Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/212

where $l$ denotes the wave-length, $p$ the period of oscillation, $u$ the distance of the point considered from the wave-plane passing through the origin, $\alpha _{1},\beta _{1},\gamma _{1}$ the amplitudes of the displacements $\xi ,\eta ,\zeta$ in the wave-plane passing through the origin, and $\alpha _{2},\beta _{2},\gamma _{2}$ the amplitudes in a wave-plane one-quarter of a wave-length distant and on the side toward which $u$ increases. If we also write ${\text{L, M, N}}$ for the direction-cosines of the wave-normal drawn in the direction in which $u$ increases, we shall have the following necessary relations:

${\text{L}}^{2}+{\text{M}}^{2}+{\text{N}}^{2}=1,$

(2)

$u={\text{L}}x+{\text{M}}y+{\text{N}}z,$

(3)

${\text{L}}\alpha _{1}+{\text{M}}\beta _{1}+{\text{N}}\gamma _{1}=0,$⁠${\text{L}}\alpha _{2}+{\text{M}}\beta _{2}+{\text{N}}\gamma _{2}=0.$

(3)

4. That the irregular part of the displacement $(\xi ',\eta ',\zeta ')$ at any given point is a simple harmonic function of the time, having the same period and phase as the regular part of the displacement $(\xi ,\eta ,\zeta ),$ may be proved by the single principle of superposition of motions, and is therefore to be regarded as exact in a discussion of this kind. But the further conclusion of the preceding paper (§ 4), "that the values of $\xi ',\eta ',\zeta '$ at any given point in the medium are capable of expression as linear functions of $\xi ,\eta ,\zeta$ in a manner which shall be independent of the time and of the orientation of the wave-planes and the distance of a nodal plane from the point considered, so long as the period of oscillation remains the same," is evidently only approximative, although a very close approximation. A very much closer approximation may be obtained, if we regard $\xi ',\eta ',\zeta ',$ at any given point of the medium and for light of a given period, as linear functions of $\xi ,\eta ,\zeta$ and the nine differential coefficients

${\frac {d\xi }{dx}},\,\,{\frac {d\eta }{dx}},\,\,{\frac {d\zeta }{dx}},\,\,{\frac {d\xi }{dy}},\,\,{\text{etc.}}$

We shall write $\xi ,\eta ,\zeta ,$ and diff. coeff. to denote these twelve quantities.

From this it follows immediately that with the same degree of approximation ${\dot {\xi '}},{\dot {\eta '}},{\dot {\zeta }},$ may be regarded, for a given point of the medium and light of a given period, as linear functions of ${\dot {\xi }},{\dot {\eta }},{\dot {\zeta }}$ and the differential coefficients of ${\dot {\xi }},{\dot {\eta }},{\dot {\zeta }}$ with respect to the coordinates. For these twelve quantities we shall write ${\dot {\xi }},{\dot {\eta }},{\dot {\zeta }}$ and diff. coeff.

5. Let us now proceed to equate the statical energy of the medium at an instant of no velocity with its kinetic energy at an instant of no displacement. It will be convenient to estimate each of these quantities for a unit of volume.

6. The statical energy of an infinitesimal element of volume may be represented by $\sigma dv,$ where $\sigma$ is a quadratic function of the components of displacement ${\xi +\xi ',\eta +\eta ',\zeta +\zeta '}.$ Since for that element of volume $\xi ',\eta ',\zeta '$ may be regarded as linear functions of $\xi ,\eta ,\zeta ,$ and diff. coeff.,