DR. J. CASEY ON CYCLIDES AND SPHERO-QTTARTICS. 
627 
responding radius as the constant of inversion (see equation 83), then in this case each 
of the four circles a, (3, y, cJ will be inverted into itself. Hence we have the following 
theorem : — 
Every sphero-quartic lias in general four circles and eight centres or poles of inversion. 
84. If we are given a focal sphero-conic F of the sphero-quartic and its corresponding 
circle of inversion a, the remaining circles and centres of inversion may be constructed 
as follows. Through the four points in which a cuts F draw three pairs of great circles, 
each pair will intersect in two points diametrically opposite ; the six points thus deter- 
it can be shown that the line joining the other pair of nodes is such that any point of it is a centre of 
inversion ; and from the two remaining roots the two other centres of inversion can be found. 
X. If +.4 — X = 0 be an elliptic paraboloid, and J the sphere which touches it at the umbilics, then 
a 2 b 2 c 
we find, as in article IY., the equation of the cubic cyclide with four nodes to be 
3,’ p. 234) is 
dx , 
fy , 
dz, 
L , 
M , 
cTL, 
car, 
<m, 
The section of this surface by the plane xz consists of a line passing through the two nodes, and a circle whose 
centre is the focus of the section of the paraboloid by the same plane ; and the section in like manner by the 
plane yz consists of the line joining the two other nodes, and a circle whose centre is the focus of the parabola, 
which is the section of the paraboloid by the plane yz. 
XI. The sections of the surface by the planes xz and yz are lines of curvature of the surface. 
Demonstration. The section by the plane xz is such that the centres of the generating spheres which touch 
the cyclide along it lie on the section of the paraboloid made by the plane of xz ■ hence, taking any two con- 
secutive points on the section, it is evident that the normals at these points will lie in the plane of the section, 
since they pass through the centres of the generating spheres. Hence they intersect, and the proposition is 
proved. 
"We can verify this analytically; for the differential equation of the lines of curvature of any surface (see 
= 0 ; 
and it is easy to see that this is satisfied by the equations .r=C + and y — C + — ^ combined with 
the equation of the surface, C, C' being any constants. Hence any plane passing through a line joining either 
of the pairs of nodes will he a line of curvature. 
XII. The inverse of a line of curvature on any surface is a line of curvature on the inverse surface. This is 
easily inferred from Salmon’s ‘Geometry of Three Dimensions,’ articles 294, 479. Now, since trinodal and 
quadrinodal cyclides are the inverses of quadrics of revolution, we easily infer the following theorems : — 
1°. Any section of a quadrinodal surface through two nodes, either both real or both imaginary, will consist of 
two circles, which will be lines of curvature. 
2°. A section of a cubic cyclide through two nodes will consist of a line and a circle, which will be lines of curvature. 
3°. If through any point P on a trinodal or quadrinodal cyclide we draw two planes passing through the nodal 
axes, the circles which are the sections at P intersect orthogonally . 
4°. Every line of curvature on a trinodal or quadrinodal cyclide consists of two circles, or a line and, a circle. 
5°. The locus of the generating spheres which touch a trinodal or quadrinodal cyclide at cdl the points of a line 
of curvature is a plane conic on the corresponding focal quadric . — January 1872.] 
MDCCCLXXI. 4 R 
