560 
MR. R, LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
156. And further, the power of the two circles, 
will he given by 
where 
ax + %+ cz +dw =0, 
ax-\- b 'y + cz + d'w — 0, 
a l,l» 
%, 2> 
a i,3’ 
«1,4> 
a 
1) 
Ctc) O; 
<4?! 3’ 
%,4> 
3 
a 3,2> 
®3,3’ 
a 3,r> 
c 
( \ U 
<^4, 0 , 
«4,3> 
°b, n 
d 
a', 
v. 
/ 
c, 
Mtt 
M = - [ak 1 + bh + ck 3 -j- dk 4 )(a'k i + b'k. 2 -f c'k., -fi d'k\). 
Whence, if the two circles cut at an angle <(>, we have 
where 
cos <f>-- 
, ,3¥ , 7/ o'P 
3 a 3 b 3 c dd 
2y /y i r (a, 1), c, d)M r (a, b, c, d) 5 
"^(a, b, c. d)= — 
a l, 1> 
^1,2 j 
a l, 3> 
«i,n 
a 
^2,1’ 
3 
«s,]> 
«3,2> 
«3,3> 
tt 3,4> 
c 
c b,n 
«4,3> 
%.3> 
Ct 44> 
d 
a, 
6 , 
G 
0 
(174) 
(175) 
157. The coordinates of 0, the imaginary circle at infinity, are evidently k v k. : , k 3 , ky 
and the equation of this circle is 
Sf, yt + S±, J ±_ 0 . / 176 ) 
dk Y ++ Si A 7*, ’.' 1 
its radius is equal to tan x \/ — 1. 
The Great Circle .—§§ 158, 159. 
158. The equation of the first degree 
ax~\~ b y —(- cz-\~dw := 0, 
will represent a great circle on the sphere, when 
o/i'| -f- bk 2 fi- ok 3 "T dk^ — 0 ; 
(177) 
