MR, R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
571 
may represent a “ Cartesian ” is given by 
cosec 2 'i\ 
a—cl 
30t 2 r. 
b-d 
+ 
cot 2 r 3 
c — cl 
= 0 , 
(196; 
Sphevi-quadrics having a Third Node .—§§ 185-189. 
185. The equation of the curve is of the form 
ax 1 +btf -f- cz~ = 0, 
the equation of the absolute being' 
x -y~=zw ; 
where the coordinates of the circle at infinity are cot r x , cot r 2 , cosec e, cosec e; i\, r 2 
being the radii of the two principal circles, and 2e the arc between their points of 
intersection. 
186. The coordinates of any circle which touches the curve at the point 
(i x'y'z'w') are given by 
J2£ _ 2rj __ -co __ 
(a + 2 Jc)od (b + 2 k)i/ cz' — lav' —IN' 
The equation of the tangent at the point will be, 
( 2xx -f- 2y'y — w'z — z'w)(ax cot r x -\-by' cot r 2 -\-cz cosec e) 
— {ctxx-\-hy'y-\-czz){2x cot r x -\-2y cot r z — w' cosec e—z cosec e). . (197) 
The equation of the normal at (xy'z'iv) will be, 
2x, 
2y. 
— w, 
— z 
2x, 
2 y', 
— w, 
— z 
ax', 
fy, 
cz, 
0 
2 cot r x , 
2 cot r 2 , 
— cosec e, 
— cosec e 
187. The systems of bitangent circles will be given by 
= 0. . . (198) 
4 D 2 
