DB. J. CASEY ON CYCLIDES AND SPIIEEO-QUAETICS. 
643 
have the system of sixteen foci common to both. The proof is exactly the same as that 
of the corresponding theorem for two cyclides. 
140. Two sphero-guartics having four concyclic common foci can he described through 
any point, and they intersect orthogonally in their eight points of intersection. 
Demonstration. Let P be the given point, P' the inverse of P with respect to the 
circle through the four common foci, then through the four common foci can be 
described two sphero-conics touching the great circle which bisects PP' perpendicularly ; 
these will be the focal sphero-conics of the required sphero-quartics, and the proposi- 
tion is evident. 
141. The construction in art. 124 may be proved as follows: from art. 131 we see 
that from any point can be drawn concyclic planes which will intersect the planes of 
the nodal conics of 2 in four tetragrams circumscribed to the nodal conics. Now if the 
points from which the four concyclic tangent planes are drawn be the pole of the plane 
of one of the nodal conics (that is, in fact, if it be one of the four centres of inversion of 
the sphero-quartic), the proposition is evident. 
CHAPTEB VIII. 
Anharmonic Properties of the Developable % and its Beciprocal. 
142. Let us consider the cyclide W-j-#U 2 =0, where 
W = a<B -f- bf + cf + dh 2 , 
IP— « 2 + / + P; 
then the tangential equation of the focal quadric of W-f-#U 2 is 
(a+&)L 2 d-(£+#)/P+(c+#)r-L(^+#)§ 2 ==0, 
and this in tetrahedral coordinates is 
Q 9 9 Q 
■ - , w 0 . 
a + k'b + k'c-\-k'd-\-k ’ 
the discriminant of this with respect to Jc will be the developable 2) circumscribed to U 
along the sphero-quartic WU. 
143. The differential of (106) with respect to Jc, gives 
a o 0 O 
oc^ 2’*' 
(T+Ap + {b + /c) 2+ {c + kf^ (d+kf =0 ( 107 ) 
and the intersection of the quadrics (106) and (107) will be the locus of the centres of 
the generating spheres passing through the sphero-quartic WU of the cyclide W-f/rU 2 ; 
and this curve, namely the intersection of (106) and (407), is a cuspidal edge on the 
surface of centres of W Hence we see that the locus of all the cuspidal edges for 
all the surfaces W+£U 2 is the developable % circumscribed to U along WU. 
Cor. 1. The cuspidal edge of the surface of centres of any cyclide of the system 
W-f-UU 2 is a quartic of the first family. (See Salmon, p. 274.) 
mdccclxxi. 4 T 
