562 
MR. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
161. The power of two points ( xyziv ), ( x'y'z'w ') is, by equation (168), given bv 
tt , ,<V 
7J"lv— X ~~—p 1/ ~zr~ -f 2 ~ -\-W ; 
ox J oy oz dw 
which, since 
xjj (xyziv) = 0, \p(x'y'ziv') — 0, 
may be written 
— 2ttK=\]j{(x—x), y—y', z—z', w—iv'} .(180) 
16‘2. If It be the radius of the sphere, and (A, B, C) any three points, we have by 
equation (141), 
IK 
^a.b:c)=-|-H(a.b,c)F ; 
and by § 132 
n 
e, a, b, c 
\0, A, B, C 
,xn 
1, 2, 3, 4 
1, 2, 3, 4 
= n 
0, A, B, CM 3 
1, 2, 3, 4. 
Hence, if (aqyjzptq), (x 2 y. 2 z 2 w 2 ), (x 3 y 3 z 3 w 3 ) be the coordinates of A, B, C, referred to 
the system (l, 2, 3, 4), we shall have 
V(ABC)=p, 
®1, 
Xq> 
GO 
Vi> 
y* 
z l, 
Z 3> 
u\, 
w 2 , 
iv 3 , 
h 
( 181 ) 
w r here 
30AW.’-n(J; I] l h+E'=o. 
Coordinate Systems of Reference. —§§ 163, 164. 
163. There are two convenient systems of reference : (i.) four mutually orthotomic 
circles, called the orthogonal system ; (ii.) two orthogonal circles and their two points 
of intersection, called the semi-orthogonal system. A particular case of the former 
would be three great circles cutting orthogonally and the imaginary circle at infinity. 
(i.) If the system (1, 2, 3, 4) be an orthogonal system, we shall fiud it most 
convenient to take Jc 1} k 2 , Jc 3 , Iq equal respectively to the cotangents of the radii 
of the circles (1, 2, 3, 4). So that the equation of the absolute will be 
xfj(x, y, z, w)=x~ + if-\-z-+w~ — Q ; 
and referring to § 137, we see that we shall have 
K=-2, 
and 
T'(a, b , c, d) = a J +6 3 -fc~ + # ; 
also we have 
cot 2 r, -f- cot 2 r 3 -J- cot 2 r 3 + cot : r 4 = — L. 
