MR. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
533 
94. There are two finite foci on each principal circle, their coordinates being 
respectively proportional to 
But they are all imaginary. 
11 ,1 
°’ r,’ < 
p 0, p ±p 
r s r 2 r 3 
-> 7 ’ °> 
'l '3 
r, r, 
iii. Bicircular Quartics having a Third Node, —§§ 95-102. 
95. The equation of the curve may be reduced (by § 83) to the form 
ax* + by 2 -f cz* = 0, 
the equation of the absolute being 
x i -\-y 2 = Azw ; 
and if r l5 r 3 are the radii of the two principal circles, e the distance between their 
points of intersection, then the coordinates of the line at infinity are -. 
96. The coordinates of any circle touching the curve at the point ( x'y'z'iv) must be 
proportional to (£, y, £, w), where 
£ r] — 2a) — 2 £ 
{a + ky (&+%' cd-2kv; -2 Jed’ ' 
and, by § 72, the equation to the tangent line at (x'y'z'iv') will be 
( 112 ) 
-f-— \(xx -\-yy'—2zw'— 2iof) = ^ + y — 2—p—^)(axx -j- byy -j- czz '); (113) 
also the equation to the normal at the point (x'y'z'iv) will be, by equation (10L), 
= 0.(114) 
X, 
y, 
— 2 w, 
— 2 z 
x', 
y'> 
— 2 w, 
— 2 z 
ax', 
W> 
cz', 
0 
1 
i 
2 
2 
j 
M 
r 2 
e 
e 
97. The circle (£qlp>) given by (112) will be a bitangent circle, if k=— a or —b. 
In the former case we have 
