602 MR. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
V(P, Q, R, s )=n 
x v 
Vn 
z i> 
tv 1, 
V 1 
x 2> 
y^ 
W. 2 , 
x 3 , 
y 3 > 
Z 3> 
tv s , 
V 3 
y±> 
w 4) 
% 
h x , 
^3) 
^4> 
h 
where 288(j?k 1 z k 2 z k 3 z kffi 5 z . II 
( 1 , 2 , 3 , 4 , 
\ 1 , 2 , 3 , 4 , 
(270) 
Coordinate Systems of Reference .—§§ 239, 240. 
239. The most convenient system of reference is five spheres which are mutually 
orthotomic : this may be called the orthogonal system. If r x , r z , r 3 , r 4 , r 5 are the 
radii of five such spheres, it is simplest to take k x , Jc 2 , £ 3 , h x , Jc 5 , the coordinates of the 
plane at infinity, as inversely proportional to them. In this case we shall have, by 
§212, 
2 ,\p(x, y, z, w, v)=x°-\-y~-\-z°-\-id l -\-v i 
2 4?(x, y, z, w, v)=x^-\-y Zj r z~-\-iv i -\-v 2 
Thus the angle between the spheres 
ax ~\~by + cz fi -dw -\-ev =0, 
ax + b'y -f- cz + d!w + ev =0, 
will be given by 
. aa' + W + cc' + eld’ + ee' 
cus </> = — + 6 a + c s + d t + e 2yam y 3 + p* + d >i + pay 
240. It is, however, very often convenient to take as the system of reference three 
spheres cutting orthogonally, and their two points of intersection. And then if 
r x , r 2 , r 3 be the radii of the spheres, e the distance between their points of intersec¬ 
tion, it is simplest to take h x , Jc 2 , & 3 , Jc x , the coordinates of the plane at infinity, as 
inversely proportional to r x , r 2 , r 3 , e, e : so that we shall have by § 214, 
2xp(x, y, z, iv, v)=x l +y~-\-z~ — 4viv 
24 r {x, y , z, iv, v) = x l -{-y°-\-z~— viv 
so that the angle between the spheres 
ax -{-by + cz -\-dw -}-ev =0, 
ax + b'y -f- cz + d'w+ev = 0, 
aa' + bb' + cc r — 4{ed' + e'd) 
f (a? + b~ + c i —de){a'~ -f h" 1 -f c'~ — d'e')’ 
will be given by 
COS (f) — 
