208 Bootu—On Fresnel’s Wave Surface and Surfaces related thereto. 
with two other equations, multiplying the first by 2’, the second by y’, and the 
third by 2’, and adding, we have 
? = 2P\/(¢?—#). Hence 2\=??—?; 
; at? (t? — a’) 
— QPP PP 
Substituting this value of 2’ in 2° + 9° + 2 — aa'— yy'—ze'=0, and similarly 
for y’ and 2, we have 
a), fOo¥) a(n) 
2 
pip a Ue = Paes OPPS PS e Bali 
a 2 (2t? — a’) y (2t? — 0°) B(Qe te) 1 
apa See Dee aerear | ©) 
and if we substitute ’y'2’ in 2? /(¢? — a’) + &. =0, we get 
a(t? — a’) op (t2 — B) z(t? — ¢) e 5 
(2¢? SSeS uy (2702 Sy ce (2072? ee yp ee ( ) 
Now (2) is the derived equation of (1) with respect to ¢’. It follows, therefore, 
that putting ¢?= 0 that the second positive pedal of W=0 is written down by 
equating to zero the discriminant of 
Sate = _=0, (3) 
regarded as a quintic in ‘ and 
e° e Z 
- ; 2 * for —_— 5 
A® stands for 9 La” B? for 295” and (@? for ae? 
we can also, by taking the inverse of this second pedal, obtain the polar reciprocal 
of V=0. Now, geometrically, the polar reciprocal of V = 0 is the first negative 
pedal of the polar reciprocal of W=0. Hence, changing a’ into = and similarly 
for 0? and c’, and putting X for in (3), and similarly for Y and Z, it 
Zz 
C+ Pre 
appears that the first negative pedal of W= 0, that is, of Fresnel’s Wave Surface, 
is obtained by equating to zero the discriminant of 
2 pre 2 yd 
— at eee 2 2 y-_—— =0 (4) 
PpEPt ee r++ 7—-B P+ /4+2—-¥ 
ce as a quantic in ¢, where a’a’ stands for 2a’ — ¢’, 08? for 20°¢ — ¢’, and 
y’ for 2c — ¢’. 
