MR. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
561 
and if this condition be satisfied, its coordinates will be given by 
Bifr Brjr cjrjr 
B% _ dr) __ Bar 
abed 
Byjr 
2KA 
-'I' - 
by equation (167), where d/' has the same meaning as in § 156, and where 
(178) 
a l,l 
a l,3 
a l,3 
a \A 
S 
to 
1—' 
a 2,2 
a % 3 
a 2A 
CO 
$ 
a 3.2 
a 3, 3 
a 3 A 
«4,1 
a 4 2 
a 4,3 
a i, 4 
159. The power of the great circle (£r)£,a>) with respect to the small circle 
ax-{-bycz -f div=■ 0, 
is given by 
7TK = 
a% + br) + c£+ da) 
a/q + bk g -t- ck 3 -p dkj 
as in § 155. 
But if the equation ax-\-by-\-cz-\-dw= 0 represent a great circle, then the power of 
any other circle (£t?£&>) with respect to it is 
— /\J : ^^-{a£ J rbr)-\-c£,-\-d(o). 
(179) 
The angle between two great circles whose equations are given is the same as that 
given by equation (175). 
The Point. —^ 160-162. 
160. The power of the point ( xyziv) with respect to the circle 
ax -f by +ca-f- dw= 0, 
ax + by + cz + dw 
c</q + bk 3 + ck 3 + dk 3 
is equal to 
or 
(' ax-\-by-\-cz-\-dw ) 
2AK 
according as the equation represents a small circle or a great circle. 
MDCCCLXXXYI. 4 C 
