Polarization in perfectly transparent Media. 461 
wave-length distant and on the side toward which w increases. 
If we also write L, M,.N for the direction-cosines of the wave- 
normal drawn in ‘the direction in which w increases, we shall 
have the following necessary relations: 
L’+M’+N’=1, (2) 
u=Le+My+ Naz, (3) 
La,+Mf,+Ny,=0, La,+Mf,+Ny.=0. (4) 
4. That the irregular part of the displacement (&’, 4,’ €’) at 
any given point is a simple harmonic function o the time, 
having the same period and phase as the regular part of the 
displacement (€, 7, €), 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 preene paper (§ 4), “that the values of &, 7’, ¢’ at 
any given point in the medium are capable of expression as 
linear “unetions of €, 7, € in a manner which shall be inde- 
pendent of the time and of the orientation of the wave-planes 
and ie distance of a nodal plane from the point considered, so 
long as the period of oscillation remains the same,” is evi- 
dently only ho areca although a very close approxima- 
tion. A very much closer approximation may be obtained, if 
we regard &’, 9’, ¢', at any given point of the medium and for 
light of a given penoeus as ae fencer of &, 7, € and the 
nine differential coéfficie 
dE dn de dé : 
dx? de’ de’? dy’ a 
We pn write &, 7, € ond diff. coéff. to denote these twelve 
uan 
“ Pon “his it follows immediately that with the same degree 
of approximation & 1, a may be regarded, for a given point 
of the medium and light of a given period, as linear functions 
of , ” t and the differential coéfficients of &, ” * with respect 
to the codrdinates. For these twelve quantities we shall write 
ag a ieee L 
mate each of these quantities for a unit of 
6. The statical energy : an infinitesimal acne of volume 
may be represented by odv, where a is a quadratic function of 
the components of iisplaseginat E+E’ n+n’, C+0'. Since for 
