568 
MR. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
in which case the equation reduces to 
ax 2 -\-by~ cz z + civ z = 0, 
which represents two imaginary circles. 
But if {x, y) be both real, then by taking for system of reference the two principal 
circles [x, y), and their two points of intersection (z, tv) we can show that the equation 
may be reduced to the form 
ax z + by 1 + cz~ = 0, 
the equation of the absolute being 
x z -\-y z =4:Zu\ 
A spheri-quadric represented by an equation of this form has a finite node, viz., the 
point z — 0 . 
179. Now, let us suppose the discriminating quartic to have three equal roots, then, 
as in § 84, we can show that if we take as system of reference the principal circle ( x ) 
corresponding to the unequal root, the node (z), and the circle (y), passing through 
(z), cutting (x) orthogonally, and passing through the other point (w), in which (#) 
cuts the curve : the equation may be reduced to the form 
x 3 =2 fyz. 
The point z is clearly a cusp, the circle x— 0 being the cuspidal edge. 
Observation.—If we suppose two of our circles of reference to be great circles, the 
curves degenerate into sphero-conics. As from § 128, it is clear that inversion is 
merely equivalent to a linear transformation, nodal and cuspidal spheri-quadrics are 
the inverse curves of sphero-conics. 
Chapter VI.— Classification of Spheri-quadrics. 
The method followed in Part I. for the classification of bicircular quartics is not 
suited for a systematic classification of spheri-quadrics, for which see Casey, “On 
Cyclides and Sphero-Quartics,” chap. xi. In this memoir it is only proposed to 
discuss the chief properties of the curves, following the order of chap, vii., Part I. 
General Spheri-quadric. —§§ 180-184"“. 
180. The equation of the curve is of the form 
ax z -f by 1 -fi cz 2 + dw z =0, 
and the equation of the absolute may be taken as 
x~ A //++ w ~—0 ; 
