Electromagnetic Wave-surface. 409 
the left members become unity, by the conjugate property ; 
hence 
userid, cons Bole 2s oe (42) 
are two other forms of the index equation. (41) and (42) 
are the simplest forms. More complex forms are created with 
that surprising ease which is characteristic of these operators; 
but we do not want any more. When expanded, the different 
forms look very different, and no one would think they repre- 
sented the same surface. This is also true of the corresponding 
Fresnel surface, which is comparatively simple in expression. 
In any equation we may exchange the operators pw and e. 
Put c=Nv—' in any form of index equation, and we have 
the velocity equation, a quadratic in v” giving the two velo- 
cities of the wave-froni. Andif we put Nv=p, making thus 
p a vector parallel to the normal of length equal to the velocity, 
it will be the vector of the surface which is the locus of the 
- foot of the perpendicular from the origin upon the tangent- 
plane to the wave-surface. 
By (83a), remembering that a is parallel to the normal, we 
see that 
cH, wH, and uN are in one plane; 
or, (43) 
HK, N, and w-'cH are in one plane. 
And by (84a), 7 
#H, cN, and cH are in one plane; 
or, (44) 
H, N, and c—'uH are in one plane. 
These conditions expanded, give us the directions of the elec- 
tric force and displacement, the magnetic force and induction, 
for a given normal. We may write the second of (43) thus, 
NV D D =): e s . e e s (45) 
cf 
and the second of (44) thus, 
& Ly = e e ° 2 . ® (46) 
Cf 
and as these differ only in the substitution of B for D, we see 
that the induction of either ray is parallel to the displacement 
of the other ; that is, the two directions of induction in the 
wave-front are the two directions of displacement. 
The Wave-Surface.—Since the velocity-surface with the 
~ vector p=UN is the locus of the foot of the perpendicular on 
the tangent-plane to the wave-surface, we have, if p be the 
