Electromagnetic Wave-surface. 407 
wave. Using this, and (23), in the equations of induction 
(19), (20), they become 
dH adi 
VN DIG my! 
AD} dH 
ee = oh 
Here, since the z differentiation is scalar, and occurs on both 
sides, it may be dropped, giving us 
RNS re Bee ti) ea) 
MOND 2eo/ rc BE 7 ee Mian a 5) 
The induction and the displacement are therefore necessarily 
in the wave-front, by the definition of a vector product, being 
perpendicular to N. Also the displacement is perpendicular 
to the magnetic force, and the induction is perpendicular to 
the electric force. 
Index Surface.—Let ear _ eee Ro Hoary 
be a vector parallel to the normal, whose length is the reci- 
procal of the normal velocity v. It is the vector of the index 
surface. By (25) and (26) we have 
cH=—VcH, therefore —H=c"'VoH; . . (28) 
and 
ge Vol), therefore H=e Vol... (9) 
Now use the theorem (17). Then, if 
M= fy Mobs, N=CiCol3 + © «© © (30) 
be the products of the principal permeabilities and capacities, 
the theorem gives, applied to (28) and (29), 
a V CoCr et i Ae ye), Come) 
Mill NOP Rh Kat ot QOZp) 
Putting the value of H given by (32) in (28) first, and then 
the value of E given by (31) in (29), we have 
Ome ON poplin. |» init oon eh OD 
—nH=p"VoVeocH. . .. . (84) 
To these apply the transformation-formula (18), giving 
—mcH=po(cpH)—pH(opo) . . . (88a) 
and 
—npH=co(ocH) —cH(eco), . . . (34a) 
