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Add DISPLAYTITLE: title the page by its section rather than its path

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			{{DISPLAYTITLE:Part I — Prop. II}}
	''[[On Physical Lines of Force]] → [[On Physical Lines of Force/Part I\|Part I]] → Prop. II''		''[[On Physical Lines of Force]] → [[On Physical Lines of Force/Part I\|Part I]] → Prop. II''

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	''Next: [[On Physical Lines of Force/Part I/Prop. III\|Prop. III]] · Previous: [[On Physical Lines of Force/Part I/Prop. I\|Prop. I]]''		''Next: [[On Physical Lines of Force/Part I/Prop. III\|Prop. III]] · Previous: [[On Physical Lines of Force/Part I/Prop. I\|Prop. I]]''

BFP Rauch: 1 revision imported: Maxwell, On Physical Lines of Force — initial import

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1 revision imported: Maxwell, On Physical Lines of Force — initial import

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''[[On Physical Lines of Force]] → [[On Physical Lines of Force/Part I|Part I]] → Prop. II''

== Maxwell's statement ==

<blockquote>
'''Prop. II.''' — If the direction-cosines of the axes of the vortices with respect to the axes of ''x'', ''y'', and ''z'' be ''l'', ''m'', and ''n'', to find the normal and tangential stresses on the coordinate planes.
</blockquote>

== The decomposition ==

Maxwell first resolves the state of the medium into two pieces, which is the whole trick of the proposition:

<blockquote>
a simple hydrostatic pressure acting in all directions, and a simple tension … acting along the axis of stress
</blockquote>

[[File:OPL-fig-maxwell-stress.svg|thumb|center|620px|Resolving the stress on an element of the medium]]

A hydrostatic pressure is isotropic and contributes the same amount to every normal stress and nothing to any tangential stress. The tension is the interesting part: it acts along one direction only, the axis of the vortices, and it is that single directed tension which carries all the magnetic behaviour.

If the magnetic force has components <math>\alpha,\beta,\gamma</math> along the coordinate axes, then the axis direction-cosines are

<math display="block">l = \frac{\alpha}{\sqrt{\alpha^{2}+\beta^{2}+\gamma^{2}}},\qquad m = \frac{\beta}{\sqrt{\alpha^{2}+\beta^{2}+\gamma^{2}}},\qquad n = \frac{\gamma}{\sqrt{\alpha^{2}+\beta^{2}+\gamma^{2}}}</math>

and the tension along that axis is proportional to <math>\mu(\alpha^{2}+\beta^{2}+\gamma^{2})</math> — the density of the vortex substance times the square of its circumferential velocity, exactly as [[On Physical Lines of Force/Part I/Prop. I|Prop. I]] requires.

== The six stresses ==

Projecting a tension <math>T</math> along the direction <math>(l,m,n)</math> onto the coordinate planes, and adding a uniform pressure <math>p_{0}</math>, gives three normal and three tangential stresses:

<math display="block">p_{xx} = \frac{\mu}{4\pi}\alpha^{2} - p_{0}, \qquad p_{yy} = \frac{\mu}{4\pi}\beta^{2} - p_{0}, \qquad p_{zz} = \frac{\mu}{4\pi}\gamma^{2} - p_{0}</math>

<math display="block">p_{xy} = \frac{\mu}{4\pi}\alpha\beta, \qquad p_{yz} = \frac{\mu}{4\pi}\beta\gamma, \qquad p_{zx} = \frac{\mu}{4\pi}\gamma\alpha</math>

''Provenance: the printed equation numbers for Prop. II could not be established, and these bodies are reconstructions in Maxwell's notation. The content is standard and is corroborated by Maxwell's own term-by-term gloss on the result of [[On Physical Lines of Force/Part I/Prop. III|Prop. III]]. See [[On Physical Lines of Force/Sources and editorial method|Sources]].''

Read the pair of lines together and a pattern appears: every component has the form

<math display="block">p_{ij} = \frac{\mu}{4\pi}\,H_{i}H_{j} \;-\; p_{0}\,\delta_{ij}</math>

== What this is, in modern terms ==

This is '''the Maxwell stress tensor of the magnetic field''', and Part I is where it enters physics. In modern notation, with <math>\mathbf{B}=\mu\mathbf{H}</math>,

<math display="block">T_{ij} = \frac{1}{4\pi}\left( B_{i}H_{j} - \tfrac{1}{2}\delta_{ij}\,\mathbf{B}\cdot\mathbf{H} \right)</math>

the <math>-\tfrac12 \delta_{ij}\mathbf{B}\cdot\mathbf{H}</math> being Maxwell's hydrostatic term, once the part of <math>p_{0}</math> that comes from the vortex motion is separated from the part that is ordinary fluid pressure.

The physical content is the sentence that survives every later reformulation: '''along a line of force the medium is in tension; across it, in compression.''' The tensor is just that statement written six times.

See [[On Physical Lines of Force/Modern derivations/The Maxwell stress tensor|the modern derivation]] for the conversion in full, including where the 4π goes and why the momentum flux interpretation only became available later.

== A note on what Maxwell has and has not proved ==

Nothing so far requires vortices. Prop. II would hold for ''any'' mechanism producing a tension along one axis and a pressure across it. Maxwell is careful about this at the opening of [[On Physical Lines of Force/Part II|Part II]]:

<blockquote>
Such a state of stress, if assumed to exist in the medium, and to be arranged according to the known laws regulating lines of force, will act upon the magnets, currents, &c. in the field with precisely the same resultant forces as those calculated on the ordinary hypothesis of direct action at a distance. This is true independently of any particular theory as to the cause of this state of stress, or the mode in which it can be sustained in the medium.
</blockquote>

The stress result is secure. The vortices are the ''explanation offered'' for the stress, and are held to a lower standard — Maxwell calls the vortex hypothesis a "probable answer", against the "mechanical deductions" which he regards as settled.

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''Next: [[On Physical Lines of Force/Part I/Prop. III|Prop. III]] · Previous: [[On Physical Lines of Force/Part I/Prop. I|Prop. I]]''

On Physical Lines of Force/Part I/Prop. II - Revision history

BFP Rauch: Add DISPLAYTITLE: title the page by its section rather than its path

BFP Rauch: 1 revision imported: Maxwell, On Physical Lines of Force — initial import

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