Electromagnetic Wave-surface. 403 
line-integral round the closed curve is expressed in (9) as the 
surface-integral over any surface bounded by the curve of 
another vector B, which =VVA._ BdS is the scalar product 
of B and the vector element of surface dS, whose direction is 
defined by its unit normal. 
Finally, passing from an unclosed to a closed surface, (10) 
expresses the surface-integral of any vector C over the closed 
surface (normal positive outward), as the volume-integral of 
its divergence within the included space. 
Linear Vector Operators.—Ilf H be the magnetic force at a 
point, B the induction, Ei the electric force, and D the dis- 
placement, all vectors, then 
Bape and D=ch fag 64. C1) 
express the relation of B to H and of D to HE in a dielectric 
medium. If it be isotropic as regards displacement, c is the 
electric capacity ; and if it be isotropic as regards induction, 
# is the magnetic permeability ; ¢ and uw are then constants, if 
the medium be homogeneous, or scalar functions of position, if 
it be heterogeneous. 
We shall not alter the form of the above equations in the 
case of eolotropy, when ¢ and w become linear operators. For 
instance, the induction will always be wH, to be understood 
as a definite vector, got from H another vector, in a manner 
fully defined by (in case we want the developments) the fol- 
lowing equations (not otherwise needed). Let Hy,..., and 
B,,--., be the components of H and B referred to any rect- 
angular axes. Then 
By = er Hy + pag He + ass, 
By = Mor Hy + fool, + pall (12) 
By = ps1 Hy + MoH, + bes Hs, 
where fy, Ke. are constants, which may have any values not 
making HB negative ; with the identities “y.= 1, dc. 
Or 
By= 4,44, Bo=2H,, Bs=psH3;; . . . (13) 
when the components are those referred to the principal axes 
of permeability, 441, 442, #3 being the principal permeabilities, 
all positive. 
Inverse Operators.—Since B=4H,we have H=,~'B, where 
u-) is the operator inverse to ». When referred to the prin- 
cipal axes, we have 
bt ay Ee aS 
FS ee? Phe ear M3 lbs 
Tether 
