MR. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
515 
In particular the radius of the circle is given by 
~ 2 ’ • K -^+^+Cf + “a„' 
or 
r 2 .K=xft(g, rj, to) .( 77 ) 
53. Hence the radius of the circle 
is given by 
where 
ax + bycz+dw = 0 , 
a l,l’ 
0, 
a l,3J 
a i,n 
a 
Q 
to 
i— 1 
^2,2? 
Po, 3> 
^2,45 
b 
a 3,1; 
%,2> 
tt 3,3’ 
ffl 3,4J 
c 
a 4,25 
a 4, 3’ 
%, 4’ 
d 
a, 
6, 
c, 
rf. 
Mr 3 
M= -1 (. a\ + bh+ck 3 +dk t y, 
( 78 ) 
and a ]tl , a li2 , &c., are the coefficients in the equation to the absolute, so that 
i }j(xyzw) = a hl x 2 +a %z y 2 + . . . +2 a h2 xy+ . . . =0. 
54. Again the power of the circle (£??£&>) with respect to the circle 
is clearly given by 
ax -\-by-\-cz-\- di v=0, 
ag + brj + c^ + da) 
77 = «fc l +6i,+(*, + (», 
(79) 
55. And again the power of the two circles 
ax -{-by -f -cz -\-dw =0, 
ax d -b'y-\-cz-\-d‘ ’w— 0, 
is given by 
where 
^l.lS 
a h2 , 
CO 
rH 
<0 
«1.45 
a 
®2,U 
a % 25 
U- 2i 3, 
^2,45 
a 3,l> 
°&3,2’ 
O3,3’ 
tt 3,45 
c 
«4,n 
*^4,25 
s 
CO 
a 4,45 
d 
a, 
f 
C, 
M77 
2 
i\I—“ (ak^ -f- bJc.2 ~b ck^ -f- dk^j[a h^-\-b A^d - d, /q). 
(SO) 
3 u 2 
