418 On the Electromagnetic Wave-surface. 
When the pw and caxes are parallel, and their principal axes 
are those of reference, we have 
NuN NeN ae 
n 
m 
—0°{N2(08, + 02) + N3(08, + 0%) 
+NAv2,+02)}, . . (89) 
O0= 
where 
NN =p, Ni + woN+ uN, 
with a similar expression for NcN, and v3==(¢p3)~?, &c., as 
before. The solution is 
v NG (v9 +1 V30) a er al Vi3) Fh = (Gas at vn) ter VX, (90) 
where 
X= Niui + Nous + Njug—2(Nj Noe + NGNgustts + Na Aece 
in which 
SEER 2 ome 
CO UF Uap 2 = Can mma 
Ug=v2,—- 02.) eee 
Take 7,=0, or cy / #:=c3/ 33 the two velocities are then 
Nivea a N3v a 15 N3Y 125 and ING o5 = Novis == N3P 21) 
reducing to one velocity v3; when N,=1. 
If, further, U.=0, or ug=0, making ¢ / wy=cy / Me=Cs3 | M3, 
5 always, and 
v= Neste Noon + Nevis, = ue (92) 
is the single value of the square of velocity of wave-front. 
Directions of HE, H, D, and B—We may expand (45) to 
obtain an equation for the two directions of the induction and 
displacement. Thus, since 
pe é 
ris 1(6/41Dy + ¢y2D2 + ¢13D3) +9( ED + C29D 2 + C93Ds) 
+ k(¢e's,D + 6’32D. + ¢'33Ds), 
— =01(/) Dy + 49D + 43 D3) +9 (H/o Di + p'29Do + f23D5) 
[i 
+ k's: Dy + f’52De + f’33Ds), 
N=iN, + N, + kNs, 
the determinant of the coefficients of 7, 7, & equated to zero 
gives the required equation. When the principal axes of w 
and ¢ are parallel, the equation greatly simplifies, being then 
Nyy News, Notts 
eine Tok NS ° . ° ° (93) 
where %,..., are the same differences of squares of principal 
Co 
