Part III — Prop. XII

On Physical Lines of Force → Part III → Prop. XII

Maxwell's statement

Prop. XII. — To find the conditions of equilibrium of an elastic sphere whose surface is exposed to normal and tangential forces, the tangential forces being proportional to the sine of the distance from a given point on the sphere.

Printed on p. 15.

What is being asked

This is a problem in the theory of elasticity, with no electricity in it at all. Take a solid elastic sphere. Press on its surface — some force normal to the surface, some tangential. Find the shape it settles into.

The tangential force is specified to be proportional to the sine of the angular distance from a given point on the sphere. That is not an arbitrary choice: it is the distribution produced by a uniform push applied to the layer of particles all round the cell. The push is greatest at the "equator" of the cell as seen from the direction of the push, and vanishes at the two "poles" — which is what $\sin θ$ describes.

Why Maxwell needs it

He has asserted in the introductory pages that the displacement is proportional to the force, with a constant depending on the dielectric. To turn that assertion into a quantity he must compute, he needs the actual deformation of a cell under the actual surface traction — and then read off how far the particles at the surface have moved.

The chain runs:

surface traction on the cell, proportional to $\sin θ$ — this proposition;
deformation of the cell in response — this proposition;
the displacement of the particles at the surface, integrated over the sphere — equation (101);
the relation between force and displacement — equation (105), in Prop. XIII.

The elasticity, and a choice Maxwell has to make

A homogeneous isotropic elastic solid has two independent moduli. How they are related is a physical question about the material, and in 1861 it was a contested one.

Maxwell notes the two positions:

For a medium whose elasticity arises from forces between pairs of particles, the ratio of the two moduli is fixed — he gives it as 6 to 5. This is the "rari-constant" theory of Navier, Poisson and Cauchy, in which a single elastic constant suffices.
A different value applies where the resistance to compression vastly exceeds the resistance to distortion — he offers the illustration of "a liquid rendered slightly elastic by gum or jelly."

He adopts the first, the "perfect solid" case, citing Rankine.

The consequence appears in equation (108), and matters more than a technical footnote should: the numerical coefficient linking the elasticity of the cells to the dielectric constant depends on this choice, and so, through Prop. XVI, does the computed velocity of light.

A load-bearing assumption. Maxwell's velocity comes out right to within about 1.3 per cent of Fizeau's figure. That agreement rests in part on adopting the 6:5 ratio. Had he taken the jelly case instead, the coefficient would differ and the agreement would not have been as striking. The rari-constant theory was later abandoned; real isotropic solids need two independent constants. That the answer came out right anyway is one of the luckier episodes in the paper — and a reason to read Prop. XVI as a strong hint rather than a proof.

Provenance: the proposition statement and the 6:5 remark, the jelly illustration and the citation of Rankine are attested in the surviving prose. No equation from Prop. XII could be recovered. See Sources.

The sphericity idealisation

Maxwell has already conceded the point in the introductory pages:

In the following investigation I have considered the relation between the displacement and the force producing it, on the supposition that the cells are spherical. The actual form of the cells probably does not differ from that of a sphere sufficiently to make much difference in the numerical result.

Spheres do not fill space; the cells must really be some space-filling polyhedron, which is why Maxwell's own figure draws them as hexagons. He computes with spheres because the elasticity problem is tractable for a sphere and not for a hexagonal prism, and he says so.

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