Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/372

mechanical momentum, which could be yielded up to the conductor. It is readily seen that such momentum must be directed at right angles to the tube and to the magnetic induction—a result which suggests that the momentum stored in unit volume of the aether may be proportional to the vector-product of the electric and magnetic vectors.

For this conjecture reasons of a more definite kind may be given.^[1] We have already seen^[2] that the ponderomotive forces on material bodies in the electromagnetic field may be accounted for by Maxwell's supposition that across any plane in the aether whose unit normal is N, there is a stress represented by

$\mathbf {P_{N}} =(\mathbf {D.N} )\mathbf {E} -{\tfrac {1}{2}}(\mathbf {D.E} )\mathbf {N} +{\tfrac {1}{4\pi }}(\mathbf {B.N} )\mathbf {H} -{\tfrac {1}{8\pi }}(\mathbf {B.H} )\mathbf {N}$.

So long as the field is steady (i.e. electrostatic or magnetostatic) the resultant of the stresses acting on any element of volume of the aether is zero, so that the element is in equilibrium. But when the field is variable, this is no longer the case. The resultant stress on the aether contained within a surface S is

$\textstyle \iint \mathbf {P_{N}} .dS$

integrated over the surface: transforming this into a volume- integral, the term (D.N)E gives a term div D.E + (D.∇)E, where ∇ denotes the vector operator (∂/∂x, ∂/∂y, ∂/∂z); and the first of these terms vanishes, since D is a circuital vector; the term - 1/2(D.E)N gives in the volume-integral a term 1/2 grad (D.E); and the magnetic terms give similar results. So the resultant force on unit-volume of the aether is

$(\mathbf {D.} \nabla )\mathbf {E} +{\tfrac {1}{2}}{\text{ grad}}(\mathbf {D.E} )+(1/r\pi )(\mathbf {B.} \nabla )\mathbf {E} +(1/8\pi ){\text{ grad}}(\mathbf {B.H} )$,

which may be written

$[{\text{curl }}\mathbf {E.D} ]+(1/4\pi )[{\text{curl }}\mathbf {H.B} ]$;

1. The hypothesis that the aether is a store house of mechanical momentum, which was first advanced by J. J. Thomson (Recent Researches in Elect, and Mug. (1893), p. 13), was afterwards developed by H. Poincaré, Archives Neérl. (2) v (1900), p. 252, and by M. Abraham, Gött, Nach., 1902, p. 20.
2. Cf. p. 802.