Electromagnetic Wave-surface. 415 
The vector rate of transfer of energy being VEH/47 in 
general, when a ray is solitary, its direction is that of the 
transfer of energy. It seems reasonable, then, to define the 
direction of a ray, whether the wave is plane or not, as per- 
pendicular to the electric and the magnetic forces. On this 
understanding, we do not need the preliminary investigation 
of the index-surface, but may proceed at once to the wave- 
surface by the investigation (69) to (77), following equations 
(25) and (26). 
The following additional useful relations are easily dedu- 
cible :-—F rom (25) and (26) we get 
_ Veo 
Pen ep) 
and from (72) and (73), 
VEH 4 
p= Eek * : . . . ° : . (81) 
Also, from either set, 
[OVA Die a 97a ieaevencg Dea ome acta be <S-} 
expressing the equality of the electric to the magnetic energy 
per unit volume (strictly, at a point). 
Some Cartesian Expansions.—In the important case of 
parallelism of the principal axes of capacity and permeability, 
the full expressions for the index or the wave-surface equa- 
tions may be written down at once from the scalar product 
abbreviated expressions. Thus, taking any equation to the 
wave, as the first of (76), for example, py,=0, y being given 
in (74), take the axes of coordinates parallel to the common 
principal axes of c and w; so that we can employ q, ¢, ¢3, the 
principal capacities, and wy, M2, “3 the principal permeabilities 
in the three components of y,;. We then have, 2, y, z being 
the coordinates of p, 
a y? oe 
- a ————— —- aie Ee se Ee a  () 83 
(ex 'p)er— mp — (pe *p)ez—mpty  (pu'p)es—epeg (88) 
where Ma nee a 
ee —e 
aa od Sa ay 3 
In (83) we may exchange the c’s and yp’s, getting the 
second of (76). Similarly the first of (77) gives 
Pye" be *cay" bs lesz" 
Se Ses ee Oe pe Sea Eg 
(pe—"p)er— mp, (pup )co— Mpa (pue"p 3 — MUL oe) 
-as another form, in which, again, the yw’s and c’s may be ex- 
changed (not forgetting to change m into n) to give a fourth 
form. 
2h 2 
