Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/228

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212

EQUATIONS OF MONOCHROMATIC LIGHT IN MEDIA

electrical phenomena,^[1] we shall not introduce at first the hypothesis of Maxwell that electrical fluxes are solenoidal.^[2] Our results, however, will be such as to require us to admit the substantial truth of this hypothesis, if we regard the processes involved in the transmission of light as electrical.

With regard to the undetermined questions of electrodynamic induction, we shall adopt provisionally that hypothesis which appears the most simple, yet proceed in such a manner that it will be evident exactly how our results must be altered, if we prefer any other hypothesis.

Electrical quantities will be treated as measured in electromagnetic units.

2. We must distinguish, as before, between the actual electrical displacements, which are too complicated to follow in detail with analysis, and which in their minutiæ elude experimental demonstration, and the displacements as averaged for spaces which are large enough to smooth out their minor irregularities, but not so large as to obliterate to any sensible extent those more regular features of the electrical motion, which form the subject of optical experiment These spaces must therefore be large as measured by the least distances between molecules, but small as measured by a wave-length of light. We shall also have occasion to consider similar averages for other quantities, as electromotive force, the electrostatic potential, etc. It will be convenient to suppose that the space for which the average is taken is the same in all parts of the field,^[3] say a sphere of uniform radius having its center at the point considered. Whatever may be the quantities considered, such averages will be represented by the notation

[\,\,\,\,]_{\text{Ave}}.

↑ It has, perhaps, retarded the aooeptanoe of the electromagnetic theory of light that it was presented in connection with a theory of electrical action, which is probably more difficult to prove or disprove, and certainly presents more difficulties of comprehension, than the connection of optical and electrical phenomena, and which, as resting largely on a priori considerations, must naturally appear very differently to different minds. Moreover, the mathematical method by which the subject was treated, while it will remain a striking monument of its author's originality of thought, and profoundly modify the development of mathematical physics, must nevertheless, by its wide departure from ordinary methods, have tended to repel such as might not make it a matter of serious study.

↑ A flux is said to be solenoidal when it satisfies the conditions which characterise the motion of an incompressible fluid,—in other words, if

u,v,w

are the rectangular components of the flux, when

{\frac {du}{dx}}+{\frac {dv}{dy}}+{\frac {dw}{dz}}=0,

and the normal component of the flux is the same on both sides of any surfaces of discontinuity which may exist.

↑ This is rather to fix our ideas, than on account of any mathematical necessity. For the space for which the average is taken may in general be considerably varied without sensibly affecting the value of the average.

[1] It has, perhaps, retarded the aooeptanoe of the electromagnetic theory of light that it was presented in connection with a theory of electrical action, which is probably more difficult to prove or disprove, and certainly presents more difficulties of comprehension, than the connection of optical and electrical phenomena, and which, as resting largely on a priori considerations, must naturally appear very differently to different minds. Moreover, the mathematical method by which the subject was treated, while it will remain a striking monument of its author's originality of thought, and profoundly modify the development of mathematical physics, must nevertheless, by its wide departure from ordinary methods, have tended to repel such as might not make it a matter of serious study.

[2] A flux is said to be solenoidal when it satisfies the conditions which characterise the motion of an incompressible fluid,—in other words, if $u,v,w$ are the rectangular components of the flux, when

${\frac {du}{dx}}+{\frac {dv}{dy}}+{\frac {dw}{dz}}=0,$
and the normal component of the flux is the same on both sides of any surfaces of discontinuity which may exist.

[3] This is rather to fix our ideas, than on account of any mathematical necessity. For the space for which the average is taken may in general be considerably varied without sensibly affecting the value of the average.

[1]

[2]

[3]