On Physical Lines of Force — Index of numbered equations

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On Physical Lines of Force → Equations

Maxwell numbers his equations continuously across all four Parts. This index lists every number that could be established for this edition, with what it expresses and where it is discussed.

Read this first. No equation in the paper could be transcribed from a page scan for this edition. The equation numbers below are attested — Maxwell cites them by number in running prose, and those references survived transcription. The equation bodies printed on this wiki are reconstructions in his notation. See Sources and editorial method for the full account and for what would be needed to fix it.

Attested equation numbers

no.	Part	what it expresses	modern form	page
(9)	I	the equations of electric currents; the current is the curl of the magnetic force	$\mathrm{\nabla }\times 𝐇=4\pi 𝐉$	Prop. III
(12)	I	force per unit volume on an element of the medium, x component	five terms — see the page	Prop. III
(13)	I	the same, y component		Prop. III
(14)	I	the same, z component		Prop. III
(15)	I	condition for a complete differential	cited from Part II	Prop. V
(16)	I	the same	cited from Part II	Prop. V
(18)	I	the condition holding at a magnetic pole		Applications
(19)	I	the condition holding everywhere else		Applications
(33)	II	the transference of particles across unit area — identical with (9)	$\mathrm{\nabla }\times 𝐇=4\pi 𝐉$	Prop. V
(34)	II	the superficial density of the particles, chosen to make (33) reproduce (9)		Prop. V
(46)	II	the energy of a vortex of volume V	$u=𝐁\cdot 𝐇/8\pi$	Prop. VI
(47)	II	work expended on an element of vortex surface		Prop. VII
(50)	II	whole work done on a vortex in unit time		Prop. VII
(54)	II	changing magnetic field and the electromotive forces brought into play	$\mathrm{\nabla }\times 𝐄=-\partial 𝐁/\partial t$	Prop. VIII
(55)	II	the electrotonic state F, G, H	$𝐁=\mathrm{\nabla }\times 𝐀$	Prop. VIII
(56)	II	the magnetic induction is solenoidal	$\mathrm{\nabla }\cdot 𝐁=0$	Prop. VIII
(57)	II	from differentiating (55) and comparing with (54)		Prop. VIII
(58)	II	electromotive force in terms of the potentials	$𝐄=-\partial 𝐀/\partial t-\mathrm{\nabla }\phi$	Prop. VIII
(77)	II	electromotive force in a moving body	$𝐄=𝐯\times 𝐁-\partial 𝐀/\partial t-\mathrm{\nabla }\phi$	Prop. XI
(101)	III	the displacement, taken over the whole surface of a cell		Prop. XIII
(105)	III	electromotive force and electric displacement	$𝐃=\epsilon 𝐄$	Prop. XIII
(108)	III	the value of the elastic coefficient adopted — the "perfect solid" case		Prop. XIII
(112)	III	the displacement current — the corrected equations of currents	$\mathrm{\nabla }\times 𝐇=𝐉+\partial 𝐃/\partial t$	Prop. XIV
(113)	III	the equation of continuity for free electricity	$\mathrm{\nabla }\cdot 𝐃=\rho$	Prop. XIV
(115)	III	cited on p. 20; content not established		Prop. XV
(116)–(125)	III	energy in the medium; integration by parts; Green's theorem; the inverse-square force	Coulomb's law	Prop. XV

The eight numbers in bold are the ones a reader is most likely to want.

Established gaps

• Part I before (9) and after (19). The numbering is not established. Part I appears to run from (1) to about (26), but that is inference from where formulae sit in the prose.
• Part II between (58) and (77). Unresolved; contradictory readings were obtained.
• Part III before (101). Where Part III's own numbering begins is unknown.
• Prop. XVI carries no printed equation numbers at all. This is firmly established, and it is a striking fact: the velocity-of-light calculation, the most famous passage in the paper, is set in unnumbered display.
• Part IV. No numbered equation could be recovered, and the final equation number of the paper is unknown. Whether Part IV contains numbered Propositions is also unestablished; proposition numbering certainly reaches XVII at the end of Part III.

Propositions

Prop.	Part	subject
I	I	the pressure of a vortex is independent of its size — page
II	I	the stress resolved on the coordinate planes — page
III	I	the force on an element of the medium — page
IV	II	the motion of a layer of particles between two vortices — page
V	II	the transference of particles across unit area — page
VI	II	the energy of the vortices — page
VII	II	the work done on a vortex by the particles — page
VIII	II	induction, and the electrotonic state — page
IX	II	an element strained — page
X	II	an element rotated — page
XI	II	electromotive force in a moving body — page
XII	III	the equilibrium of an elastic sphere — page
XIII	III	electromotive force and displacement — page
XIV	III	the displacement current — page
XV	III	the force between two electrified bodies — page
XVI	III	the velocity of transverse undulations — page
XVII	III	the capacity of a Leyden jar — page

Part I contains Props. I–III only. Part II contains IV–XI. Part III contains XII–XVII. Part IV's propositions, if any, are not established.

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See also: Notation and concordance · Modern derivations · Sources