644 
DR. J. CASEY ON CYCLIDES AND SPHERO-QTTARTICS. 
Cor. 2. These cuspidal edges have another and a more important geometrical signi- 
fication ; they are the curves in which the quadrics of the system 
PC* , ?/ 2 I 1 w ' A 
(a + k) [b + k) (c + 1c) ' (d-\-k) 
touch the envelope ; on this account I shall call them curves of taction. 
144. The envelope of the quadric (107) is Clebsch’s Surface of Centres (see Salmon, 
page 399). If we form the tangential equation of this quadric, we get 
and this is the focal quadric of the cyclide 
W'4-2£AV+TTJ 2 =0, (108) 
where 
w '=av + ^ 2 +6-y + 
The envelope of (108) is the cyclide 
W'U 2 =W 2 , (109) 
a surface of the eighth degree. This is the envelope of all the spheres whose centres 
move on Clebsch’s Surface of Centres, and which cut a given sphere orthogonally. 
145. The lines of % are cut homographically by its curves of taction. 
Demonstration. ^ is the envelope of all the quadrics, 
(cod + b[D -f cr + df) -f- liftd -f gr -\-C J r f) ; 
and by giving four different values to k, say k\ k u , &c., the anharmonic ratio of the 
four points in which any line of ^ is divided by the corresponding lines of taction is 
(lc , —lc")(Jc , "—/c "") : (K -¥'){!!'-¥") (110) 
Hence the proposition is proved. 
A particular case is that the anharmonic ratio is constant of the four points in which 
any line of % is divided by its four nodal conics ; the value of this anharmonic ratio is 
( a—b)(c—d):(a—c)(b — d ) (HI) 
146. The envelope of tangent lines to the curves of taction of % at all the points where 
any line of A meets them is a plane conic which touches the cuspidal edge. 
Let L, L' be two consecutive lines of 5) ; then, since L, U are divided homographically 
by the curves of taction, the proposition is evident. 
147. If L, L' be two non-consecutive lines of the lines joining the points where they 
meet the curves of taction generate a ruled quadric ; this is evident, since the curves of 
taction divide L, L' homographically. 
Cor. The lines joining the four pairs of points in which L, L' meet the double lines of 
5) are generators of a ruled quadric. 
148. The reciprocal of the developable % is the developable formed by the tangent 
lines of the sphero-quartic WU. We shall denote this latter developable by A. 
