Boorn—On Fresnel’s Wave Surface and Surfaces related thereto. 209 
An ellipsoid is also a surface whose polar reciprocal (with reference to a con- 
centric sphere) is of the same type as itself; hence it would follow, geometrically, 
that its negative pedal may be written down from its second positive pedal. The 
determination of the second positive pedal of the ellipsoid is the same thing as 
finding the envelope of the sphere 
eP+Pt+2—xr' —yy — 227 =0, 
where (2? + y? + 2?) = wa? + By? + C2”, 
Now, proceeding in precisely the way adopted above, it will appear, with- 
out difficulty, that the first positive pedal of Fresnel’s ‘‘ Surface of Elasticity ” is 
obtained by equating to zero the discriminant of 
io y D 
9278 908 286 
regarded as a quartic in 9. Thisalso gives the polar pe ocal of the same surface, 
-O(’+7+2VP=0, (5) 
and changing @ into 2 &ec., and writing X for =—; Z 3 eat it appears that the 
negative pedal of the ellipsoid is obtained by equating to zero the discriminant of 
x 2, 2 a 
gta tz -9=9, (6) 
2-3 2-5 2-5 
a b° Ge 
i 
regarded as a quartic in 6. This is the result obtained, otherwise originally, by 
Professor Cayley. See Salmon’s ‘“ Surfaces,” Art. 517. Moreover, in the same 
way as above, the first positive pedal of the surface 
(e+ y+ BNNs = Cx i By? =e ez? 
is written down by equating to zero the discriminant of 
a(n—1) y(n—1 n—1 i ah Sowkind L 
n { eee aes ae (eH +y+2°P = 0, (7) 
regarded as a quantic in 6. The result also provides the polar reciprocal of the 
same surface for all values of by putting X for #/(#+y’?+2*), and similarly for 
y and z. In all such cases the traces of the several surfaces on the principal planes 
may be directly obtained after the manner indicated in Salmon’s “ Surfaces,” 
Art. 202, Ex. 2, page 157. 
Again, the parallel surface to Fresnel’s surface of elasticity may be obtained 
by the same method: this is the same thing as determine the envelope of the sphere 
aN’ y'\ ee ee ee 
(*-5) +(9-5) +(-- 5) =( +9): 
J\ 2 I\ 2 N\ 2 
where | + i - (=) =1, taVa7*+y?+ 23, 
