On Physical Lines of Force/Part III/Prop. XVI

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

Maxwell's statement

Prop. XVI. — To find the rate of propagation of transverse vibrations through the elastic medium of which the cells are composed, on the supposition that its elasticity is due entirely to forces acting between pairs of particles.

Printed on p. 22. The qualifying clause at the end is the "perfect solid" assumption adopted in Prop. XII, and it is load-bearing.

The calculation, in Maxwell's own words

By the ordinary method of investigation we know that [the velocity is the square root of the coefficient of transverse elasticity divided by the density], where [m] is the coefficient of transverse elasticity, and [ρ] is the density. By referring to the equations of Part I., it will be seen that if [ρ] is the density of the matter of the vortices, and [μ] is the "coefficient of magnetic induction," [the density equals μ], and by (108), [the elasticity is E²]. In air or vacuum [μ] = 1, and therefore [V = E].

The bracketed material stands where the printed formulae appear; the surrounding words are verbatim. Prop. XVI carries no printed equation numbers — this is one of the few things about the mathematics of the paper that could be firmly established. See Sources.

Written out, the argument is three steps:

$$V=\sqrt{\frac{m}{\rho }}$$

$$\rho =\mu \text{(from Part I)},m=E^2\text{(from equation 108)}$$

$$\therefore V=\frac{E}{\sqrt{\mu }},\text{and in air or vacuum }\mu =1,\text{ so }V=E$$

Transverse undulations of the electromagnetic medium

What has just happened

$E$ is the constant from Prop. XIII — the elasticity of the cells, which is also, by Prop. XV, the ratio between the two measures of a quantity of electricity. It had been measured, by Weber and Kohlrausch, in an experiment with no optics in it.

So the speed of transverse waves in Maxwell's medium is a number already known from electrical measurements alone. He does not have to fit it. He looks it up.

The numbers

The velocity of light in air, as determined by M. Fizeau, is 70,843 leagues per second (25 leagues to a degree) which gives [the velocity in millimetres per second].

The footnote to Fizeau reads:

Comptes Rendus, vol. xxix. (1849), p. 90. In Galbraith and Haughton's 'Manual of Astronomy,' M. Fizeau's result is stated at 169,944 geographical miles of 1000 fathoms, which gives 193,118 statute miles; the value deduced from aberration is 192,000 miles.

	value	from
velocity of transverse undulations in the medium	310,740,000,000 mm/s	Kohlrausch and Weber's electrical measurements
velocity of light	314,858,000,000 mm/s	Fizeau's optical measurement

They differ by about 1.3 per cent — well within what either measurement could claim in 1862.

Provenance: the two numerals are strongly corroborated but were not read directly from the page; Maxwell restates the same two figures in metres per second in the 1865 Dynamical Theory, §96. The Wikisource transcription of the Fizeau sentence reads "lenses per second", an evident OCR corruption of "leagues". See Sources.

The inference

The velocity of transverse undulations in our hypothetical medium, calculated from the electro-magnetic experiments of MM. Kohlrausch and Weber, agrees so exactly with the velocity of light calculated from the optical experiments of M. Fizeau, that we can scarcely avoid the inference that light consists in the transverse undulations of the same medium which is the cause of electric and magnetic phenomena.

This sentence is conventionally printed wholly in italics. Whether the 1862 printing italicised it has not been verified for this edition, so it is set here in roman.

On the phrasing

It is worth noticing how carefully hedged the most famous claim in the paper is. Not light is an electromagnetic wave. Instead:

• "in our hypothetical medium" — the model is still labelled as a hypothesis in the same sentence that delivers its greatest success;
• "we can scarcely avoid the inference" — the conclusion is presented as forced on him rather than claimed;
• "the same medium" — the claim made is about identity of media, not about the nature of light as such.

The same restraint appears in the recapitulation earlier in Part III, where the identification is offered as a subordinate clause:

… that the elasticity of the magnetic medium in air is the same as that of the luminiferous medium, if these two coexistent, coextensive, and equally elastic media are not rather one medium.

What was and was not established

Established. Two electrical measurements, combined through the mechanical model, yield a velocity equal to the measured velocity of light. Optics and electromagnetism are quantitatively linked.

Not established. That light is an electromagnetic disturbance in any sense independent of the vortex model. The derivation runs through the elasticity of the cells, the 6-to-5 modulus ratio of Prop. XII, and the identification of the density with $\mu$. Remove the model and the argument as given does not stand.

Maxwell knew this, and it is why the 1865 Dynamical Theory exists. There he derives the wave equation directly from the field equations, with no cells, no idle wheels and no elasticity of figure — and the result survives. The modern derivation is on the wave equation page.

A remark on luck and judgement

Two things had to go right for this proposition to work, and only one of them was in Maxwell's control.

• The 1.3 per cent agreement depended on Weber and Kohlrausch's number and Fizeau's number both being close to correct, and on the 6-to-5 modulus ratio, a theory of elasticity later abandoned.
• The structural claim — that the ratio of the electrical units must have the dimensions of a velocity, and that this velocity is the propagation speed of disturbances in the field — is exactly right, and survives every later reformulation.

Maxwell could not fully distinguish the two at the time. That he pressed on the coincidence rather than dismissing it is the judgement the paper is remembered for.

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Next: Prop. XVII · Previous: Prop. XV

Modern derivation: The electromagnetic wave equation · Plain-language version: How Maxwell found the speed of light