016 
MR. R, LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
the coordinates of the plane at infinity being —, —, —, where t\, r 2 , r 3 are the radii 
^ ^ 
of (x, y, z ), and e the distance between (tv, v). The point w— 0 is a cnfc-node. 
266. The coordinates (£, y, £, co, ttj) of any tangent sphere at the point (x'y'z'tv'v') 
must satisfy 
£ 7) % —2cr — 2a> 
(a + k)x'~ (6 + %' = (c + k)z' = dw’- 2 kv'~ - 2 kvf 
So that the equation of the tangent plane at any point (xy'z'ivv) is 
ax' by' cz' dw'\, , , , , , . 
+ +;yH- \(xx-\-yy-\-zz—2wv—2vw) 
x* y f z f w f "h v f \ / / / / 
-- + ' +——2-- Max x -f- hy'y -fi czz -f- cIw'w ); 
\ r l r 2 r 3 e / 
(289) 
and the equations of the normal at (x'y'z'tv'v') are 
X, 
y> 
z, 
1 
to 
— 2w 
x’, 
y\ 
2', 
1 
to 
c5^ 
— 2iv' 
ax, 
by', 
cz', 
dw, 
0 
1 
i 
1 
<M 1 
1 
2 
r ’ 
r i 
5 
6 
e 
267. The three systems of bitangent spheres are given by : 
a . r . ? 3 , 4ww d o A 
g= 0; - -(- 4-- -o)' = 0 
b—a'c—a' a a~ 
f 3 4 &)d d 0 
^ =0; + 73;+^7 r* 
£=0; 
a — b'c — b' b b° 
f 2 , 7 ? 2 , d 3 _ 
the focal curves are given by : 
;r=0 ; 
b o I ® Q d 0 A 
7 ?/~H- Z"—— W~=0 
o—a u c—a a 
tf-\-z~— 4ht=0 
h ; 
(290) 
(291) 
(292) 
and similar equations. Thus the focal curves are nodal spberi-quadrics on the three 
principal spheres. 
