On Physical Lines of Force/Notation and concordance

On Physical Lines of Force → Notation

Maxwell wrote in 1861. Vector notation did not yet exist — Heaviside and Gibbs produced it in the 1880s — so every relation on these pages is written three times, once for each coordinate. Reading the paper is much easier once you can see the vector hiding behind each triple.

A concordance of notation

The main symbols

Maxwell	Meaning in his words	Modern	Notes
α, β, γ	the components of the magnetic force; in Part I, the circumferential velocity of a vortex times the direction-cosines of its axis	$𝐇$	the speed of the spin is the strength of the field
μ	the "capacity for magnetic induction"; also the density of the substance of the vortices	$\mu$	equal to 1 in air; greater in paramagnetic, less in diamagnetic media
μα, μβ, μγ	—	$𝐁=\mu 𝐇$	the magnetic induction
p, q, r	the quantity of electric current per unit of area perpendicular to the axes of x, y, z	$𝐉$	in Part III these acquire the displacement term
P, Q, R	the components of electromotive force	$𝐄$	force on unit of the particles, not force per unit charge in the modern sense
F, G, H	the components of the "electrotonic state", or electromagnetic momentum	$𝐀$	Maxwell's name for Faraday's electrotonic state
ψ	the electric tension, or potential	$\phi$
h	the electric displacement	$𝐃/4\pi$	see the note on 4π below
E	a coefficient depending on the nature of the dielectric — the elasticity of the cells	$\sqrt{1/\epsilon }$	not the modern electric field
e	the quantity of free electricity	$\rho$
m	the coefficient of transverse elasticity of the cell substance	—
ρ	the density of the cell substance	—	in Part I; do not confuse with modern charge density
V	the velocity of transverse undulations	$c$

Two collisions to watch for. Maxwell's E is a dielectric coefficient, not the electric field — the electric field is his P, Q, R. And his H is the third component of the vector potential, not the magnetic force — the magnetic force is his α, β, γ. Both letters mean something else today.

Derivatives

Maxwell writes total-derivative notation throughout, where a modern author would use partial derivatives:

• He writes $\frac{d\gamma }{dy}$; we write $\frac{\partial \gamma }{\partial y}$.
• He writes $\frac{d}{dt}$ both for a rate of change at a fixed point and, in Prop. XI, for a rate of change following a moving element. The distinction is carried by the prose, not the symbol. Prop. XI is precisely where it matters.

On this wiki, quotations and reconstructed 1861 equations keep Maxwell's d; the modern derivation pages use ∂.

The factor 4π

Maxwell works in a Gaussian-style system where 4π appears explicitly in the field equations rather than being absorbed into the constants. So where a modern SI textbook has

$$\mathrm{\nabla }\times 𝐇=𝐉$$

Maxwell has

$$4\pi p=\frac{d\gamma }{dy}-\frac{d\beta }{dz},4\pi q=\frac{d\alpha }{dz}-\frac{d\gamma }{dx},4\pi r=\frac{d\beta }{dx}-\frac{d\alpha }{dy}\mathrm{(}\mathrm{9}\mathrm{)}$$

The 4π is not decoration. It is the surface area of a unit sphere, and it is there because his unit of current is defined by the magnetic effect of a closed circuit. When you convert his results to modern form, tracking the 4π is most of the work.

Sign conventions

Maxwell's sign for the electromotive force in Prop. VIII follows from the direction he chose, "perfectly arbitrarily", for the rotation of the vortices:

We also assumed, perfectly arbitrarily, that the direction of these vortices is such that, on looking along a line of force from south to north, we should see the vortices revolving in the direction of the hands of a watch.

Everything downstream — which way the idle wheels turn, which way an induced current runs — is fixed by that one choice. It happens to agree with the modern convention.

Vocabulary

Several of Maxwell's terms have drifted. See the glossary for the full list; the ones that trip readers most often:

• electromotive force — for Maxwell this is a force acting on the particles at every point of the medium, not a voltage round a circuit
• electric tension — potential
• quantity vs. intensity — quantity is what flows, intensity is what drives it
• capacity for magnetic induction — permeability
• electrotonic state — Faraday's name for a condition of the medium under induction; Maxwell gives it the components F, G, H and it becomes the vector potential

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See also: Index of numbered equations · From components to vectors