DE. J. CASEY ON CYCLIDES AND SPHEEO-QTJAETICS. 
711 
cyclides, each of the twelfth degree, on which the cuspidal edge also lies. These equa- 
tions will he similar in form to the sextic cones containing the cuspidal edge of the 
developable circumscribed about two quadrics (see art. 182). 
Cor. If any of these hinodals he inverted from one of its nodes, it becomes one of the 
sextic cones of art. 182. 
334. The equation (207), cleared of fractions, becomes 
A'W+IcI+tfT'+IdAW (212) 
A' 
If in this equation we put h— Ave get 
A 2 W+aAT+x 2 AT+a 3 A ,2 W'=0. . . (213) 
Compare Salmon’s ‘Geometry of Three Dimensions,’ art. 206. The value of T is 
A '\a 
and T' is got from T by interchanging accented and unaccented letters. 
In terms of the cyclides T, T' can be expressed all the cyclides having permanent 
relations to W, W r . Thus if 
S be the reciprocal of W with respect to W', 
S' be the reciprocal of W' with respect to W, 
then 
T=©'W-S', (215) 
T'=©W'-S (216) 
Hence W, S', T have a common curve of intersection. 
335. The discriminant of (212) is 
27 A 2 A ,2 W 2 W ,2 +4(A'WT' 3 -f A W'T 3 ) — TT'(TT+ ISA A'WW'), . . (217) 
an equation of the sixteenth degree, since it contains a, j3, y, § in the eighth degree. 
The imaginary circle at infinity is on this surface a multiple curve of the eighth degree, 
so that it is an octavic cyclide. 
By making W=0, we see that the surface touches W along the curve WT, and that 
it meets W again in the curve of intersection of W with T' 2 — 4 A W'T ; this represents a 
system of eight circles which are generators of W. The sections of (217) by the spheres 
of reference are easily obtained ; for, by a known process, the section of the discriminant 
of (207) by the sphere c> will be the sphero-quartic squared, 
/ aa'cd bh'l 3 2 cc'y 2 \ 2 
yid' —a'd^~ bd'—b'd' ca'—c'd) ’ 
multiplied by the discriminant of 
at / 3 2 y 2 
a~ l + ka'~ 1 b^ 1 + kb 1 * 1 ^ c~ l + kc'* 1 ’ 
