572 
MR. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
and the coordinates of the foci by 
x= 0 , y^-zw- 
y— 0 , x^—zw- 
(b-a)c g ~1 
ab 
(a — b)c 
ab " S 
(200) 
188. From the form of these equations we see that every curve whose equation is 
of the form 
is confocal with the curve 
and subtracting we have 
—— -L—-- r -L- — o 
2 i I 02 i . I o ? 
«“ + K p +fC 7 " 
X" ?/ 2 3 
— 4 - J +— — 0 • 
2 I 02 I 0 v J 
« P 7' 
r 
a 2 (a 3 + /c) y8 2 (/3 3 + «r) 
: 0 . 
Hence two such curves intersect orthogonally. 
We infer that through any point on a sphere two spheri-quadrics can be drawn 
confocal with a given nodal-spheri-quadric ; and these curves will cut orthogonally. 
7T 
189. If r z = ~ and the coefficients a, b, c in the equation 
satisfy the relation 
ax~ + by- -f- cz 1 = 0 , 
(a — b)c = cib, . 
( 201 ) 
then one of the foci on the principal circle y— 0 coincides with the centre of the 
other principal circle, and the curve becomes a Cartesian, having a third node. 
Cuspidal Spheri-quadrics .—§§ 190-192. 
190. The equation of the curve is of the form 
a: 3 =2 ayz, 
the system of reference being the principal circle, the circle orthogonal to it through 
the cusp and the other point, in which the principal circle cuts the curve, the cusp 
and the other point common to the two circles. 
If r v r % be the radii of the circles, 2e the arc between their" points of intersection, 
we may take the equation of the absolute as 
y' 2 =zw, 
the coordinates of the circle at infinity being cot r L , cot r 2 , cosec e, cosec e. 
