MR. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
513 
48. If 6 denote as previously the line at infinity, and the system of reference he 
denoted by (1, 2, 3, 4), then P being any point; since 7r Pi0 =l, 7r r>1 >=0, we see that 
the coordinates of P must satisfy :— 
i. a homogeneous quadric relation, 
'P, 1, 2, 3, 4N 
vP 1, 2, 3, 4 , 
n 
= 0; 
= 0. 
ii. a non-homogeneous linear relation, 
/P, 1, 2, 3, 4\ 
1, 2, 3, 4y 
Let us suppose that xyziv , the coordinates of P, are given by 
■ 1 l ] • TT] , y Z/n. 77* Pi 2 > £ &g. 7 T P) g , IV Iv^,7Tp ^^ , 
then xyziv must satisfy the relation, which is called the absolute :— 
xfj(x, y, z, w) = 
0, 
X 
2/ 
Z 
W 
V 
V 
V 
h 
a? 
V 
7r l, 1» 
^1,2: 
- ^l, 3’ 
”1 ,4 
y 
V 
^il) 
71"2,2 : 
1 ^2,3’ 
'7 r 2>4 
z 
V 
U 
• ^3,35 
w 
17> 
7r ‘i, n 
' 73_ 4,2: 
» 35 
^1,1 
be written 
P/r df ,df 
+ *J£ = 
K ’ 1 
= 0. 
(69) 
or 
0/q + dk J + dk~ K ’ 
(70) 
where K is some constant.* 
49. Similarly if S denote any circle, since ir s>e =l, we see that its coordinates 
(£, t), £, (o) must satisfy the linear relation 
II 
S, 1, 2, 3, 4 
e , 1, 2, 3, 4 
= 0, 
or 
df bf Of . . 
%, + w+^; + ^- K . (n) 
* [If we write (®, y, z, w;) == (a L1 , rr, li2 , . . .) (#, y, s, m>) 2 , then we shall have, 
K 2a lyl . ly _ 
n (1) 2, 3, 4\ /2, 3, 4\ 
VI, 2, 3, 4/ \2, 3, 4/ 
;&c.—October, 1886.] 
3 U 
MDCCCLXXXVI. 
