MR. R. LACHLAN - ON SYSTEMS OF CIRCLES AND SPHERES. 
531 
small, the coordinates of the four foci belonging to the system, given by equation (10G), 
will be given by 
y 
i=0, 
and 
whence 
a—cl b—d c—cl 
x 3 -f-2/ 2 +2 3 =0 ; 
?/ 
(b — c)(a — d ) (c — a)(b — d) (a — b)(c—d )’ 
(108) 
91.. From the form of these equations it follows, that all curves whose equations are 
of the form 
oooo 
rpA yi* g'"' r UJ' J 
W+lW 0 ’ ( 109 ) 
and 
ooo o 
, ?r , z* . , 
- 3 + / 3 3 +' 3 +^ = ° 5 .( ll °) 
will have the same foci. 
Subtracting these equations, we have 
w‘ 
■ 0 . 
«V + *) 1 /3 2 (/3 2 +/c) 1 7 s (t 9 +A) 1 b 3 (S 3 -L 7c)' 
Hence the circles whose coordinates are respectively proportional to 
x y z w 
2 ? /O05 03 £0 3 
cr p 7 - er 
y 
w 
«* + *’ /3 2 + F 7 3 + /d g 2 ft’’ 
must cut orthogonally; but these circles touch the curves given by (109), (110) at 
their common points; hence confocal bicircular quartics cut orthogonally. 
Through any point, two quartics can be drawn confocal with a given bicircular quartic, 
since the equation (109) is a quadratic in Jc. We see, too, that two circular cubics can 
be drawn confocal with a given bicircular quartic. 
92. Let (^£w) be the coordinates of any circle S; this will cut orthogonally one of 
the bitangent circles, at the point (x'y'z'iv) on the curve 
ax "-(- by 2 -+ cz z +di v" = 0, 
if 
(a — d)x'g- j- (b — d)y'rj + (c — d)z'l,— 0. 
It follows that two bitangent circles belonging to this system can be drawn to cut 
S orthogonally; and their four points of contact lie on the circle 
(a—d)£x-{-(b—d)yy +(c — d) £2 = 0. 
3 y 2 
