416 Mr. Oliver Heaviside on the 
These reduce to the Fresnel surface if either w,;=p.=p3 or 
Cy = Co == C2 
Let e=0 to find the sections in the plane yz. The first 
denominator in (83) gives 
waves 
CIE pete 1 2 2 
H + — }o,— — =0, or y%cyp3 + 2°C\fg=1 
(E is) Oe ie ae Ges 1ig= 4, 
representing an ellipse, semiaxes 
013=(C4M@3)—? and vj9=(cyMlg)~?. 
The other terms give 
2 2 2 
ui e ) 2 DN fie Y z 
— + — J(C3y° + C92") = 5- ===5 
( fig) rl as 
Y fye3 + 2? oy Co = 1. 
a 
An ellipse, semiaxes v3;=(¢34,)7? and vp1=(¢,4,)—%. Simi- 
larly, in the plane zx the sections are ellipses whose semiaxes 
ATE Voy, Vo3, AN V4, V32, Where for brevity v,.=(c,u5)~2 ; and 
in the plane zy, the ellipses have semiaxes V31, V32, and 043, Uj. 
In one of the principal planes two of the ellipses intersect, 
giving four places where the two members of the double surface 
unite. 
If ey | py=C2 | o=C3 / M3, We have a single ellipsoidal wave- 
surface whose equation is 
2 
Or 
ied 2 2 
S+5+5e1 2... (885) 
23 31 Vie 
Now, of course, vj9= V9, &e. 
When the w and ¢ axes are not parallel, we cannot imme- 
diately write down the full expansion of the wave-surface 
equation. Proceed thus:—Taking py,=0 as the equation, 
let 
R= m(pp='p). and a=m—y; 
then, by (74) and (76), 
ue au 
De eer oot or pa=0, 
where 
p=(Re— pate 8st on i ee 
Risascalar. If a@,, «,, & are the three components of a 
referred to any rectangular axes, and 2, y, z the components 
of p, we have, by (86) and (12), 
w= (Rey — py )e1 + (Rejo—py2 2 a (Rej3 —py3)3, 
Y = (Reo, — fear 1 + (Reog — fig) + (Roos — fy) 3; 
Z= (Reg, — fog) + (Rego — go Jeo + (Regs — fg es 5 
