530 
MR. R, LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
The equation to the tangent at the point is, by equation (98), 
/ / . / I / | , J ax' by' cz' dw'\ 
(xx+yy-\-zz+ww )I + —+ - +~J 
\ '1 '2 ' 3 '4 / 
= (<xcc 'x + by y+cz'z-j- d w'i d) ( y + 7 + — 
? ’l r 2 y 4 
(104) 
The equation to the normal at the point (x'y'z'v/) is, by equation (101), 
x, y, z, iv 
x, y', z', id 
ax, by', cz, did 
1 1 1 1 
r l r 2 r 3 V i 
= 0 . 
(105) 
88. The coordinates of any bitangent circle being (£, y, £, 0) we must have 
Hence we must have 
(a — d)x' (b—d)y f ( c—d)z ’ 
2 V2 
? , y , & 
a—d 5 —d c —d 
fr^=o 
(106) 
89. The pair of double tangents which belong to this system of bitangent circles 
are given by 
v + jh + A =0 
T 7 ,7~ ^ 
where 
a—d &—d c —d 
x +-+ -=0. 
r l r 2 r 3 
If <]> be the angle between them, we can deduce at once from § G5, remembering 
that 
- 2 +-i+- 9 +A=0; 
r i tv ry r i 
abed 
l f O "l „ O “1 
x , r 4 [(a-d)(b—d)(c-d)\rd ry ry ?y 
“♦= > '> !+: L )+r LA + I> ■ 
a — d\r 2 2 1 r*J 1 b — d\r,y 1 ryj ' c— d\i'd r 3 2 
• • (10?) 
90. Since the foci may be considered as bitangent circles whose radii are indefinitely 
