution 
dif]ldl = (dl/dQ)H/(dI/dH)Q + Q the total expansion is rj' = yj Q/M ; 
so that on writing as usual k for (dI/dH)Q, we have d-q’/'dl = — I (d/c~7f^Q)n + 
2 Q d'M~^ldl. This is the exact equation which should be directly satisfied by series 
of observations of 17 ', I, and M, formed with different constant values of H and Q, 
provided hysteresis is negligible. As 77' must be an even function of I, it follows that 
when I and Q are small, y] — — (dK:“7<^^Q)n, or J-I~ (d«:“\/dQ)i, as in § 67. 
For the case of a ring magnetized by a coil, there can be no free polarity except 
at an air-gap ; tlius there is no stress of magnetic origin in the material. The 
alterations of longitudinal and transverse dimensions of rings of iron and iiickeH 
are thus wholly intrinsic changes due to the magnetic polarity and in no part due 
to mechanical stiess such as Q. In the neighbourhood of the origin, where f] is 
proportional to I“, the curves given by Bidwell expressing the relation between 
y] and I should be parabolic, as in fact they are. At the magnetization corre¬ 
sponding to a maximum or minimum ordinate t] of the curve, the effect of a 
very small imposed tension on the magnetization should change sign, being null for 
that particular magnetization ; the summit of the curve is therefore the Villaei 
critical point. But if there is a tension Q so considerable that change of the elastic 
modulus by magnetization contributes appreciably to the elongation, the Villaei 
})oint will be displaced from the summit of the curve, backwards when magnetization 
increases the modulus. It appears from the experiments of BiowELLt that for iron 
tension increases the intrinsic elongation, for nickel it at first increases then 
diminishes and finally for stronger fields increases it, while for cobalt there is no 
sensible effect. 
84. Mechaiiiccd Stress in a Polarized Solid Sphere .—The mechanical stress 
sustained by a sphere of soft iron situated in a uniform magnetic field H can be 
simply expressed. The well-known analysis of Poissox gives a constant field 
* Shelfokd Bidvell, ‘PLil. Trans.,’ A, 1888, p. 228 ; ‘ Roy, Soc. Proc., 1894. 
t ‘ Roy. Soc. Pioc.,’ vol. 47, 1890, p. 480. 
2 Q 2 
