206 Bootu—On Fresnel’s Wave Surface and Surfaces related thereto. 
is the inverse of the polar reciprocal of W=0. Use was also made of the prin- 
ciples of the cyclic sections of a cone of the second order. There is a circular 
section of (= 0 parallel to L=0, which is one of the four circles on the reciprocal 
of W=0; its equation is 
Vs=ca/@—P+a/P—e@+/Ve—e=0. 
This latter section is the inverse of M = 0, and they both lie on a sphere; its 
equation is 
ac + I'M = ac (a — ¢) 8, = 0, 
where 8S; is erp terize(os.) ft ste(u47) | _ 
i 6 QaaGe 
and the ‘‘ power” of the origin with reference to S, = 0 is unity as it ought to be. 
If a, B, y be the coordinates of the centre of the sphere, then 
a=3(c+2) [Ss [si (0) 
1\ |? -— @& 
=a = : 
and y=4(a+7) lz e 
and if R be the radius, then 
a+y-R=1, 
The plane 1/=0 cuts V=0 in a circle, and the sphere o,= 0 touches V= 0 all 
along this circle, so that if a tangent plane be drawn to V = 0 at every point on 
this circle, the developable so generated is a cone of the second order and of 
revolution. This cone stands out or away from the origin, and the portion of the 
surface V in its neighbourhood is a small protuberance corresponding to what 
Professor Tait calls a “basin” on the wave surface. There are, of course, three 
other real cones—in fact, as there are sixteen ‘“‘ basins” on the wave surface, there 
are sixteen corresponding cones touching V=0. We shall only notice the case 
corresponding to the four imaginary “basins” at infinity on the wave surface. 
The equation V = 0 can be written in the form P,P,P,;P, + J? = 0 where 
J stands for 
P(P+e +P (e+a)+2(7+0)—2(e4+74+2), 
P, for eV —P+y/ 0 —C +278 — a, 
P, for tVe—B-—y/@—C+2eV8 —a, 
P; for —-2/—P+y/VO—O+2/P— a, 
and P, for aVe—P+y/@—C-2VP—@; 
that is to say, the imaginary spheres degenerate in this case to imaginary planes, 
