MR, R, LACHLAN ON SYSTEMS OE CIRCLES AND SPHERES. 599 
In particular the radius of the sphere (£r)£a jct) will be given by 
2r z =xb(i, r], £, co, ct).(2G0) 
230. Hence the radius of the sphere 
ax + % + cz-\-dw-\-ev— 0, 
is given by 
C b,2J ®1,3’ 
a 2,l> Cl 2,2’ ft 2,3> 
®3.1’ C<, 3,2> C<, 3,3> 
<^ 4,15 ^ 4 , 2 > ^ 4 , 3 ’ 
Cl 5, 1> a 5,2J %,3> 
a, b, c, 
where 
w 2v~(cik>T L -f- ciio -j~ dk±-\- chd)~, 
We shall find it convenient to denote the bordered Hessian of i// by T*'; and suppose 
the coefficients so determined that we may express equation (261) thus— - 
u—^(a, b, c, d, e) .(262) 
a \A> 
«1,5> 
a 
4 , 
%,5> 
0 
a 3,n 
«3 )6! 
c 
<^4,4» 
«4,5> 
d 
®5, 4’ 
a 5,5> 
e 
d, 
e, 
u 
231. The power of the sphere (^(wnr) with respect to the sphere 
ax-\- by-\-cz-\-div-\-ev= 0, 
a^ + br/ 4- -f d(o + em 
is clearly given by 
7 T- 
cdy + bh 4- cAg + dk i + ek* 
and the power of the two spheres 
ax -\-by -\-cz -\-dw -\-ev =0, 
ax + b'y+cz + d'w -f ev =0, 
is clearly given by 
,0'P 7 />P , p y Y , i&P , 
+ & 05 +C 0c * d dd +C de 
7 T- 
(cik-^ T - 8/<*2 4“ ek s 4- dk± 4" ck^)(cbk-^ 4~ b'k^ 4~ dk^ 4“ d ly 4“ o 84 ) 
And so the angle of intersection of the spheres will be given by 
, 0 ^ 7 / 0 ^ , 0 ¥ 
a V + & V + c V + dVr + e V 
, i da do oc od oe . 
cos cb — -—_ ___ . . — 
~ ■y/'SHa, b, c, d, e).' v P(« / , b', c\ d', c'j 
. (263) 
(264) 
