618 
DR. J. CASEY ON CYCLIDES AND SPHERO-QUARTICS. 
we have two circles on the surface of a sphere, the condition that the pole with respect 
to the sphere of the plane of one may lie on the plane of the other is expressed by the 
invariant 1— E=0. Hence it is evident that the same invariant relation will express 
for two conics on a quadric that the pole of the plane of each with respect to the qua- 
dric lies on the plane of the other ; and as circles so related cut orthogonally, we shall 
extend the term as in art. 9, and say that two conics so related cut orthogonally. In 
fact the relation 1— R=0 may be called (see art. 9) the harmonic invariant of the qua- 
drics or conics whose equations are connected by it. 
67. It is evident from the last article that, being given four plane conics on a quadric, 
the condition that the four conics should be tangential to a fifth is the determinant 
0 , (12), (13), (14), 
(12) , 0 , (23), (24), 
(13) , (23), 0 , (34), 
(14) , (24), (34), 0, 
and in fact Dr. Salmon’s direct proof of this theorem in case of conics on a plane will 
apply verbatim to the more general case here considered (see Salmon’s ‘ Conics,’ 5th 
edition, page 366). 
Cor. From the equation (71) we can find, as Dr. Salmon has done for conics on a plane, 
the equations of the pairs of conics which touch three conics on a quadric ; the equa- 
tion is 
x/(23)(S*— L)±->/(31)(S*— N)±v/(12)(S*-N)=0; .... (72) 
or this equation may be inferred also from art. 59. 
68. If we are given any three conics, S* — L, S* — M, S* — N, on the surface of a qua- 
dric, we get, precisely the same as in art. 51, the equation of the conic J which cuts 
them orthogonally. And so in general, being given any homogeneous function of the 
second degree ( a , b, c,f \ g, L, S* — M, S'— N), we see that it represents a twisted 
quartic of the first family, and that all the properties of sphero-quartics may be applied 
projectively to it (see observations, art. 26). 
CHAPTER V.— INVERSION AND CENTRES OF INVERSION. 
Section I. — Cyclides. 
69. If a, (3, P be any three spheres, & 15 & 2 , their diameters, DE, EG common 
tangents to a, P, ; /3, P respectively ; then if the system be inverted from any arbitrary 
point, and denoting the inverse system by the same letters accented, we have (see Sal- 
mon’s ‘ Conics,’ fifth edition, page 114), 
DE 2 
Si 
fg 2 _d'e ' 2 _ f'g ' 2 
°2 Sj S 2 
(73) 
Now this result holds whatever he the magnitude of P ; it will be true in the limit when 
P reduces to a point, in which case DE 2 , EG 2 become the result of substituting the 
