THE ELECTRIC AND LUMINIFEROUS MEDIUM. 
789 
incidentally extend our view to the case in which these functions have not the same 
principal axes. The variational equation of motion is represented by the vanishing of 
the time-integral of the expression 
cW 
dy dll 
— —, 
dz dg j ^ \ dz df 
d f?U d fm\ . Id dJ] d dV\ , 
“ TxaI ^ ^ * 
+ 
d\l fZU\ ... / dV 
dh 
dV 
_ r M /yZ_ diu _ . d / d d\]' _ dU'\ . 
M d,t \dy dh dz dg ) dt\dz df dx dh) ^ dt \dx dg 
d / d d\]' d dU 
dy d/ 
dr 
Ijd/ dU' 
dU'\ . d/ dU' ;dUU . , d/,dU' 
K !- c/S. 
d/y 
The equations of propagation are therefore of type 
d^^ d dUi _ d dU^ _ 
d dt- dy dh dz dg ~ 
where 
U, = U + ^U' 
The boundary condition demands in general the continuity of the expression 
S^+ (n 
dUi 
df 
in crossing the interface; for the special case of {fm, n) = (1, 0, 0), this involves 
continuity in rj, dTJJdg and dJJ^jdh. 
Thus, under the most general circumstances, the inclusion of opacity is made 
analytically by changing the potential energy-function from U to U^, where is 
still a quadratic function, but with complex coefficients. If U and U' have their 
principal axes in the same directions, a change of the principal indices of refraction of 
the medium from real to complex values suffices to deduce the circumstances both 
of propagation and of reflexion of light in partially opacpie substances from the ones 
that obtain for perfectly transparent media. In all cases however the function 
has three principal axes of its own, whose position depends on the period of the light. 
Dynamical Equations of the Primordial Medium. 
97. The medium by means of wliich we have been attempting to co-ordinate 
inanimate phenomena is of uniform density, if there be excepted the small volumes 
