Electromagnetic Wave-surface. 417 
from which «;, «, a3 may be solved in terms of w, y, z; thus 
y= ALT Aygy + A432, 
he = Ag + Ago + Ao32, 
03 = Ag LF AzoY 1 A332 5 
where, by using (15), 
oe: (Rees — fy )( Ress — fs) — (Reo; Tas) 
A 
41 
Rae (Re;3— p13 )( Re23— 23) — (Rey2 — M2) (Res3 — M33) . 
i= ne A inden 
and the rest by symmetry. Then, since 
pe=Xta, + ya,+za;=0, 
we get the full expansion. A need not be written fully, as it 
goes out. The equation may be written symmetrically, thus, 
O=1+mn(py'p)(po~'p) — {U°( Confess + Casfto2— 2Cosfes) +. - 
+ Qary (Cy3flo3 + Cosptis— Crofl33— Ca3fti2) +++}, ~ (87) 
where the coefficients of y”, 2”, yz, and zx are omitted. Here 
N= py boflg and N=C4Cy¢3; Whilst 
Set / ve / 2 {i 2 / / / 
po p= C2" + C'o9y° + C'332° + 20 ay + 2c'osyz + 2e'g 24, 
where ¢’;,.--,are the inverse coefficients. See equation (15). 
The expansion of pu~'p is exactly similar, using the inverse wu 
coefficients. 
If in (87) we for every ¢ or w write the reciprocal coeffi- 
cients, we obtain the equation to the index-surface; that is, 
supposing 2, y, z then to be the components of o instead of p. 
And, since cv=N, the unit wave normal, we have the velocity 
equation as follows, in the general case, 
put NuN NeN 
ee + vt —v? {Nic o9ft’33 + C33}0 00 — 2! aft 93) +e +s 
+ 2NiNo(¢'13f4'03 + C2313 C124" 33 — C3312) +... . (88) 
in which N,, N., N; are the components of N, or the direc- 
tion-cosines of the normal. To show the dependence of v’ 
upon the capacity and permeability perpendicular to N, take 
N,=1, N,=0, N3;=0, which does not destroy generality, 
because in (88) the axes of reference are arbitrary. Then (88) 
reduces to 
vt —(¢' soft! 33 + € sabt’29— 2¢' 23/423) 0” , 
+ (69033 — C35) o2kt’ 33 = aul 
