MR. R, LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
597 
0 
£C 
y 
z> 
w 
V 
— 0 
V 
h! 
V 
K 
h 
V 
7r l, 1? 
7r i, a» 
^l, 3? 
7r l,4> 
77 1, a 
y 
v 
1, 
7ro,2, 
ir 2,35 
^ 5 
iC 
V 
^3,1, 
77 *3.2’ 
n 3,3’ 
7r 3,4<5 
^3.5 
w 
V 
15 
7T 4,25 
77 '4,3, 
7r 4,45 
n.s 
V 
,r 5,l» 
^5,2’ 
CO 
^*5,4’ 
77 5,5 
(252) 
This is called the equation of the absolute : we will denote it by 
xfj( x , y, z, w, v) = {a lA , « lt2 , a h3 , a lA , . . .)(x, y, z, iv, v) : =0, . (253) 
where 
-nf 2 ’ 3 ’ 4 ’ 5 
1 \2, 3, 4, 5 
_n f 1 ’ 3 ’ 4 ’ 
1 \2, 3, 4, 5/ 
'V jj (b 2, 3, 4, 5V /l, 2, 3, 4, 5V ; 
\b 2, 3, 4, 5/ VI, 2, 3, 4, 5/ 
and then the linear relation which (xyzivv) must satisfy, may be expressed thus 
h yy = “ 2 - ..(254) 
mo 
226. If (f. v , C, a), 7x) be the coordinates of any sphere, S, we see that, since 7r Sj0 =l, 
the coordinates of S must satisfy a non-homogeneous linear relation 
which may be written 
Til 6, L 2 ’ 3 ’ 4 ’ 5 W 
VS, 1, 2, 3, 4, 5/ U ’ 
7. Vf , JL 7, fy — 
^ 0 £ + 077 - 
— 2 . 
. . (255) 
227. Again, if L denote any plane whose coordinates are (\, y, v, p, cr), since 7 t Li0 =O, 
7 r LiL = —2, we see that the coordinates of L must satisfy a linear homogeneous equation 
and a lion-homogeneous quadric relation, viz., 
