MR. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
569 
where the coordinates of the circle at infinity are cot i\, cot r 2 , cot r 3 , cot r x , r 3 * r g , r 4 
being the radii of the principal circles. 
181. The coordinates of any circle touching the curve at the point ( x'y'z'w') must be 
proportional to 
(a-\-Jc)x', (b-\-h)y', (a-\-h)z', (d-\-k)w r . 
The equation to the tangent at the point will be, 
(x'x + y’y + dz -fi iu'w) (ax' cot r x -\-by' cot r 2 -\-cz cot r^-\-dw cot r 4 ) 
= (cix'x-\-by'y-\-czz-\-dw'io)(x cot r L -\-y' cot i\-\-z' cot r 3 -\-iv' cot r 4 ). 
The equation to the normal at the point (x'y'z'iv) will be, 
X, 
y> 
IV 
X , 
y> 
w' 
ax', 
fy. 
cz, 
div 
cot 1 \, 
cot r. 2 , 
cot r 3 , 
cot r 
182. The systems of bitangent circles will be given by, 
(189) 
(190) 
y= 0 , 
{= 0 . 
(0 = 0 , 
^ ^ 
O 
1 "" - 
b — cc 1 c — a 
1 d-a 
r e , c 
0 
L ^ 
r 
1 
'O 
h 
rO 
1 
r cl-b 
C-1 
OJLn 
o 
©- 
I 
1 ° 
I 
[ ^ 
h 
1 
53 
r d—c 
«Yf> 
<> 
to 
, 
1: 
1 
H 
1 
Rj 
1 
* c-d 
and the coordinates of the single foci will be given by, 
(191) 
a'= 0 , 
y— o, 
z = 0, 
w= 0, 
70 " 
( c—d)(b — ci ) ( d—i)(c — a ) (b — c)(d—co) 
x - 
70" 
(c — cl) (a —b) (cl — ct) (c — b) (a — c) (cl — b) I 
r 
70" 
>■ 
(b — d)(a — c) (d — a)(b — c) (< %—b)(cl—c ) 
O O o 
& _ r_ _ __ ~~ 
(b — c ) (a — d) (c — a)(b — d) (a — b)(c — cl) 
(192) 
The curve has also six double foci (see Casey, “ Cy elides,” § 130), and thus the 
twenty-eight points of intersection of the eight common tangents of the curve and 
the circle at infinity are accounted for. 
MDCCCLXXX VI. 4 D 
