placed in a field of magnetic force whose components are a, ß, y, is -(Aa + B B+ Cy) dx dy dz. Hence, if the force urging the element to move without rotation in the direction of x is X, dx dy dz, da dB (1) and if the moment of the couple tending to turn the element about the axis of a from y towards z is L dx dy dz, The forces and the moments corresponding to the axes of y and z may be written down by making the proper substitutions. 640.] If the magnetized body carries an electric current, of which the components are u, v, w, then, by equations C, Art. 603, there will be an additional electromagnetic force whose components are X2, Y, Z2, of which X, is Hence, the total force, X, arising from the magnetism of the molecule, as well as the current passing through it, is The quantities a, b, c are the components of magnetic induction, and are related to a, ß, y, the components of magnetic force, by the equations given in Art. 400, The components of the current, u, v, w, can be expressed in terms of a, ß, y by the equations of Art. 607, Multiplying this equation, (8), by a, and dividing by 47, we may add the result to (7), and we find d dz [aa− 1 (a2 + p2 + y2)] + 1/ [ba] + 1/2 [ca]}, (9) X = d dy where X is the force referred to unit of volume in the direction of x, and I is the moment of the forces about this axis. On the Explanation of these Forces by the Hypothesis of a Medium in a State of Stress. h 641.] Let us denote a stress of any kind referred to unit of area by a symbol of the form Ph, where the first suffix, indicates that the normal to the surface on which the stress is supposed to act is parallel to the axis of h, and the second suffix,, indicates that the direction of the stress with which the part of the body on the positive side of the surface acts on the part on the negative side is parallel to the axis of k. The directions of h and k may be the same, in which case the stress is a normal stress. They may be oblique to each other, in which case the stress is an oblique stress, or they may be perpendicular to each other, in which case the stress is a tangential stress. The condition that the stresses shall not produce any tendency to rotation in the elementary portions of the body is Phk=Pkh⋅ In the case of a magnetized body, however, there is such a tendency to rotation, and therefore this condition, which holds in the ordinary theory of stress, is not fulfilled. Let us consider the effect of the stresses on the six sides of the elementary portion of the body dx dy dz, taking the origin of coordinates at its centre of gravity. On the positive face dy dz, for which the value of x isda, the forces are The forces acting on the opposite side, -X_x, — Y_z, and — Z_x, may be found from these by changing the sign of da. We may express in the same way the systems of three forces acting on each of the other faces of the element, the direction of the force being indicated by the capital letter, and the face on which it acts by the suffix. If X dadyde is the whole force parallel to ≈ acting on the element, Xdx dy dz = x+x+X+y+X+z+X_x+ X_μ+X_z9 If Ldædydz is the moment of the forces about the axis of tending to turn the element from y to z, whence Ldx dy dz=dy (Z+,—Z_) — dz (Y+2-Y_2), § z' = (P-P) dx dydz, L = P12-Pay บะ (14) Comparing the values of X and L given by equations (9) and (11) with those given by (13) and (14), we find that, if we make the force arising from a system of stress of which these are the components will be statically equivalent, in its effects on each 642.] MAGNETIC STRESS. 255 element of the body, with the forces arising from the magnetization and electric currents. 642.] The nature of the stress of which these are the components may be easily found, by making the axis of a bisect the angle between the directions of the magnetic force and the magnetic induction, and taking the axis of y in the plane of these directions, and measured towards the side of the magnetic force. If we put for the numerical value of the magnetic force, B for that of the magnetic induction, and 2e for the angle between their Hence, the state of stress may be considered as compounded of— 1 (1) A pressure equal in all directions = H2. 8 π (2) A tension along the line bisecting the angle between the directions of the magnetic force and the magnetic induction (3) A pressure along the line bisecting the exterior angle between (4) A couple tending to turn every element of the substance in the plane of the two directions from the direction of magnetic induction to the direction of magnetic force = BH sin 2 €. 1 When the magnetic induction is in the same direction as the magnetic force, as it always is in fluids and non-magnetized solids, then = 0, and making the axis of a coincide with the direction of the magnetic force, x and the tangential stresses disappear. The stress in this case is therefore a hydrostatic pressure Balong the lines of 643.] When there is no magnetization, B = H, and the stress is still further simplified, being a tension along the lines of force equal to H2, combined with a pressure in all directions at right angles 1 8 π to the lines of force, numerically equal also to The force arising from these stresses on an element of the medium where m is the density of austral magnetic matter referred to unit |