DR. J. CASEY ON CYCLIDES AND SPIIERO-QUARTICS. 
715 
equation of the surface becomes 
%{ab'-a'b)\cd'-c'd)y'h i 
+ 2 t{aV - a'b)(ac' - a'c)(cd' - c'd)\bd' - b'dfPyd 
+2cc 2 (3yV- { ( aV - a'b)(cd' - dd) - {ad! - a!d){bd - b'c ) } ■ 
X {{ad 1 — a! d){bd — b'c) — {bd!—b'd){cc<! — c'a)} 
X {(bd'—b'd)(ca'—c'a)—(ab'—a'b)(cd'—c'd)}. I 
(227) 
The imaginary circle at infinity is a multiple curve of the order eight on this surface. 
Cor. 1. When we make § = 0 in equation (227) we get a perfect square. TIence each 
of the four spheres of reference meets the surface in a double line on the surface. 
These double lines correspond to the double lines of the developable A (see Chapter 
VIII.), and each of them has six double points. Thus the sphero-octavic in which § 
intersects the surface is expressed in terms of a, (3, y, and is of the form 
(be 1 — b'c) 2 (cci'-da) 2 (ab'—a'b) 2 fooq\ 
bed- cV/3 2 a'b'y 2 
Hence the three pairs of points (a/3), (|3y), (ya) are double points. 
Cor. 2. The equation (227), expressed in terms of the covariant cyclides, is given by 
the determinant 
2(0WW'-T'W-AW' 2 ), OWW'-TW — T'W', I 
= 0. . (229) 
OWW'-TW-TW', 2(0'WW'-TW'-A'W 2 ), ; v ' 
Cor. 3. The surface also meets W in the curve of intersection of W with T' 2 — 4ATW', 
which we have shown represents a system of eight circles which are generators of W. 
Cor. 4. Any arbitrary circle orthogonal to U meets eight generating circles of the 
surface, and the spheres determined by the arbitrary circle and the eight meeting circles 
are generators of a cyclide. 
346. If a cyclide W be given by the equation 
acd + bfi- -j- cy 2 + db 2 — 0, 
and also by this other equation referred to different spheres, 
a'd 2 +b'P' 2 +cy 2 +dr=0, 
then we can infer, as in Salmon’s ‘ Geometry of Three Dimensions,’ art. 192, the fol- 
lowing theorems : — 
1°. The eight spheres a, (3, y, c>, a', (3', y', V are generators of the same cyclide. 
2°. The two quartets of pairs of inverse points 
(a (3 y), (a (3 h), (a y b ), (/ 3 y b ), 
(«'3'y'), («'/ 3'*), {d y cf), {(3'y'd), 
lie on a cyclide ; this theorem (2°) may be inferred by reciprocation from 1°. 
mdccclxxi. 5 e 
