584 
MR. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
gives us 
77. 
y.i» 
TT-i^fc TT -I', r t> 73 "-', 9> 
7T„ ,, 7T JiC , 0, 0, 0, 
77,^0, d, d? d, 
7r .'/,3> d, d, ^3,8> 0, 
TTy t 4) t), 0, 0, 
7T, /)5 , 0, 0, 0, 
7T 
4, 9> 
0, 
77. 
0 
0 
0 
0 
77- 
*,10 
5 > 10 
= 0 ; 
which may be written 
77, 
' + 
1 1,6 
1 *,r" yp 
^" 2,7 
77^', 8’77y,? * 77,f q77 
0,8 
‘ 4,9 
[ *, io-"y. 
77 
5,10 
(227) 
Hence a denoting any sphere, radius r x , 
1 = 7A 6 _pAh 7 | " r .» | 77*. 9 |_ 
1 1,6 
2,7 
77, 
3,8 
1 1,9 
77 
‘a-, 10 
1 5,10 
0 _ 2_77^0.772^1 | 7 T. l ' )7 . 77 .r i 2 1 77.rg.77.r_3 ( 7 T. r 3.77^4 . 77 .,- m. 77 .r 
1 — / - 
1 1,6 
77 r, 
‘ + 
n-,io- "i-,5 
0,9 
77- 
5,10 
J 
(228) 
Or again, if a; denote any plane, 
77, 
0 = 7r A®_ l — 
77], 6 7T : 
77.r 
,8^^9 1 77 "' r ' 10 
' 3,8 
77 
4,9 
77-, 
5,10 
_^_77*. 6*77*. 1 _|_ 77*,y.TT^ g ^ 77.rg.77.r, ^ 77jrcj.7r.r_ , ^ 77.r_ i () -77.r_ 5 
77l,6 772,7 77 3i g 77 1i9 77 5|10 
V 
(229) 
Also, taking a; and y as coinciding with the plane at infinity, 
0= 1 +- + 1 + J + ' 
1 1,6 
77 0 
7T.; 
3,8 
77 
4,9 
77, 
,10 
(230) 
A particular case would be any five points in space, and the five spheres circum¬ 
scribing the five tetrahedra; thus by equation (230) we see that the sum of the 
squares of the reciprocals of the tangents from each of five points to the spheres 
passing through the remaining four is zero. 
214. A system of three spheres and their two points of intersection constitute an 
important system, which may be called a “ semi-orthogonal ” system. Denoting the 
three spheres by (1, 2, 3), and their points of intersection by (4, 5), then it (x, y) 
denote any other spheres, the equation 
