DR. J. CASEY ON CYCLIDES AND SPHEEO-QUARTICS. 
589 
and denoting this by II, we infer from equation (13) that the condition that should cut 
orthogonally two quadrics, S — (\x-\-py-\-vz-\-%w) 2 , S — ('/Ae+y/vy + +++c) 2 ,both inscribed 
in the same quadric S given by its general equation, is the invariant relation 
A + n = 0 (14) 
11. To find the equation of a quadric cutting four given quadrics orthogonally. Let 
S*+A, S^+B, s^+c, s=±d 
be the four given quadrics. It follows from equation (13) that we must have 
(Ca + a'(jj -j- a"v -}- a'"g +1 = 0, 
fa+b'p+b"»+b"' s ± 1 = 0, 
C A + c' [Jj + c" V + c’"g +1 = 0, 
d7,+d'[A+d ,! v +d’"g± 1=0, 
A, v, § being the coordinates of the pole of the plane of contact of the sought quadric 
with respect to S, and that this quadric will be then — 0. 
Hence, eliminating A, v, § from these five equations, we get 
s 1 , 
X, 
+ 
z , 
w , 
+1> 
«, 
a 1 , 
a", 
a'", 
+ 
1— l 
h 
V, 
v'. 
V", 
+1, 
C, 
c\ 
d', 
c 1 ", 
1 — 1 
1+ 
d, 
d\ 
d", 
d'", 
where the double signs of the first column answer to those of the binomial, S^+A. 
Hence if we denote for shortness by the notation (S= a V c" d 1 ") the determinant (1G), in 
the case where all the units in the first column are positive, we shall have eight ortho- 
gonal quadrics, whose equations are as follows : — 
(S* 
a 
V 
c" 
eF) = 0, 
( 17 ) 
(Si 
—a 
V 
c" 
d'")= 0, 
....... (18) 
(Si 
a 
-V 
d' 
d’")= 0, 
( 19 ) 
(Si 
a 
V 
—c" 
d'")= 0, 
(20) 
(Si 
a 
V 
c" 
-d"') = 0, 
(21) 
(Si 
— a 
-V 
c" 
d'")= 0, 
(22) 
(Si 
— a 
V 
-c" 
d'") = 0, 
(23) 
(Si 
— a 
V 
c" 
4 m 2 
-d'")= 0, 
....... (24) 
