DR. J. CASEY ON CYCLIDES AND SPHERO-QUARTICS. 
597 
CHAPTER III. 
Section I. — Different forms of the Equations of Cy elides. 
27. Let T be a tangent plane to the focal quadric F ; P, P' the corresponding points 
of the cy elide ; then (art. 18) P, P' are the limiting points of the sphere U and the plane T. 
Hence 
1 -6 
1 — E + c 2 + d 2 f 11 1 — (6 2 + c 2 + <F)’ 
— c — cl 
y 1 — (b 2 + c 2 + d 2 )’ -(6*+c* + <Py 
Hence forming the condition that the polar plane of the point (X, y, v, f with respect to the quadric S + LN is 
(6 — b')y + (c — c')z + (<7 — d')tv = 0, 
we get 
bb' + cc'+clcl'= 1. 
This condition I propose to call the orthotomic invariant of the two quadrics. 
If we take the more general forms, 
ar + y 2 + z” + ltd + 2{ax + by + cz + clw)x, 
X s + y- + zr + mt + 2 (a!x -\-b'y + dz + cVw)x, 
for S + LM, S + LN, these may, without loss of generality, he written in the more compact forms 
ax 2 -\-y~ -{-z 2 -{-ru 1 -\-2(by -\-cz +div )x, 
a! x 1 + if + z 2 + w 2 + 2 (b'y + c'z + d'w)x ; 
and we find, as before, the orthotomic invariant to be 
2W + 2cc' + 2 dd' = a + a'. 
Compare equation (1), article 1. 
The two quadrics related, as here considered, have many important properties. Thus the poles of the plane L 
with respect to the quadrics, and the four points in which the line of connexion of these poles meets the quadrics, 
form a system of six points in involution. 
Def . Two quadrics related as in this proposition may he said to cut orthogonally. 
II. Given five quadrics, S + LM, S+LN, &c., where 
kl = ci'x + b'y + c'z + el'w = 0, 
N = a"x +b"y-\- c"z + d"w = 0, 
then the condition that the five quadrics should he coorthogonal is the determinant 
a' , 
V , 
c' 
, dj , 
1 , 
a" , 
V , 
c" 
, cl" , 
L 
a'" , 
b"' , 
o'" 
, cl'" , 
l, 
a"", 
b"", 
o'" 
", cl"", 
1, 
a""', 
b""', 
o'" 
", d'"", 
1, 
Hence we infer the following theorem: — If a, ft, y, $ be any four quadrics of the form S + LM=0, 
S + LN=0, &c., then the quadric Aa+jn/3 + yy +g8 is coorthogonal with a, ft, y, 3, and the pole of the plane 
L=0 with respect to Xa + p./3 + vy + ^3 will be a point whose tetrahedral coordinates are proportional to 
X, y., v, f, the angular points of the tetrahedron of reference being the poles of L with respect to a, ft, y, S 
respectively. — January 1872.] 
4 N 2 
